Kimia, B.B., Tannenbaum, A.R., & Zucker, S.W. (1990). Toward a computational theory of shape: An overview. Lecture Notes in Computer Science, 427, 402--407.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and surfaces with curvature based velocities. In this area, the level set numerical method developed by Osher and Sethian [70] was very influential and crucial. Early developments on this idea were provided in [68] and their equations were first suggested for shape analysis in computer vision in [46]. The basic idea is to represent the deforming curve, surface, or image, as the level set of a higher dimensional hypersurface. This technique, not only provides more accurate numerical implementations, but also solves topological issues which were very difficult to treat before. The ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes in Computer Science 427, pp. 402-407, Springer-Verlag, New York, 1990.

....in section 3 for problems in image processing and computer vision, in particular for image segmentation. One of the most popular applications of level set methods in image processing and com puter vision is for image segmentation. The contributions in this area started shortly after the work in [96] (which is one of the first papers in computer vision using the level set method) 39 Figure 4.6: Texture synthesis via intrinsic reaction diffusion flows on implicit surfaces. Left: isotropic. Right: anisotropic. by the works in [19, 114, 115] These authors showed how to embed in the level set ....

Kimia, B. B., Tannenbaum, A. and Zucker, S. W. Toward a computational theory of shape: An overview, Lecture Notes in Computer Science 427, 402-407, Springer-Verlag, New York, 1990. 52

....a pure 1 D diffusion, where a minimal amount of isotropic smoothing is added only for regularization purposes. Other anisotropic partial differential equations for smoothing images rely on morphological methods such as the mean curvature motion (geometric heat equation, Euclidean shortening flow) [30, 4] t u = u = jruj curv(u) 1) with being the direction perpendicular to ru. Since mean curvature motion propagates isophotes in inner normal direction with curvature dependent speed, we should not expect such completely local methods to be capable of closing interrupted line like structures. ....

B.B. Kimia, A. Tannenbaum, S.W. Zucker, Toward a computational theory of shape: An overview, O. Faugeras (Ed.), Computer vision -- ECCV '90, Lecture Notes in Comp. Science, Vol. 427, Springer, Berlin, 402--407, 1990.

....Divergence and Flux, Blood Vessel Segmentation. 1 Introduction Level set based numerical methods for hyperbolic conservation laws developed by Osher and Sethian [20] for curvature dependent ame propagation were introduced to the computer vision community for shape analysis by Kimia et al. [12]. Such models were later adapted to the problem of shape segmentation independently by Caselles et al. 5] and Malladi et al. 18] Here the essential idea was to halt an evolving curve in the presence of intensity edges by multiplying the evolution equation with an image gradient based stopping ....

....of the level sets of the image. The geometric heat equation has been extensively studied in the mathematics literature and has been shown to have remarkable smoothing properties, particularly in 2D [9, 10] It is also the basis for several nonlinear geometric scale spaces such as those studied in [2, 1, 12, 13]. It is important to emphasize that despite this history, the ux maximizing ow is a distinct partial di erential equation since the vector eld is not in general a normalized gradient. 4. The use of a gradient vector eld as a static external force for a parametric snake model has been proposed ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. Lecture Notes in Computer Science, 427:402-407, 1990.

....PDE s. Another key contribution in the PDE formalism has been the general segmentation framework developed by Mumford and Shah [20] Their work has unified a large number of image segmentation approaches, and opened as well a large number of theoretical and practical problems (see [19] Next in [16], Kimia et al. introduced curve evolution methods into computer vision for a computational theory of planar shape. For some of the key mathematical works in curvature driven flows upon which this work is founded (see [11] 12] and [24] and the references therein) They defined a ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," in Lecture Notes in Computer Science, vol. 427. New York: Springer-Verlag, 1990.

....discrete sampling of the scanner has skipped over the fine detail of the elbow joint. A crease is detected around the elbow, but it is not continuous enough to completely sever the arm data into two surface patches. It can be argued that a more detailed analysis could handle this situation, e.g. [9 11, 24], but there will always be times when it is just not possible to segment smoothly joined, articulated objects at such a low level. Consider the out stretched human arm how is the boundary that separates it into the upper and lower arms precisely delineated Instead we have to rely on more ....

B. B. Kimia, A. Tannebaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, 1990.

....and surfaces with curvature based velocities. In this area, the level set numerical method developed by Osher and Sethian [70] was very influential and crucial. Early developments on this idea were provided in [68] and their equations were first suggested for shape analysis in computer vision in [46]. The basic idea is to represent the deforming curve, surface, or image, as the level set of a higher dimensional hypersurface. This technique, not only provides more accurate numerical implementations, but also solves topological issues which were very difficult to treat before. The ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes in Computer Science 427, pp. 402-407, SpringerVerlag, New York, 1990. Numerical implementation of intrinsic heat flow 95

....Scale space representations One important aspect of shape representation is the characterisation of shape at different spatial scales. A scale space adds an extra dimension to a shape representation by spanning a range of spatial scales (Koenderink [75] Lindeberg [83] For example, Kimia et al. [72, 73] model the evolution of boundary curves using a reaction diffusion equation, which allows shocks (shape singularities) to subdivide a shape hierarchically as scale space is traversed. Gaussian scale space is computed by propagating the linear diffusion equation over an initial image or curve, ....

B. Kimia, A. Tannenbaum, and S. Zucker. Toward a computational theory of shape: an overview. In Lecture Notes in Computer Science, volume 427, pages 402--407. SpringerVerlag, 1990.

....image (see figure 2) In the following sections we outline a possible solution to these problems. We note that this work on interface motion and hyperbolic conservation laws as discussed in [19, 23, 24] has been applied in the area of computer vision for shape characterization by Kimia et al. [11, 12], who unify many diverse aspects of shape by defining a continuum of shapes (reaction diffusion space) which places shapes within a neighborhood of other similar shapes. This leads to a hierarchical description of a shape which is suitable for its recognition. The key distinguishing feature of ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker, "Toward a Computational Theory of Shape: An Overview," in Proceedings of ECCV, Antibes, France, 1990. 29

....properties, or other factors independent of the curve. If c = 1 or c = 1, then #(t) is the dilation or erosion of the initial curve #(0) by a disk of radius t. The speed model c = 1##, # # 1, has been extensively studied in [16, 22] for general evolution of boundaries and interfaces and in [6] for shape analysis in computer vision. To overcome the topological problem of splitting and merging and numerical problems with the Lagrangian formulation (3) an Eulerian formulation was proposed in [16] where the original curve #(0) is first embedded in the surface of an arbitrary 2D Lipschitz ....

B. Kimia, A. Tannenbaum, and S. Zucker, "Toward a Computational Theory of Shape: An Overview", Proc. ECCV-90, France, April 1990.

.... observed by Besl, Boulanger, Witkin and many others) The surface descriptors we extract are selected to have simple curvature properties since a substantial literature suggests that curvature properties are closely associated with the necessary cues for many types of generic object recognition [1, 11, 13, 8, 15, 21, 10]. The key aspects of our approach are: ffl initial surface segmentation at multiple scales, ffl refinement of the segmentation into regions with uniform curvature properties, ffl property estimation and quality estimation of the resulting segments, ranking of the segments across all scales. ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, May 1990.

.... in [1] and extensions to triple points in [3] The fundamental Eulerian perspective presented by this approach has since been adopted in many theoretical analyses of mean curvature flow, in particular, see [13] In computer vision, a model for shape theory based on this work has been presented in [16]. 3 Shape Recovery with Front Propagation In this section, we describe how the level set formulation for the front propagation problem discussed in the previous section can be used for shape recovery. First, note that the front represents the boundary of an evolving shape. Since the idea is to ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," in Proceedings of ECCV, Antibes, France, 1990.

....28 7 Bent ellipsoid with non uniform tesselation 30 8 Ellipsoid with sparse tessellation 32 9 Ellipsoid with noise of high spatial frequency 34 10 Ellipsoid with noise of low spatial frequency 36 Bibliography 36 Chapter 1 Introduction 1.1 Growth models 1.1. 1 Diffusive growth Work by Kimia, Tannenbaum Zucker (1990), Pizer Burbeck (1994) Amit, Grenander Piccioni (1991) and others suggest diffusion systems as the most general approach to modeling shape. Zucker and Pizer follow shape change through scale and image space in analogy to theories of human visual perception, whereas Grenander formulates a more ....

Kimia, B., Tannenbaum, A. & Zucker, S. (1990), Toward a computational theory of shape: an overview, in `1st ECCV 90', Vol. 427 of Lecture Notes in Computer Science, Springer Verlag, pp. 23--27.

....follows that the stationary state in Figure 15 is practically independent of the initial shape of the patch. We notice that in general, the iterations at the beginning of the migration are a combination of constant and curvature dependent migration terms (reaction and diffusion components; see [17, 18]) If the patch becomes elongated (e.g. as in Figure 15) the migration gradually turns into a purely curvature dependent one (diffusion) as in the case of an arc. Figure 15: DMP s of a planar patch: free migration, migration subject to a single fixed point constraint (middle) migration subject ....

B.B. Kimia, A. Tannenbaum, and S.W. Zucker. Toward a computational theory of shape: An overview. Technical Report TR-CIM-89-13, McGill University, Montreal, Canada, June 1989.

....the evolution of higher dimensional sets (surfaces) using an analog evolution equation. He showed that a convex surface shrinks to a spherical point; however, he also pointed out that non convex surfaces develop singularities and thus change their topology 7 . Based on these ideas Kimia et al. [48, 49] studied a generalization of (5) where the curve evolves as a function of its local geometry: C t = a(u; t) Delta T b(u; t) Delta N ; 6) where T is the tangent to the curve, a(u; t) b(u; t) are arbitrary functions, and u is a parameter of the curve. In [33] it is shown that ....

....t) Delta N ; 6) where T is the tangent to the curve, a(u; t) b(u; t) are arbitrary functions, and u is a parameter of the curve. In [33] it is shown that this latter equation is in fact equivalent to C t = ff( Delta N ; which can be obtained by a reparameterization of (6) In [48, 49] the authors analyze planar curve evolution by a linear speed function ff( C t = ff 0 ff 1 ) Delta N ; 7) where ff 0 ; ff 1 are constants. They show that a curve evolved by (7) satisfies a hyperbolic conservation law [63] with a viscosity solution 8 [79] The term defined by ....

[Article contains additional citation context not shown here]

B.B. Kimia, A. Tannenbaum, and S.W. Zucker. Toward a computational theory of shape: An overview. Technical Report TR-CIM-89-13, McGill University, Montreal, Canada, June 1989.

.... subject to a single fixed point constraint (middle) migration subject to a set of fixed points (bottom) We notice that in general, the iterations at the beginning of the migration are a combination of constant and curvature dependent migration terms (reaction and diffusion components; see [12], 13] If the patch becomes elongated (e.g. as in Figure 6) the migration gradually turns into a purely curvature dependent one (diffusion) as in the case of an arc. As we see from these figures, the curvaturedependent component has smoothing effects on the patch, while the constant term ....

B.B. Kimia, A. Tannenbaum, and S.W. Zucker. Toward a computational theory of shape: An overview. Technical Report TR-CIM-89-13, McGill University, Montreal, Canada, June 1989.

.... unit normal, one considers families of plane curves evolving according to the geometric heat equation C t = N : 1) This equation has a number of properties which make it very useful in image processing, and in particular, the basis of a nonlinear scale space for shape representation [1, 3, 24, 25, 32]. Indeed, 1) is the Euclidean curve shortening flow, in the sense that the Euclidean perimeter shrinks as quickly as possible when the curve evolves according to (1) 16, 17, 21] Since, we will need a similar argument for the snake model we discuss in the next section, let us work out the ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview", Lecture Notes in Computer Science 427, pp. 402-407, SpringerVerlag, New York, 1990.

....Size is Time heuristic is closely related to the concept of blurring. It constrains the order of processes so that the object evolves under the grammar in the same way it would be revealed by deblurring. The derivation in Figure 1.8b respects this heuristic. Processes Based on Curvature Kimia s [36] process model is based on the idea that an arbitrary deformation of a curve C can be described as a linear sum of two local deformations along the normal N : C t = fi 0 fi 1 ) N ; where t is time, is curvature, fi 0 is constant motion along the normal to the boundary of the curve ....

B. Kimia, A. Tannenbaum, and S.W. Zucker. Toward a computational theory of shape: An overview. Technical report, McGill Research Centre for Intelligent Machines, Montreal, Quebec, Canada, 1989.

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B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes Comput. Sci., vol. 427, pp. 402--407, 1990.

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B. B. Kimia, Allen Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes in Computer Science, 427:402-407, Springer-Verlag: New York, 1990.

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B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes in Computer Science 427, pp. 402-407, Springer-Verlag, New York, 1990.

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B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes in Computer Science 427, pp. 402-407, SpringerVerlag, New York, 1990. 39

....for representing shape based on shape analysis via boundary evolution equations. As we will see, such evolution equations again solve certain variational problems, and so t into the optimal control (calculus of variations) paradigm being discussed here. The approach given here is based on [30, 31, 34, 32]. See also the nice results of [43] One of the main ideas in this work, is to consider an explicit treatment of singularities, which is founded on a series of conservation laws. In this way we consolidate the two 13 major approaches to shape the one based on boundary and the other on interior ....

....N : T (47) Z 2 (s; t)ds: 48) Finally, we let (t) 2 j (s; t)j (s; t)ds (49) denote the total absolute curvature. The behavior of the classical solutions equation (40) can be rather thoroughly analyzed and one can prove useful results (from the applied point of view) of the kind [4, 30, 31, 34, 32]. Let C(s; t) be a classical solution of (40) for t 2 [0; t ) and ( M for all 2 R (regarding as a function of ) Then, L(t) L(0)e Mt : 50) In case ( a 1, we get L(t) min(L(0) 2 t; L(0)e 4a ) 51) Let C(s; t) be a classical solution of (40) for t 2 [0; t ) ....

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B. B. Kimia, A. Tannenbaum, and S. W. Zucker, \Toward a computational theory of shape: An overview", Lecture Notes in Computer Science 427, pp. 402-407, Springer-Verlag, New York, 1990.

.... and N the inward unit normal, one considers families of plane curves evolving according to the geometric heat equation = This equation has a number of properties which make it very useful in image processing, and in particular, the basis of a nonlinear scale space for shape representation [1, 3, 22, 23]. In particular, 1) is the Euclidean curve shortening flow, in the sense that the Euclidean perimeter shrinks as quickly as possible when the curve evolves according to (1) 14, 16] Since, we will need a similar argument for the snake model we discuss in the next section, let us work out the ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview", Lecture Notes in Computer Science 427, pp. 402-407, SpringerVerlag, New York, 1990.

.... and N the inward unit normal, one considers families of plane curves evolving according to the geometric heat equation This equation has a number of properties which make it very useful in image processing, and in particular, the basis of a nonlinear scale space for shape representation [1, 3, 24, 25, 32]. Indeed, 1) is the Euclidean curve shortening flow, in the sense that the Euclidean perimeter shrinks as quickly as possible when the curve evolves according to (1) 16, 17, 21] Since, we will need a similar argument for the snake model we discuss in the next section, let us work out the ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview", Lecture Notes in Computer Science 427, pp. 402-407, SpringerVerlag, New York, 1990.

....of image intensity for scale [28] for shape from shading [8] etc. In addition, more recently curve evolution applications have used an embedding surface to represent the evolving curve, thus utilizing the additional dimension to regularize computations [41, 14, 11] e.g. shape representation [22, 24, 23], shape from shading [47, 25] image smoothing [10, 2, 21] affine invariant curve evolution [48, 1, 3] shape modeling [9, 34, 54] optical flow [29] and stereo [30] In the above examples the process of locating the curve, and computing its geometric properties, e.g. orientation and curvature, ....

....(ii) shapes developing topological splits, iii) the collapse of one curve segment onto another, and finally (iv) the collapse of an entire shape onto a point. These four discontinuous events, or shocks, are the key elements of a representation of shape for recognition, as first proposed in [22, 24, 23] and numerically detected in [52] Figure 12 depicts the original shapes, along with their embedding surfaces, which are then evolved using fi( 1, although other choices are possible as well. The GENO interpolation is compared with discrete binarization of OE, as well as with bilinear ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, 1990. Springer Verlag.

....for a number of qualitative shape properties to be extracted via the axis curvature, the velocity along the axis (object angle) and its discontinuities, etc. Curiously, an evolutionary approach to shape description supports and complements this view, and gives it a sound mathematical foundation [28, 29, 31]. To elaborate, Kimia et al. suggest a representation based on deformations of the shape s boundary: 8 : C t = fi( N C(s; 0) C 0 (s) 1) where C is the boundary vector of coordinates, N is the outward normal, s is the path parameter, t is the time duration (magnitude) of ....

....is that the reaction axis corresponds to a grassfire simulation. Thus, the set of shocks which form along the reaction axis, fi 1 = 0, is indeed Blum s skeleton [4] 2 , see Appendix A. Shocks form along other axes of the reaction diffusion space as well, and are classified into four types [28, 31], Figure 2) Definition 1 A first order shock is a discontinuity in orientation of the boundary of a shape. 2 In his grassfire formulation Blum called these generalized corners [5] 2 Second Order Shock P P Fourth Order Shock P Third Order Shocks First Order Shock P Q d d Figure 2: ....

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B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, 1990. Springer Verlag.

....describes the long range coupling via the skeleton and the subpixel implementation of SCDM. To see how long range coupling is mediated by the inter region skeleton, consider Figure 7. Let R and R Gamma be the regions and RB be the background. Let S denote the shocks 1 of RB as defined in [7, 8, 13, 18]. Consider a point A Gamma 2 R Gamma , the boundary of R Gamma , and its corresponding shock point S(A Gamma ) A, and let A be such that S(A ) S(A Gamma ) A, where skeletal point A is viewed as coupling the deformable model s boundary points, A and A Gamma . Let ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In ECCV, pages 402--407, France, 1990.

....contour which is the level set of some surface. The use of evolving contours as level sets and the relevant numerical implementation was earlier proposed by Osher and Sethian for flame propagation [42, 49] and used by Kimia et al. in constructing a representation for shape in computer vision [23, 24]. These active contours handle topological changes in that they can split without additional computational difficulty due to the level set representation. Malladi et al. illustrated the effectiveness of this approach to capture interesting structures in medical images. While the difficulty of ....

....an approach that resolves some of these difficulties. The basic idea is to construct a representation for a shape before it is segmented and as such, its representation directly derives from the image information. A complete representation of a given shape in the shape from deformation framework [24, 27, 29, 25] is obtained by detecting first , second , third , and fourth order shocks which are formed when the shape is evolved by a reaction diffusion process. This shock based morphogenetic representation views shape dynamically: A set of seeds, or fourth order shocks, are born which then grow to join ....

[Article contains additional citation context not shown here]

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, 1990. Springer Verlag.

....contour which is the level set of some surface. The use of evolving contours as level sets and the relevant numerical implementation was earlier proposed by Osher and Sethian for flame propagation [42, 49] and used by Kimia et al. in constructing a representation for shape in computer vision [23, 24]. These active contours handle topological changes in that they can split without additional computational difficulty due to the level set representation. Malladi et al. illustrated the effectiveness of this approach to capture interesting structures in medical images. While the difficulty of ....

....form. This results in a view of shape as a morphogenetic sequence 1 beginning with the birth of fourth order shocks which grow, and in the process are modified by first , second , and third order shocks, to finally reconstruct the original shape. This view of shape as a morphogenetic process [31, 30, 23, 24] is key to the present approach, and therefore we will review it first. Our current approach is presented as follows: First, we will propose that growing fourth order shocks, or bubbles, represent hypotheses for multiple objects in the image. Second, we show that various low level visual processes ....

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B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. CIM-89-13, McGill Centre for Intelligent Machines, McGill University, Montreal, Canada, 1989.

....contour which is the level set of some surface. The use of evolving contours as level sets and the relevant numerical implementation was earlier proposed by Osher and Sethian for flame propagation [20, 23] and used by Kimia et al. in constructing a representation for shape in computer vision [15]. While Malladi et al. illustrated the effectiveness of this approach to capture interesting structures in some medical images, symmetric initialization is required for each object of interest, thus leading to extensive user interaction. In addition, gaps and narrow regions present difficulties ....

....be initiated to mediate between pixels and models. The basic idea is to construct a representation for a shape before it is segmented and as such, its representation directly derives from the image information. A complete representation of a given shape in the shape from deformation framework [15, 16, 17] is obtained by detecting first , second , third , and fourth order shocks which are formed when the shape is evolved by a reaction diffusion process. This shock based morphogenetic representation views shape dynamically: A set of seeds, or fourthorder shocks, are born which then grow to join ....

[Article contains additional citation context not shown here]

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In ECCV:90.

....of image intensity for scale [31] for shape from shading [9] etc. In addition, more recently curve evolution applications have used an embedding surface to represent the evolving curve, thus utilizing the additional dimension to regularize computations [47, 15, 12] e.g. shape representation [25, 27, 26], shape from shading [52, 28] image smoothing [11, 2, 24] affine invariant curve evolution [53, 1, 3] shape modeling [10, 39, 61] optical flow [32] and stereo [33] In the above examples the process of locating the curve, and computing its geometric properties, e.g. orientation and curvature, ....

....the fronts are within the same pixel, GENO is able to recover the fronts as distinct and to place discontinuities (circles) that lie along the threads of the screw. 6. 2 Shape Representation The second application we consider is of that of shape representation in the context of deformations of it [25, 27, 26]. The key to this representation is a classification of the shocks or discontinuities that occur in the course of evolution, into four types. The details of a numerical algorithm for shock detection based on GENO appear in [59] The accurate representation of multiple curves per pixel, and the ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, 1990. Springer Verlag.

....a depth segregated edge map and use the notion of a gap skeleton to group certain nearby endpoints. 2 Shock detection and classification Shape can be completely described as the collection of four types of shocks which form in the course of deformations of shape in the reaction diffusion space [4, 5, 9] C t = fi 0 Gamma fi 1 ) N : 1) The four types of shocks correspond to intuitive elements of shape, namely, parts, protrusions, and bends [3] The deformations are implemented via the curve evolution paradigm by embedding the curves C(s; t) as the level set of a surface f (x; y; t) 0g ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In ECCV, Antibes, France, 1990.

.... unit normal, one considers families of plane curves evolving according to the geometric heat equation C t = N : 1) This equation has a number of properties which make it very useful in image processing, and in particular, the basis of a nonlinear scale space for shape representation [1, 3, 22, 23]. In particular, 1) is the Euclidean curve shortening flow, in the sense that the Euclidean perimeter shrinks as quickly as possible when the curve evolves according to (1) 14, 16] Since, we will need a similar argument for the snake model we discuss in the next section, let us work out the ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview", Lecture Notes in Computer Science 427, pp. 402-407, SpringerVerlag, New York, 1990.

....an approach that resolves some of these difficulties. The basic idea is to construct a representation for a shape before it is segmented and as such, its representation directly derives from the image information. A complete representation of a given shape in the shape from deformation framework [5, 7, 6] is obtained by detecting first , second , third , and fourth order shocks which are formed when the shape is evolved by a reaction diffusion process. This shock based morphogenetic representation views shape dynamically: Figure 1: This figure illustrates how bubbles capture a binary object in ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, 1990. Springer Verlag.

....rather are subject to shifts under the action of deformations to the bounding contour. When such a deformation alters the topological shock structure sufficiently, new perceptual categories can arise. To order the possibilities, we arrange the earlier sequences along the sides of a shape triangle (Kimia, 1990). Informally, the idea is that the interpretation of each shape lies on a continuum between distinct categories represented by the parts, protrusions and bends nodes, Figure 7. For example, along the parts bends axis the percept changes from that of Bends Parts Protrusions The Shape Triangle ....

....and bends nodes, Figure 7. For example, along the parts bends axis the percept changes from that of Bends Parts Protrusions The Shape Triangle Figure 7: The sides of the shape triangle represent continua of shapes; the extremes correspond to the parts , protrusions and bends nodes (Kimia, 1990). a dumbbell shape with two distinct parts to that of a worm shape with a single part. The sides of the triangle reflect these continua and capture the tension between object composition (parts) boundary deformation (protrusions) and region deformation (bends) Most importantly, we know from the ....

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Kimia, B. B., Tannenbaum, A., and Zucker, S. W. (1990). Toward a computational theory of shape: An overview. Proceedings of the First European Conference on Computer Vision, pages 402--407.

No context found.

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview", Lecture Notes in Computer Science 427, pp. 402-407, Springer-Verlag, New York, 1990.

.... grammar [23] and for extraction from realistic grey scale images [27, 28] Shape can be completely described as the collection of four types of shocks that arise in the course of deformations of shape, canonically captured in the reaction diffusion space C t = fi 0 Gamma fi 1 ) Gamma N [10, 13, 23]. The four types of shocks, Figure 1, correspond to intuitive elements of shape, namely, parts, protrusions, and bends [12] and augment traditional skeletal representations with notions of type, direction, and velocity, grouping via a shock grammar Table 1, saliency, and a shock hierarchy based ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. ECCV, pages 402--407, Antibes, France, 1990. Springer Verlag.

....Gaussian operator [17] isophotes of image intensity for scale [14] etc. In addition, more recently, curve evolution applications have used an embedding surface to represent the evolving curve, thus utilizing the additional dimension to regularize computations [18, 5] e.g. shape representation [10, 11], shape from shading [12] image smoothing [3] affine invariant curve evolution [22, 1] shape modeling [2, 16, 26] optical flow [15] etc. In the above examples, the process of locating the curve and computing its geometric properties, e.g. orientation and curvature, can benefit from the ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In ECCV '90, France, 1990.

....topological changes. Therefore, for images with multiple objects, the snake or balloon methods require extensive user interaction. This problem can be resolved by the use of the level set evolution, proposed by Osher and Sethian for flame propagation [24, 25, 2] introduced to computer vision in [12, 15] for shape representation, and first applied to active contours in [4, 20] The latter approach considers a curve C as the zero level set of a surface, OE(x; y) 0, evolving under constant and curvature deformation modulated by an image based speed term: OE t S(x; y) fi 0 Gamma fi 1 (x; ....

....an approach that resolves some of these difficulties. The basic idea is to construct a representation for a shape before it is segmented and as such, its representation directly derives from the image information. A complete representation of a given shape in the shape from deformation framework [12, 14, 15, 13] is obtained by detecting first , second , third , and fourthorder shocks which are formed when the shape is evolved by a reaction diffusion process. This shock based morphogenetic representation views shape dynamically: A set of seeds, or fourth order shocks, are born which then grow to join ....

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B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In Proceedings of the First European Conference on Computer Vision, Antibes, France, 1990. Springer Verlag.

....boundary collapses into a single point. The key idea is to view shape as a morphogenetic sequence 1 in reverse time beginning with the birth of fourth order shocks which grow, and in the process are modified by first , second , and third order shocks, to finally reconstruct the original shape [17, 16, 11, 12]. C t = fi 0 Gamma fi 1 )N; 4) Fig. 1. This figure illustrates a morphogenetic view of shape as being reconstructed from its shock based representation. First two first order shocks are born, a) then these shocks grow and are modified by first order shocks, b) c) and (d) These two ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. CIM-89-13, McGill Centre for Intelligent Machines, McGill University, Montreal, Canada, 1989.

....dilation with a circle of radius 10, B(10) 2 H USEYIN TEK AND BENJAMIN B. KIMIA A T N A B Initial Curve Deformed Curve Fig. 1. The points on the initial curve A move to B to generate a new curve. The direction and magnitude of this motion is arbitrary in order to capture general deformations [15]. N N T N T N T Y X N T Structuring Element (w, f (w) w New Boundary Original Boundary Fig. 2. Morphological operations in the extrinsic and intrinsic coordinates for an arbitrary convex structuring element. The class of structuring elements B for which this property holds is exactly the set ....

....approaches [9] Thus, key applications of mathematical morphology, such as, i) smoothing of shapes and images, ii) skeletal representation of shape, and (iii) segmentation by watersheds computation can all be formulated in the curve evolution framework. For the connection to smoothing case, see [15, 16, 17, 14] where Gaussian and curvaturedriven smoothing of curves and images are represented together with morphological operations in a unified reaction diffusion framework driven by C t = fi 0 Gamma fi 1 ) N . The extraction of shocks, whose geometric locus is the crest lines of the distance ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. In ECCV, pages 402--407, 1990.

....1 y 1 x 2 y 2 denotes the usual Euclidean inner product. 2. 1 Planar curve evolution The theory of planar curve evolution has been considered in a variety of fields such as differential geometry [26, 27, 50, 51, 59] theory of parabolic equations [3] numerical analysis [13, 53] computer vision [21, 22, 34, 35, 36, 37, 54, 56, 58, 60, 70, 75], viscosity solutions [12, 19, 66] phase transitions [30] and image processing [2, 52, 61, 64] One of the most important of such flows is derived when a planar curve deforms in the direction of the Euclidean normal, with speed equal to the Euclidean curvature. Formally, let C(p; t) S 1 ....

....to a round point via the flow given in (1) The flow (1) which is non linear since v is a time dependent curve parametrization, is also called the Euclidean geometric heat flow. It has been utilized for the definition of a geometric, Euclidean invariant, multiscale representation of planar shapes [1, 34, 35]. As we will show below, this flow is also important for image enhancement applications. Note than in contrast with the classical heat flow given by C t = C pp , the Euclidean geometric heat flow 1 We will consistently use v to denote Euclidean arc length, reserving s for the affine arc length ....

[Article contains additional citation context not shown here]

B. B. Kimia, A. Tannenbaum, and S. W. Zucker, "Toward a computational theory of shape: An overview," Lecture Notes in Computer Science 427, pp. 402-407, SpringerVerlag, New York, 1990.

....our proposed geometrical method is based on curvature dependent differential deformations of shape. To motivate this model for 3D shape, we first review deformations of curves and their application to 2D shape. Curve Evolution and Smoothing: The shape from deformation framework of Kimia et al. [4, 5, 6, 7] proposes to understand shape in the context of a topology of it built around deformations: C t = fi( Delta) N; C(s; t) x(s; t) y(s; t) 1) This research was supported by NSF grant IRI 9305630. The authors thank Kaleem Siddiqi for technical assistance, and Kenneth Herndon, Graphics ....

B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. ECCV 1990.

....a set of parts will produce an undulating surface. For example, how much does a pile of rocks have to erode before it becomes a set of hills and valleys We take a unified position with respect to qualitative shape and maintain that a global theory is necessary to refine the distinction [12]. With this background, we show in this paper that it is often possible to detect qualitative features of the surface shape directly from the image intensities. The idea is that the two notions of qualitative local shape described Figure 1 have something important in common. Hills and parts both ....

Kimia BB and Tannenbaum A and Zucker SW, "Toward a Computational Theory of Shape : An Overview", Computer Vision - ECCV 90 in Lecture Notes in Computer Science (Springer Verlag 1990)

....the shocks (or singularities) of a curve evolution process, acting on a simple closed curve in the plane, into a shock tree. We begin by providing some background on the representation (for details see [20, 38] and then present experimental results on matching shock trees. 4. 1 The Shock Tree In [19, 20] the following evolution equation was proposed for visual shape analysis: C t = 1 ff)N C(p; 0) C 0 (p) 11) Here C(p; t) is the vector of curve coordinates, N (p; t) is the inward normal, p is the curve parameter, and t is the evolutionary time of the deformation. The constant ff 0 ....

B. B. Kimia, A. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. Lecture Notes in Computer Science, 427:402--407, 1990.

....drawing of figure from Studies for the Libyan Sibyl, Michelangelo (Nicolaides [1941] erties such as curvature sign, concavities, convexities, narrowings and bulges. These properties play a critical role in computational theories of planar shape perception (Blum [1973] Hoffman Richards [1985] Kimia, Tannenbaum Zucker [1990]; Leyton [1989] and moreover impose strong constraints on the shape of the three dimensional surfaces in the scene (Biederman [1988] Koenderink [1984] Lowe [1985] Marr [1982] In topology there is no concept of partial contour closure (a curve is either closed or it is not) Such a view has ....

B. Kimia, A. Tannenbaum & S. W. Zucker [1990], "Toward a computational theory of shape: An overview," Proc. 1 st European Conf. on Computer Vision 427, 402--407.

No context found.

Kimia, B.B., Tannenbaum, A.R., & Zucker, S.W. (1990). Toward a computational theory of shape: An overview. Lecture Notes in Computer Science, 427, 402--407.

No context found.

B.B. Kimia, A. Tannenbaum, and S.W. Zucker, "Toward a Computational Theory of Shape: An Overview," Proc. European Conf. Computer Vision, Lecture Notes in Computer Science, vol. 427, pp. 402-407, 1990.

No context found.

B. B. Kimia, A. Tannenbaum, and S. W. Zucker. Toward a computational theory of shape: An overview. Lecture Notes in Computer Science, 427:402--407, 1990.

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