When we are interested to the detection of the roughness features by means of the 3D reconstruction, based on photometric stereo techniques,
an important problem is the elimination of the brightness variation due to different light conditions which can alter the response.
This paper will concentrate on presenting results of a new method for eliminating this problem.
Every pixel of a picture gives only one number: the brightness of the corresponding point on the object, whereas the surface orientation is
described by a normal vector that has two degrees of freedom. The level of brightness depends on many factors as well as the homogeneity
of reflection properties of the material or its physical continuity and the surface smoothness or roughness.
In this work we will show how the application of the Discrete Wavelet Transform (DWT) to the processing of some images, captured
on different light conditions, permits to solve the problem of emphasizing roughness features of a metallic surface. Wavelet transforms can
model irregular data patterns such as sharp changes, better than the Fourier transforms and standard statistical procedures (e.g., parametric
and non-parametric regressions) and provide a multiresolution approximation to the data.
Here we propose, also, a non-parametric method, based on the wavelet theory, for the estimation of the threshold level of a gray levels
distribution, obtained from the intensity image matrix

NASTI, G.

NIOLA, VINCENZO

QUAREMBA, G.

Archivio della ricerca - Università degli studi di Napoli Federico II

Journal of Materials Processing Technology 164–165 (2005) 1410–1415A problem of emphasizing features of a surface roughness bymeans the Discrete Wavelet TransformVincenzo Niola a, ∗, Gennaro Nasti b, Giuseppe Quaremba ba Department of Mechanical Engineering for Energetics, University of Naples “Federico II”, Italyb University of Naples “Federico II”, ItalyAbstractWhen we are interested to the detection of the roughness features by means of the 3D reconstruction, based on photometric stereo techniques,an important problem is the elimination of the brightness variation due to different light conditions which can alter the response.This paper will concentrate on presenting results of a new method for eliminating this problem.Every pixel of a picture gives only one number: the brightness of the corresponding point on the object, whereas the surface orientation isdescribed by a normal vector that has two degrees of freedom. The level of brightness depends on many factors as well as the homogeneityof reflection properties of the material or its physical continuity and the surface smoothness or roughness.omad©K1rttFfbbtcc(0dIn this work we will show how the application of the Discrete Wavelet Transform (DWT) to the processing of some images, capturedn different light conditions, permits to solve the problem of emphasizing roughness features of a metallic surface. Wavelet transforms canodel irregular data patterns such as sharp changes, better than the Fourier transforms and standard statistical procedures (e.g., parametricnd non-parametric regressions) and provide a multiresolution approximation to the data.Here we propose, also, a non-parametric method, based on the wavelet theory, for the estimation of the threshold level of a gray levelsistribution, obtained from the intensity image matrix.2005 Elsevier B.V. All rights reserved.eywords: Image processing; Surface roughness; Wavelet transform; Optimum threshold. IntroductionThe study of criteria for evaluating the surface roughnessepresents, to day, one of the most important problem forhe production of some critical mechanical organs in ordero confer them some specific and functional characteristics.or that reason many authors consider the roughness as theourth dimension of the design.Since, in many cases, the roughness determines the level ofrightness of a metallic surface, its reflectance can be studiedy a specific model, based on some hypotheses and solvinghe image irradiance equation using some algorithms [1,2].In this paper we will show a new method of image pro-essing for emphasizing the features of a surface roughness,aptured on different light conditions. It is based on the prop-∗ Corresponding author. Tel.: +39 0817 683 482.E-mail addresses: vniola@unina.it (V. Niola), gennaronasti@libero.itG. Nasti), quaremba@unina.it (G. Quaremba).erties of the Wavelet Transform to detect the presence of de-tails which, usually, are lost when we apply to the signal afiltering process in order to reduce the noise.The Wavelet Transform gives a time-frequency represen-tation of a signal that has two main advantages over usualmethods: an optimal resolution even in the time and fre-quency domains and lack of the requirements of stationarityof the signal. It is defined as the convolution between the sig-nal x(t) and the wavelet functions ψa,b(t)which are dilated orcontracted and shifted versions of a unique wavelet functionψ(t).Contracted versions of the wavelet function will matchthe high frequency components of the original signal andon the other hand, the dilated versions will match low fre-quency oscillations. Then, by correlating the original signalwith wavelet functions of different sizes, we can obtain thedetails of the signal at different scales. These correlationswith the different wavelet functions can be arranged in a hi-erarchical scheme called multiresolution decomposition [3].924-0136/$ – see front matter © 2005 Elsevier B.V. All rights reserved.oi:10.1016/j.jmatprotec.2005.02.169V. Niola et al. / Journal of Materials Processing Technology 164–165 (2005) 1410–1415 1411The multiresolution decomposition separates the signalinto “details” at different scales, the remaining part, beinga coarser representation of the signal, is called “approxima-tion”. Moreover, it was shown [3] that each detail (Dj) andapproximation signal (Aj) can be obtained from the previ-ous approximation Aj−1 via a convolution with high-pass andlow-pass filters, respectively.When wavelets are used to encode two-dimensional sig-nals (i.e., pictures or images), often this is done by using“separable products” of a one-dimensional wavelet and a one-dimensional scaling function. This makes it possible to usethe Wavelet Transform and in this case each new waveletmeasures variations in the image along three different di-rections: horizontal, vertical and diagonal edges. The Dis-crete Wavelet Transform (DWT) of images is then calculatedessentially by applying the one-dimensional Wavelet Trans-form along the rows and the columns of the image [4–6].Therefore, an algorithm similar to the one-dimensional caseis possible for two-dimensional wavelets and scaling func-tions obtained from one-dimensional ones by tensor product[3,7,8].In mathematics, a singularity is a point at which a func-tion is not differentiable although it is differentiable in aneighbourhood of that point. Singularities record large signalchanges over very short time changes. Crests of roughnessmay be though of as a smoother version of singularities whereltaul2tIfwaϕaψFhwavelet) orthonormal to ψ(x), according to the L2 norm. Inour work we choose:ϕ(x) ={1, x∈ (0, 1]0, x /∈ (0, 1] and ψ(x) =−1, x∈[0, 12]1, x∈(12 , 1]0, x /∈ [0, 1]They are called Haar Wavelets. It is easy to prove that, ifψ(x)is a mother wavelet, then also ψj,k is a mother wavelet.Moreover, in this case, the systems of functions{{ϕj0k}, {ψjk}, k ∈Z, j ∈ [j0, j1] ∩ Z}is an orthonormal system in L2(R). The definition of the esti-mator (1) is based on the Parseval Theorem. In fact, accordingto this result, any L2(R) can be represented as a convergentseriesh(x) =∑kaj0kϕj0k(x) +j1∑j=j0∑kbjkψjk(x) (2)whereaj0k =∫ +∞t=−∞h(t)ϕj0k(t) dt andbjk =∫ +∞h(t)ψjk(t) dtNf∑iffToeMwdENqarge signal changes occur over slightly broader time changes.A theory for examining the singularities of functions usinghe Wavelet Transform was developed [9], which has beenpplied here to identify and characterise the crests that makep the surface roughness of a metallic test piece on differentight conditions.. A threshold estimator based on Haar WaveletWe follow an approach similar to [10].Let X1,X2, . . .,XN be N independent identically dis-ributed random variables whose density (unknown) is f(x).n this case, a wavelet estimator of f is given by [11,12]ˆ =∑kaˆj0ϕj0k(x) +j1∑j=j0∑kˆbjkψjk(x) (1)here j0, j1 ∈Zˆj0k =1NN∑i=1ϕj0k(Xi), ˆbjk =1NN∑i=1ψjk(Xi)jk(x) = 2j/2ϕ(2jxk), k ∈Zndjk(x) = 2j/2ψ(2jxk), k ∈Zinally, ψ(x) is a function (mother wavelet) whose first(h∈N) moments are zero and ϕ(x) is a function (fathert=−∞ow assume that ϕ and ψ are compactly supported. Then,ollowing [11], by utilising the relation belowkϕjk(x) · ϕjk(Xi) +∑kψjk(x) · (Xi)=∑kϕj+1k(x) · ϕj+1k(Xi)t follows that the estimator (1) can be written in the equivalentormˆj1 (x) =1NN∑i=1∑kϕj1+1k(x)ϕj1+1k(Xi) (1’)he choice of j1 is discussed in statistic literature. The choicef j1 is made in order to minimize the mean integrated squaredrrorISE = E||f − ˆfj1 ||22= E|| ˆfj1 − E( ˆfj1 )||22 + E||f − E( ˆfj1 )||22here E(·) is the expected value of the observations and ||·||2enotes the usual L2 norm. Note that( ˆfj1 (x)) =∫ +∞y=−∞ϕj1+1k(x)ϕj1+1k(y)f (y) dyow, let(x, y) =+∞∑k=−∞φ(x− k)φ(y − k)1412 V. Niola et al. / Journal of Materials Processing Technology 164–165 (2005) 1410–1415Fig. 1. Image of surface roughness of the same metallic test piece obtained on different light conditions.and q(x, y) = 2nq(2nx, 2ny) < 2n. It follows thatE|| ˆf − E( ˆfj1 )||22 ≤1N∫ +∞−∞∫ +∞−∞q2j1+1 (y, x)f (y) dx dy=∫ +∞−∞qj1+1 (y, y)f (y) dy ≤2j1+1NNow, let Wn be the space spanned by the orthonormal basis{2n/2ψ(2n/2x− k), k∈Z}. Furthermore, let V0 be the spacespanned by the orthonormal basis {ψ(x− k), k∈Z} and letVn =Vn−1 +Wn−1 (i∈N). By construction,∪n∈N0Vn is densein L2(R).SinceE(fj1 ) is the projection of f on the space Vj1 , we candeduce that E||f − E( ˆfj1 )||22 converges to zero as j1 tends toinfinity, for any f∈L2(R).An alternative measure of error is the mean square errorMSQ = E[f (x) − ˆfj1 (x)]2= E[E( ˆfj1 (x) − ˆfj1 (x)]2 + E[f (x) − E ˆfj1 (x)]2However, in a regression estimation (i.e., when Xi −Xi−1 =1/N = const.) it is accepted that j1 = log2 N− ln(log2 N).Therefore, in our work we have chosenj(1)wxmfcsfposition∫ suˆf (x)(x− u) dx =: ˆE(X− u,X > u)where s is given by s= sup{x:F(x) < 1}.Fig. 2. (a) Gray levels histogram referred to the first image (left). (b) Graylevels histogram referred to the second image (right).1 = log2 median(xi − xi−1) − ln(log2(1median(xi − xi−1)))(3)herei = Xi√∑Ni=1XiThe determination of an optimal threshold is a compro-ise of many factors.The ˆf (x) can be calculated by utilising (1) or (1′). Then,or any i∈ {1, 2, . . .,N}, we evaluate the ˆF (Xi) by numeri-al quadrature of ˆf (x) by setting ∫ Xi0 ˆf (x) dx:= ˆF (Xi). Nowuppose that {Xi} are sorted in ascending order. Therefore,or any u∈ [X1,XN], we can estimate E(X− u,X> u) by theV. Niola et al. / Journal of Materials Processing Technology 164–165 (2005) 1410–1415 1413Fig. 3. An example of binary images obtained by applying usual thresholding method.Finally, we evaluate E(X− u|X> u) by settingˆE(X− u)|X > u):=ˆE(X− u,X > u)1 − ˆF (x)Fo3. Application of DWT and resultsFig. 1 shows the images of the same metallic test piececaptured on different light conditions. It is evident the differ-ence of brightness caused by the position of the lamp. Theimages are converted into intensity matrix containing the val-ues in the range 0 (black) to 1 (full intensity or white) (Fig. 2aand b).The evidence of prevalence of darkness on the first image(left) with respect to the second one (right) is illustrated bycomparing their histograms bar plot of gray levels.The importance of converting the intensity image to bi-nary image by an optimum thresholding is illustrated in Fig. 3obtained by applying to the gray levels histogram the usualmethod [13–15]. The output binary image has values of 0(black) for all pixels in the input image with luminance lessthan threshold parameter and 1 (white) for all other pixels.Note that we specify the threshold in the range [0,1], regard-less of the class of the input image.Since exists a large difference in brightness level for thesame image it would be difficult to base a quantitative mea-sure of surface roughness on such data by means of the 3Dreconstruction by photometric stereo techniques.We have verified, also, that the threshold obtained by ap-plying the DWT to the gray levels histogram provides thebest response in terms of emphasizing the surface roughnessfig. 4. Grand average of the approximation coefficients calculated on imagesf Fig. 3.Fig. 5. An example of the better perfoeatures.rmance obtained with DWT.1414 V. Niola et al. / Journal of Materials Processing Technology 164–165 (2005) 1410–1415Fig. 6. Grand average of the approximation coefficients calculated on imagesof Fig. 5.The output of the wavelet transform is a set of coefficientswhich give a measure of how the properties of the chosenwavelet “fit” the function at the particular translation andscale.The results by applying the DWT to the original imagesare presented in Fig. 4, in which are presented the mean dis-tribution of the approximation wavelet coefficients.Note that the shape of the curves, in terms of qualitativeresponse, is quite similar, while, in terms of entropy, a com-mon concept in many fields, mainly in signal processing [16],is more different: 24.0990 and 54.9523 respectively for thefirst (left) and the second (right) image.The improvement of the response obtained by choosinga good thresholding level is illustrated in Figs. 5 and 6,where the value of the entropy is of 48.5104 and 48.3762respectively for the first (left) and the second (right)image.The data presentation shown by Fig. 6 appears to be thebest way of displaying the ability shown by DWT for elimi-nating the error due to brightness influence.4. ConclusionAs a general remark we can state that with wavelets abetter resolution and localization of the features of the imageiebfTtfsmsists of relatively low-frequency oscillations which are as-sociated with the energy content of the image as well ashis entropy. Both the information are useful for detectingthe resolution in terms of denoising and enhancement of theimage.Future research shall identify correlations between indi-vidual features across a given sample set. This will allow amore selective averaging to take place with the effect of re-ducing the required number of trials necessary to obtain thefeatures of a surface roughness. By understanding the under-lying mechanism that generate the brightness variation andreducing the number of trials necessary to calculate it, theoverall objective is to develop a method to extract the fea-tures from a single trial.The preliminary results demonstrate that the signals, ob-tained from the images, can be decomposed into a finite set ofdistinct crests without any loss of overall information, whenaveraged across all signals. The crest detection method hasgenerated a set of peaks that are characterised by position,scale and amplitude. This allows them to be analysed in morerefined ways than just the usual methods and, of course, it as-sumes that the method used in this paper represents the firststep in order to eliminate the error due to the brightness in-fluence to measure the surface roughness.In this manner the progress of metal machining, grindingand polishing operations can be monitored, in real time, untiltAsRs achieved.There are many wavelet packet libraries. They differ fromach other by their generating filters ϕ andψ, the shape of theasic waveforms and the frequency content. It was importantor our investigation to have refined frequency resolution.herefore, we have chosen the wavelet packets derived fromhe splines of the eighth order.It was discovered that the brightness of a metallic sur-ace produces disturbances of two types. The first type con-ists of the high-frequency oscillations which determine theain features of the processed image. The second type con-he desired surface finish is achieved.cknowledgementsWe are grateful to Dr. Roberto Fusco for the computingupport.eferences[1] V. Niola, G. Quaremba, G. Nasti, A method for 3D reconstructionbased on photometric stereo techniques, in: International Conferenceon Advanced Optical Diagnostics in Solids, Fluids and Combustion(VSJ-SPIE’04), University of Tokyo, Tokyo, Japan, 2004.[2] V. Niola, G. Quaremba, G. Nasti, Wear recognition in antifrictionbearings by Stereo-Photometric Technique and DWT, in: Interna-tional Conference on Advanced Optical Diagnostics in Solids, Fluidsand Combustion (VSJ-SPIE’04), University of Tokyo, Tokyo, Japan,2004.[3] S. Mallat, A theory for multiresolution signal decomposition: thewavelet representation, IEEE Patt. Anal. Mach. Intell. 11 (7) (1989)674–693.[4] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press,Boston, 1998.[5] G. Strang, T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA.[6] M. Vetterli, J. Kovacevic, Wavelets and Subband Coding, Prentice-Hall, Englewood Cliffs, 1995.[7] I. Daubechies, Ten Lectures on Wavelets, SIAM, 1992.[8] Y. Meyer, Wavelets and Operators, Cambridge University Press,1993.V. Niola et al. / Journal of Materials Processing Technology 164–165 (2005) 1410–1415 1415[9] S. Mallat, W.L. Hwang, Singularity detection and processing withwavelets, IEEE Trans. Inform. Theory 38 (2) (1992) 617–643.[10] V. Niola, G. Quaremba, R. Oliviero, A method for estimating theshape parameter of a Weibull p.d.f. based on wavelet analysis, Int.J. Modell. Simul., submitted for publication.[11] W. Ha¨rdle, G. Kerkyacharian, D. Picard, A. Tsybakov, Lecture Notesin Statistics: Wavelets, Approximation and Statistical Applications,Springer, New York, 1998.[12] R.T. Ogden, Essential Wavelets for Statistical Applications and DataAnalysis, Birkha¨user, Boston, 1997.[13] S.J. Lim, Two-Dimensional Signal and Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1990.[14] K.A. Jain, Fundamentals of Digital Image Processing, Prentice-Hall,Englewood Cliffs, NJ, 1989, pp. 150–153.[15] W.B. Pennebaker, J.L. Mitchell, JPEG: Still Image Data CompressionStandard, Van Nostrand Reinhold, New York, 1993.[16] R.R. Coifman, M.V. Wickerhauser, Entropy-based algorithms forbest basis selection, IEEE Trans. Inf. Theory 38 (2) (1992) 713–718.

A problem of emphasizing features of a surface roughness by means the Discrete Wavelet Transform

Abstract: When we are interested to the detection of the roughness features by means the 3-D reconstruction, based on photometric stereo techniques, an important problem is the elimination of the brightness variation due to different light conditions which can alter the response. This paper will concentrate on presenting results of a new method for eliminating this problem. In this work we will show how the application of the Discrete Wavelet Transform (DWT) to the processing of some images, captured on different light conditions, permits to solve the problem of emphasizing roughness features of a metallic surface. Here we propose, also, a non-parametric method, based on the wavelet theory, for the estimation of the threshold level of a gray levels distribution, obtained from the intensity image matrix

A problem of emphasizing features of a surface roughness by means the Discrete Wavelet Transform

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