Urbach, Erik R.

Wilkinson, Michael H.F.

English

An efficient algorithm is presented for the computation of gray-scale morphological operations with 2-D structuring elements (S.E.). The required computing time is independent of the image content and of the number of gray levels used. For circular S.E.s, it always outperforms the only existing comparable method, which was proposed earlier, by a factor between 1.8 and 8.6, depending on the image type. So far, filtering using multiple S.E.s is always done by performing the operator for each size and shape of the S.E. separately. With our method filtering with multiple S.E.s can be performed by a single operator for a reduced computational cost per size or shape, which makes this method more suitable for use in granulometries, dilation-erosion scale spaces, and template matching using the hit-or-miss transform

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   University of GroningenEfficient 2-D Gray-Scale Dilations and Erosions with Arbitrary Flat Structuring ElementsUrbach, Erik R.; Wilkinson, Michael H.F.Published in:2006 IEEE INTERNATIONAL CONFERENCE ON IMAGE PROCESSING, ICIP 2006, 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:2006Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Urbach, E. R., & Wilkinson, M. H. F. (2006). Efficient 2-D Gray-Scale Dilations and Erosions with ArbitraryFlat Structuring Elements. In 2006 IEEE INTERNATIONAL CONFERENCE ON IMAGE PROCESSING,ICIP 2006, PROCEEDINGS (pp. 1573-1576). (IEEE International Conference on Image Processing (ICIP)).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).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: 20-06-2022EFFICIENT 2-D GRAY-SCALE DILATIONS AND EROSIONS WITH ARBITRARY FLATSTRUCTURING ELEMENTSErik R. Urbach and Michael H.F. WilkinsonInstitute of Mathematics and Computing ScienceUniversity of GroningenP.O. Box 800, 9700 AV GroningenThe NetherlandsABSTRACTAn efficient algorithm is presented for the computation ofgray-scale morphological operations with 2-D structuring el-ements (S.E.). The required computing time is independent ofthe image content and of the number of gray levels used. Forcircular S.E.s, it always outperforms the only existing com-parable method, which was proposed by Van Droogenbroeckand Talbot [1], by a factor between 1.8 and 8.6, dependingon the image type. So far, filtering using multiple S.E.s isalways done by performing the operator for each size andshape of the S.E. separately. With our method filtering withmultiple S.E.s can be performed by a single operator for areduced computational cost per size or shape, which makesthis method more suitable for use in granulometries, dilation-erosion scale spaces, and template matching using the hit-or-miss transform.Index Terms— Morphological operators, erosions, dila-tions, structuring elements, mathematical morphology, imagefiltering, erosion-dilation scale spaces1. INTRODUCTIONMorphological operators [2] like dilation and erosion withstructuring elements (S.E.) are the most fundamental opera-tors in mathematical morphology and have become commontools for both image filtering and analysis [3] of binary andgray-scale images since the development of efficient algo-rithms [4, 5]. Usually, these algorithms are limited to lin-ear S.E. and shapes like rectangles that can be decomposedin a series of linear S.E.s. A recent overview of efficientalgorithms for morphological operators with linear S.E. and2-D S.E. decompositions can be found in [6]. All methodsbased on decomposition of 2-D S.E.s into linear S.E.s sharethe same limitation: many shapes either cannot be decom-posed efficiently or they cannot be decomposed at all. In thebinary case, efficient algorithms for some 2-D shapes like cir-cles do exist, but these cannot efficiently be extended to thegray-scale case, for which polygonal approximations [7,8] ofcircles usually are used instead. For larger circles these ap-proximations tend be either too coarse or too computationallyintensive, since the number or linear S.E.s required is propor-tional to the diameter of the circle. Van Droogenbroeck andTalbot [1] proposed an efficient algorithm for computing mor-phological operations with arbitrary 2-D shapes using a his-togram, which makes the computing time of their algorithmdependent on the number of gray levels used. Their idea is tocompute for one pixel p of the image the complete histogrambased on the intensities of the pixels around p correspond-ing to elements of the S.E. The value of p after erosion is theminimum intensity in the histogram which has a value > 0.For all succeeding pixels of the image (by moving around theS.E. over the image), the histogram is efficiently updated andthe position of its minimum intensity changes only if i) a newminimum value is shifted into the histogram, which can bekept track when the histogram is updated, or ii) when the cur-rent minimum is shifted out of the histogram, in which casethe algorithm searches for the first following (brighter) inten-sity which is now represented in the histogram.In this paper, we present a new method for performingmorphological operators with any 2-D structuring element thatalways outperforms existing algorithms for arbitrary structur-ing elements, and which has two advantages: i) it is inde-pendent of both image content and the number of gray lev-els used, and ii) application of a single operator using manydifferent S.E.s can be computed somewhat more efficiently,which may be useful for granulometries [2, 3] and erosion-dilation scale spaces [9]. Compared to Van Droogenbroeckand Talbot’s method, it has the further advantage that it alsoworks on floating point data. Our method is compared withthe method of Van Droogenbroeck and Talbot in section 3.2. ALGORITHMFor the sake of simplicity and clarity, we limit our discussionhere to discrete 2-D gray-scale images f and erosions with2-D flat S.E.s B. It should be noted that our method can beeasily adapted for 3-D images and other morphological opera-15731­4244­0481­9/06/$20.00 ©2006 IEEE ICIP 2006Fig. 1. Decomposition of S.E. B into chordstions. It is assumed that all images f have their origin top-leftand that images are processed in scanline order.Like existing methods, our approach to improve the com-putational efficiency of erosions is by reducing the number theredundant comparisons performed by a direct implementationof its definition:(f  B)(p) = minz∈Bf(p + z) (1)Our algorithm decomposes an arbitrary S.E. into a seriesof chords, i.e., runs of foreground pixels of maximum extent,as demonstrated in Fig. 1 for a letter H. Each chord is rep-resented by a triple containing: (i) its y-offset with respect tothe origin of the S.E., (ii) its minimal x-position, and (iii) itslength l. For each S.E. we store an array of Nc chords, theminimum and maximum y-offsets ymin and ymax, and mini-mum and maximum x-values xmin and xmax, and the maxi-mum chord length lmax occurring in B.Obviously, if we can compute the minimum value withineach chord, for each shifted S.E., and compute the minima ofall chord-minima, we have computed the minimum of all pix-els within the shifted S.E. One way to do this when processingimage row y, is by creating an auxiliary 2-D array My′(i, x)for each image row y′ between y − ymin and y − ymax. Eacharray index i runs from 0 to lmax, inclusive, and x from xminto X +xmax−1, inclusive, with X the X dimension of imagef . Each value My′(i, x) is defined asMy′(i, x) = minx′∈[x,x+i]f(x′, y′) (2)Each array can be computed in lmax comparisons per pixel.We can now compute the minimum over B in Nc − 1 furthercomparisons per pixel. For each value of x we compute(f B)(x, y) = minj∈{0,1,...,Nc−1}My+yj (lj , x+xmin,j), (3)in which yj , xmin,j , and lj denote the y-offset, minimum xposition and length l of chord j. This version has a compu-tational complexity per pixel of O(lmax + Nc). However, itsmemory complexity is quite high O(lmax(ymax − ymin)X),because computing arrays M ′y only needs to be done once forevery value of y′./* Copy image data into My′(0, :)*/for (x = 0; x < X; x + +)My′(0, x) = f(x, y′);/* Pad copied data on either side ofcopied row */for (x = xmin; x < 0; x + +)My′(0, x) = f(0, y′);for (x = X; x < X + xmax − 1; x + +)My′(0, x) = f(X − 1, y′);/* Compute minima of runs of length 2n,starting at each x */for (i = 1; i ≤ log2(lmax − 1); i + +)for (x = xmin; x < X + xmax − 1 − 2i−1; x + +)My′(i, x) = min(My′(i− 1, x), My′(i− 1, x+2i−1));Fig. 2. Pseudocode for computing My′ in time proportionalto log(lmax)We can reduce both complexities using the following ob-servation. We can compute the minimum of any chord lengthlj from the minima of two (possibly overlapping) run lengthsof 2nj with nj = log2(lj − 1). If we modify our chorddescription to contain n, and two x values x1,j = xmin,j ,and x2 = xmin,j + lj − 2nj , we can compute the minimumof each chord by just one additional comparison per chord.We only need to store log2(lmax − 1) minimum values perpixel, which can be computed in log2(lmax − 1) − 1 com-parisons per pixel, as shown in Figure 2. Our computation forthe erosion now becomes(f  B)(x, y) =minj∈{0,1,...,Nc−1}(min(My+yj (nj , x + x1,j),My+yj (nj , x + x2,j))),(4)in which x1,j and x2,j are the x1 and x2 values of chord j.This means we can compute any erosion (or by duality dila-tion) for any structuring element in time and memory com-plexity O(Nc + log lmax) per pixel.Furthermore, any structuring element B′ ⊆ B can becomputed using the same precomputed values, which couldyield further improvements when, e.g., computing erosion-dilation scale spaces. If we have NB nested S.E. with maxi-mal chord length lmax,B and numbers of chords Nc,b the timecomplexity becomes O(log lmax,B +∑Nc,b), rather than atime complexity of O(∑log lmax,b +∑Nc,b), which rep-resents a modest gain. Figure 3 shows the pseudocode forcomputing the erosion on a single line.It is obvious that the algorithm described here is inde-pendent in its time complexity of the image content, unlikethe method of Van Droogenbroek and Talbot [1]. Both theirmethod and our method extend readily to 3-D. In our case weneed to augment the S.E. with a zmin and zmax, each chord1574/* Compute the minima for the firstchord */for (x = 0; x < X; x + +)g(x, y) = min(My+y0(n0, x + x1,0),My+y0(n0, x + x2,0));/* Compute for all other chords */for (j = 1; j ≤ Nc; j + +)for (x = 0; x < X; x + +)g(x, y) = min(g(x, y), My+y0(n0, x + x1,j),My+y0(n0, x + x2,j));Fig. 3. Pseudocode for performing the erosion on one imageline, in which the output image line is given by g(x, y). Thisshows that one comparison per pixel for the first chord of theS.E. is needed, and two per pixel for each next one.with a z-offset, and the arrays My′ need to be replaced byMy′,z′ . Otherwise the extension is straightforward.3. EXPERIMENTSWe compared the computation time of our method with a opti-mized naive implementation, where only the foreground pix-els of the S.E. are considered, and with the algorithm of VanDroogenbroeck and Talbot [1] using circular and H-shapedstructuring elements on two 8-bit 2160x1440 gray-scale im-ages: a natural image and a generated image with uniformlydistributed noise. In order to measure the influence of thenumber of gray levels on the computation time, 16-bit ver-sions of these two images were computed by scaling the grayvalues in both images. The C source code of both algorithmsis available on request.Fig. 4 shows the computation times of these methods for a2.8 GHz Pentium 4 processor based PC, with 1 GB of RAM.A thin letter H and circular S.E.s of increasing width wereused as S.E. shapes. As noted in the previous section, whenan image is to be eroded with multiple S.E.s, our approachprovides a way to compute these in less time than would berequired when each of them would be computed separately.To measure the speed gain of this strategy (called “New multierode” in Fig. 4), it was compared with the total time neededfor computing the same erosions using the existing [1] (re-ferred to as “DT many erode”) and our method (“New manyerode”).From the figures it is clear that the method proposed isalways faster than the existing methods. On the 8-bit naturalimage, the computation time was, for a single erosion witha circle of diameter 49, 2.03s for the DT method, 10.41s forthe naive method, and 1.15s for our new method. On the 8-bit noise image the computation times for the same S.E. wererespectively 2.14s, 10.14s, and 1.15s, while the DT methodneeded 9.89s for the 16-bit version of the same image and S.E.Since the computation time of our method is independent onthe number of gray levels in the image, 16-bit and 8-bit im-ages are computed equally fast. Even on floating point data,only a 30 % speed penalty was incurred. The histogram usedby the existing method becomes extremely sparse when verysmall S.E.s are used with a 16-bit image, which explains thelong computing time for the smallest circular S.E.s shown inFig. 4. Both the proposed and the existing DT method havethe highest speed gain compared with the naive implemen-tation when S.E. shapes with a low perimeter area ratio areused, such as circles and squares. The computation timeswhen a thin letter H shaped S.E. was used instead of circle isalso shown in Fig. 4. The computation times for a single ero-sion using that letter H of width 49 on the 8-bit natural imageare 3.96s, 3.31s, and 5.51s for respectively the DT method,our new method, and the naive implementation, while the DTmethod needed 10.90s for the 16-bit version of the same im-age. Finally, our “multi-erode” approach currently offers onlya small improvement of about 10 % over the “many-erode”version, as expected from the differences in complexity. Itshould be noted that when a set of equally sized S.E.s withdifferent shapes is used instead of the increasingly sized S.E.sused here, a stronger reduction in computing time is to be ex-pected.Finally, our algorithm showed no change in computingtime between different images of the same size. The apparentdifferences between the plots of Fig. 4 are due to differentscales being used due to differences in computing time forthe Droogenbroeck and Talbot algorithm.4. CONCLUSIONSA new method for computing morphological operations witharbitrary 2-D flat structuring elements was proposed. It hasa computational complexity that is independent of the num-ber of gray levels in the image. The proposed method hasa clear computational performance advantage over existingmethods when S.E.s are used that cannot be easily decom-posed into linear structuring elements. A further improve-ment is achieved when multiple S.E.s are used with a singleoperator, such as the computation of granulometries and di-lation or erosion scale spaces [9], since the results of manycomparisons are computed once and stored in an auxiliary ar-ray, which can be reused for filtering of all succeeding S.E.s.Like the existing method of Van Droogenbroeck and Tal-bot [1] arbitrary S.E.s can be used. However, our methodalways outperforms the existing method, especially when im-ages with higher bit depth as common in applications such asmedical imaging, are used. Besides, our method can handlefloating point images easily, which is impossible in the VanDroogenbroeck and Talbot method. The Van Droogenbroeckand Talbot method can however be easily adapted to otherpercentiles.We are currently working on relatively small speed im-provements of the method proposed by further reducing the15750 10 20 30 40 50024681012Diameter of S.E. (pixels)Computing time (s)8−bit DT erode16−bit DT erodeNew erodeNaive erode0 10 20 30 40 5002468101214Diameter of S.E. (pixels)Computing time (s)8−bit DT erode16−bit DT erodeNew erodeNaive erode0 10 20 30 40 50024681012Diameter of S.E. (pixels)Computing time (s)8−bit DT erode16−bit DT erodeNew erodeNaive erodeCircular S.E. on natural image Circular S.E. on noise image H-shaped S.E. on natural image0 10 20 30 40 50010203040506070Max. diameter of S.E. (pixels)Computing time (s)8−bit DT many erode16−bit DT many erodeNew many erodeNew multi erode0 10 20 30 40 50020406080100120Max. diameter of S.E. (pixels)Computing time (s)8−bit DT many erode16−bit DT many erodeNew many erodeNew multi erode0 10 20 30 40 50020406080100120140Max. diameter of S.E. (pixels)Computing time (s)8−bit DT many erode16−bit DT many erodeNew many erodeNew multi erodeCircular S.E.s on natural image Circular S.E.s on noise image H-shaped S.E.s on natural imageFig. 4. Required computing time for erosions with circular (left,middle) and H-shaped (right) S.E.s using the slow naive, theVan Droogenbroeck-Talbot (DT), and our proposed (new) algorithm on a natural image and on a noise image. Top row: singleerosions with width k; bottom row: series of erosions with width l = 3, 5, 7, ..., k.number of comparisons needed. Further comparisons to theDroogenbroeck and Talbot, and other algorithms, on a widerrange of S.E. types are also being studied, especially for theapplication of template matching using the hit-or-miss trans-form, where large sets of S.E. shapes at many scales and ori-entations can be computed more efficiently5. REFERENCES[1] Marc Van Droogenbroeck and Hugues Talbot, “Fast com-putation of morphological operations with arbitrary struc-turing elements,” Pattern Recogn. Lett., vol. 17, pp.1451–1460, 1996.[2] J. Serra, Image Analysis and Mathematical Morphology,vol. 1, Academic Press, New York, 2 edition, 1982.[3] S. Batman and E. R. Dougherty, “Size distributions formultivariate morphological granulometries: texture clas-sification and statistical properties,” Optical Engineering,vol. 36, no. 5, pp. 1518–1529, May 1997.[4] M. van Herk, “A fast algorithm for local minimum andmaximum filters on rectangular and octagonal kernels,”Pattern Recogn. Lett., vol. 13, pp. 517–521, 1992.[5] J. Gil and R. Kimmel, “Efficient dilation, erosion, open-ing and closing algorithms,” IEEE Trans. on PAMI, vol.24, no. 12, pp. 1606–1617, 2002.[6] M. Van Droogenbroeck and M.J. Buckley, “Morpholog-ical erosions and openings: Fast algorithms based on an-chors,” Journal of Mathematical Imaging and Vision, vol.22, pp. 121–142, 2005.[7] P. Soille, E. Breen, and R. Jones, “Recursive implemen-tation of erosions and dilations along discrete lines at ar-bitrary angles,” IEEE Transactions on Pattern Analysisand Machine intelligence, vol. 18, no. 5, pp. 562–567,May 1996.[8] P. Soille and H. Talbot, “Directional morphological fil-tering,” IEEE Transactions on Pattern Analysis and Ma-chine Intelligence, vol. 23, pp. 1313–1329, 2001.[9] P. T. Jackway and M. Deriche, “Scale-space propertiesof the multiscale morphological dilation-erosion,” IEEETransactions on Pattern Analysis and Machine Intelli-gence, vol. 18, pp. 38–51, 1996.1576

Efficient 2-D Gray-Scale Dilations and Erosions with Arbitrary Flat Structuring Elements

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Efficient 2-D Gray-Scale Dilations and Erosions with Arbitrary Flat Structuring Elements

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