Texture feature extraction operators, which comprise linear filtering, eventually followed by post-processing, are considered. The filters used are Laws’ masks, filters derived from well-known discrete transforms, and Gabor filters. The post-processing step comprises non-linear point operations and/or local statistics computation. The performance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture operators.

Grigorescu, S.E.,

Kruizinga, P.,

Petkov, N.,

Texture feature extraction operators, which comprise linear filtering, eventually followed by post-processing, are considered. The filters used are Laws’ masks, filters derived from well-known discrete transforms, and Gabor filters. The post-processing step comprises non-linear point operations and/or local statistics computation. The performance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture operators

Grigorescu, S.E.

Petkov, N.

Kruizinga, P.

Proceedings - University of Groningen

   University of GroningenA comparative study of filter based texture operators using Mahalanobis distanceGrigorescu, S.E.; Petkov, N.; Kruizinga, P.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operatorsusing Mahalanobis distance. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). IEEE (The Institute ofElectrical and Electronics Engineers).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 16-07-2023A comparative study of filter based texture operators using Mahalanobis distance S. E. Grigorescu, N. Petkov, and P. Kruizinga Institute of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl Abstract Texture feature extraction operators, which comprise lin­ear filtering. eventually followed by post-processing, are considered. The filters used are Laws' masks. filters de­rived from well-known discrete transforms. and Gabor fil­ters. The post-processing step comprises non- linear point operations and/or local statistics computation. The pelfor­mance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture oper­ators. 1. Introduction A number of texture feature extraction operators have been proposed in the literature which are similar in that they comprise two processing steps: (i) linear filtering followed by (ii) post-processing (Fig. 1). The post-processing step typically involves a non-linear point operation followed by the computation of some local statistics. Laws proposed a specific type of linear filtering followed by smoothing based on local averaging applied to the ab­solute values of the filter output [5]. Unser used for filter­ing various well-known transforms like discrete sine (DST), discrete cosine (DCT), discrete even sine (DEST), discrete real even Fourier (DREFT) and discrete real odd Fourier (DROFT) transforms, and for post-processing the computa­tion of channel variances [9]. In [2, 6], a Gabor filter bank followed by thresholding was used. For further references on texture operators, which comprise filtering followed by post-processing, see [8]. Different filters and different types of post-processing are used in the methods mentioned above. Since the re­sults achieved with these methods are also different, a natu­ral question arises of which filter scheme and which type of post-processing give the best results. 0-7695-0750-6/00 $10.00 © 2000 IEEE 885 Texture operator �F'lt _I Post- I:  Iprocessingl � :--. -, I Post- I f\�;:; #� I FIlter l -IProcessingl -r L-/ i i /: Feature ; ; : vectors . . . . �F'l: I P�st- I i : .. �.�.��t.��:=:!����_���!���nj Figure 1. General structure of filter based tex­ture operators. Previous studies in this direction typically focus on the joint performance of the linear filtering, the post -processing and a subsequent classification or segmentation stage (for references see [4, 8]). In a very thorough recent work, Ran­den and HUS0Y [8] bring systematics into the matter by fix­ing the type of post-processing and comparing the effect of different linear filtering schemes. Similar to previous stud­ies they evaluate the performance of texture operators on the basis of the classification and segmentation results that are obtained when the feature vectors are fed into a given clas­sifier. Such an evaluation has the drawback that it does not measure the performance of the texture operator only, but the joint performance of the texture operator and a subse­quent classifier [4]. Moreover these studies do not evaluate the contribution of each individual processing step to the overall performance of the texture feature operator. For evaluation of the performance of texture operators, we use a new method that was proposed elsewhere [4]. It is based on a statistical approach (Fisher criterion and Ma­halanobis distance) to evaluate the capability of a feature operator to generate discriminable feature vectors for dif­ferent textures. This method can be used to compare the texture operators only, regardless of any subsequent classi­fication or segmentation operation. Furthermore, this eval-uation method can be applied after each processing step so that the contribution of each step to the overall performance can be assessed. The rest of the paper is organized as follows. In Section 2 an overview of the filtering schemes used in our exper­iments is given. In Section 3 we discuss different types of post-processing. Comparison criteria and experimental results are presented in Section 4. Finally, in Section 5 conclusions are drawn. 2. Filter banks For linear filtering Laws [5] used a set of square convolu­tion kernels obtained as the outer product of pairs of vectors from a set of n n-dimensional vectors. He conducted exper­iments for n = 3, 5, and 7 and obtained similar results. In our experiments we used Laws' convolution kernels derived from the following set of n = 5 vectors [51: l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 [-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 [1, -4,6, -4, 1]: resulting in a bank of 25 filters. Unser [9] used several different filter banks. Similar to Laws' approach, the convolution kernels proposed by Unser are computed as the outer product of pairs of vectors. Each set of vectors used in Unser's experiments corresponds to one of the following transforms: identity (shift) transform, DST, DCT, DEST, DREFT,and DROFT. More details about these transforms and the filter banks derived from them can be found in ll, 9J. We conducted experiments with all the filter banks derived from the transforms mentioned above. In all cases a set of 25 convolution kernels of size 5 x 5 was used for conformity with the experiments with Laws' oper­ator. Such a filter bank leads to a 25-dirnensional feature space. In [2, 6], filter banks based on Gabor function kernels were used. In our experiments we used two sets of Gabor filters: one employing symmetric Gabor function kernels (1) and the other employing antisymmetric Gabor function kernels (2). In both cases, we used 24 Gabor filters tuned to three preferred spatial frequencies (f = 2-ky'2, k = 0,1, and 2) and eight preferred orientations (e = m � , m = ° . . . 7). For all Gabor filters the product of the preferred spatial frequency f and the standard deviation (7 of the Gaussian factor was kept constant (f(7 = 2) which means that the filters have a constant spatial frequency bandwidth. More details about these types of Gabor filter banks can be found in [4, 7]. Besides these Gabor filters, the filter banks used in our experiments also comprise a Gaussian filter (3) for DC com­ponent computation. 886 ( X2 + (2y)2 ) g}� (i,j) = exp -2(72 . cos(27rfx) (a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 2172 • sm(27rfx where: . . ( i2+j2) 9(�,}) = exp -2ar ' i,j -2 ... 2 x i cos e + j sin e (71 = 3 Y = -i sin e + j cos e (1) (2) (3) This combination of Gabor and Gaussian filters results in banks of 25 filters with a coverage of the spatial frequency domain similar to that of Laws' and Unser's filter banks. 3. Post-processing We applied several types of post-processing to the out­put of the filter banks described in Section 2. The post­processing is applied individually to the output of each channel in a filter bank. The types of post-processing used in our experiments fall in two categories: one-step post-processing and two-step post-processing. The one-step post-processing comprises either a nonlinear point opera­tion (NLPO) or computation of local statistics. The two­step post-processing comprises an NLPO followed by com­putation of local statistics. In our experiments we used as NLPO thresholding and modulus computation. For local statistics we used mean, variance and standard deviation computation. In previous studies only the mean and the variance were used as statis­tical measures of local texture properties. We performed ex­periments with standard deviation as well because it offers a statistical characterization similar to that given by the vari­ance and, at the same time, it takes values in the same range as the one which is characteristic of averaging. All the lo­cal statistics are computed in a square neighborhood of size 17 x 17 around each concerned pixel (similar to [5, 8]). Below, we present the results obtained with texture oper­ators having one of the following structures: linear filtering step only (see the first column in Table 1), linear filtering followed by one-step post processing (see the second and third columns in Table 1 and the first columns in Tables 2, 3, and 4), and linear filtering followed by two-step post­processing (see the second and third columns in Tables 2, 3, and 4). In all tables, X and 1.1 designate thresholding and modu­lus computation, respectively. T ransforms/filters NLPO X 1·1 Identity transform 1.31 1.31 1.31 Laws' filters 1.31 2.10 2.62 DST 1.3 1 1.85 2.59 DCT 1.31 1.84 2.29 DEST 1.31 1.64 2.24 DREFT 1.31 2.11 3.04 DROFT 1.31 1.83 2.43 Lin. sym. Gabor filters 1.31 1.54 1.80 Lin. antisym. Gabor filters 1.21 2.03 2.14 Table 1. Average Mahalanobis distances for features obtained by linear filtering only (first column) and for features obtained by linear fil­tering followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 2.29 2.29 2.29 Laws' filters 2.30 9.42 10.13 DST 2.26 9.10 9.98 DCT 2.28 8.92 9.30 DEST 2.27 8.21 9.08 DREFT 2.28 9.62 10.23 DROFT 2.28 9.19 9.95 Lin. sym. Gabor filters 2.42 11.80 11.96 Lin. antisym. Gabor filters 2.61 15.26 16.28 Table 2. Average Mahalanobis distances for features obtained by computing the local mean of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). 4. Comparison The feature vectors computed in different points of a tex­ture image using a given operator are not identical; they rather form a cluster in the multi-dimensional feature space. It is desirable that feature clusters, which correspond to dif­ferent textures, be separable. The larger this separability the better the operator used for texture feature computation. As a measure of cluster separability we use the Mahalanobis distance [3, 4]. We evaluated the performance of the various operators presented in the previous sections by looking at the pair­wise separability of the clusters of feature vectors obtained from nine test images, each containing a single oriented tex­ture (Fig. 2). To build a cluster, one thousand feature vectors 887 Transforms/filters NLPO X 1·1 Identity transform 1.47 1.47 1.47 Laws ' filters 6.54 6.14 5.84 DST 6.72 6.64 6.78 DCT 6.80 6.81 6.84 DEST 6.60 6.35 6.37 DREFT 6.77 6.36 6.02 DROFT 6.63 6.41 6.53 Lin. sym. Gabor filters 10.76 10.14 11.16 Lin. antisym. Gabor filters 10.87 9.65 9.55 Table 3. Average Mahalanobis distances for features obtained by computing the local vari­ance of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 1.56 1.56 1.56 Laws' filters 8.99 8.52 8.00 DST 9.28 9.12 9.18 DCT 9.14 9.23 8.94 DEST 8.95 8.35 8.39 DREFT 9.33 8.81 8.29 DROFT 9.08 8.60 8.75 Lin. sym. Gabor filters 15.50 14.43 15.23 Lin. antisym. Gabor filters 15.38 13.74 13.61 Table 4. Average Mahalanobis distances for features obtained by computing the local standard dev. of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). were taken at random positions from each texture image. The Mahalanobis distance was computed for every pair of texture images and for each operator. For brevity, only the averages of the 36 Mahalanobis distances computed with each operator are given here: each number in the Tables 1 to 4 is the average of 36 Mahalanobis distances between nine clusters of feature vectors computed with a given method. Regarding the statistical interpretation of the given val­ues in terms of misclassification probability, two clusters having normal distributions with the same standard devi­ation and a value of the Mahalanobis distance of 1.31 -see, for example, the first column of Table 1 - overlap for 25.89%. For Mahalanobis distance of 2 the overlap is 14.76%, while for values of the Mahalanobis distance greater than 5 the overlap is smaller than 0.1 % which means that the clusters are well separable. Figure 2. Test images used in the experi­ments. 5. Conclusions From Tables I, 2, 3, and 4 one can deduce the sig­nificance of each processing step of the concerned texture operators. The main conclusion that can be drawn is that post-processing can improve considerably the performance of filter based texture operators. Post-processing schemes, which include the computation of local statistics, perform better than those comprising an NLPO only do. Depending on the type of local statistics, the presence of an NLPO in a post-processing scheme can improve or degrade results. Computing the local mean, for instance, gives very good results if it is preceded by an NLPO (compare second and third column of Table 2 with the first columns of Tables 1 and 2). Similar improvement is achieved by local stan­dard deviation computation. In the latter case, no NLPO is needed between the filtering step and the local standard de­viation computation: such an intermediate step has negative effect on the results (compare the first column of Table 4 with the second and third columns of the same table). Computing the standard deviation gives better results than computing the variance (compare Tables 3 and 4). In both cases, the use of an intermediate NLPO degrades the results (compare first columns with the second and the third columns in Tables 3 and 4). On average, taking the modulus as a point operation and computing the local mean gives the best results (see last column of Table 2). 888 With a given type of post-processing the different filter­ing schemes lead to different results. The identity trans­form gives the worst results: the chance of misclassification for two normally distributed clusters with a Mahalanobis distance of 1.56 is 21. 79%. With an average Mahalanobis distance of 9 which corresponds to a misclassification of 10-7%, all sinusoidal transforms and the Laws' filter bank give comparable results. Still, they are worse than those obtained with Gabor filters, where the average Mahalanobis distance of 15 corresponds to a misclassification probability of 10-12%. The differences between the results obtained with Ga­bor filter banks and the other filter banks can be due to the differences in the tiling of the frequency domain. All of the sinusoidal transforms result in filters with a rectan­gular power spectrum with only two possible orientations (0°, 90°) while Gabor filters have an elliptical power spec­trum with eight possible orientations. Taking in consider­ation that the test material contains oriented textures only, it seems plausible to ex plain the good results obtained with Gabor filters by their orientation selectivity. References [1) A. Jain. A sinusoidal family of unitary transforms. IEEE­PAMI, 1(4):356--365, 1979. [2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation using gabor filters. Pattern Recognition, 24(12):1167-1186, 1991. [3) P. Kruizinga and N. Petkov. Grating cell operator features for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 1010--1014, Brisbane Australia, 1998. [4) P. Kruizinga and N. Petkov. Non-linear operator for oriented texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 1999. [51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on Image Processing for Missile Guidance, volume 238, pages 376--380, 1980. (6) J. Malik and P. Perona. Pre attentive texture discrimination with early vision mechanisms. Journal a/the Optical Society of America, A, 7(5):923-932, 1990. [7] N. Petkov and P. Kruizinga. Computational models of visual neurons specialized in the detection of periodic and aperiodic oriented visual stimuli: Bar and grating cells. Biological Cy­bernetics, 76:83-96, 1997. [8] T. Randen and 1. H. Husoy. Filte ring for texture classification: a comparative study. IEEE-PAMl, 21(4):291-310, 1999. [9] M. Unser. Local lin ear transfoffils for texture measurements. Signal Processing, 11(1):61-79 , 1986. 

English

A comparative study of filter based texture operators using Mahalanobis distance

University of Groningen Digital Archive

ARTS repository - University of Groningen

   University of GroningenA comparative study of filter based texture operators using Mahalanobis distanceGrigorescu, S.E.; Petkov, N.; Kruizinga, P.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operatorsusing Mahalanobis distance. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). IEEE (The Institute ofElectrical and Electronics Engineers).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 20-06-2022A comparative study of filter based texture operators using Mahalanobis distance S. E. Grigorescu, N. Petkov, and P. Kruizinga Institute of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl Abstract Texture feature extraction operators, which comprise lin­ear filtering. eventually followed by post-processing, are considered. The filters used are Laws' masks. filters de­rived from well-known discrete transforms. and Gabor fil­ters. The post-processing step comprises non- linear point operations and/or local statistics computation. The pelfor­mance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture oper­ators. 1. Introduction A number of texture feature extraction operators have been proposed in the literature which are similar in that they comprise two processing steps: (i) linear filtering followed by (ii) post-processing (Fig. 1). The post-processing step typically involves a non-linear point operation followed by the computation of some local statistics. Laws proposed a specific type of linear filtering followed by smoothing based on local averaging applied to the ab­solute values of the filter output [5]. Unser used for filter­ing various well-known transforms like discrete sine (DST), discrete cosine (DCT), discrete even sine (DEST), discrete real even Fourier (DREFT) and discrete real odd Fourier (DROFT) transforms, and for post-processing the computa­tion of channel variances [9]. In [2, 6], a Gabor filter bank followed by thresholding was used. For further references on texture operators, which comprise filtering followed by post-processing, see [8]. Different filters and different types of post-processing are used in the methods mentioned above. Since the re­sults achieved with these methods are also different, a natu­ral question arises of which filter scheme and which type of post-processing give the best results. 0-7695-0750-6/00 $10.00 © 2000 IEEE 885 Texture operator �F'lt _I Post- I:  Iprocessingl � :--. -, I Post- I f\�;:; #� I FIlter l -IProcessingl -r L-/ i i /: Feature ; ; : vectors . . . . �F'l: I P�st- I i : .. �.�.��t.��:=:!����_���!���nj Figure 1. General structure of filter based tex­ture operators. Previous studies in this direction typically focus on the joint performance of the linear filtering, the post -processing and a subsequent classification or segmentation stage (for references see [4, 8]). In a very thorough recent work, Ran­den and HUS0Y [8] bring systematics into the matter by fix­ing the type of post-processing and comparing the effect of different linear filtering schemes. Similar to previous stud­ies they evaluate the performance of texture operators on the basis of the classification and segmentation results that are obtained when the feature vectors are fed into a given clas­sifier. Such an evaluation has the drawback that it does not measure the performance of the texture operator only, but the joint performance of the texture operator and a subse­quent classifier [4]. Moreover these studies do not evaluate the contribution of each individual processing step to the overall performance of the texture feature operator. For evaluation of the performance of texture operators, we use a new method that was proposed elsewhere [4]. It is based on a statistical approach (Fisher criterion and Ma­halanobis distance) to evaluate the capability of a feature operator to generate discriminable feature vectors for dif­ferent textures. This method can be used to compare the texture operators only, regardless of any subsequent classi­fication or segmentation operation. Furthermore, this eval-uation method can be applied after each processing step so that the contribution of each step to the overall performance can be assessed. The rest of the paper is organized as follows. In Section 2 an overview of the filtering schemes used in our exper­iments is given. In Section 3 we discuss different types of post-processing. Comparison criteria and experimental results are presented in Section 4. Finally, in Section 5 conclusions are drawn. 2. Filter banks For linear filtering Laws [5] used a set of square convolu­tion kernels obtained as the outer product of pairs of vectors from a set of n n-dimensional vectors. He conducted exper­iments for n = 3, 5, and 7 and obtained similar results. In our experiments we used Laws' convolution kernels derived from the following set of n = 5 vectors [51: l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 [-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 [1, -4,6, -4, 1]: resulting in a bank of 25 filters. Unser [9] used several different filter banks. Similar to Laws' approach, the convolution kernels proposed by Unser are computed as the outer product of pairs of vectors. Each set of vectors used in Unser's experiments corresponds to one of the following transforms: identity (shift) transform, DST, DCT, DEST, DREFT,and DROFT. More details about these transforms and the filter banks derived from them can be found in ll, 9J. We conducted experiments with all the filter banks derived from the transforms mentioned above. In all cases a set of 25 convolution kernels of size 5 x 5 was used for conformity with the experiments with Laws' oper­ator. Such a filter bank leads to a 25-dirnensional feature space. In [2, 6], filter banks based on Gabor function kernels were used. In our experiments we used two sets of Gabor filters: one employing symmetric Gabor function kernels (1) and the other employing antisymmetric Gabor function kernels (2). In both cases, we used 24 Gabor filters tuned to three preferred spatial frequencies (f = 2-ky'2, k = 0,1, and 2) and eight preferred orientations (e = m � , m = ° . . . 7). For all Gabor filters the product of the preferred spatial frequency f and the standard deviation (7 of the Gaussian factor was kept constant (f(7 = 2) which means that the filters have a constant spatial frequency bandwidth. More details about these types of Gabor filter banks can be found in [4, 7]. Besides these Gabor filters, the filter banks used in our experiments also comprise a Gaussian filter (3) for DC com­ponent computation. 886 ( X2 + (2y)2 ) g}� (i,j) = exp -2(72 . cos(27rfx) (a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 2172 • sm(27rfx where: . . ( i2+j2) 9(�,}) = exp -2ar ' i,j -2 ... 2 x i cos e + j sin e (71 = 3 Y = -i sin e + j cos e (1) (2) (3) This combination of Gabor and Gaussian filters results in banks of 25 filters with a coverage of the spatial frequency domain similar to that of Laws' and Unser's filter banks. 3. Post-processing We applied several types of post-processing to the out­put of the filter banks described in Section 2. The post­processing is applied individually to the output of each channel in a filter bank. The types of post-processing used in our experiments fall in two categories: one-step post-processing and two-step post-processing. The one-step post-processing comprises either a nonlinear point opera­tion (NLPO) or computation of local statistics. The two­step post-processing comprises an NLPO followed by com­putation of local statistics. In our experiments we used as NLPO thresholding and modulus computation. For local statistics we used mean, variance and standard deviation computation. In previous studies only the mean and the variance were used as statis­tical measures of local texture properties. We performed ex­periments with standard deviation as well because it offers a statistical characterization similar to that given by the vari­ance and, at the same time, it takes values in the same range as the one which is characteristic of averaging. All the lo­cal statistics are computed in a square neighborhood of size 17 x 17 around each concerned pixel (similar to [5, 8]). Below, we present the results obtained with texture oper­ators having one of the following structures: linear filtering step only (see the first column in Table 1), linear filtering followed by one-step post processing (see the second and third columns in Table 1 and the first columns in Tables 2, 3, and 4), and linear filtering followed by two-step post­processing (see the second and third columns in Tables 2, 3, and 4). In all tables, X and 1.1 designate thresholding and modu­lus computation, respectively. T ransforms/filters NLPO X 1·1 Identity transform 1.31 1.31 1.31 Laws' filters 1.31 2.10 2.62 DST 1.3 1 1.85 2.59 DCT 1.31 1.84 2.29 DEST 1.31 1.64 2.24 DREFT 1.31 2.11 3.04 DROFT 1.31 1.83 2.43 Lin. sym. Gabor filters 1.31 1.54 1.80 Lin. antisym. Gabor filters 1.21 2.03 2.14 Table 1. Average Mahalanobis distances for features obtained by linear filtering only (first column) and for features obtained by linear fil­tering followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 2.29 2.29 2.29 Laws' filters 2.30 9.42 10.13 DST 2.26 9.10 9.98 DCT 2.28 8.92 9.30 DEST 2.27 8.21 9.08 DREFT 2.28 9.62 10.23 DROFT 2.28 9.19 9.95 Lin. sym. Gabor filters 2.42 11.80 11.96 Lin. antisym. Gabor filters 2.61 15.26 16.28 Table 2. Average Mahalanobis distances for features obtained by computing the local mean of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). 4. Comparison The feature vectors computed in different points of a tex­ture image using a given operator are not identical; they rather form a cluster in the multi-dimensional feature space. It is desirable that feature clusters, which correspond to dif­ferent textures, be separable. The larger this separability the better the operator used for texture feature computation. As a measure of cluster separability we use the Mahalanobis distance [3, 4]. We evaluated the performance of the various operators presented in the previous sections by looking at the pair­wise separability of the clusters of feature vectors obtained from nine test images, each containing a single oriented tex­ture (Fig. 2). To build a cluster, one thousand feature vectors 887 Transforms/filters NLPO X 1·1 Identity transform 1.47 1.47 1.47 Laws ' filters 6.54 6.14 5.84 DST 6.72 6.64 6.78 DCT 6.80 6.81 6.84 DEST 6.60 6.35 6.37 DREFT 6.77 6.36 6.02 DROFT 6.63 6.41 6.53 Lin. sym. Gabor filters 10.76 10.14 11.16 Lin. antisym. Gabor filters 10.87 9.65 9.55 Table 3. Average Mahalanobis distances for features obtained by computing the local vari­ance of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 1.56 1.56 1.56 Laws' filters 8.99 8.52 8.00 DST 9.28 9.12 9.18 DCT 9.14 9.23 8.94 DEST 8.95 8.35 8.39 DREFT 9.33 8.81 8.29 DROFT 9.08 8.60 8.75 Lin. sym. Gabor filters 15.50 14.43 15.23 Lin. antisym. Gabor filters 15.38 13.74 13.61 Table 4. Average Mahalanobis distances for features obtained by computing the local standard dev. of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). were taken at random positions from each texture image. The Mahalanobis distance was computed for every pair of texture images and for each operator. For brevity, only the averages of the 36 Mahalanobis distances computed with each operator are given here: each number in the Tables 1 to 4 is the average of 36 Mahalanobis distances between nine clusters of feature vectors computed with a given method. Regarding the statistical interpretation of the given val­ues in terms of misclassification probability, two clusters having normal distributions with the same standard devi­ation and a value of the Mahalanobis distance of 1.31 -see, for example, the first column of Table 1 - overlap for 25.89%. For Mahalanobis distance of 2 the overlap is 14.76%, while for values of the Mahalanobis distance greater than 5 the overlap is smaller than 0.1 % which means that the clusters are well separable. Figure 2. Test images used in the experi­ments. 5. Conclusions From Tables I, 2, 3, and 4 one can deduce the sig­nificance of each processing step of the concerned texture operators. The main conclusion that can be drawn is that post-processing can improve considerably the performance of filter based texture operators. Post-processing schemes, which include the computation of local statistics, perform better than those comprising an NLPO only do. Depending on the type of local statistics, the presence of an NLPO in a post-processing scheme can improve or degrade results. Computing the local mean, for instance, gives very good results if it is preceded by an NLPO (compare second and third column of Table 2 with the first columns of Tables 1 and 2). Similar improvement is achieved by local stan­dard deviation computation. In the latter case, no NLPO is needed between the filtering step and the local standard de­viation computation: such an intermediate step has negative effect on the results (compare the first column of Table 4 with the second and third columns of the same table). Computing the standard deviation gives better results than computing the variance (compare Tables 3 and 4). In both cases, the use of an intermediate NLPO degrades the results (compare first columns with the second and the third columns in Tables 3 and 4). On average, taking the modulus as a point operation and computing the local mean gives the best results (see last column of Table 2). 888 With a given type of post-processing the different filter­ing schemes lead to different results. The identity trans­form gives the worst results: the chance of misclassification for two normally distributed clusters with a Mahalanobis distance of 1.56 is 21. 79%. With an average Mahalanobis distance of 9 which corresponds to a misclassification of 10-7%, all sinusoidal transforms and the Laws' filter bank give comparable results. Still, they are worse than those obtained with Gabor filters, where the average Mahalanobis distance of 15 corresponds to a misclassification probability of 10-12%. The differences between the results obtained with Ga­bor filter banks and the other filter banks can be due to the differences in the tiling of the frequency domain. All of the sinusoidal transforms result in filters with a rectan­gular power spectrum with only two possible orientations (0°, 90°) while Gabor filters have an elliptical power spec­trum with eight possible orientations. Taking in consider­ation that the test material contains oriented textures only, it seems plausible to ex plain the good results obtained with Gabor filters by their orientation selectivity. References [1) A. Jain. A sinusoidal family of unitary transforms. IEEE­PAMI, 1(4):356--365, 1979. [2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation using gabor filters. Pattern Recognition, 24(12):1167-1186, 1991. [3) P. Kruizinga and N. Petkov. Grating cell operator features for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 1010--1014, Brisbane Australia, 1998. [4) P. Kruizinga and N. Petkov. Non-linear operator for oriented texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 1999. [51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on Image Processing for Missile Guidance, volume 238, pages 376--380, 1980. (6) J. Malik and P. Perona. Pre attentive texture discrimination with early vision mechanisms. Journal a/the Optical Society of America, A, 7(5):923-932, 1990. [7] N. Petkov and P. Kruizinga. Computational models of visual neurons specialized in the detection of periodic and aperiodic oriented visual stimuli: Bar and grating cells. Biological Cy­bernetics, 76:83-96, 1997. [8] T. Randen and 1. H. Husoy. Filte ring for texture classification: a comparative study. IEEE-PAMl, 21(4):291-310, 1999. [9] M. Unser. Local lin ear transfoffils for texture measurements. Signal Processing, 11(1):61-79 , 1986. 

University of Groningen Research Database

  
 University of Groningen
A comparative study of filter based texture operators using Mahalanobis distance
Grigorescu, S.E.; Petkov, Nicolai; Kruizinga, P.
Published in:
15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS
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Citation for published version (APA):
Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operators
using Mahalanobis distance. In A. Sanfeliu, J. J. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra
(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS
(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). LOS ALAMITOS: IEEE
(The Institute of Electrical and Electronics Engineers).
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A comparative study of filter based texture operators using Mahalanobis distance 
S. E. Grigorescu, N. Petkov, and P. Kruizinga 
Institute of Mathematics and Computing Science 
University of Groningen 
P.O. Box 800 9700 AV Groningen The Netherlands 
simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl 
Abstract 
Texture feature extraction operators, which comprise lin­
ear filtering. eventually followed by post-processing, are 
considered. The filters used are Laws' masks. filters de­
rived from well-known discrete transforms. and Gabor fil­
ters. The post-processing step comprises non- linear point 
operations and/or local statistics computation. The pelfor­
mance is measured by means of the Mahalanobis distance 
between clusters of feature vectors derived from different 
textures. The results show that post-processing improve 
considerably the performance of filter based texture oper­
ators. 
1. Introduction 
A number of texture feature extraction operators have 
been proposed in the literature which are similar in that they 
comprise two processing steps: (i) linear filtering followed 
by (ii) post-processing (Fig. 1). The post-processing step 
typically involves a non-linear point operation followed by 
the computation of some local statistics. 
Laws proposed a specific type of linear filtering followed 
by smoothing based on local averaging applied to the ab­
solute values of the filter output [5]. Unser used for filter­
ing various well-known transforms like discrete sine (DST), 
discrete cosine (DCT), discrete even sine (DEST), discrete 
real even Fourier (DREFT) and discrete real odd Fourier 
(DROFT) transforms, and for post-processing the computa­
tion of channel variances [9]. In [2, 6], a Gabor filter bank 
followed by thresholding was used. For further references 
on texture operators, which comprise filtering followed by 
post-processing, see [8]. 
Different filters and different types of post-processing 
are used in the methods mentioned above. Since the re­
sults achieved with these methods are also different, a natu­
ral question arises of which filter scheme and which type of 
post-processing give the best results. 
0-7695-0750-6/00 $10.00 © 2000 IEEE 885 
Texture operator 
�F'lt _I Post- I:  Iprocessingl � 
:--. -, I Post- I f\�;:; #� 
I FIlter l -IProcessingl -r L-/ i i /: Feature 
; ; : vectors . . . . 
�F'l: I P�st- I i 
: .. �.�.��t.��:=:!����_���!���nj 
Figure 1. General structure of filter based tex­
ture operators. 
Previous studies in this direction typically focus on the 
joint performance of the linear filtering, the post -processing 
and a subsequent classification or segmentation stage (for 
references see [4, 8]). In a very thorough recent work, Ran­
den and HUS0Y [8] bring systematics into the matter by fix­
ing the type of post-processing and comparing the effect of 
different linear filtering schemes. Similar to previous stud­
ies they evaluate the performance of texture operators on the 
basis of the classification and segmentation results that are 
obtained when the feature vectors are fed into a given clas­
sifier. Such an evaluation has the drawback that it does not 
measure the performance of the texture operator only, but 
the joint performance of the texture operator and a subse­
quent classifier [4]. Moreover these studies do not evaluate 
the contribution of each individual processing step to the 
overall performance of the texture feature operator. 
For evaluation of the performance of texture operators, 
we use a new method that was proposed elsewhere [4]. It 
is based on a statistical approach (Fisher criterion and Ma­
halanobis distance) to evaluate the capability of a feature 
operator to generate discriminable feature vectors for dif­
ferent textures. This method can be used to compare the 
texture operators only, regardless of any subsequent classi­
fication or segmentation operation. Furthermore, this eval-
uation method can be applied after each processing step so 
that the contribution of each step to the overall performance 
can be assessed. 
The rest of the paper is organized as follows. In Section 
2 an overview of the filtering schemes used in our exper­
iments is given. In Section 3 we discuss different types 
of post-processing. Comparison criteria and experimental 
results are presented in Section 4. Finally, in Section 5 
conclusions are drawn. 
2. Filter banks 
For linear filtering Laws [5] used a set of square convolu­
tion kernels obtained as the outer product of pairs of vectors 
from a set of n n-dimensional vectors. He conducted exper­
iments for n = 3, 5, and 7 and obtained similar results. In 
our experiments we used Laws' convolution kernels derived 
from the following set of n = 5 vectors [51: 
l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 
[-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 
[1, -4,6, -4, 1]: resulting in a bank of 25 filters. 
Unser [9] used several different filter banks. Similar to 
Laws' approach, the convolution kernels proposed by Unser 
are computed as the outer product of pairs of vectors. Each 
set of vectors used in Unser's experiments corresponds to 
one of the following transforms: identity (shift) transform, 
DST, DCT, DEST, DREFT,and DROFT. More details about 
these transforms and the filter banks derived from them can 
be found in ll, 9J. We conducted experiments with all the 
filter banks derived from the transforms mentioned above. 
In all cases a set of 25 convolution kernels of size 5 x 5 was 
used for conformity with the experiments with Laws' oper­
ator. Such a filter bank leads to a 25-dirnensional feature 
space. 
In [2, 6], filter banks based on Gabor function kernels 
were used. In our experiments we used two sets of Gabor 
filters: one employing symmetric Gabor function kernels 
(1) and the other employing antisymmetric Gabor function 
kernels (2). In both cases, we used 24 Gabor filters tuned to 
three preferred spatial frequencies (f = 2-ky'2, k = 0,1, 
and 2) and eight preferred orientations (e = m � , m = 
° . . . 7). For all Gabor filters the product of the preferred 
spatial frequency f and the standard deviation (7 of the 
Gaussian factor was kept constant (f(7 = 2) which means 
that the filters have a constant spatial frequency bandwidth. 
More details about these types of Gabor filter banks can be 
found in [4, 7]. 
Besides these Gabor filters, the filter banks used in our 
experiments also comprise a Gaussian filter (3) for DC com­
ponent computation. 
886 
( X2 + (2y)2 ) 
g}� (i,j) 
= 
exp -
2(72 
. cos(27rfx) 
(a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 
2172 • sm(27rfx 
where: 
. . 
( 
i2+j2
) 9(�,}) = exp - 2ar ' 
i,j -2 ... 2 
x i cos e + j sin e 
(71 = 3 
Y = -i sin e + j cos e 
(1) 
(2) 
(3) 
This combination of Gabor and Gaussian filters results in 
banks of 25 filters with a coverage of the spatial frequency 
domain similar to that of Laws' and Unser's filter banks. 
3. Post-processing 
We applied several types of post-processing to the out­
put of the filter banks described in Section 2. The post­
processing is applied individually to the output of each 
channel in a filter bank. The types of post-processing 
used in our experiments fall in two categories: one-step 
post-processing and two-step post-processing. The one-step 
post-processing comprises either a nonlinear point opera­
tion (NLPO) or computation of local statistics. The two­
step post-processing comprises an NLPO followed by com­
putation of local statistics. 
In our experiments we used as NLPO thresholding and 
modulus computation. For local statistics we used mean, 
variance and standard deviation computation. In previous 
studies only the mean and the variance were used as statis­
tical measures of local texture properties. We performed ex­
periments with standard deviation as well because it offers a 
statistical characterization similar to that given by the vari­
ance and, at the same time, it takes values in the same range 
as the one which is characteristic of averaging. All the lo­
cal statistics are computed in a square neighborhood of size 
17 x 17 around each concerned pixel (similar to [5, 8]). 
Below, we present the results obtained with texture oper­
ators having one of the following structures: linear filtering 
step only (see the first column in Table 1), linear filtering 
followed by one-step post processing (see the second and 
third columns in Table 1 and the first columns in Tables 2, 
3, and 4), and linear filtering followed by two-step post­
processing (see the second and third columns in Tables 2, 
3, and 4). 
In all tables, X and 1.1 designate thresholding and modu­
lus computation, respectively. 
T ransforms/filters NLPO 
X 1·1 
Identity transform 1.31 1.31 1.31 
Laws' filters 1.31 2.10 2.62 
DST 1.3 1 1.85 2.59 
DCT 1.31 1.84 2.29 
DEST 1.31 1.64 2.24 
DREFT 1.31 2.11 3.04 
DROFT 1.31 1.83 2.43 
Lin. sym. Gabor filters 1.31 1.54 1.80 
Lin. antisym. Gabor filters 1.21 2.03 2.14 
Table 1. Average Mahalanobis distances for 
features obtained by linear filtering only (first 
column) and for features obtained by linear fil­
tering followed by an NLPO (second and third 
columns). 
Transforms/filters NLPO 
X 1·1 
Identity transform 2.29 2.29 2.29 
Laws' filters 2.30 9.42 10.13 
DST 2.26 9.10 9.98 
DCT 2.28 8.92 9.30 
DEST 2.27 8.21 9.08 
DREFT 2.28 9.62 10.23 
DROFT 2.28 9.19 9.95 
Lin. sym. Gabor filters 2.42 11.80 11.96 
Lin. antisym. Gabor filters 2.61 15.26 16.28 
Table 2. Average Mahalanobis distances for 
features obtained by computing the local 
mean of the output of: (i) linear filters (first 
column) and (ii) linear filters followed by an 
NLPO (second and third columns). 
4. Comparison 
The feature vectors computed in different points of a tex­
ture image using a given operator are not identical; they 
rather form a cluster in the multi-dimensional feature space. 
It is desirable that feature clusters, which correspond to dif­
ferent textures, be separable. The larger this separability the 
better the operator used for texture feature computation. As 
a measure of cluster separability we use the Mahalanobis 
distance [3, 4]. 
We evaluated the performance of the various operators 
presented in the previous sections by looking at the pair­
wise separability of the clusters of feature vectors obtained 
from nine test images, each containing a single oriented tex­
ture (Fig. 2). To build a cluster, one thousand feature vectors 
887 
Transforms/filters NLPO 
X 1·1 
Identity transform 1.47 1.47 1.47 
Laws ' filters 6.54 6.14 5.84 
DST 6.72 6.64 6.78 
DCT 6.80 6.81 6.84 
DEST 6.60 6.35 6.37 
DREFT 6.77 6.36 6.02 
DROFT 6.63 6.41 6.53 
Lin. sym. Gabor filters 10.76 10.14 11.16 
Lin. antisym. Gabor filters 10.87 9.65 9.55 
Table 3. Average Mahalanobis distances for 
features obtained by computing the local vari­
ance of the output of: (i) linear filters (first 
column) and (ii) linear filters followed by an 
NLPO (second and third columns). 
Transforms/filters NLPO 
X 1·1 
Identity transform 1.56 1.56 1.56 
Laws' filters 8.99 8.52 8.00 
DST 9.28 9.12 9.18 
DCT 9.14 9.23 8.94 
DEST 8.95 8.35 8.39 
DREFT 9.33 8.81 8.29 
DROFT 9.08 8.60 8.75 
Lin. sym. Gabor filters 15.50 14.43 15.23 
Lin. antisym. Gabor filters 15.38 13.74 13.61 
Table 4. Average Mahalanobis distances for 
features obtained by computing the local 
standard dev. of the output of: (i) linear filters 
(first column) and (ii) linear filters followed by 
an NLPO (second and third columns). 
were taken at random positions from each texture image. 
The Mahalanobis distance was computed for every pair of 
texture images and for each operator. For brevity, only the 
averages of the 36 Mahalanobis distances computed with 
each operator are given here: each number in the Tables 1 to 
4 is the average of 36 Mahalanobis distances between nine 
clusters of feature vectors computed with a given method. 
Regarding the statistical interpretation of the given val­
ues in terms of misclassification probability, two clusters 
having normal distributions with the same standard devi­
ation and a value of the Mahalanobis distance of 1.31 -
see, for example, the first column of Table 1 - overlap 
for 25.89%. For Mahalanobis distance of 2 the overlap 
is 14.76%, while for values of the Mahalanobis distance 
greater than 5 the overlap is smaller than 0.1 % which means 
that the clusters are well separable. 
Figure 2. Test images used in the experi­
ments. 
5. Conclusions 
From Tables I, 2, 3, and 4 one can deduce the sig­
nificance of each processing step of the concerned texture 
operators. The main conclusion that can be drawn is that 
post-processing can improve considerably the performance 
of filter based texture operators. Post-processing schemes, 
which include the computation of local statistics, perform 
better than those comprising an NLPO only do. Depending 
on the type of local statistics, the presence of an NLPO in 
a post-processing scheme can improve or degrade results. 
Computing the local mean, for instance, gives very good 
results if it is preceded by an NLPO (compare second and 
third column of Table 2 with the first columns of Tables 
1 and 2). Similar improvement is achieved by local stan­
dard deviation computation. In the latter case, no NLPO is 
needed between the filtering step and the local standard de­
viation computation: such an intermediate step has negative 
effect on the results (compare the first column of Table 4 
with the second and third columns of the same table). 
Computing the standard deviation gives better results 
than computing the variance (compare Tables 3 and 4). 
In both cases, the use of an intermediate NLPO degrades 
the results (compare first columns with the second and the 
third columns in Tables 3 and 4). 
On average, taking the modulus as a point operation and 
computing the local mean gives the best results (see last 
column of Table 2). 
888 
With a given type of post-processing the different filter­
ing schemes lead to different results. The identity trans­
form gives the worst results: the chance of misclassification 
for two normally distributed clusters with a Mahalanobis 
distance of 1.56 is 21. 79%. With an average Mahalanobis 
distance of 9 which corresponds to a misclassification of 
10-7%, all sinusoidal transforms and the Laws' filter bank 
give comparable results. Still, they are worse than those 
obtained with Gabor filters, where the average Mahalanobis 
distance of 15 corresponds to a misclassification probability 
of 10-12%. 
The differences between the results obtained with Ga­
bor filter banks and the other filter banks can be due to 
the differences in the tiling of the frequency domain. All 
of the sinusoidal transforms result in filters with a rectan­
gular power spectrum with only two possible orientations 
(0°, 90°) while Gabor filters have an elliptical power spec­
trum with eight possible orientations. Taking in consider­
ation that the test material contains oriented textures only, 
it seems plausible to ex plain the good results obtained with 
Gabor filters by their orientation selectivity. 
References 
[1) A. Jain. A sinusoidal family of unitary transforms. IEEE­
PAMI, 1(4):356--365, 1979. 
[2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation 
using gabor filters. Pattern Recognition, 24(12):1167-1186, 
1991. 
[3) P. Kruizinga and N. Petkov. Grating cell operator features 
for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, 
editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 
1010--1014, Brisbane Australia, 1998. 
[4) P. Kruizinga and N. Petkov. Non-linear operator for oriented 
texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 
1999. 
[51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on 
Image Processing for Missile Guidance, volume 238, pages 
376--380, 1980. 
(6) J. Malik and P. Perona. Pre attentive texture discrimination 
with early vision mechanisms. Journal a/the Optical Society 
of America, A, 7(5):923-932, 1990. 
[7] N. Petkov and P. Kruizinga. Computational models of visual 
neurons specialized in the detection of periodic and aperiodic 
oriented visual stimuli: Bar and grating cells. Biological Cy­
bernetics, 76:83-96, 1997. 
[8] T. Randen and 1. H. Husoy. Filte ring for texture classification: 
a comparative study. IEEE-PAMl, 21(4):291-310, 1999. 
[9] M. Unser. Local lin ear transfoffils for texture measurements. 
Signal Processing, 11(1):61-79 , 1986. 


   University of GroningenA comparative study of filter based texture operators using Mahalanobis distanceGrigorescu, S.E.; Petkov, N.; Kruizinga, P.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operatorsusing Mahalanobis distance. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). IEEE (The Institute ofElectrical and Electronics Engineers).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 03-06-2022A comparative study of filter based texture operators using Mahalanobis distance S. E. Grigorescu, N. Petkov, and P. Kruizinga Institute of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl Abstract Texture feature extraction operators, which comprise lin­ear filtering. eventually followed by post-processing, are considered. The filters used are Laws' masks. filters de­rived from well-known discrete transforms. and Gabor fil­ters. The post-processing step comprises non- linear point operations and/or local statistics computation. The pelfor­mance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture oper­ators. 1. Introduction A number of texture feature extraction operators have been proposed in the literature which are similar in that they comprise two processing steps: (i) linear filtering followed by (ii) post-processing (Fig. 1). The post-processing step typically involves a non-linear point operation followed by the computation of some local statistics. Laws proposed a specific type of linear filtering followed by smoothing based on local averaging applied to the ab­solute values of the filter output [5]. Unser used for filter­ing various well-known transforms like discrete sine (DST), discrete cosine (DCT), discrete even sine (DEST), discrete real even Fourier (DREFT) and discrete real odd Fourier (DROFT) transforms, and for post-processing the computa­tion of channel variances [9]. In [2, 6], a Gabor filter bank followed by thresholding was used. For further references on texture operators, which comprise filtering followed by post-processing, see [8]. Different filters and different types of post-processing are used in the methods mentioned above. Since the re­sults achieved with these methods are also different, a natu­ral question arises of which filter scheme and which type of post-processing give the best results. 0-7695-0750-6/00 $10.00 © 2000 IEEE 885 Texture operator �F'lt _I Post- I:  Iprocessingl � :--. -, I Post- I f\�;:; #� I FIlter l -IProcessingl -r L-/ i i /: Feature ; ; : vectors . . . . �F'l: I P�st- I i : .. �.�.��t.��:=:!����_���!���nj Figure 1. General structure of filter based tex­ture operators. Previous studies in this direction typically focus on the joint performance of the linear filtering, the post -processing and a subsequent classification or segmentation stage (for references see [4, 8]). In a very thorough recent work, Ran­den and HUS0Y [8] bring systematics into the matter by fix­ing the type of post-processing and comparing the effect of different linear filtering schemes. Similar to previous stud­ies they evaluate the performance of texture operators on the basis of the classification and segmentation results that are obtained when the feature vectors are fed into a given clas­sifier. Such an evaluation has the drawback that it does not measure the performance of the texture operator only, but the joint performance of the texture operator and a subse­quent classifier [4]. Moreover these studies do not evaluate the contribution of each individual processing step to the overall performance of the texture feature operator. For evaluation of the performance of texture operators, we use a new method that was proposed elsewhere [4]. It is based on a statistical approach (Fisher criterion and Ma­halanobis distance) to evaluate the capability of a feature operator to generate discriminable feature vectors for dif­ferent textures. This method can be used to compare the texture operators only, regardless of any subsequent classi­fication or segmentation operation. Furthermore, this eval-uation method can be applied after each processing step so that the contribution of each step to the overall performance can be assessed. The rest of the paper is organized as follows. In Section 2 an overview of the filtering schemes used in our exper­iments is given. In Section 3 we discuss different types of post-processing. Comparison criteria and experimental results are presented in Section 4. Finally, in Section 5 conclusions are drawn. 2. Filter banks For linear filtering Laws [5] used a set of square convolu­tion kernels obtained as the outer product of pairs of vectors from a set of n n-dimensional vectors. He conducted exper­iments for n = 3, 5, and 7 and obtained similar results. In our experiments we used Laws' convolution kernels derived from the following set of n = 5 vectors [51: l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 [-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 [1, -4,6, -4, 1]: resulting in a bank of 25 filters. Unser [9] used several different filter banks. Similar to Laws' approach, the convolution kernels proposed by Unser are computed as the outer product of pairs of vectors. Each set of vectors used in Unser's experiments corresponds to one of the following transforms: identity (shift) transform, DST, DCT, DEST, DREFT,and DROFT. More details about these transforms and the filter banks derived from them can be found in ll, 9J. We conducted experiments with all the filter banks derived from the transforms mentioned above. In all cases a set of 25 convolution kernels of size 5 x 5 was used for conformity with the experiments with Laws' oper­ator. Such a filter bank leads to a 25-dirnensional feature space. In [2, 6], filter banks based on Gabor function kernels were used. In our experiments we used two sets of Gabor filters: one employing symmetric Gabor function kernels (1) and the other employing antisymmetric Gabor function kernels (2). In both cases, we used 24 Gabor filters tuned to three preferred spatial frequencies (f = 2-ky'2, k = 0,1, and 2) and eight preferred orientations (e = m � , m = ° . . . 7). For all Gabor filters the product of the preferred spatial frequency f and the standard deviation (7 of the Gaussian factor was kept constant (f(7 = 2) which means that the filters have a constant spatial frequency bandwidth. More details about these types of Gabor filter banks can be found in [4, 7]. Besides these Gabor filters, the filter banks used in our experiments also comprise a Gaussian filter (3) for DC com­ponent computation. 886 ( X2 + (2y)2 ) g}� (i,j) = exp -2(72 . cos(27rfx) (a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 2172 • sm(27rfx where: . . ( i2+j2) 9(�,}) = exp -2ar ' i,j -2 ... 2 x i cos e + j sin e (71 = 3 Y = -i sin e + j cos e (1) (2) (3) This combination of Gabor and Gaussian filters results in banks of 25 filters with a coverage of the spatial frequency domain similar to that of Laws' and Unser's filter banks. 3. Post-processing We applied several types of post-processing to the out­put of the filter banks described in Section 2. The post­processing is applied individually to the output of each channel in a filter bank. The types of post-processing used in our experiments fall in two categories: one-step post-processing and two-step post-processing. The one-step post-processing comprises either a nonlinear point opera­tion (NLPO) or computation of local statistics. The two­step post-processing comprises an NLPO followed by com­putation of local statistics. In our experiments we used as NLPO thresholding and modulus computation. For local statistics we used mean, variance and standard deviation computation. In previous studies only the mean and the variance were used as statis­tical measures of local texture properties. We performed ex­periments with standard deviation as well because it offers a statistical characterization similar to that given by the vari­ance and, at the same time, it takes values in the same range as the one which is characteristic of averaging. All the lo­cal statistics are computed in a square neighborhood of size 17 x 17 around each concerned pixel (similar to [5, 8]). Below, we present the results obtained with texture oper­ators having one of the following structures: linear filtering step only (see the first column in Table 1), linear filtering followed by one-step post processing (see the second and third columns in Table 1 and the first columns in Tables 2, 3, and 4), and linear filtering followed by two-step post­processing (see the second and third columns in Tables 2, 3, and 4). In all tables, X and 1.1 designate thresholding and modu­lus computation, respectively. T ransforms/filters NLPO X 1·1 Identity transform 1.31 1.31 1.31 Laws' filters 1.31 2.10 2.62 DST 1.3 1 1.85 2.59 DCT 1.31 1.84 2.29 DEST 1.31 1.64 2.24 DREFT 1.31 2.11 3.04 DROFT 1.31 1.83 2.43 Lin. sym. Gabor filters 1.31 1.54 1.80 Lin. antisym. Gabor filters 1.21 2.03 2.14 Table 1. Average Mahalanobis distances for features obtained by linear filtering only (first column) and for features obtained by linear fil­tering followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 2.29 2.29 2.29 Laws' filters 2.30 9.42 10.13 DST 2.26 9.10 9.98 DCT 2.28 8.92 9.30 DEST 2.27 8.21 9.08 DREFT 2.28 9.62 10.23 DROFT 2.28 9.19 9.95 Lin. sym. Gabor filters 2.42 11.80 11.96 Lin. antisym. Gabor filters 2.61 15.26 16.28 Table 2. Average Mahalanobis distances for features obtained by computing the local mean of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). 4. Comparison The feature vectors computed in different points of a tex­ture image using a given operator are not identical; they rather form a cluster in the multi-dimensional feature space. It is desirable that feature clusters, which correspond to dif­ferent textures, be separable. The larger this separability the better the operator used for texture feature computation. As a measure of cluster separability we use the Mahalanobis distance [3, 4]. We evaluated the performance of the various operators presented in the previous sections by looking at the pair­wise separability of the clusters of feature vectors obtained from nine test images, each containing a single oriented tex­ture (Fig. 2). To build a cluster, one thousand feature vectors 887 Transforms/filters NLPO X 1·1 Identity transform 1.47 1.47 1.47 Laws ' filters 6.54 6.14 5.84 DST 6.72 6.64 6.78 DCT 6.80 6.81 6.84 DEST 6.60 6.35 6.37 DREFT 6.77 6.36 6.02 DROFT 6.63 6.41 6.53 Lin. sym. Gabor filters 10.76 10.14 11.16 Lin. antisym. Gabor filters 10.87 9.65 9.55 Table 3. Average Mahalanobis distances for features obtained by computing the local vari­ance of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 1.56 1.56 1.56 Laws' filters 8.99 8.52 8.00 DST 9.28 9.12 9.18 DCT 9.14 9.23 8.94 DEST 8.95 8.35 8.39 DREFT 9.33 8.81 8.29 DROFT 9.08 8.60 8.75 Lin. sym. Gabor filters 15.50 14.43 15.23 Lin. antisym. Gabor filters 15.38 13.74 13.61 Table 4. Average Mahalanobis distances for features obtained by computing the local standard dev. of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). were taken at random positions from each texture image. The Mahalanobis distance was computed for every pair of texture images and for each operator. For brevity, only the averages of the 36 Mahalanobis distances computed with each operator are given here: each number in the Tables 1 to 4 is the average of 36 Mahalanobis distances between nine clusters of feature vectors computed with a given method. Regarding the statistical interpretation of the given val­ues in terms of misclassification probability, two clusters having normal distributions with the same standard devi­ation and a value of the Mahalanobis distance of 1.31 -see, for example, the first column of Table 1 - overlap for 25.89%. For Mahalanobis distance of 2 the overlap is 14.76%, while for values of the Mahalanobis distance greater than 5 the overlap is smaller than 0.1 % which means that the clusters are well separable. Figure 2. Test images used in the experi­ments. 5. Conclusions From Tables I, 2, 3, and 4 one can deduce the sig­nificance of each processing step of the concerned texture operators. The main conclusion that can be drawn is that post-processing can improve considerably the performance of filter based texture operators. Post-processing schemes, which include the computation of local statistics, perform better than those comprising an NLPO only do. Depending on the type of local statistics, the presence of an NLPO in a post-processing scheme can improve or degrade results. Computing the local mean, for instance, gives very good results if it is preceded by an NLPO (compare second and third column of Table 2 with the first columns of Tables 1 and 2). Similar improvement is achieved by local stan­dard deviation computation. In the latter case, no NLPO is needed between the filtering step and the local standard de­viation computation: such an intermediate step has negative effect on the results (compare the first column of Table 4 with the second and third columns of the same table). Computing the standard deviation gives better results than computing the variance (compare Tables 3 and 4). In both cases, the use of an intermediate NLPO degrades the results (compare first columns with the second and the third columns in Tables 3 and 4). On average, taking the modulus as a point operation and computing the local mean gives the best results (see last column of Table 2). 888 With a given type of post-processing the different filter­ing schemes lead to different results. The identity trans­form gives the worst results: the chance of misclassification for two normally distributed clusters with a Mahalanobis distance of 1.56 is 21. 79%. With an average Mahalanobis distance of 9 which corresponds to a misclassification of 10-7%, all sinusoidal transforms and the Laws' filter bank give comparable results. Still, they are worse than those obtained with Gabor filters, where the average Mahalanobis distance of 15 corresponds to a misclassification probability of 10-12%. The differences between the results obtained with Ga­bor filter banks and the other filter banks can be due to the differences in the tiling of the frequency domain. All of the sinusoidal transforms result in filters with a rectan­gular power spectrum with only two possible orientations (0°, 90°) while Gabor filters have an elliptical power spec­trum with eight possible orientations. Taking in consider­ation that the test material contains oriented textures only, it seems plausible to ex plain the good results obtained with Gabor filters by their orientation selectivity. References [1) A. Jain. A sinusoidal family of unitary transforms. IEEE­PAMI, 1(4):356--365, 1979. [2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation using gabor filters. Pattern Recognition, 24(12):1167-1186, 1991. [3) P. Kruizinga and N. Petkov. Grating cell operator features for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 1010--1014, Brisbane Australia, 1998. [4) P. Kruizinga and N. Petkov. Non-linear operator for oriented texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 1999. [51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on Image Processing for Missile Guidance, volume 238, pages 376--380, 1980. (6) J. Malik and P. Perona. Pre attentive texture discrimination with early vision mechanisms. Journal a/the Optical Society of America, A, 7(5):923-932, 1990. [7] N. Petkov and P. Kruizinga. Computational models of visual neurons specialized in the detection of periodic and aperiodic oriented visual stimuli: Bar and grating cells. Biological Cy­bernetics, 76:83-96, 1997. [8] T. Randen and 1. H. Husoy. Filte ring for texture classification: a comparative study. IEEE-PAMl, 21(4):291-310, 1999. [9] M. Unser. Local lin ear transfoffils for texture measurements. Signal Processing, 11(1):61-79 , 1986. 

   University of GroningenA comparative study of filter based texture operators using Mahalanobis distanceGrigorescu, S.E.; Petkov, N.; Kruizinga, P.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operatorsusing Mahalanobis distance. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). 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For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 08-09-2023A comparative study of filter based texture operators using Mahalanobis distance S. E. Grigorescu, N. Petkov, and P. Kruizinga Institute of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl Abstract Texture feature extraction operators, which comprise lin­ear filtering. eventually followed by post-processing, are considered. The filters used are Laws' masks. filters de­rived from well-known discrete transforms. and Gabor fil­ters. The post-processing step comprises non- linear point operations and/or local statistics computation. The pelfor­mance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture oper­ators. 1. Introduction A number of texture feature extraction operators have been proposed in the literature which are similar in that they comprise two processing steps: (i) linear filtering followed by (ii) post-processing (Fig. 1). The post-processing step typically involves a non-linear point operation followed by the computation of some local statistics. Laws proposed a specific type of linear filtering followed by smoothing based on local averaging applied to the ab­solute values of the filter output [5]. Unser used for filter­ing various well-known transforms like discrete sine (DST), discrete cosine (DCT), discrete even sine (DEST), discrete real even Fourier (DREFT) and discrete real odd Fourier (DROFT) transforms, and for post-processing the computa­tion of channel variances [9]. In [2, 6], a Gabor filter bank followed by thresholding was used. For further references on texture operators, which comprise filtering followed by post-processing, see [8]. Different filters and different types of post-processing are used in the methods mentioned above. Since the re­sults achieved with these methods are also different, a natu­ral question arises of which filter scheme and which type of post-processing give the best results. 0-7695-0750-6/00 $10.00 © 2000 IEEE 885 Texture operator �F'lt _I Post- I:  Iprocessingl � :--. -, I Post- I f\�;:; #� I FIlter l -IProcessingl -r L-/ i i /: Feature ; ; : vectors . . . . �F'l: I P�st- I i : .. �.�.��t.��:=:!����_���!���nj Figure 1. General structure of filter based tex­ture operators. Previous studies in this direction typically focus on the joint performance of the linear filtering, the post -processing and a subsequent classification or segmentation stage (for references see [4, 8]). In a very thorough recent work, Ran­den and HUS0Y [8] bring systematics into the matter by fix­ing the type of post-processing and comparing the effect of different linear filtering schemes. Similar to previous stud­ies they evaluate the performance of texture operators on the basis of the classification and segmentation results that are obtained when the feature vectors are fed into a given clas­sifier. Such an evaluation has the drawback that it does not measure the performance of the texture operator only, but the joint performance of the texture operator and a subse­quent classifier [4]. Moreover these studies do not evaluate the contribution of each individual processing step to the overall performance of the texture feature operator. For evaluation of the performance of texture operators, we use a new method that was proposed elsewhere [4]. It is based on a statistical approach (Fisher criterion and Ma­halanobis distance) to evaluate the capability of a feature operator to generate discriminable feature vectors for dif­ferent textures. This method can be used to compare the texture operators only, regardless of any subsequent classi­fication or segmentation operation. Furthermore, this eval-uation method can be applied after each processing step so that the contribution of each step to the overall performance can be assessed. The rest of the paper is organized as follows. In Section 2 an overview of the filtering schemes used in our exper­iments is given. In Section 3 we discuss different types of post-processing. Comparison criteria and experimental results are presented in Section 4. Finally, in Section 5 conclusions are drawn. 2. Filter banks For linear filtering Laws [5] used a set of square convolu­tion kernels obtained as the outer product of pairs of vectors from a set of n n-dimensional vectors. He conducted exper­iments for n = 3, 5, and 7 and obtained similar results. In our experiments we used Laws' convolution kernels derived from the following set of n = 5 vectors [51: l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 [-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 [1, -4,6, -4, 1]: resulting in a bank of 25 filters. Unser [9] used several different filter banks. Similar to Laws' approach, the convolution kernels proposed by Unser are computed as the outer product of pairs of vectors. Each set of vectors used in Unser's experiments corresponds to one of the following transforms: identity (shift) transform, DST, DCT, DEST, DREFT,and DROFT. More details about these transforms and the filter banks derived from them can be found in ll, 9J. We conducted experiments with all the filter banks derived from the transforms mentioned above. In all cases a set of 25 convolution kernels of size 5 x 5 was used for conformity with the experiments with Laws' oper­ator. Such a filter bank leads to a 25-dirnensional feature space. In [2, 6], filter banks based on Gabor function kernels were used. In our experiments we used two sets of Gabor filters: one employing symmetric Gabor function kernels (1) and the other employing antisymmetric Gabor function kernels (2). In both cases, we used 24 Gabor filters tuned to three preferred spatial frequencies (f = 2-ky'2, k = 0,1, and 2) and eight preferred orientations (e = m � , m = ° . . . 7). For all Gabor filters the product of the preferred spatial frequency f and the standard deviation (7 of the Gaussian factor was kept constant (f(7 = 2) which means that the filters have a constant spatial frequency bandwidth. More details about these types of Gabor filter banks can be found in [4, 7]. Besides these Gabor filters, the filter banks used in our experiments also comprise a Gaussian filter (3) for DC com­ponent computation. 886 ( X2 + (2y)2 ) g}� (i,j) = exp -2(72 . cos(27rfx) (a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 2172 • sm(27rfx where: . . ( i2+j2) 9(�,}) = exp -2ar ' i,j -2 ... 2 x i cos e + j sin e (71 = 3 Y = -i sin e + j cos e (1) (2) (3) This combination of Gabor and Gaussian filters results in banks of 25 filters with a coverage of the spatial frequency domain similar to that of Laws' and Unser's filter banks. 3. Post-processing We applied several types of post-processing to the out­put of the filter banks described in Section 2. The post­processing is applied individually to the output of each channel in a filter bank. The types of post-processing used in our experiments fall in two categories: one-step post-processing and two-step post-processing. The one-step post-processing comprises either a nonlinear point opera­tion (NLPO) or computation of local statistics. The two­step post-processing comprises an NLPO followed by com­putation of local statistics. In our experiments we used as NLPO thresholding and modulus computation. For local statistics we used mean, variance and standard deviation computation. In previous studies only the mean and the variance were used as statis­tical measures of local texture properties. We performed ex­periments with standard deviation as well because it offers a statistical characterization similar to that given by the vari­ance and, at the same time, it takes values in the same range as the one which is characteristic of averaging. All the lo­cal statistics are computed in a square neighborhood of size 17 x 17 around each concerned pixel (similar to [5, 8]). Below, we present the results obtained with texture oper­ators having one of the following structures: linear filtering step only (see the first column in Table 1), linear filtering followed by one-step post processing (see the second and third columns in Table 1 and the first columns in Tables 2, 3, and 4), and linear filtering followed by two-step post­processing (see the second and third columns in Tables 2, 3, and 4). In all tables, X and 1.1 designate thresholding and modu­lus computation, respectively. T ransforms/filters NLPO X 1·1 Identity transform 1.31 1.31 1.31 Laws' filters 1.31 2.10 2.62 DST 1.3 1 1.85 2.59 DCT 1.31 1.84 2.29 DEST 1.31 1.64 2.24 DREFT 1.31 2.11 3.04 DROFT 1.31 1.83 2.43 Lin. sym. Gabor filters 1.31 1.54 1.80 Lin. antisym. Gabor filters 1.21 2.03 2.14 Table 1. Average Mahalanobis distances for features obtained by linear filtering only (first column) and for features obtained by linear fil­tering followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 2.29 2.29 2.29 Laws' filters 2.30 9.42 10.13 DST 2.26 9.10 9.98 DCT 2.28 8.92 9.30 DEST 2.27 8.21 9.08 DREFT 2.28 9.62 10.23 DROFT 2.28 9.19 9.95 Lin. sym. Gabor filters 2.42 11.80 11.96 Lin. antisym. Gabor filters 2.61 15.26 16.28 Table 2. Average Mahalanobis distances for features obtained by computing the local mean of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). 4. Comparison The feature vectors computed in different points of a tex­ture image using a given operator are not identical; they rather form a cluster in the multi-dimensional feature space. It is desirable that feature clusters, which correspond to dif­ferent textures, be separable. The larger this separability the better the operator used for texture feature computation. As a measure of cluster separability we use the Mahalanobis distance [3, 4]. We evaluated the performance of the various operators presented in the previous sections by looking at the pair­wise separability of the clusters of feature vectors obtained from nine test images, each containing a single oriented tex­ture (Fig. 2). To build a cluster, one thousand feature vectors 887 Transforms/filters NLPO X 1·1 Identity transform 1.47 1.47 1.47 Laws ' filters 6.54 6.14 5.84 DST 6.72 6.64 6.78 DCT 6.80 6.81 6.84 DEST 6.60 6.35 6.37 DREFT 6.77 6.36 6.02 DROFT 6.63 6.41 6.53 Lin. sym. Gabor filters 10.76 10.14 11.16 Lin. antisym. Gabor filters 10.87 9.65 9.55 Table 3. Average Mahalanobis distances for features obtained by computing the local vari­ance of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 1.56 1.56 1.56 Laws' filters 8.99 8.52 8.00 DST 9.28 9.12 9.18 DCT 9.14 9.23 8.94 DEST 8.95 8.35 8.39 DREFT 9.33 8.81 8.29 DROFT 9.08 8.60 8.75 Lin. sym. Gabor filters 15.50 14.43 15.23 Lin. antisym. Gabor filters 15.38 13.74 13.61 Table 4. Average Mahalanobis distances for features obtained by computing the local standard dev. of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). were taken at random positions from each texture image. The Mahalanobis distance was computed for every pair of texture images and for each operator. For brevity, only the averages of the 36 Mahalanobis distances computed with each operator are given here: each number in the Tables 1 to 4 is the average of 36 Mahalanobis distances between nine clusters of feature vectors computed with a given method. Regarding the statistical interpretation of the given val­ues in terms of misclassification probability, two clusters having normal distributions with the same standard devi­ation and a value of the Mahalanobis distance of 1.31 -see, for example, the first column of Table 1 - overlap for 25.89%. For Mahalanobis distance of 2 the overlap is 14.76%, while for values of the Mahalanobis distance greater than 5 the overlap is smaller than 0.1 % which means that the clusters are well separable. Figure 2. Test images used in the experi­ments. 5. Conclusions From Tables I, 2, 3, and 4 one can deduce the sig­nificance of each processing step of the concerned texture operators. The main conclusion that can be drawn is that post-processing can improve considerably the performance of filter based texture operators. Post-processing schemes, which include the computation of local statistics, perform better than those comprising an NLPO only do. Depending on the type of local statistics, the presence of an NLPO in a post-processing scheme can improve or degrade results. Computing the local mean, for instance, gives very good results if it is preceded by an NLPO (compare second and third column of Table 2 with the first columns of Tables 1 and 2). Similar improvement is achieved by local stan­dard deviation computation. In the latter case, no NLPO is needed between the filtering step and the local standard de­viation computation: such an intermediate step has negative effect on the results (compare the first column of Table 4 with the second and third columns of the same table). Computing the standard deviation gives better results than computing the variance (compare Tables 3 and 4). In both cases, the use of an intermediate NLPO degrades the results (compare first columns with the second and the third columns in Tables 3 and 4). On average, taking the modulus as a point operation and computing the local mean gives the best results (see last column of Table 2). 888 With a given type of post-processing the different filter­ing schemes lead to different results. The identity trans­form gives the worst results: the chance of misclassification for two normally distributed clusters with a Mahalanobis distance of 1.56 is 21. 79%. With an average Mahalanobis distance of 9 which corresponds to a misclassification of 10-7%, all sinusoidal transforms and the Laws' filter bank give comparable results. Still, they are worse than those obtained with Gabor filters, where the average Mahalanobis distance of 15 corresponds to a misclassification probability of 10-12%. The differences between the results obtained with Ga­bor filter banks and the other filter banks can be due to the differences in the tiling of the frequency domain. All of the sinusoidal transforms result in filters with a rectan­gular power spectrum with only two possible orientations (0°, 90°) while Gabor filters have an elliptical power spec­trum with eight possible orientations. Taking in consider­ation that the test material contains oriented textures only, it seems plausible to ex plain the good results obtained with Gabor filters by their orientation selectivity. References [1) A. Jain. A sinusoidal family of unitary transforms. IEEE­PAMI, 1(4):356--365, 1979. [2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation using gabor filters. Pattern Recognition, 24(12):1167-1186, 1991. [3) P. Kruizinga and N. Petkov. Grating cell operator features for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 1010--1014, Brisbane Australia, 1998. [4) P. Kruizinga and N. Petkov. Non-linear operator for oriented texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 1999. [51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on Image Processing for Missile Guidance, volume 238, pages 376--380, 1980. (6) J. Malik and P. Perona. Pre attentive texture discrimination with early vision mechanisms. Journal a/the Optical Society of America, A, 7(5):923-932, 1990. [7] N. Petkov and P. Kruizinga. Computational models of visual neurons specialized in the detection of periodic and aperiodic oriented visual stimuli: Bar and grating cells. Biological Cy­bernetics, 76:83-96, 1997. [8] T. Randen and 1. H. Husoy. Filte ring for texture classification: a comparative study. IEEE-PAMl, 21(4):291-310, 1999. [9] M. Unser. Local lin ear transfoffils for texture measurements. Signal Processing, 11(1):61-79 , 1986. 

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   University of GroningenA comparative study of filter based texture operators using Mahalanobis distanceGrigorescu, S.E.; Petkov, N.; Kruizinga, P.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operatorsusing Mahalanobis distance. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). IEEE (The Institute ofElectrical and Electronics Engineers).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 04-06-2022A comparative study of filter based texture operators using Mahalanobis distance S. E. Grigorescu, N. Petkov, and P. Kruizinga Institute of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl Abstract Texture feature extraction operators, which comprise lin­ear filtering. eventually followed by post-processing, are considered. The filters used are Laws' masks. filters de­rived from well-known discrete transforms. and Gabor fil­ters. The post-processing step comprises non- linear point operations and/or local statistics computation. The pelfor­mance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture oper­ators. 1. Introduction A number of texture feature extraction operators have been proposed in the literature which are similar in that they comprise two processing steps: (i) linear filtering followed by (ii) post-processing (Fig. 1). The post-processing step typically involves a non-linear point operation followed by the computation of some local statistics. Laws proposed a specific type of linear filtering followed by smoothing based on local averaging applied to the ab­solute values of the filter output [5]. Unser used for filter­ing various well-known transforms like discrete sine (DST), discrete cosine (DCT), discrete even sine (DEST), discrete real even Fourier (DREFT) and discrete real odd Fourier (DROFT) transforms, and for post-processing the computa­tion of channel variances [9]. In [2, 6], a Gabor filter bank followed by thresholding was used. For further references on texture operators, which comprise filtering followed by post-processing, see [8]. Different filters and different types of post-processing are used in the methods mentioned above. Since the re­sults achieved with these methods are also different, a natu­ral question arises of which filter scheme and which type of post-processing give the best results. 0-7695-0750-6/00 $10.00 © 2000 IEEE 885 Texture operator �F'lt _I Post- I:  Iprocessingl � :--. -, I Post- I f\�;:; #� I FIlter l -IProcessingl -r L-/ i i /: Feature ; ; : vectors . . . . �F'l: I P�st- I i : .. �.�.��t.��:=:!����_���!���nj Figure 1. General structure of filter based tex­ture operators. Previous studies in this direction typically focus on the joint performance of the linear filtering, the post -processing and a subsequent classification or segmentation stage (for references see [4, 8]). In a very thorough recent work, Ran­den and HUS0Y [8] bring systematics into the matter by fix­ing the type of post-processing and comparing the effect of different linear filtering schemes. Similar to previous stud­ies they evaluate the performance of texture operators on the basis of the classification and segmentation results that are obtained when the feature vectors are fed into a given clas­sifier. Such an evaluation has the drawback that it does not measure the performance of the texture operator only, but the joint performance of the texture operator and a subse­quent classifier [4]. Moreover these studies do not evaluate the contribution of each individual processing step to the overall performance of the texture feature operator. For evaluation of the performance of texture operators, we use a new method that was proposed elsewhere [4]. It is based on a statistical approach (Fisher criterion and Ma­halanobis distance) to evaluate the capability of a feature operator to generate discriminable feature vectors for dif­ferent textures. This method can be used to compare the texture operators only, regardless of any subsequent classi­fication or segmentation operation. Furthermore, this eval-uation method can be applied after each processing step so that the contribution of each step to the overall performance can be assessed. The rest of the paper is organized as follows. In Section 2 an overview of the filtering schemes used in our exper­iments is given. In Section 3 we discuss different types of post-processing. Comparison criteria and experimental results are presented in Section 4. Finally, in Section 5 conclusions are drawn. 2. Filter banks For linear filtering Laws [5] used a set of square convolu­tion kernels obtained as the outer product of pairs of vectors from a set of n n-dimensional vectors. He conducted exper­iments for n = 3, 5, and 7 and obtained similar results. In our experiments we used Laws' convolution kernels derived from the following set of n = 5 vectors [51: l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 [-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 [1, -4,6, -4, 1]: resulting in a bank of 25 filters. Unser [9] used several different filter banks. Similar to Laws' approach, the convolution kernels proposed by Unser are computed as the outer product of pairs of vectors. Each set of vectors used in Unser's experiments corresponds to one of the following transforms: identity (shift) transform, DST, DCT, DEST, DREFT,and DROFT. More details about these transforms and the filter banks derived from them can be found in ll, 9J. We conducted experiments with all the filter banks derived from the transforms mentioned above. In all cases a set of 25 convolution kernels of size 5 x 5 was used for conformity with the experiments with Laws' oper­ator. Such a filter bank leads to a 25-dirnensional feature space. In [2, 6], filter banks based on Gabor function kernels were used. In our experiments we used two sets of Gabor filters: one employing symmetric Gabor function kernels (1) and the other employing antisymmetric Gabor function kernels (2). In both cases, we used 24 Gabor filters tuned to three preferred spatial frequencies (f = 2-ky'2, k = 0,1, and 2) and eight preferred orientations (e = m � , m = ° . . . 7). For all Gabor filters the product of the preferred spatial frequency f and the standard deviation (7 of the Gaussian factor was kept constant (f(7 = 2) which means that the filters have a constant spatial frequency bandwidth. More details about these types of Gabor filter banks can be found in [4, 7]. Besides these Gabor filters, the filter banks used in our experiments also comprise a Gaussian filter (3) for DC com­ponent computation. 886 ( X2 + (2y)2 ) g}� (i,j) = exp -2(72 . cos(27rfx) (a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 2172 • sm(27rfx where: . . ( i2+j2) 9(�,}) = exp -2ar ' i,j -2 ... 2 x i cos e + j sin e (71 = 3 Y = -i sin e + j cos e (1) (2) (3) This combination of Gabor and Gaussian filters results in banks of 25 filters with a coverage of the spatial frequency domain similar to that of Laws' and Unser's filter banks. 3. Post-processing We applied several types of post-processing to the out­put of the filter banks described in Section 2. The post­processing is applied individually to the output of each channel in a filter bank. The types of post-processing used in our experiments fall in two categories: one-step post-processing and two-step post-processing. The one-step post-processing comprises either a nonlinear point opera­tion (NLPO) or computation of local statistics. The two­step post-processing comprises an NLPO followed by com­putation of local statistics. In our experiments we used as NLPO thresholding and modulus computation. For local statistics we used mean, variance and standard deviation computation. In previous studies only the mean and the variance were used as statis­tical measures of local texture properties. We performed ex­periments with standard deviation as well because it offers a statistical characterization similar to that given by the vari­ance and, at the same time, it takes values in the same range as the one which is characteristic of averaging. All the lo­cal statistics are computed in a square neighborhood of size 17 x 17 around each concerned pixel (similar to [5, 8]). Below, we present the results obtained with texture oper­ators having one of the following structures: linear filtering step only (see the first column in Table 1), linear filtering followed by one-step post processing (see the second and third columns in Table 1 and the first columns in Tables 2, 3, and 4), and linear filtering followed by two-step post­processing (see the second and third columns in Tables 2, 3, and 4). In all tables, X and 1.1 designate thresholding and modu­lus computation, respectively. T ransforms/filters NLPO X 1·1 Identity transform 1.31 1.31 1.31 Laws' filters 1.31 2.10 2.62 DST 1.3 1 1.85 2.59 DCT 1.31 1.84 2.29 DEST 1.31 1.64 2.24 DREFT 1.31 2.11 3.04 DROFT 1.31 1.83 2.43 Lin. sym. Gabor filters 1.31 1.54 1.80 Lin. antisym. Gabor filters 1.21 2.03 2.14 Table 1. Average Mahalanobis distances for features obtained by linear filtering only (first column) and for features obtained by linear fil­tering followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 2.29 2.29 2.29 Laws' filters 2.30 9.42 10.13 DST 2.26 9.10 9.98 DCT 2.28 8.92 9.30 DEST 2.27 8.21 9.08 DREFT 2.28 9.62 10.23 DROFT 2.28 9.19 9.95 Lin. sym. Gabor filters 2.42 11.80 11.96 Lin. antisym. Gabor filters 2.61 15.26 16.28 Table 2. Average Mahalanobis distances for features obtained by computing the local mean of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). 4. Comparison The feature vectors computed in different points of a tex­ture image using a given operator are not identical; they rather form a cluster in the multi-dimensional feature space. It is desirable that feature clusters, which correspond to dif­ferent textures, be separable. The larger this separability the better the operator used for texture feature computation. As a measure of cluster separability we use the Mahalanobis distance [3, 4]. We evaluated the performance of the various operators presented in the previous sections by looking at the pair­wise separability of the clusters of feature vectors obtained from nine test images, each containing a single oriented tex­ture (Fig. 2). To build a cluster, one thousand feature vectors 887 Transforms/filters NLPO X 1·1 Identity transform 1.47 1.47 1.47 Laws ' filters 6.54 6.14 5.84 DST 6.72 6.64 6.78 DCT 6.80 6.81 6.84 DEST 6.60 6.35 6.37 DREFT 6.77 6.36 6.02 DROFT 6.63 6.41 6.53 Lin. sym. Gabor filters 10.76 10.14 11.16 Lin. antisym. Gabor filters 10.87 9.65 9.55 Table 3. Average Mahalanobis distances for features obtained by computing the local vari­ance of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 1.56 1.56 1.56 Laws' filters 8.99 8.52 8.00 DST 9.28 9.12 9.18 DCT 9.14 9.23 8.94 DEST 8.95 8.35 8.39 DREFT 9.33 8.81 8.29 DROFT 9.08 8.60 8.75 Lin. sym. Gabor filters 15.50 14.43 15.23 Lin. antisym. Gabor filters 15.38 13.74 13.61 Table 4. Average Mahalanobis distances for features obtained by computing the local standard dev. of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). were taken at random positions from each texture image. The Mahalanobis distance was computed for every pair of texture images and for each operator. For brevity, only the averages of the 36 Mahalanobis distances computed with each operator are given here: each number in the Tables 1 to 4 is the average of 36 Mahalanobis distances between nine clusters of feature vectors computed with a given method. Regarding the statistical interpretation of the given val­ues in terms of misclassification probability, two clusters having normal distributions with the same standard devi­ation and a value of the Mahalanobis distance of 1.31 -see, for example, the first column of Table 1 - overlap for 25.89%. For Mahalanobis distance of 2 the overlap is 14.76%, while for values of the Mahalanobis distance greater than 5 the overlap is smaller than 0.1 % which means that the clusters are well separable. Figure 2. Test images used in the experi­ments. 5. Conclusions From Tables I, 2, 3, and 4 one can deduce the sig­nificance of each processing step of the concerned texture operators. The main conclusion that can be drawn is that post-processing can improve considerably the performance of filter based texture operators. Post-processing schemes, which include the computation of local statistics, perform better than those comprising an NLPO only do. Depending on the type of local statistics, the presence of an NLPO in a post-processing scheme can improve or degrade results. Computing the local mean, for instance, gives very good results if it is preceded by an NLPO (compare second and third column of Table 2 with the first columns of Tables 1 and 2). Similar improvement is achieved by local stan­dard deviation computation. In the latter case, no NLPO is needed between the filtering step and the local standard de­viation computation: such an intermediate step has negative effect on the results (compare the first column of Table 4 with the second and third columns of the same table). Computing the standard deviation gives better results than computing the variance (compare Tables 3 and 4). In both cases, the use of an intermediate NLPO degrades the results (compare first columns with the second and the third columns in Tables 3 and 4). On average, taking the modulus as a point operation and computing the local mean gives the best results (see last column of Table 2). 888 With a given type of post-processing the different filter­ing schemes lead to different results. The identity trans­form gives the worst results: the chance of misclassification for two normally distributed clusters with a Mahalanobis distance of 1.56 is 21. 79%. With an average Mahalanobis distance of 9 which corresponds to a misclassification of 10-7%, all sinusoidal transforms and the Laws' filter bank give comparable results. Still, they are worse than those obtained with Gabor filters, where the average Mahalanobis distance of 15 corresponds to a misclassification probability of 10-12%. The differences between the results obtained with Ga­bor filter banks and the other filter banks can be due to the differences in the tiling of the frequency domain. All of the sinusoidal transforms result in filters with a rectan­gular power spectrum with only two possible orientations (0°, 90°) while Gabor filters have an elliptical power spec­trum with eight possible orientations. Taking in consider­ation that the test material contains oriented textures only, it seems plausible to ex plain the good results obtained with Gabor filters by their orientation selectivity. References [1) A. Jain. A sinusoidal family of unitary transforms. IEEE­PAMI, 1(4):356--365, 1979. [2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation using gabor filters. Pattern Recognition, 24(12):1167-1186, 1991. [3) P. Kruizinga and N. Petkov. Grating cell operator features for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 1010--1014, Brisbane Australia, 1998. [4) P. Kruizinga and N. Petkov. Non-linear operator for oriented texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 1999. [51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on Image Processing for Missile Guidance, volume 238, pages 376--380, 1980. (6) J. Malik and P. Perona. Pre attentive texture discrimination with early vision mechanisms. Journal a/the Optical Society of America, A, 7(5):923-932, 1990. [7] N. Petkov and P. Kruizinga. Computational models of visual neurons specialized in the detection of periodic and aperiodic oriented visual stimuli: Bar and grating cells. Biological Cy­bernetics, 76:83-96, 1997. [8] T. Randen and 1. H. Husoy. Filte ring for texture classification: a comparative study. IEEE-PAMl, 21(4):291-310, 1999. [9] M. Unser. Local lin ear transfoffils for texture measurements. Signal Processing, 11(1):61-79 , 1986. 

   University of GroningenA comparative study of filter based texture operators using Mahalanobis distanceGrigorescu, S.E.; Petkov, N.; Kruizinga, P.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operatorsusing Mahalanobis distance. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). IEEE (The Institute ofElectrical and Electronics Engineers).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 09-09-2023A comparative study of filter based texture operators using Mahalanobis distance S. E. Grigorescu, N. Petkov, and P. Kruizinga Institute of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl Abstract Texture feature extraction operators, which comprise lin­ear filtering. eventually followed by post-processing, are considered. The filters used are Laws' masks. filters de­rived from well-known discrete transforms. and Gabor fil­ters. The post-processing step comprises non- linear point operations and/or local statistics computation. The pelfor­mance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture oper­ators. 1. Introduction A number of texture feature extraction operators have been proposed in the literature which are similar in that they comprise two processing steps: (i) linear filtering followed by (ii) post-processing (Fig. 1). The post-processing step typically involves a non-linear point operation followed by the computation of some local statistics. Laws proposed a specific type of linear filtering followed by smoothing based on local averaging applied to the ab­solute values of the filter output [5]. Unser used for filter­ing various well-known transforms like discrete sine (DST), discrete cosine (DCT), discrete even sine (DEST), discrete real even Fourier (DREFT) and discrete real odd Fourier (DROFT) transforms, and for post-processing the computa­tion of channel variances [9]. In [2, 6], a Gabor filter bank followed by thresholding was used. For further references on texture operators, which comprise filtering followed by post-processing, see [8]. Different filters and different types of post-processing are used in the methods mentioned above. Since the re­sults achieved with these methods are also different, a natu­ral question arises of which filter scheme and which type of post-processing give the best results. 0-7695-0750-6/00 $10.00 © 2000 IEEE 885 Texture operator �F'lt _I Post- I:  Iprocessingl � :--. -, I Post- I f\�;:; #� I FIlter l -IProcessingl -r L-/ i i /: Feature ; ; : vectors . . . . �F'l: I P�st- I i : .. �.�.��t.��:=:!����_���!���nj Figure 1. General structure of filter based tex­ture operators. Previous studies in this direction typically focus on the joint performance of the linear filtering, the post -processing and a subsequent classification or segmentation stage (for references see [4, 8]). In a very thorough recent work, Ran­den and HUS0Y [8] bring systematics into the matter by fix­ing the type of post-processing and comparing the effect of different linear filtering schemes. Similar to previous stud­ies they evaluate the performance of texture operators on the basis of the classification and segmentation results that are obtained when the feature vectors are fed into a given clas­sifier. Such an evaluation has the drawback that it does not measure the performance of the texture operator only, but the joint performance of the texture operator and a subse­quent classifier [4]. Moreover these studies do not evaluate the contribution of each individual processing step to the overall performance of the texture feature operator. For evaluation of the performance of texture operators, we use a new method that was proposed elsewhere [4]. It is based on a statistical approach (Fisher criterion and Ma­halanobis distance) to evaluate the capability of a feature operator to generate discriminable feature vectors for dif­ferent textures. This method can be used to compare the texture operators only, regardless of any subsequent classi­fication or segmentation operation. Furthermore, this eval-uation method can be applied after each processing step so that the contribution of each step to the overall performance can be assessed. The rest of the paper is organized as follows. In Section 2 an overview of the filtering schemes used in our exper­iments is given. In Section 3 we discuss different types of post-processing. Comparison criteria and experimental results are presented in Section 4. Finally, in Section 5 conclusions are drawn. 2. Filter banks For linear filtering Laws [5] used a set of square convolu­tion kernels obtained as the outer product of pairs of vectors from a set of n n-dimensional vectors. He conducted exper­iments for n = 3, 5, and 7 and obtained similar results. In our experiments we used Laws' convolution kernels derived from the following set of n = 5 vectors [51: l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 [-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 [1, -4,6, -4, 1]: resulting in a bank of 25 filters. Unser [9] used several different filter banks. Similar to Laws' approach, the convolution kernels proposed by Unser are computed as the outer product of pairs of vectors. Each set of vectors used in Unser's experiments corresponds to one of the following transforms: identity (shift) transform, DST, DCT, DEST, DREFT,and DROFT. More details about these transforms and the filter banks derived from them can be found in ll, 9J. We conducted experiments with all the filter banks derived from the transforms mentioned above. In all cases a set of 25 convolution kernels of size 5 x 5 was used for conformity with the experiments with Laws' oper­ator. Such a filter bank leads to a 25-dirnensional feature space. In [2, 6], filter banks based on Gabor function kernels were used. In our experiments we used two sets of Gabor filters: one employing symmetric Gabor function kernels (1) and the other employing antisymmetric Gabor function kernels (2). In both cases, we used 24 Gabor filters tuned to three preferred spatial frequencies (f = 2-ky'2, k = 0,1, and 2) and eight preferred orientations (e = m � , m = ° . . . 7). For all Gabor filters the product of the preferred spatial frequency f and the standard deviation (7 of the Gaussian factor was kept constant (f(7 = 2) which means that the filters have a constant spatial frequency bandwidth. More details about these types of Gabor filter banks can be found in [4, 7]. Besides these Gabor filters, the filter banks used in our experiments also comprise a Gaussian filter (3) for DC com­ponent computation. 886 ( X2 + (2y)2 ) g}� (i,j) = exp -2(72 . cos(27rfx) (a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 2172 • sm(27rfx where: . . ( i2+j2) 9(�,}) = exp -2ar ' i,j -2 ... 2 x i cos e + j sin e (71 = 3 Y = -i sin e + j cos e (1) (2) (3) This combination of Gabor and Gaussian filters results in banks of 25 filters with a coverage of the spatial frequency domain similar to that of Laws' and Unser's filter banks. 3. Post-processing We applied several types of post-processing to the out­put of the filter banks described in Section 2. The post­processing is applied individually to the output of each channel in a filter bank. The types of post-processing used in our experiments fall in two categories: one-step post-processing and two-step post-processing. The one-step post-processing comprises either a nonlinear point opera­tion (NLPO) or computation of local statistics. The two­step post-processing comprises an NLPO followed by com­putation of local statistics. In our experiments we used as NLPO thresholding and modulus computation. For local statistics we used mean, variance and standard deviation computation. In previous studies only the mean and the variance were used as statis­tical measures of local texture properties. We performed ex­periments with standard deviation as well because it offers a statistical characterization similar to that given by the vari­ance and, at the same time, it takes values in the same range as the one which is characteristic of averaging. All the lo­cal statistics are computed in a square neighborhood of size 17 x 17 around each concerned pixel (similar to [5, 8]). Below, we present the results obtained with texture oper­ators having one of the following structures: linear filtering step only (see the first column in Table 1), linear filtering followed by one-step post processing (see the second and third columns in Table 1 and the first columns in Tables 2, 3, and 4), and linear filtering followed by two-step post­processing (see the second and third columns in Tables 2, 3, and 4). In all tables, X and 1.1 designate thresholding and modu­lus computation, respectively. T ransforms/filters NLPO X 1·1 Identity transform 1.31 1.31 1.31 Laws' filters 1.31 2.10 2.62 DST 1.3 1 1.85 2.59 DCT 1.31 1.84 2.29 DEST 1.31 1.64 2.24 DREFT 1.31 2.11 3.04 DROFT 1.31 1.83 2.43 Lin. sym. Gabor filters 1.31 1.54 1.80 Lin. antisym. Gabor filters 1.21 2.03 2.14 Table 1. Average Mahalanobis distances for features obtained by linear filtering only (first column) and for features obtained by linear fil­tering followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 2.29 2.29 2.29 Laws' filters 2.30 9.42 10.13 DST 2.26 9.10 9.98 DCT 2.28 8.92 9.30 DEST 2.27 8.21 9.08 DREFT 2.28 9.62 10.23 DROFT 2.28 9.19 9.95 Lin. sym. Gabor filters 2.42 11.80 11.96 Lin. antisym. Gabor filters 2.61 15.26 16.28 Table 2. Average Mahalanobis distances for features obtained by computing the local mean of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). 4. Comparison The feature vectors computed in different points of a tex­ture image using a given operator are not identical; they rather form a cluster in the multi-dimensional feature space. It is desirable that feature clusters, which correspond to dif­ferent textures, be separable. The larger this separability the better the operator used for texture feature computation. As a measure of cluster separability we use the Mahalanobis distance [3, 4]. We evaluated the performance of the various operators presented in the previous sections by looking at the pair­wise separability of the clusters of feature vectors obtained from nine test images, each containing a single oriented tex­ture (Fig. 2). To build a cluster, one thousand feature vectors 887 Transforms/filters NLPO X 1·1 Identity transform 1.47 1.47 1.47 Laws ' filters 6.54 6.14 5.84 DST 6.72 6.64 6.78 DCT 6.80 6.81 6.84 DEST 6.60 6.35 6.37 DREFT 6.77 6.36 6.02 DROFT 6.63 6.41 6.53 Lin. sym. Gabor filters 10.76 10.14 11.16 Lin. antisym. Gabor filters 10.87 9.65 9.55 Table 3. Average Mahalanobis distances for features obtained by computing the local vari­ance of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 1.56 1.56 1.56 Laws' filters 8.99 8.52 8.00 DST 9.28 9.12 9.18 DCT 9.14 9.23 8.94 DEST 8.95 8.35 8.39 DREFT 9.33 8.81 8.29 DROFT 9.08 8.60 8.75 Lin. sym. Gabor filters 15.50 14.43 15.23 Lin. antisym. Gabor filters 15.38 13.74 13.61 Table 4. Average Mahalanobis distances for features obtained by computing the local standard dev. of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). were taken at random positions from each texture image. The Mahalanobis distance was computed for every pair of texture images and for each operator. For brevity, only the averages of the 36 Mahalanobis distances computed with each operator are given here: each number in the Tables 1 to 4 is the average of 36 Mahalanobis distances between nine clusters of feature vectors computed with a given method. Regarding the statistical interpretation of the given val­ues in terms of misclassification probability, two clusters having normal distributions with the same standard devi­ation and a value of the Mahalanobis distance of 1.31 -see, for example, the first column of Table 1 - overlap for 25.89%. For Mahalanobis distance of 2 the overlap is 14.76%, while for values of the Mahalanobis distance greater than 5 the overlap is smaller than 0.1 % which means that the clusters are well separable. Figure 2. Test images used in the experi­ments. 5. Conclusions From Tables I, 2, 3, and 4 one can deduce the sig­nificance of each processing step of the concerned texture operators. The main conclusion that can be drawn is that post-processing can improve considerably the performance of filter based texture operators. Post-processing schemes, which include the computation of local statistics, perform better than those comprising an NLPO only do. Depending on the type of local statistics, the presence of an NLPO in a post-processing scheme can improve or degrade results. Computing the local mean, for instance, gives very good results if it is preceded by an NLPO (compare second and third column of Table 2 with the first columns of Tables 1 and 2). Similar improvement is achieved by local stan­dard deviation computation. In the latter case, no NLPO is needed between the filtering step and the local standard de­viation computation: such an intermediate step has negative effect on the results (compare the first column of Table 4 with the second and third columns of the same table). Computing the standard deviation gives better results than computing the variance (compare Tables 3 and 4). In both cases, the use of an intermediate NLPO degrades the results (compare first columns with the second and the third columns in Tables 3 and 4). On average, taking the modulus as a point operation and computing the local mean gives the best results (see last column of Table 2). 888 With a given type of post-processing the different filter­ing schemes lead to different results. The identity trans­form gives the worst results: the chance of misclassification for two normally distributed clusters with a Mahalanobis distance of 1.56 is 21. 79%. With an average Mahalanobis distance of 9 which corresponds to a misclassification of 10-7%, all sinusoidal transforms and the Laws' filter bank give comparable results. Still, they are worse than those obtained with Gabor filters, where the average Mahalanobis distance of 15 corresponds to a misclassification probability of 10-12%. The differences between the results obtained with Ga­bor filter banks and the other filter banks can be due to the differences in the tiling of the frequency domain. All of the sinusoidal transforms result in filters with a rectan­gular power spectrum with only two possible orientations (0°, 90°) while Gabor filters have an elliptical power spec­trum with eight possible orientations. Taking in consider­ation that the test material contains oriented textures only, it seems plausible to ex plain the good results obtained with Gabor filters by their orientation selectivity. References [1) A. Jain. A sinusoidal family of unitary transforms. IEEE­PAMI, 1(4):356--365, 1979. [2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation using gabor filters. Pattern Recognition, 24(12):1167-1186, 1991. [3) P. Kruizinga and N. Petkov. Grating cell operator features for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 1010--1014, Brisbane Australia, 1998. [4) P. Kruizinga and N. Petkov. Non-linear operator for oriented texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 1999. [51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on Image Processing for Missile Guidance, volume 238, pages 376--380, 1980. (6) J. Malik and P. Perona. Pre attentive texture discrimination with early vision mechanisms. Journal a/the Optical Society of America, A, 7(5):923-932, 1990. [7] N. Petkov and P. Kruizinga. Computational models of visual neurons specialized in the detection of periodic and aperiodic oriented visual stimuli: Bar and grating cells. Biological Cy­bernetics, 76:83-96, 1997. [8] T. Randen and 1. H. Husoy. Filte ring for texture classification: a comparative study. IEEE-PAMl, 21(4):291-310, 1999. [9] M. Unser. Local lin ear transfoffils for texture measurements. Signal Processing, 11(1):61-79 , 1986. 

University of Groningen

   University of GroningenA comparative study of filter based texture operators using Mahalanobis distanceGrigorescu, S.E.; Petkov, N.; Kruizinga, P.Published in:15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGSIMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2000Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Grigorescu, S. E., Petkov, N., & Kruizinga, P. (2000). A comparative study of filter based texture operatorsusing Mahalanobis distance. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, & J. Serra(Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3, PROCEEDINGS(pp. 885-888). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). LOS ALAMITOS: IEEE(The Institute of Electrical and Electronics Engineers).CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019A comparative study of filter based texture operators using Mahalanobis distance S. E. Grigorescu, N. Petkov, and P. Kruizinga Institute of Mathematics and Computing Science University of Groningen P.O. Box 800 9700 AV Groningen The Netherlands simona@cs.rug.nl, petkov@cs.rug.nl, peterkr@cs.rug.nl Abstract Texture feature extraction operators, which comprise lin­ear filtering. eventually followed by post-processing, are considered. The filters used are Laws' masks. filters de­rived from well-known discrete transforms. and Gabor fil­ters. The post-processing step comprises non- linear point operations and/or local statistics computation. The pelfor­mance is measured by means of the Mahalanobis distance between clusters of feature vectors derived from different textures. The results show that post-processing improve considerably the performance of filter based texture oper­ators. 1. Introduction A number of texture feature extraction operators have been proposed in the literature which are similar in that they comprise two processing steps: (i) linear filtering followed by (ii) post-processing (Fig. 1). The post-processing step typically involves a non-linear point operation followed by the computation of some local statistics. Laws proposed a specific type of linear filtering followed by smoothing based on local averaging applied to the ab­solute values of the filter output [5]. Unser used for filter­ing various well-known transforms like discrete sine (DST), discrete cosine (DCT), discrete even sine (DEST), discrete real even Fourier (DREFT) and discrete real odd Fourier (DROFT) transforms, and for post-processing the computa­tion of channel variances [9]. In [2, 6], a Gabor filter bank followed by thresholding was used. For further references on texture operators, which comprise filtering followed by post-processing, see [8]. Different filters and different types of post-processing are used in the methods mentioned above. Since the re­sults achieved with these methods are also different, a natu­ral question arises of which filter scheme and which type of post-processing give the best results. 0-7695-0750-6/00 $10.00 © 2000 IEEE 885 Texture operator �F'lt _I Post- I:  Iprocessingl � :--. -, I Post- I f\�;:; #� I FIlter l -IProcessingl -r L-/ i i /: Feature ; ; : vectors . . . . �F'l: I P�st- I i : .. �.�.��t.��:=:!����_���!���nj Figure 1. General structure of filter based tex­ture operators. Previous studies in this direction typically focus on the joint performance of the linear filtering, the post -processing and a subsequent classification or segmentation stage (for references see [4, 8]). In a very thorough recent work, Ran­den and HUS0Y [8] bring systematics into the matter by fix­ing the type of post-processing and comparing the effect of different linear filtering schemes. Similar to previous stud­ies they evaluate the performance of texture operators on the basis of the classification and segmentation results that are obtained when the feature vectors are fed into a given clas­sifier. Such an evaluation has the drawback that it does not measure the performance of the texture operator only, but the joint performance of the texture operator and a subse­quent classifier [4]. Moreover these studies do not evaluate the contribution of each individual processing step to the overall performance of the texture feature operator. For evaluation of the performance of texture operators, we use a new method that was proposed elsewhere [4]. It is based on a statistical approach (Fisher criterion and Ma­halanobis distance) to evaluate the capability of a feature operator to generate discriminable feature vectors for dif­ferent textures. This method can be used to compare the texture operators only, regardless of any subsequent classi­fication or segmentation operation. Furthermore, this eval-uation method can be applied after each processing step so that the contribution of each step to the overall performance can be assessed. The rest of the paper is organized as follows. In Section 2 an overview of the filtering schemes used in our exper­iments is given. In Section 3 we discuss different types of post-processing. Comparison criteria and experimental results are presented in Section 4. Finally, in Section 5 conclusions are drawn. 2. Filter banks For linear filtering Laws [5] used a set of square convolu­tion kernels obtained as the outer product of pairs of vectors from a set of n n-dimensional vectors. He conducted exper­iments for n = 3, 5, and 7 and obtained similar results. In our experiments we used Laws' convolution kernels derived from the following set of n = 5 vectors [51: l-'I [1,4,6,4,1], V2 [-1,-2,0,2,1], V3 [-1,0,2,0,-1], V4 = [-1,2,0,-2,1], and V5 [1, -4,6, -4, 1]: resulting in a bank of 25 filters. Unser [9] used several different filter banks. Similar to Laws' approach, the convolution kernels proposed by Unser are computed as the outer product of pairs of vectors. Each set of vectors used in Unser's experiments corresponds to one of the following transforms: identity (shift) transform, DST, DCT, DEST, DREFT,and DROFT. More details about these transforms and the filter banks derived from them can be found in ll, 9J. We conducted experiments with all the filter banks derived from the transforms mentioned above. In all cases a set of 25 convolution kernels of size 5 x 5 was used for conformity with the experiments with Laws' oper­ator. Such a filter bank leads to a 25-dirnensional feature space. In [2, 6], filter banks based on Gabor function kernels were used. In our experiments we used two sets of Gabor filters: one employing symmetric Gabor function kernels (1) and the other employing antisymmetric Gabor function kernels (2). In both cases, we used 24 Gabor filters tuned to three preferred spatial frequencies (f = 2-ky'2, k = 0,1, and 2) and eight preferred orientations (e = m � , m = ° . . . 7). For all Gabor filters the product of the preferred spatial frequency f and the standard deviation (7 of the Gaussian factor was kept constant (f(7 = 2) which means that the filters have a constant spatial frequency bandwidth. More details about these types of Gabor filter banks can be found in [4, 7]. Besides these Gabor filters, the filter banks used in our experiments also comprise a Gaussian filter (3) for DC com­ponent computation. 886 ( X2 + (2y)2 ) g}� (i,j) = exp -2(72 . cos(27rfx) (a) . .  ( X2 + (2y)2) . ) 9f,e(t,}) = exp 2172 • sm(27rfx where: . . ( i2+j2) 9(�,}) = exp - 2ar ' i,j -2 ... 2 x i cos e + j sin e (71 = 3 Y = -i sin e + j cos e (1) (2) (3) This combination of Gabor and Gaussian filters results in banks of 25 filters with a coverage of the spatial frequency domain similar to that of Laws' and Unser's filter banks. 3. Post-processing We applied several types of post-processing to the out­put of the filter banks described in Section 2. The post­processing is applied individually to the output of each channel in a filter bank. The types of post-processing used in our experiments fall in two categories: one-step post-processing and two-step post-processing. The one-step post-processing comprises either a nonlinear point opera­tion (NLPO) or computation of local statistics. The two­step post-processing comprises an NLPO followed by com­putation of local statistics. In our experiments we used as NLPO thresholding and modulus computation. For local statistics we used mean, variance and standard deviation computation. In previous studies only the mean and the variance were used as statis­tical measures of local texture properties. We performed ex­periments with standard deviation as well because it offers a statistical characterization similar to that given by the vari­ance and, at the same time, it takes values in the same range as the one which is characteristic of averaging. All the lo­cal statistics are computed in a square neighborhood of size 17 x 17 around each concerned pixel (similar to [5, 8]). Below, we present the results obtained with texture oper­ators having one of the following structures: linear filtering step only (see the first column in Table 1), linear filtering followed by one-step post processing (see the second and third columns in Table 1 and the first columns in Tables 2, 3, and 4), and linear filtering followed by two-step post­processing (see the second and third columns in Tables 2, 3, and 4). In all tables, X and 1.1 designate thresholding and modu­lus computation, respectively. T ransforms/filters NLPO X 1·1 Identity transform 1.31 1.31 1.31 Laws' filters 1.31 2.10 2.62 DST 1.3 1 1.85 2.59 DCT 1.31 1.84 2.29 DEST 1.31 1.64 2.24 DREFT 1.31 2.11 3.04 DROFT 1.31 1.83 2.43 Lin. sym. Gabor filters 1.31 1.54 1.80 Lin. antisym. Gabor filters 1.21 2.03 2.14 Table 1. Average Mahalanobis distances for features obtained by linear filtering only (first column) and for features obtained by linear fil­tering followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 2.29 2.29 2.29 Laws' filters 2.30 9.42 10.13 DST 2.26 9.10 9.98 DCT 2.28 8.92 9.30 DEST 2.27 8.21 9.08 DREFT 2.28 9.62 10.23 DROFT 2.28 9.19 9.95 Lin. sym. Gabor filters 2.42 11.80 11.96 Lin. antisym. Gabor filters 2.61 15.26 16.28 Table 2. Average Mahalanobis distances for features obtained by computing the local mean of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). 4. Comparison The feature vectors computed in different points of a tex­ture image using a given operator are not identical; they rather form a cluster in the multi-dimensional feature space. It is desirable that feature clusters, which correspond to dif­ferent textures, be separable. The larger this separability the better the operator used for texture feature computation. As a measure of cluster separability we use the Mahalanobis distance [3, 4]. We evaluated the performance of the various operators presented in the previous sections by looking at the pair­wise separability of the clusters of feature vectors obtained from nine test images, each containing a single oriented tex­ture (Fig. 2). To build a cluster, one thousand feature vectors 887 Transforms/filters NLPO X 1·1 Identity transform 1.47 1.47 1.47 Laws ' filters 6.54 6.14 5.84 DST 6.72 6.64 6.78 DCT 6.80 6.81 6.84 DEST 6.60 6.35 6.37 DREFT 6.77 6.36 6.02 DROFT 6.63 6.41 6.53 Lin. sym. Gabor filters 10.76 10.14 11.16 Lin. antisym. Gabor filters 10.87 9.65 9.55 Table 3. Average Mahalanobis distances for features obtained by computing the local vari­ance of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). Transforms/filters NLPO X 1·1 Identity transform 1.56 1.56 1.56 Laws' filters 8.99 8.52 8.00 DST 9.28 9.12 9.18 DCT 9.14 9.23 8.94 DEST 8.95 8.35 8.39 DREFT 9.33 8.81 8.29 DROFT 9.08 8.60 8.75 Lin. sym. Gabor filters 15.50 14.43 15.23 Lin. antisym. Gabor filters 15.38 13.74 13.61 Table 4. Average Mahalanobis distances for features obtained by computing the local standard dev. of the output of: (i) linear filters (first column) and (ii) linear filters followed by an NLPO (second and third columns). were taken at random positions from each texture image. The Mahalanobis distance was computed for every pair of texture images and for each operator. For brevity, only the averages of the 36 Mahalanobis distances computed with each operator are given here: each number in the Tables 1 to 4 is the average of 36 Mahalanobis distances between nine clusters of feature vectors computed with a given method. Regarding the statistical interpretation of the given val­ues in terms of misclassification probability, two clusters having normal distributions with the same standard devi­ation and a value of the Mahalanobis distance of 1.31 -see, for example, the first column of Table 1 - overlap for 25.89%. For Mahalanobis distance of 2 the overlap is 14.76%, while for values of the Mahalanobis distance greater than 5 the overlap is smaller than 0.1 % which means that the clusters are well separable. Figure 2. Test images used in the experi­ments. 5. Conclusions From Tables I, 2, 3, and 4 one can deduce the sig­nificance of each processing step of the concerned texture operators. The main conclusion that can be drawn is that post-processing can improve considerably the performance of filter based texture operators. Post-processing schemes, which include the computation of local statistics, perform better than those comprising an NLPO only do. Depending on the type of local statistics, the presence of an NLPO in a post-processing scheme can improve or degrade results. Computing the local mean, for instance, gives very good results if it is preceded by an NLPO (compare second and third column of Table 2 with the first columns of Tables 1 and 2). Similar improvement is achieved by local stan­dard deviation computation. In the latter case, no NLPO is needed between the filtering step and the local standard de­viation computation: such an intermediate step has negative effect on the results (compare the first column of Table 4 with the second and third columns of the same table). Computing the standard deviation gives better results than computing the variance (compare Tables 3 and 4). In both cases, the use of an intermediate NLPO degrades the results (compare first columns with the second and the third columns in Tables 3 and 4). On average, taking the modulus as a point operation and computing the local mean gives the best results (see last column of Table 2). 888 With a given type of post-processing the different filter­ing schemes lead to different results. The identity trans­form gives the worst results: the chance of misclassification for two normally distributed clusters with a Mahalanobis distance of 1.56 is 21. 79%. With an average Mahalanobis distance of 9 which corresponds to a misclassification of 10-7%, all sinusoidal transforms and the Laws' filter bank give comparable results. Still, they are worse than those obtained with Gabor filters, where the average Mahalanobis distance of 15 corresponds to a misclassification probability of 10-12%. The differences between the results obtained with Ga­bor filter banks and the other filter banks can be due to the differences in the tiling of the frequency domain. All of the sinusoidal transforms result in filters with a rectan­gular power spectrum with only two possible orientations (0°, 90°) while Gabor filters have an elliptical power spec­trum with eight possible orientations. Taking in consider­ation that the test material contains oriented textures only, it seems plausible to ex plain the good results obtained with Gabor filters by their orientation selectivity. References [1) A. Jain. A sinusoidal family of unitary transforms. IEEE­PAMI, 1(4):356--365, 1979. [2J A. Jain and F. Farrokhnia. Unsupervised texture segmentation using gabor filters. Pattern Recognition, 24(12):1167-1186, 1991. [3) P. Kruizinga and N. Petkov. Grating cell operator features for oriented texture. In A. Jain, S. Venkatesh, and B. Lovell, editors, Pmc. o/the Int. Conf. on Pattern Recognition, pages 1010--1014, Brisbane Australia, 1998. [4) P. Kruizinga and N. Petkov. Non-linear operator for oriented texture. IEEE Trans. on Image Processing, 8( 1 0): 1 395-1 407, 1999. [51 K. Laws. Rapid texture identification. In Proc. SPIE Conf on Image Processing for Missile Guidance, volume 238, pages 376--380, 1980. (6) J. Malik and P. Perona. Pre attentive texture discrimination with early vision mechanisms. Journal a/the Optical Society of America, A, 7(5):923-932, 1990. [7] N. Petkov and P. Kruizinga. Computational models of visual neurons specialized in the detection of periodic and aperiodic oriented visual stimuli: Bar and grating cells. Biological Cy­bernetics, 76:83-96, 1997. [8] T. Randen and 1. H. Husoy. Filte ring for texture classification: a comparative study. IEEE-PAMl, 21(4):291-310, 1999. [9] M. Unser. Local lin ear transfoffils for texture measurements. Signal Processing, 11(1):61-79 , 1986. 

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A comparative study of filter based texture operators using Mahalanobis distance

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