{"title": "Salient Boundary Detection using Ratio Contour", "book": "Advances in Neural Information Processing Systems", "page_first": 1571, "page_last": 1578, "abstract": "", "full_text": "Salient Boundary Detection using Ratio Contour\n\nSong Wang, Toshiro Kubota\n\nJeffrey Mark Siskind\n\nDept. Computer Science & Engineering\n\nSchool Electrical & Comput. Engr.\n\nUniversity of South Carolina\n\nColumbia, SC 29208\n\nPurdue University\n\nWest Lafayette, IN 47906\n\nfsongwang|kubotag@cse.sc.edu\n\nqobi@purdue.edu\n\nAbstract\n\nThis paper presents a novel graph-theoretic approach, named ratio con-\ntour, to extract perceptually salient boundaries from a set of noisy bound-\nary fragments detected in real images. The boundary saliency is de\ufb01ned\nusing the Gestalt laws of closure, proximity, and continuity. This pa-\nper \ufb01rst constructs an undirected graph with two different sets of edges:\nsolid edges and dashed edges. The weights of solid and dashed edges\nmeasure the local saliency in and between boundary fragments, respec-\ntively. Then the most salient boundary is detected by searching for an\noptimal cycle in this graph with minimum average weight. The proposed\napproach guarantees the global optimality without introducing any biases\nrelated to region area or boundary length. We collect a variety of images\nfor testing the proposed approach with encouraging results.\n\n1\n\nIntroduction\n\nHuman vision and neural systems possess very strong capabilities of identifying salient\nstructures from various images. Implementing such capabilities on a computer is an im-\nportant but extremely challenging problem for arti\ufb01cial intelligence, computer vision, and\nmachine learning. The main challenges come from two closely related aspects: (a) the def-\ninition of the structural saliency, and (b) the design of ef\ufb01cient algorithms for \ufb01nding the\nsalient structures. On one hand, we expect very comprehensive and advanced de\ufb01nitions\nof the saliency so that it models accurately the human perceptual and visual process. On\nthe other hand, we expect simple de\ufb01nitions of saliency so that the global optimum can be\nfound in polynomial time.\n\nPrevious methods for salient-structure detection can be grouped into two classes. The \ufb01rst\nclass of methods aims to directly group or segment all the image pixels into some dis-\njoint regions, which are expected to coincide with the underlying salient structures. Earlier\nefforts include the region-merging/splitting methods, watershed methods, and the active-\ncontour-like methods. Those methods usually have dif\ufb01culties in \ufb01nding the globally op-\ntimal boundaries in terms of the selected saliency de\ufb01nitions. Recently we have witnessed\nsome advanced methods, like ratio region [5], minimum cut[17], normalized cut [14], glob-\nally optimal region/cycle [9], and ratio cut [15], which aim to produce globally optimal\nboundaries. However, those pixel-grouping based methods usually have dif\ufb01culties in ef-\nfectively incorporating perceptual rules, such as boundary smoothness, into their saliency\nde\ufb01nitions.\n\n\fInstead of operating directly on the image pixels, another class of methods is designed\nbased on some pre-extracted boundary fragments (or for brevity, fragments) 1, which can\nbe obtained using some standard edge-detection methods like Canny detectors. As shown\nin Fig. 1(a), although those fragments are disconnected and contain serious noise, they pro-\nvide abundant information on boundary length, tangent directions, and curvatures, which\ncan greatly facilitate the incorporation of advanced perceptual rules like boundary smooth-\nness. Shashua and Ullman [13] presents a parallel network model for detecting salient\nboundary based on fragment proximity, boundary length, and boundary smoothness. Re-\ncent development in this class includes Alter and Basri [2], Jacobs [8], Sarkar and Boyer\n[12], Guy and Medioni [7], Williams and Thornber [16, 11], and Amir and Lindenbaum\n[3]. However, many of them still have dif\ufb01culty in \ufb01nding the closed boundaries in a sense\nof global optimality with respect to the given boundary-saliency measure. Elder and Zucker\n[6] use the shortest-path algorithm to connect fragments to form salient closed boundaries.\nHowever, the results have a bias to produce boundaries with shorter length.\n\nThis paper presents a new graph based approach to extract salient closed boundaries from\na set of fragments detected from real images. This approach seeks a good balance between\nthe complexity of the saliency de\ufb01nition and the complexity of the optimization algorithm.\nThe boundary saliency is based on the well-known Gestalt laws of closure, proximity, and\ncontinuity. To avoid the various biases as in Elder and Zucker [6], this paper de\ufb01nes the\nboundary saliency as the average saliency along the whole boundary. We \ufb01nally formulate\nthe salient-boundary detection problem into a problem for \ufb01nding an optimal cycle in an\nundirected graph. We show this problem is of polynomial time-complexity and give an\nalgorithm to solve it. The proposed algorithm is then tested on a variety of real images.\n\n2 Problem Formulation\n\n(a)\n\n(b)\n\n(c)\n\n(d)\n\nFigure 1: An illustration of detecting salient boundaries from some fragments. (a) Bound-\nary fragments, (b) salient boundary by connecting some fragments with dashed curves, (c)\na solid-dashed graph, and (d) an alternate cycle in (c).\n\nThe basic primitives in the ratio-contour approach are a set of noisy (boundary) fragments\nextracted by edge detection. For simplicity, here we assume each detected fragment is a\ncontinuous open curve segment with two endpoints, as shown in Fig. 1(a). Our goal is\nto identify and connect a subset of fragments to form the most salient structural bound-\nary as shown in Fig. 1(b). In this paper, we measure the boundary saliency using sim-\nple Gestalt laws of closure, proximity, and continuity. The closure means that the salient\nboundary must be a closed contour. The proximity implies that we desire relatively small\ngaps (dashed curves in Fig. 1(b)) in connecting the fragments. The continuity indicates that\nthe resulting contour should be continuous and suf\ufb01ciently smooth.\n\nLet the parametric form of a boundary B be v(t); 0 (cid:20) t (cid:20) 1. We have v(0) = v(1) as the\nboundary is closed. Considering the boundary proximity and the continuity, we de\ufb01ne its\n\n1Most literatures use the terminology edge instead of fragment. However, in this paper edge has\n\nother speci\ufb01ed meaning in a graph model.\n\n\fcost, which is negatively related to the boundary saliency, as\n\nR(B) ,\n\nT (B)\nL(B)\n\n= RB[(cid:27)(t) + (cid:21) (cid:1) (cid:20)2(t)]dt\n\nRB dt\n\n;\n\n(1)\n\nwhere (cid:27)(t) = 1 if v(t) is in the gap and (cid:27)(t) = 0, otherwise. (cid:20)(t) is the curvature at v(t).\nWe can see that the un-normalized cost T (B) combines the total gap-length and curvature\nalong the boundary B and has bias to produce a short boundary. The issue is addressed in\n(1) through normalizing T (B) by the boundary length L(B). The most salient boundary\nB is then the one with the minimum cost R(B). The parameter (cid:21) > 0 is set to balance the\nweight between proximity and continuity.\n\nWe can formulate the above cost into an undirected graph G = (V; E) with vertices V =\nfv1; v2; (cid:1) (cid:1) (cid:1) ; vng and edges E = fe1; e2; (cid:1) (cid:1) (cid:1) ; emg. A unique vertex is constructed from\neach fragment endpoint. Two different kinds of edges, solid edges and dashed edges, are\nconstructed between vertices. (a) If vi and vj correspond to the two endpoints of the same\nfragment, we construct a solid edge between vi and vj to model this fragment. (b) Between\neach possible vertex pair vi and vj, we construct a dashed edge to model the gap or a virtual\nfragment (dashed curves in Fig. 1(b)). An example is shown in Fig. 1(c), which is made up\nof 3 solid edges for three fragments and all 15 possible dashed edges. For clarity, sometimes\nwe call the boundary fragment a real fragment when both real and virtual fragments are\ninvolved.\n\nThe constructed graph always has even number of vertices, as each real fragment has two\nendpoints. More interestingly, no two solid edges are incident from the same vertex and\neach vertex has exactly one incident solid edge. We name such a graph an (undirected)\nsolid-dashed graph. We further de\ufb01ne an alternate cycle in a solid-dashed graph as a simple\ncycle that traverses the solid edges and dashed edges alternately. Examples of a solid-\ndashed graph and an alternate cycle are given in Fig. 1(c) and (d), respectively. Since\na boundary always traverses real fragments and virtual fragments alternately, it can be\ndescribed by an alternate cycle. Note that not all the cycles in a solid-dashed graph are\nalternate cycles, because two adjacent dashed edges can appear sequentially in the same\ncycle.\n\nAccording to the cost function (1), we de\ufb01ne a weight function w(e) and a length function\nl(e) for each edge e. For convenience, we de\ufb01ne B(e) as a function that gives the (real\nor virtual) fragment corresponding to an edge e. Then the weight w(e) , T (B(e)) =\n\nRB(e)[(cid:27)(t) + (cid:21) (cid:1) (cid:20)2(t)]dt is the un-normalized cost on B(e). The edge length l(e) is de\ufb01ned\n\nas the length of B(e). We can see that the most salient boundary with minimum cost (1)\ncorresponds to an alternate cycle C with minimum cycle ratio\n\nCR(C) = Pe2C w(e)\nPe2C l(e)\n\n:\n\nFragments extracted from real images usually contain noise, intersections, and even some\nclosed curves, which cause dif\ufb01culties in estimating the curve length, curvature, and there-\nfore, the edge weight and length. We will describe a spline-based method to address this\nproblem in Section 4. In the following, we \ufb01rst present a polynomial-time algorithm to\nidentify the alternate cycle with the minimum cycle ratio CR(C).\n\n3 Ratio-Contour Algorithm\n\nFor simplicity, we denote the alternate cycle with minimum cycle ratio as MRA (Minimum\nRatio Alternate) cycle. In this section, we introduce a graph algorithm for \ufb01nding the MRA\ncycle in polynomial time. This algorithm consists of three reductions. (a) Both the weight\n\n\fand edge length of the solid edges can be set to zero by merging them into the weight\nand length of their adjacent dashed edges, without changing the underlying MRA. (b) The\nproblem of \ufb01nding an MRA cycle can be reduced to a problem of detecting a negative-\nweight alternate (NWA) cycle in the same graph. (c) Finding NWA cycles in a solid-dashed\ngraph with zero solid-edge weights and zero solid-edge lengths can be reduced to \ufb01nding a\nminimum-weight perfect matching (MWPM) in the same graph. Finding MWPM has been\nshown to be of polynomial-time complexity with various ef\ufb01cient algorithms available.\n\n3.1 Setting Zero-Weight and Zero-Length to Solid Edges\n\nAs illustrated in Fig. 2(a) and (b), each solid edge e can only be adjacent to a set of dashed\nedges, say fe1; e2; (cid:1) (cid:1) (cid:1) ; eK g, in a solid-dashed graph, and no two solid edges are adjacent\nto each other. Therefore, we can directly merge the solid-edge weight and length to its\nadjacent dashed edges by\n\n( w(ek) w(ek) + w(e)\n\nl(ek) l(ek) + l(e)\n\nNk\n;\n\nNk\n\nk = 1; 2; (cid:1) (cid:1) (cid:1) K;\n\nwhere Nk = 2 if ek shares one vertex with e as in Fig. 2(a) and Nk = 1 if ek shares both\nvertices with e as in Fig. 2(b). Then we reset the weight and length of this solid edge to\nzero, i.e., w(e) = 0; l(e) = 0. This merging process is performed on all solid edges. While\nsolid and dashed edges are traversed alternately in an alternate cycle, it is not dif\ufb01cult to\nachieve the following conclusion.\n\nLemma 3.1 The above processing of edge weights and edge-lengths does not change the\ncycle ratio of any alternate cycles.\n\ne1\n\ne\n\n(a)\n\nek\n\ne1\n\ne\n\n(b)\n\n(c)\n\n(d)\n\nFigure 2: An illustration of reductions in ratio-contour algorithm. (a) Merging the weight\n(b) A special case for weight\nand length of a solid edge to its adjacent dashed edges.\nmerging.\n(d) Derived cycle from the\nperfect matching shown in (c).\n\n(c) A perfect matching in a solid-dashed graph.\n\n3.2 Reducing to Negative-Alternate-Cycle Detection\n\nThe following lemma claims that MRA cycles are invariant to some more general linear\nedge-weight transforms.\n\nLemma 3.2 The MRA cycle in a solid-dashed graph G = (V; E) is invariant to the fol-\nlowing linear transform on the edge weight\n\nw(e) w(e) (cid:0) b (cid:1) l(e); 8e 2 E:\n\n(2)\n\nThe proof for this lemma is similar to the one we gave for general ratio-cycle detection\nproblem [15]. Notice that all the edge lengths are non-negative. There always exists an\noptimal b = b(cid:3) so that after weight transform (2), the MRA cycle has the cycle ratio of\nzero. In this case, the MRA cycle is the same as the cycle with total edge weight of zero.\n\n\fThe detection of the optimal b(cid:3) and the MRA cycle can then be reduced into a problem\nof \ufb01nding the NWA cycle (negative weight alternate cycle). Basically, if we can detect an\nNWA cycle after the edge weight transform (2), we know b > b(cid:3). Otherwise, we know that\nb (cid:20) b(cid:3). With an NWA cycle detection algorithm, we can use binary or sequential search to\nlocate the optimal b(cid:3) and the desired MRA cycle. This search process is polynomial if all\nthe edge weight are integers [15]. In addition, with the \ufb01rst reduction mentioned in Section\n3.1, it is easy to see that the linear transform (2) always preserves zero weight and zero\nlength for all solid edges in this search process.\n\n3.3 Reducing to Minimum Weight Perfect Matching\n\nThe problem of detecting an NWA cycle in a solid-dashed graph can be reduced to a prob-\nlem of \ufb01nding a minimum weight perfect matching (MWPM) in the same graph. A perfect\nmatching in G denotes a subgraph that contains all the vertices in G while each vertex only\nhas one incident edge. An example is shown in Fig. 2(c), where three thick edges together\nwith their vertices form a perfect matching. The MWPM is the perfect matching with min-\nimum total edge weight. As all the solid edges form a trivial perfect matching with total\nweight zero, the MWPM in our solid-dashed graph should have non-positive total weight.\n\nWe can construct a set of cycles from a perfect matching P by (a) removing from P all\nthe solid edges and their endpoints, and (b) adding to P any solid edges in the solid-dashed\ngraph G whose two endpoints are still in P after the removal in (a). The remaining subgraph\nmust consist of a set of cycles because each remaining vertex has two incident edges: one\nis solid and the other one is dashed. This also con\ufb01rms that all the resulting cycles are\nalternate cycles. An example of this reduction is shown in Fig. 2(d), which is constructed\nfrom (c). As all the solid edges have zero weight and zero length, it is not dif\ufb01cult to see\nthat the total weight of the perfect matching is the same as the total weight of the resulting\ncycles. Therefore, the NWA detection problem is reduced into a problem of \ufb01nding a\nperfect matching with negative total weight. This is the same as the problem of \ufb01nding the\nMWPM, which is of polynomial-time complexity [1].\n\n4 Edge-Weight and Edge-Length Functions\n\nWe need to estimate the curvature and length of both real and virtual fragments for de\ufb01ning\nw(e) and l(e) of solid and dashed edges. To deal with the noise and aliasing in detected\nfragments, we impose a pre-smoothing process on those fragments. In this paper, we ap-\nproximate a fragment by a set of quadratic splines with the parametric form\n\n(cid:18) xi(ti)\nyi(ti) (cid:19) =(cid:18) xi\n\nyi (cid:19) +(cid:18) Ai Bi\n\nCi Di (cid:19)(cid:18) t2\n\nti (cid:19) ;\n\ni\n\nwhere 0 (cid:20) ti (cid:20) 1 is the parameter for the spline. We developed an iterative algorithm [10]\nto estimate the optimal parameters xi, yi, Ai, Bi, Ci, and Di minimizing a comprehensive\ncost function that measures smoothness, under the constraint of C 0 and C 1 continuities\nacross the fragment. An example is illustrated in Fig. 3 where solid curves in (a) and (b)\nare fragments before and after smoothing. More discussion and analysis on this curve-\nsmoothing method can be found in our previous work [10].\n\nWith the parametric form of quadratic splines, the total length and the curvature along a real\nfragment can be computed by summing over each spline its length and its total curvature\nas\n\nli = Z 1\ni (t)dt = Z 1\n\n0\n\n(cid:20)2\n\nZ 1\n\n0\n\ndt;\n\n0 p(2Ait + Bi)2 + (2Cit + Di)2dt;\n\n4(AiDi (cid:0) BiCi)2\n\n[(2Ait + Bi)2 + (2Cit + Di)2]3\n\n\fwhere li is the length and (cid:20)i(t) is the curvature function of the ith spline.\nHowever, estimating these quantities for a virtual fragment is not trivial, as no information\nis given on how the virtual fragment should look like. We take the following approach to\ncompute the dashed-edge weight. First, a pair of endpoints involved in forming a particular\ndashed edge is connected with a straight line. Then a new curve segment is constructed\nby connecting this straight line and adjacent fragments. The smoothing process described\nabove is applied to this new curve segment. In this smoothed curve segment, the virtual\nfragment is then the part corresponding to the straight line before the smoothing. The\ndashed curve in Fig. 3(b) shows a resulting virtual fragment used for estimating curvature,\nlength, and \ufb01nally edge weight.\n\n(a)\n\n(b)\n\n(c)\n\n(d)\n\nFigure 3: An illustration of the edge weight estimation process. (a) Two noisy fragments.\n(b) Smoothed real fragments and an estimated virtual fragment. (c) Fragments obtained\nby Canny detector. (d) Smoothed fragments after breaking undesired connections, corre-\nsponding to the portion of the box in (c). Crossings specify the endpoints and breaking\npoints.\n\nIn real implementation, another issue is that the detected fragments using edge detectors\nmay not be disjoint open curves as assumed in Section 2. It is common that some frag-\nments are connected in the form of intersections, attachments, and even undesired closure,\nas shown in Fig. 4. Therefore, we need to break those connections to construct the graph\nmodel. First, we identify the intersection points and split them to get multiple open frag-\nments. An example is shown in Fig. 4(a) and (d), where an intersection point is broken into\nthree endpoints. In the constructed graph, they (u1, u2, and u3) are connected by dashed\nedges with zero weight and zero length. Attachment speci\ufb01es the case where two fragments\nare undesirably connected into a single fragment as shown in Fig. 4(b). This greatly hurts\nthe reliability of salient boundary detection as those attached fragments may exclude many\ndesired dashed edges from the graph. We alleviate this problem by splitting all the frag-\nments at their high-curvature points, as illustrated in Figs. 4(b) and (e). Similarly, we can\nbreak closed fragments into open fragments at high-curvature points, as shown in Fig. 4(c)\nand (f). Note that the identi\ufb01cation of high-curvature points requires the smoothing of the\nnoisy fragments. We apply the same smoothing technique described above to each frag-\nment for this purpose. Figures 3(c) and (d) show an example of dealing with the above\nspecial cases.\n\n5 Experiments and Discussion\n\nIn this section, we test the proposed ratio-contour algorithm to extract the salient boundaries\nfrom real images. For initial fragment detection, we use the standard Canny edge detector\nin the Matlab software with its default threshold settings. We also adopt the Blossom4\nimplementation [4] of the minimum-weight perfect matching.\n\nOne problem in the implementation is the construction of dashed edges, which may be of\na very large number (O(n2)) if we connect every two possible vertices. In this paper, we\nconstrain the proximity to reduce the number of dashed edges. In the implementation, for\neach vertex, we only keep certain number of incident dashed edges with smallest length.\n\n\fB(e )2\n\nB(e )3\n\n2e\nu 3\n\n3e\n\nB(e )1\n\n(a)\n\nu 2\n\n(d)\n\nu 1\n\n1e\n\nB(e )2\n\n3B(e )\n\nB(e )1\n\n(b)\n\nu 2\n\n2e\n\nu 1\n\n1e\n\n3e\n\nB(e )1\n\n(c)\n\nu 1\n\n1e\n\nu 2\n\n(e)\n\n(f)\n\nFigure 4: An illustration of fragment identi\ufb01cation and graph construction in some special\ncases. (a), (b), and (c) show the detected fragments with intersections, attachments, and\nclosures. (d), (e), and (f) are the constructed graphs from (a), (b), and (c), respectively.\n\nThis number is uniformly set to 20 in all experiments. Meanwhile, we set the parameter\n(cid:21) = 50 in the edge-weight de\ufb01nition. Figure 5 shows salient boundaries detected from\nseven real images, together with the initial fragments from Canny detector. It can be seen\nthat the proposed method integrates well the Gestalt laws of proximity, continuity, and\nclosure.\n\n(a)\n\n(b)\n\n(c)\n\n(d)\n\n(e)\n\n(f)\n\n(g)\n\nFigure 5: Salient boundaries detected from some real images using the proposed ratio-\ncontour algorithm. Each sub\ufb01gure from (a) to (g) contains three images, left: original\nimages, middle: Canny detection results, and right: the detected most salient boundaries.\n\n6 Conclusions\n\nWe have presented a novel graph-theoretic approach, named ratio contour, for extracting\nperceptually salient boundaries from a set of noisy boundary fragments detected in real\nimages. The approach guarantees the global optimality without introducing any biases re-\n\n\flated to region area or boundary length, and exhibits promising performance in extracting\nsalient objects from real cluttered images. One potential extension of this research is to ex-\ntract multiple salient objects that are overlapped or share part of boundaries by performing\nratio-contour algorithm iteratively. We are currently investigating this extension and plan\non reporting the result in the future.\n\nAcknowledgements\n\nThe authors would like to thank David Jacobs and anonymous reviewers for important\ncomments. This work was funded, in part, by National Science Foundation grant EIA-\n0312861, and the USC SOM-COEIT research development fund.\n\nReferences\n\n[1] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin. Network Flows: Theory, Algorithms, & Applica-\n\ntions. Prentice Hall, Englewood Cliffs, 1993.\n\n[2] T. Alter and R. Basri. Extracting salient contours from images: An analysis of the saliency\nIn IEEE Conference on Computer Vision and Pattern Recognition, pages 13\u201320,\n\nnetwork.\n1996.\n\n[3] A. Amir and M. Lindenbaum. A generic grouping algorithm and its quantitative analysis. IEEE\n\nTransactions on Pattern Analysis and Machine Intelligence, 20(2):168\u2013185, 1998.\n\n[4] W. Cook\n\nand A. Rohe.\n\nhttp://www.or.unibonn.de/home/rohe/matching.html, Aug. 1998.\n\nComputing minimum-weight\n\nperfect matchings.\n\n[5] I. Cox, S. B. Rao, and Y. Zhong. Ratio regions: A technique for image segmentation.\n\nInternational Conference on Pattern Recognition, pages 557\u2013564, 1996.\n\nIn\n\n[6] J. Elder and S. Zucker. Computing contour closure. In European Conference on Computer\n\nVision, pages 399\u2013412, 1996.\n\n[7] G. Guy and G. Medioni. Inferring global perceptual contours from local features. International\n\nJournal of Computer Vision, 20(1):113\u2013133, 1996.\n\n[8] D. Jacobs. Robust and ef\ufb01cient detection of convex groups.\n\nAnalysis and Machine Intelligence, 18(1):23\u201337, 1996.\n\nIEEE Transactions on Pattern\n\n[9] I. H. Jermyn and H. Ishikawa. Globally optimal regions and boundaries as minimum ratio\ncycles. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(10):1075\u20131088,\n2001.\n\n[10] T. Kubota. Contextual and non-combinatorial approach to feature extraction. In Int\u2019l Workshop\n\non EMMCVPR, pages 467\u2013482, 2003.\n\n[11] S. Mahamud, L. R. Williams, K. K. Thornber, and K. Xu. Segmentation of multiple salient\nclosed contours from real images. IEEE Transactions on Pattern Analysis and Machine Intel-\nligence, 25(4):433\u2013444, 2003.\n\n[12] S. Sarkar and K. Boyer. Quantitative measures of change bvased on feature organization:\nEigenvalues and eigenvectors. In IEEE Conference on Computer Vision and Pattern Recogni-\ntion, pages 478\u2013483, 1996.\n\n[13] A. Shashua and S. Ullman. Structural saliency: The detection of globallly salient structures\nIn International Conference on Computer Vision, pages\n\nusing a locally connected network.\n321\u2013327, 1988.\n\n[14] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern\n\nAnalysis and Machine Intelligence, 22(8):888\u2013905, 2000.\n\n[15] S. Wang and J. M. Siskind. Image segmentation with ratio cut. IEEE Transactions on Pattern\n\nAnalysis and Machine Intelligence, 25(6):675\u2013690, 2003.\n\n[16] L. Williams and K. K. Thornber. A comparison measures for detecting natural shapes in clut-\n\ntered background. International Journal of Computer Vision, 34(2/3):81\u201396, 2000.\n\n[17] Z. Wu and R. Leahy. An optimal graph theoretic approach to data clustering: Theory and\nits application to image segmentation. IEEE Transactions on Pattern Analysis and Machine\nIntelligence, 15(11):1101\u20131113, 1993.\n\n\f", "award": [], "sourceid": 2482, "authors": [{"given_name": "Song", "family_name": "Wang", "institution": null}, {"given_name": "Toshiro", "family_name": "Kubota", "institution": null}, {"given_name": "Jeffrey", "family_name": "Siskind", "institution": null}]}