Bayer, Micha

Droop, Stephen

Roerdink, Jos B.T.M.

Wilkinson, Michael H.F.

English

A method for shape analysis of diatoms (single-cell algae with silica shells) based on extraction of features on the contour of the cells by multi-scale mathematical morphology is presented. After building a morphological contour curvature scale space, we present a method for extracting the most prominent features by unsupervised cluster analysis. The number of extracted features matches well with those found visually in 92 % of the 350 diatom images examined

Dissertations of the University of Groningen

   University of GroningenDiatom contour analysis using morphological curvature scale spacesWilkinson, Michael H.F.; Roerdink, Jos B.T.M.; Droop, Stephen; Bayer, MichaPublished 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):Wilkinson, M. H. F., Roerdink, J. B. T. M., Droop, S., & Bayer, M. (2000). Diatom contour analysis usingmorphological curvature scale spaces. In A. Sanfeliu, JJ. Villanueva, M. Vanrell, R. Alquezar, T. Huang, &J. Serra (Eds.), 15TH INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION, VOL 3,PROCEEDINGS (pp. 652-655). (INTERNATIONAL CONFERENCE ON PATTERN RECOGNITION). 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: 28-12-2022Diatom Contour Analysis using Morphological Curvature Scale SpacesMichael H.F. WilkinsonJos B.T.M. RoerdinkInstitute for Mathematics and Computing Science,University of Groningen,Groningen, The NetherlandsStephen DroopMicha BayerRoyal Botanical Gardens,Edinburgh, Scotland, UK.AbstractA method for shape analysis of diatoms (single-cell al-gae with silica shells) based on extraction of features on thecontour of the cells by multi-scale mathematical morphol-ogy is presented. After building a morphological contourcurvature scale space, we present a method for extractingthe most prominent features by unsupervised cluster anal-ysis. The number of extracted features matches well withthose found visually in 92 % of the 350 diatom images ex-amined.1 IntroductionIn both human and computer vision systems curvatureextrema of the contour of an object are thought to containimportant information about the shape [8]. In this paperwe present a technique to extract this information by multi-scale mathematical morphology. Several such multi-scaletechniques exist, such as size distributions, or granulome-tries, which are used to quantify the amount of detail in animage at different scales [3, 9]. A similar method, based onsequential alternating filters, has been proposed by Bang-ham and coworkers [1, 2]. In this paper we develop a dif-ferent multi-scale approach, based on work by Leymarieand Levine [7]. They developed a morphological curva-ture scale space for shape analysis, based on sequences ofmorphological top-hat or bottom-hat filters with increasingsize of the structuring element used. In this paper we willpresent one modification of their technique, and extend itby including a method by which the features in the scalespace may be clustered in an unsupervised way. The ulti-mate aim is to obtain a small set of rotation, translation andscale invariant shape parameters, which contain as much in-formation about the shapes of interest as possible.The shapes considered here are outline images of di-atoms, though the methodology is applicable to othershapes as well. Diatoms are microscopic, single-cell algae,0 0.2 0.4 0.6 0.8 1−0.500.511.520 0.2 0.4 0.6 0.8 1−0.500.511.52Figure 1. Diatom images and their corre-sponding contour curvatures.which build, often highly ornate, silica shells. Some exam-ples are shown in Figure 1. Apart from being highly orna-mental, the shapes and ornamentations of these shells allowphycologists to classify and identify these organisms. Fur-thermore, these shells are highly resistant to many outsideinfluences, and consequently fossilize well, allowing studyof fossil as well as living diatoms. The Automatic DiatomIdentification And Classification (ADIAC) project [4], ofwhich this research is a part, aims to automate the processof diatom identification by digital image analysis. In thispaper we will focus on a method to extract the most promi-nent curvature features on the contours of diatom, ratherthan on classification of diatoms shapes.2 Curvature scale spacesA common approach to shape recognition is to focus onthe curvature of the contour [10]. Since the curvature of thecontour defines a shape completely, it should in theory bepossible to classify diatom outline shapes using the curva-656ture. Methods based on mathematical morphology [1, 2, 7]have the advantage that the resulting features are localizedon the contour, rather than delocalized such as Fourier de-scriptors.Before building the morphological scale space we com-pute a contour representation of our objects using Free-man’s chain code, and obtain the curvature measure usingthe method of Leymarie and Levine [7]. They obtain curva-tures by Gaussian smoothing of the direction data, followedby differentiation. The resulting curvatures can be seen infigure 1.Once the curvature as a function of position on the con-tour has been computed, we are left with the problem ofextracting the features from the one-dimensional curvaturedata. A second step is to build a morphological scale spaceby extracting peaks and valleys using a method of Leymarieand Levine [7]. An outline of the method is given here, fortechnical details refer to [7].The morphological scale space is built by first splittingthe curvaturek(u) into a positive partk+(u), correspondingto convex parts of the contour, and a negative partk−(u),corresponding to concave contour segments. This is shownin figure 2. In the convex part of the curve, we now extractpeaks at different scales using top-hat filters with flat struc-turing elements of increasing sizes. A top-hat filter extractspeaks by first computing a morphological opening of thesignal, and then subtracting the opening from the original.Any peaks narrower than the structuring element are presentin the final result without any distortion. By performingrepeated top-hat filters at increasing scales we can build ascale space. The result of each top-hat filter is comparedto that at the preceding scale. If a feature present at theprevious scale is not any larger in the current one, its max-imum height, average height, width and location are storedin a linked list structure. After that, the peak is subtractedfrom the original curve. The only difference between ourmethod and that of Leymarie and Levine’s is that peaks maybe nested in our case. The procedure stops when the origi-nal curve has been reduced to zero.A similar procedure is used on the negative partk−(u),but in this case we are extracting valleys. The procedureonly differs from that used onk+(u) in that we use bottom-hat filters, i.e. a morphological closing with a flat structur-ing element is subtracted from the original.At the end of this procedure, we have two curvaturescale spaces: one “bottom scale space” of valleys derivedfrom k−(u), and a “top scale space” of peaks derived fromk+(u). In both scale spaces, each feature is represented byits start and end points on the contour, its extremum cur-vature value, its extremum position, and its mean curvaturevalue. The scale spaces can be visualized by plotting eachfeature as a box of either the peak or the average height atthe appropriate point in the curvature graph. If nested fea-0 0.2 0.4 0.6 0.8 1−10123Position along contourCurvature0 0.2 0.4 0.6 0.8 1−1−0.500.511.522.53Position along contourCurvature0 0.2 0.4 0.6 0.8 1−10123Position along contourCurvature0 0.2 0.4 0.6 0.8 1−10123Position along contourCurvatureFigure 2. Building curvature spaces. Top:binary images; centre: curvature plots splitinto k+ (solid) and k− (dashed); Bottom: scalespace features displayed as blocks of the cor-rect width and average height.tures are present, we can simply stack the features in theplot, as has been done in figure 2.3 Scale space feature extractionDirect use of the scale space data for classification ofshapes has several problems. The first is that the scale spacecontains spurious detail caused by noise. This increases thelength of the pattern vector needlessly. Furthermore, thepattern vectors of different shapes differ in length, which isa problem for many statistical methods. To standardize thepattern vector lengths, we either have to pad the shorter vec-tors with zeros, or we have to clip the longer vectors. Thesimplest approach is to select a numbernf of features withthe largest absolute values of their curvature from each setof scale space features. If the scale spaces contains fewerthannf features, features with zero curvature are added tothe feature vector. Alternatively, one could use the numbers,means and variances of the heights and widths of peaksabove a given threshold as a pattern vector of fixed length.We attempted this, but no satisfactory threshold could befound (data not shown). The problem is that either small,significant peaks are missed or spurious peaks are detected.657In the approach we propose, we first reduce noise by re-moving those features with a curvature smaller than somethreshold (0.3 gave the best results in our case) from thescale space. We then classify the remaining features by un-supervised, distribution free cluster analysis. This ensuresthat we make noa priori assumptions about the number ofclusters or the shape of their distribution. The mean-shiftmethod we have chosen [6] is based on kernel density esti-mation. It should be stressed that any other unsupervised,distribution free method could also be used.Kernel density estimates [11] approximate an unknownprobability density distributionp(x) fromN data elementsxi, by p̂(x), which is given byp̂(x) = h−dN−1N∑i=1K(xi−xh ), (1)whereK(u) is the called the kernel,d is the number of di-mensions ofx, andh is thewindowor smoothing width.The kernel function is a non-negative, even function, withunity integral and unity variance [11]. Because an optimalchoicehopt, which minimizes the mean integrated squareerror (MISE) can be estimated from the data [11], comput-ing the kernel density estimate can be automated fully. Theoptimal kernel function in terms of MISE is the Epanech-nikov kernelKe [5], which is given byKe(u) ={d+22cd(1− ‖u‖2) if ‖u‖2 < 10 otherwise,(2)in which cd is the volume of ad-dimensional unit sphere.UsingKe, the gradient∇p̂(x) of the kernel density estimatep̂(x) is∇p̂(x) = h−dN−1∑‖xi−x‖<hxi − x. (3)Therefore, the direction of ascent for any given locationx isgiven as the mean of all the vectors starting atx nd endingatdata pointsxi which lie within a radius ofh of x.Based on this reasoning, Fukunaga and Hostetler [6] de-rived the following clustering algorithm:1. For each data itemxi compute the centre of gravitȳxiof all data items which lie withinh of xi.2. If, for any i, x̄i 6= xi, replace allxi by x̄i, and repeatfirst step,3. else end procedure.As a final step in our curvature feature extraction, wecompute the mean curvatures, number of members, andvariances of the scale space features in each of the clustersfound. For the top and bottom curvature scale spaces, we se-lect the clusters containing the scale space features with theFigure 3. Problem cases for the detectionmethod. Top two: rounded forms yieldingspurious peaks; bottom left: blunt, slightlysquare ends may be split; bottom right: dis-similar peaks may be put into separate clus-ters.largest positive and negative curvatures respectively. Thus,the two curvature scale space are reduced to a single fea-ture vector of size 6: the numbers, mean curvatures, andvariances of the points of highest positive and negative cur-vature along the contour. Since we no longer concern our-selves with the positions of these points on the contour, ourmeasure is automatically rotation invariant.4 Diatom imagesDiatoms were photographed on a Kodak MegaPlus ES1.0 digital camera attached to a Zeiss Axiophot light micro-scope with 63x and 100x oil immersion lenses (both with anumeric aperture of 1.4), using brightfield optics only. Af-ter deshading to remove optical imperfections such as dust,the images were segmented using manual thresholding todetect outlines for each diatom. No outline smoothing wasperformed on the image. For each binary image, a numberof derived images were computed, by rotation and scalingof the images. In this way, we could automatically createmultiple images of the same cells without repeating the la-borious manual segmentation process.5 ResultsA visual classification of contour details of 350 imagesof diatoms was carried out to determine the number of mostprominent peaks and dips in the contour of each diatom im-age. For each diatom, a visual rating of the similarity (simi-lar, dissimilar, and slightly dissimilar) of the peaks and dipsin the most significant cluster was given. For the peaks,which are more prominent on most diatoms, there was dis-658agreement between visual estimation of the number of mostprominent peaks in only 24 out of 350 cases (6.86 %). In22 of these cases spurious peaks with extrema just above thethreshold of 0.3 were added. In one case a slightly squaredend was split into two features. The remaining case wasone in which the peaks were dissimilar. Examples of theseproblem shapes are shown in Figure 3.The dip detection had a similar success rate, missing afeature in only 5 cases out of 350, and adding spuriouspeaks in 26 cases. In all there were 31 mismatches in thenumber of features detected visually, and the number of fea-tures found using our algorithm. The total success rate forfeature detection (peaks and dips) is therefore 92.14 %.Of the correctly identified clusters of features, 55 wereclassified visually as containing dissimilar features. Thewithin-cluster variance of the curvatures of these clusterswas 0.089 on average, with a standard deviation of 0.155.Of the 259 correctly identified clusters with more than onevisually similar feature, the mean within-cluster variancewas 0.0057 with standard deviation 0.0098. The differencebetween these means is significant atp < 0.01 (using Stu-dent’s T-test). Despite this, significant overlap between thedistributions exists, hampering classification based on thisparameter.6 DiscussionIt appears that the morphological curvature scale space,combined with unsupervised cluster analysis is well suitedto finding major peaks and dips in the contour of shapes,provided these are strong enough. Less sharp features arereadily confused with noise. We suspect this may be causedby the way we obtain our curvature measure in the firstplace. Some form of adaptive smoothing along the contourmay improve the resolving power of the method.The noise at low curvatures also hampers the detection ofscale space features at the largest scales, which correspondto weakly curved regions between the strong peaks. Theseare often broken up into smaller regions by noise features.Subtle differences in curvature in these regions may be veryimportant to diatom classification. In future work we willstudy alternative methods to obtain the curvature data in or-der to reduce this noise. Preprocessing of the binary imagesto smooth the outline will also be explored.Finally, the current method is insensitive to differences inthe relative locations of the curvature features. This meansthat, e.g., an elongated rectangle and a square are insepara-ble. Our method shares this property with other morpholog-ical multiscale techniques, such as computing pattern spec-tra of the curvature data [9]. In our case, computing a bot-tom scale space of thek+(u) data may remedy this. Thewidths of the major dips in thek+(u) correspond to thedistances between the major peaks, so these may be usedto discriminate between some (but perhaps not all) of suchcases.Despite its shortcomings due to noise, this method showspromise in automatic detection of curvature scale space fea-tures of diatoms and other shapes, and a full-blown compar-ison of this, and other contour classification methods will bemade within the ADIAC project in the future.AcknowledgmentThis study is part of the ADIAC project, funded by theEuropean Commission MAST programme, contract MAS3-CT97-0122. A special thanks to Steve Juggins and RitchieHead of the University of Newcastle upon Tyne, Dept. ofGeography, for the use of their diatom samples.References[1] J. A. Bangham, P. Chardaire, C. J. Pye, and P. D. Ling. Mul-tiscale nonlinear decomposition: the sieve decompositiontheorem.IEEE Trans. Pattern Anal. Mach. Intell., 18:529–538, 1996.[2] J. A. Bangham, P. D. Ling, and R. Harvey. Scale-space fromnonlinear filters. IEEE Trans. Pattern Anal. Mach. Intell.,18:520–528, 1996.[3] E. J. Breen and R. Jones. Attribute openings, thinnings andgranulometries.Comp. Vis. Image Understand., 64(3):377–389, 1996.[4] H. du Buf, M. Bayer, S. Droop, R. Head, S. Juggins,S. Fischer, M. W. H. Bunke, J. Roerdink, J. Pech-Pacheco,G. Christobal, H. Shahbazkia, and A. Ciobanu. Diatom iden-tification: A double challenge called adiac. InProceedingsICIAP, pages 734–739, Venice, September 1999.[5] V. A. Epanechnikov. Nonparametric estimation of a multidi-mensioal probability density.Theor. Probab. Appl., 14:153–158, 1969.[6] K. Fukunaga and L. D. Hostetler. Estimation of the gradientof a density function with applications in pattern recogni-tion. IEEE Transactions on Information Theory, IT-21:32–40, 1975.[7] F. Leymarie and M. D. Levine. Curvature morphology.Technical Report TR-CIM-88-26, Computer Vision andRobotics Laboratory, McGill University, Montreal, Quebec,Canada, 1988.[8] M. Leyton. Symmetry-curvature duality.Comp. Vis. Graph.Image Proc., 38:327–341, 1987.[9] P. F. M. Nacken. Chamfer metrics in mathematical morphol-ogy. J. Math. Imaging Vision, 4:233–253, 1994.[10] T. Pavlidis. Algorithms for shape analysis of contours andwaveforms.IEEE Trans. Pattern Anal. Mach. Intell., 2:301–312, 1980.[11] B. W. Silverman.Density Estimation for Statistics and DataAnalysis. Chapman and Hall, London, 1986.659

Diatom contour analysis using morphological curvature scale spaces

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Diatom contour analysis using morphological curvature scale spaces

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