## Toward a Shock Grammar for Recognition (1995)

Venue: | IEEE Conf. on Computer Vision and Pattern Recognition |

Citations: | 8 - 0 self |

### BibTeX

@INPROCEEDINGS{Siddiqi95towarda,

author = {Kaleem Siddiqi and Benjamin B. Kimia},

title = {Toward a Shock Grammar for Recognition},

booktitle = {IEEE Conf. on Computer Vision and Pattern Recognition},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

The recognition of objects from their projected two-dimensional shapes is a challenging problem owing to the spectrum of possible variations reflected in the image domain, e.g., those caused by movement of parts, changes in viewing geometry, occlusion, etc. This motivates a need for quantitative as well as qualitative descriptions of shape in terms of structural relations between components; the latter remain largely invariant under the above changes. In this paper we confront the theoretical and practical difficulties of computing such a representation, based on the detection of shocks or singularities that arise as a shape is deformed, as organized in two stages. First, we develop subpixel local detectors for the detection of shocks and a classification of them into four types. Second, we show that shock patterns are not arbitrary, but obey the rules of a grammar which limits the possible shock combinations. In addition, shock patterns satisfy specific topological and geometric constraints. We develop this shock grammar and exploit the topological and geometric constraints to enforce global consistency: shock hypotheses that violate the grammar or are topologically or geometrically invalid are pruned, and survivors are organized into higher level structures. The result is a computational method for the detection, classification, and grouping of shocks. This leads to a description of shape as a hierarchical graph of shock groups. The graph is computed in the reaction-diffusion space, where diffusion plays a role of regularization to determine the significance of each shock-group. The representation is stable with rotations, scale changes, occlusion, movement of parts, noise and other variations, even at very low resolutions. We illustrate the suitability of this repres...

### Citations

961 | Front propagating with curvature dependent speed: Algorithms based on HamiltonJacobi formulations
- OSHER, SETHIAN
- 1988
(Show Context)
Citation Context ... the concepts of entropy [34] and viscosity solutions [41, 12, 11]. For theoretical as well as numerical reasons, the original curve flow is embedded in the level set evolution of an evolving surface =-=[15, 48, 2]-=-, z = OE(x; y; t), with the correspondence that the evolving shape is represented at all times by its zero level set OE(x; y; t) = 0: It can be shown that the zero level set of surfaces evolving accor... |

771 | User's guide to viscosity solutions of second order partial dierential equations
- Crandall, Ishii, et al.
- 1992
(Show Context)
Citation Context ... no longer defined. As a remedy to this problem, classical differential geometric notions are considered in the generalized or weak sense by using the concepts of entropy [34] and viscosity solutions =-=[41, 12, 11]-=-. For theoretical as well as numerical reasons, the original curve flow is embedded in the level set evolution of an evolving surface [15, 48, 2], z = OE(x; y; t), with the correspondence that the evo... |

592 |
Efficient implementation of essentially non-oscillatory shockcapturing schemes
- Shu, Osher
- 1989
(Show Context)
Citation Context ... solution of conservation laws and the propagation of fronts. These essentially nonoscillatorys(ENO) schemes were introduced by Harten et al. [20], and were later made more efficient by Shu and Osher =-=[57]-=-. The basic idea is to select between two contiguous sets of data points for interpolation the one which gives the lower variation, or coefficient of the highest derivative of an interpolation polynom... |

496 |
Representation and recognition of the spatial organization of three-dimensional shapes
- Marr, Nishihara
- 1978
(Show Context)
Citation Context ...er will assign these the labels "hawk" and "cormorant". 1 of primitive-based approaches, where shape is viewed statically as a combination of a small number of components, e.g., ge=-=neralized cylinders [42]-=-, or geons [3] , versus process-based approaches [38] where shape is explained dynamically or developmentally via a set of processes acting on a simpler shape. Returning to Blum's proposal of a sphere... |

342 |
Image selective smoothing and edge detection by nonlinear diffusion
- Alvarez, Lions, et al.
- 1992
(Show Context)
Citation Context ...tons [14], and eliminates the possibility of spurious skeletal branches. A similar idea is used in [32] to obtain the skeleton, where the computation is regularized via the geomet24 ric heat equation =-=[1, 27]. Pizer et-=- al. suggest a notion of significance based on the amount of blurring required to annihilate a particular skeletal branch [52]. Upon annihilation, the branch is identified as a "child" of th... |

309 | Motion of level sets by mean curvature
- Evans, Spruck
- 1991
(Show Context)
Citation Context ... the concepts of entropy [34] and viscosity solutions [41, 12, 11]. For theoretical as well as numerical reasons, the original curve flow is embedded in the level set evolution of an evolving surface =-=[15, 48, 2]-=-, z = OE(x; y; t), with the correspondence that the evolving shape is represented at all times by its zero level set OE(x; y; t) = 0: It can be shown that the zero level set of surfaces evolving accor... |

288 |
Elements of the Theory of Computation
- Papadimitriou
- 1981
(Show Context)
Citation Context ...sition of a group (the sequential constraints) can be concisely described via a shock grammar. Formally, a grammar G is a language generating device, which is defined by a quadruple (V; \Sigma; R; S) =-=[35]-=-. Here V is an alphabet divided into two parts, the set of terminal symbols \Sigma ` V and the set of non-terminal symbols V \Gamma \Sigma. S, the start symbol, is an element of V \Gamma \Sigma and R,... |

255 |
The heat equation shrinking convex plane curves
- Gage, Hamilton
- 1986
(Show Context)
Citation Context ...erties 14 . In particular, in the extreme case of pure diffusion, fi 1 fi 0 !1, it is known that any embedded curve must evolve to a round point without developing self-intersections or singularities =-=[16, 18]-=-, and that the number of extrema and inflection points is non-increasing [29]. These results imply that no new branches will form, and that the shape will disappear as a single fourth-order shock, wit... |

254 | Parts of recognition
- Hoffman, Richards
- 1984
(Show Context)
Citation Context ...s of a shape's contour, and Hoffman and Richard's representation based on codons, or contour segments bounded by minima of curvature, the latter motivated by a transversality-based proposal for parts =-=[22]-=-. However, it is important to note that these classifications are not mutually exclusive, nor are they unique. First, it may in fact be desirable to capture both boundary as well as region-based featu... |

251 | Uniqueness and existence of viscosity solutions of generalized mean curvature equations
- Chen, Giga, et al.
- 1991
(Show Context)
Citation Context ... no longer defined. As a remedy to this problem, classical differential geometric notions are considered in the generalized or weak sense by using the concepts of entropy [34] and viscosity solutions =-=[41, 12, 11]-=-. For theoretical as well as numerical reasons, the original curve flow is embedded in the level set evolution of an evolving surface [15, 48, 2], z = OE(x; y; t), with the correspondence that the evo... |

244 |
The heat equation shrinks embedded plane curves to round points
- Grayson
- 1987
(Show Context)
Citation Context ...erties 14 . In particular, in the extreme case of pure diffusion, fi 1 fi 0 !1, it is known that any embedded curve must evolve to a round point without developing self-intersections or singularities =-=[16, 18]-=-, and that the number of extrema and inflection points is non-increasing [29]. These results imply that no new branches will form, and that the shape will disappear as a single fourth-order shock, wit... |

239 |
Shape description using weighted symmetric axis features
- Blum, Nagel
- 1978
(Show Context)
Citation Context ...l aspects of skeletons. However, it is unfortunate that Blum's key insight that the SAT provides for qualitative shape descriptions in terms of "shape morphemes", e.g., disc, worm, wedge, fl=-=are, etc. [5, 6]-=-, is usually forgotten. More specifically, Blum points out that the grassfire formulation, where the symmetric axis is obtained as quench points of a grassfire moving parallel to the boundary, places ... |

219 |
A theory of multiscale, curvature-based shape representation for planar curves
- Mokhtarian, Mackworth
- 1992
(Show Context)
Citation Context ... enhanced [50]. 14 It should be noted that curvature deformation is equivalent to Gaussian smoothing of the shape's boundary, provided that arc-length parametrization is preserved at each step, as in =-=[45]-=-. 25 Figure 21: The survival of a first-order shock group with diffusion indicates its significance. Left to Right: fi 0 = \Gamma0:2, fi 1 = 0:0; 0:125; 0:25; 0:5. Each column depicts the shock groups... |

207 | Trace inference, curvature consistency, and curve detection
- Parent, Zucker
- 1989
(Show Context)
Citation Context ...ges: 1) a local detection stage giving rise to true as well as some spurious edges [23], and 2) a global interpretation stage which strengthens edges with lateral support through a relaxation process =-=[60, 49]-=-. Similarly, a key idea of this paper is that shock computations can be made robust by relying not only on better (subpixel) local detectors and classifiers, but also on the global interactions among ... |

185 |
Uniformly high order accurate essentially non-oscillatory schemes
- Harten, Engquist, et al.
- 1987
(Show Context)
Citation Context ...rical analysis literature, in application to the numerical solution of conservation laws and the propagation of fronts. These essentially nonoscillatorys(ENO) schemes were introduced by Harten et al. =-=[20]-=-, and were later made more efficient by Shu and Osher [57]. The basic idea is to select between two contiguous sets of data points for interpolation the one which gives the lower variation, or coeffic... |

154 |
Biological shape and visual science
- Blum
- 1973
(Show Context)
Citation Context ... emphasising salient features of the shape's boundary such as curvature extrema and inflection points. Prominent examples of region-based representations include Blum's symmetric axis transform (SAT) =-=[4, 5]-=-, motivated by the need for a geometry that captures biological shape and describes its growth, and the related smoothed local symmetries (SLS) [7] and process inferring symmetric axis [38]. Well stud... |

148 |
Partial Differential Equations
- John
- 1981
(Show Context)
Citation Context ...pe, it remains part of it forever. Second, by Huygens' Principle, under constant motion in the normal direction the evolved front is the envelope of circular waves issuing from the original wavefront =-=[24]-=-. Proof: The proof is in two stages. First, we will show that each shock point is also a skeletal point. Second, we will summarize the argument that each skeletal point is also a shock point. To show ... |

141 |
Solid shape
- Koenderink
- 1990
(Show Context)
Citation Context ...eleton [52, 36], it is not immediately clear how skeletal branches obtained at one level of smoothing can be related to those obtained at another, in a principled fashion. Blurring the shape's region =-=[33]-=- can lead to topological splits, while blurring the shape's boundary can place undue emphasis on elongated features, and can lead to self-intersections 13 . Following the theoretical development of [2... |

131 | Hierarchic voronoi skeletons
- Ogniewicz, Kubler
- 1995
(Show Context)
Citation Context ...the amount that the boundary segment lying between its associated bi-tangent points protrudes, with respect to the object width [21]. In a discrete implementation of the SAT based on Voronoi Diagrams =-=[46, 47]-=- Ogniewicz et al. characterize the prominence of a symmetric axis edge via the length of the associated boundary segment. In their notion of significance based on ridge support [36], Leymarie and Levi... |

119 |
Smoothed local symmetries and their implementation
- Brady, Asada
- 1984
(Show Context)
Citation Context ...ons include Blum's symmetric axis transform (SAT) [4, 5], motivated by the need for a geometry that captures biological shape and describes its growth, and the related smoothed local symmetries (SLS) =-=[7]-=- and process inferring symmetric axis [38]. Well studied examples of boundary-based descriptions include polygonal approximations of a shape's contour, and Hoffman and Richard's representation based o... |

113 |
Curvature and the evolution of fronts
- Sethian
- 1985
(Show Context)
Citation Context ...the zero level set of surfaces evolving according to OE t + fi()jrOEj = 0 (3) correspond to the viscosity solutions of (1) [2]. The numerical simulation of (3) was first proposed by Osher and Sethian =-=[56, 48]-=- who provided an elegant scheme for flame front propagation. We utilize this scheme for shape evolution; for convenience we take OE 0 to be the distance transform of the shape. We now describe a class... |

75 |
Towards a computational theory of shape, an overview
- Kimia, Tannenbaum, et al.
- 1990
(Show Context)
Citation Context ...y 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 i... |

75 |
Simulating the grassfire transform using an active contour model
- Leymarie, Levine
- 1992
(Show Context)
Citation Context ...d bends. In related work, Leymarie and Levine have simulated the grassfire transform using active contours moving so as to seek minimum potential energy configurations on an inverted distance surface =-=[36], an appro-=-ach which enables the use of a Euclidean metric while bypassing discretization problems in typical skeletonization algorithms. Further, a notion of "branch significance" is introduced, based... |

73 |
A process-grammar for shape
- Leyton
(Show Context)
Citation Context ...rm (SAT) [4, 5], motivated by the need for a geometry that captures biological shape and describes its growth, and the related smoothed local symmetries (SLS) [7] and process inferring symmetric axis =-=[38]-=-. Well studied examples of boundary-based descriptions include polygonal approximations of a shape's contour, and Hoffman and Richard's representation based on codons, or contour segments bounded by m... |

72 | Object representation by cores: Identifying and representing primitive spatial regions
- Burbeck, Pizer
- 1994
(Show Context)
Citation Context ...as region-based features, as is the case in Pizer et al.'s core based model, where the scale at which boundaries are detected and described is a function of the width of the object through its region =-=[10]. Second, an orthogo-=-nal classification is that 1 The avid bird-watcher will assign these the labels "hawk" and "cormorant". 1 of primitive-based approaches, where shape is viewed statically as a combi... |

64 |
Shapes, shocks, and deformations, i: The components of shape and the reaction-diffusion space
- Kimia, Tannenbaum, et al.
- 1995
(Show Context)
Citation Context ...y 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 i... |

57 |
Curves and singularities
- Bruce, Giblin
- 1992
(Show Context)
Citation Context ...i 1 = 0), provides the set of skeletal points, and its associated radius function, Appendix A. This implies that geometric and topological properties that hold for skeletons, studied by Giblin et al. =-=[17, 9, 8]-=-, Matheron [43], Meyer [44], among others, 8 In our implementation the two segments are considered parallel when their slopes are within 10 degrees. In this case a set of third-order shocks is interpo... |

52 |
On the evolution of curves via a function of curvature, I the classical case. J.Math. Analysis and Applications
- Kimia, Tannenbaum, et al.
- 1992
(Show Context)
Citation Context ...y 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 i... |

51 |
schemes with subcell resolution
- ENO
- 1989
(Show Context)
Citation Context ... OE xx OE yy \GammaOE 2 xy (1+OE 2 x +OE 2 y ) 2 ;s1 +s2 = (1+OE 2 x )OE yy \Gamma2OE x OE y OE xy +(1+OE 2 y )OE xx (1+OE 2 x +OE 2 y ) 3=2 . 6 This is in essence Harten's "sub-cell resolution&q=-=uot; idea [19]-=-, the difference being that geometric reasoning is used in place of conservation to determine the sub-cell shock location. 9 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 64 65 66 67 ... |

48 | Logical/linear operators for image curves
- Iverson, Zucker
- 1995
(Show Context)
Citation Context ... edge detection. An approach that has proven to be successful in that domain is one of dividing the task into two stages: 1) a local detection stage giving rise to true as well as some spurious edges =-=[23]-=-, and 2) a global interpretation stage which strengthens edges with lateral support through a relaxation process [60, 49]. Similarly, a key idea of this paper is that shock computations can be made ro... |

44 |
Two stages of curve detection suggest two styles of visual computation
- Zucker, Dobbins, et al.
- 1989
(Show Context)
Citation Context ...ges: 1) a local detection stage giving rise to true as well as some spurious edges [23], and 2) a global interpretation stage which strengthens edges with lateral support through a relaxation process =-=[60, 49]-=-. Similarly, a key idea of this paper is that shock computations can be made robust by relying not only on better (subpixel) local detectors and classifiers, but also on the global interactions among ... |

42 |
Hierarchical shape description via the multiresolution symmetry axis transform
- Pizer, Oliver, et al.
- 1987
(Show Context)
Citation Context ...omputation is regularized via the geomet24 ric heat equation [1, 27]. Pizer et al. suggest a notion of significance based on the amount of blurring required to annihilate a particular skeletal branch =-=[52]. Upon ann-=-ihilation, the branch is identified as a "child" of the branch into which it disappears, providing a hierarchical organization of SAT branches. Whereas the above approaches often lead to int... |

41 |
Accurate corner detection: An analytical study
- Deriche, Giraudon
- 1990
(Show Context)
Citation Context ...to apply directly to images. 2 Shock Classification and Detection: Local Operators Whereas at first glance shock detection might appear to be a straightforward task (e.g., corner detection algorithms =-=[13, 59]-=- can be used to locate first-order shocks), finding singularities, particularly in low resolution cases, remains a challenging problem. The first difficulty is to arrive at a complete classification o... |

40 | Geometric heat equation and nonlinear diffusion of shapes and images
- Kimia, Siddiqi
- 1994
(Show Context)
Citation Context ...tons [14], and eliminates the possibility of spurious skeletal branches. A similar idea is used in [32] to obtain the skeleton, where the computation is regularized via the geomet24 ric heat equation =-=[1, 27]. Pizer et-=- al. suggest a notion of significance based on the amount of blurring required to annihilate a particular skeletal branch [52]. Upon annihilation, the branch is identified as a "child" of th... |

36 | Symmetry-curvature duality
- Leyton
- 1987
(Show Context)
Citation Context ...ts; the precise quantitative description of remote regions, such as the palm of the hand, remains largely unchanged. details: every positive maxima of curvature gives rise to a distinct symmetry axis =-=[37]-=-. In principle it is not possible to discern the significance of a branch based on its length alone and a variety of alternate measures have been proposed. Blum and Nagel suggest a boundary axis weigh... |

34 | Geometric shock-capturing ENO schemes for subpixel interpolation, computation and curve evolution
- Siddiqi, Kimia, et al.
(Show Context)
Citation Context ...Siddiqi et al. have adapted these ideas to the 2D problem of locating level curves of an embedding surface while: 1) not blurring across discontinuities, and 2) explicitly and accurately placing them =-=[58]. The key -=-idea is to replace polynomials with geometric interpolants: lines, circular arcs, Euler spirals, etc., while also allowing for "breaks" in the interpolated curve 6 . This geometric essential... |

33 |
Encoding contour shape by curvature extrema
- Richards, Dawson, et al.
- 1986
(Show Context)
Citation Context ...reconstruction, Figure 24 (middle). With increased diffusion the next shock branch to annihilate represents the middle triangle, Figure 24 (right). As a second example, consider Richards' pear shapes =-=[53], Figure 2-=-5. The shocks which survive increased diffusion bring out a coarse level similarity between the shapes as "two seed-based parts with corresponding protrusions, attached at a neck", as is ref... |

31 |
Recognition by Components
- Biederman
- 1987
(Show Context)
Citation Context ...these the labels "hawk" and "cormorant". 1 of primitive-based approaches, where shape is viewed statically as a combination of a small number of components, e.g., generalized cylin=-=ders [42], or geons [3]-=- , versus process-based approaches [38] where shape is explained dynamically or developmentally via a set of processes acting on a simpler shape. Returning to Blum's proposal of a sphere as a primitiv... |

29 |
Symmetry sets
- Bruce, Giblin, et al.
- 1985
(Show Context)
Citation Context ...i 1 = 0), provides the set of skeletal points, and its associated radius function, Appendix A. This implies that geometric and topological properties that hold for skeletons, studied by Giblin et al. =-=[17, 9, 8]-=-, Matheron [43], Meyer [44], among others, 8 In our implementation the two segments are considered parallel when their slopes are within 10 degrees. In this case a set of third-order shocks is interpo... |

29 |
Multiple resolution skeletons
- Dill, Levine, et al.
- 1987
(Show Context)
Citation Context ... boundary information into the model via an initial extraction of positive maxima of curvature at which the endpoints of snakes are fixed. This allows for a multiscale approach to computing skeletons =-=[14]-=-, and eliminates the possibility of spurious skeletal branches. A similar idea is used in [32] to obtain the skeleton, where the computation is regularized via the geomet24 ric heat equation [1, 27]. ... |

25 |
Discrete Voronoi Skeletons, HartungGorre
- Ogniewicz
- 1993
(Show Context)
Citation Context ...the amount that the boundary segment lying between its associated bi-tangent points protrudes, with respect to the object width [21]. In a discrete implementation of the SAT based on Voronoi Diagrams =-=[46, 47]-=- Ogniewicz et al. characterize the prominence of a symmetric axis edge via the length of the associated boundary segment. In their notion of significance based on ridge support [36], Leymarie and Levi... |

20 |
Shock waves and entropy
- Lax
- 1971
(Show Context)
Citation Context ...o parameters: the ratio of the coefficients fi 0 =fi 1 and time, t, constituting the two axes of the reaction-diffusion space. Underlying the representation of shape in this space are a set of shocks =-=[34]-=-, or entropy-satisfying singularities, which develop during the evolution. These are points where some information is lost during the deformation process, such as discontinuities in orientation along ... |

19 |
Local symmetries of plane curves
- Giblin, Brassett
(Show Context)
Citation Context ...i 1 = 0), provides the set of skeletal points, and its associated radius function, Appendix A. This implies that geometric and topological properties that hold for skeletons, studied by Giblin et al. =-=[17, 9, 8]-=-, Matheron [43], Meyer [44], among others, 8 In our implementation the two segments are considered parallel when their slopes are within 10 degrees. In this case a set of third-order shocks is interpo... |

19 |
Examples of topological properties of skeletons
- Matheron
- 1988
(Show Context)
Citation Context ...e set of skeletal points, and its associated radius function, Appendix A. This implies that geometric and topological properties that hold for skeletons, studied by Giblin et al. [17, 9, 8], Matheron =-=[43]-=-, Meyer [44], among others, 8 In our implementation the two segments are considered parallel when their slopes are within 10 degrees. In this case a set of third-order shocks is interpolated as a line... |

14 |
Scale-Space Theory
- Lindeberg
- 1994
(Show Context)
Citation Context ...gure 23 is 15 In discussing the lifetime in scale space as a potential measure of significance, Lindeberg has pointed out that a grey-level blob will survive longer if it is isolated from other blobs =-=[40]-=-. If another blob is nearby, the two may merge into a single blob, representing their union at a coarser scale. In analogy, when two protrusions are nearby the shock branches may merge into a single b... |

13 |
Annular symmetry operators: A method for locating and describing objects
- Kelly, Levine
- 1995
(Show Context)
Citation Context ...monstrated the use of annular operators in obtaining coarse object descriptions from real imagery, containing textures, gaps, internal structures, and contrast-sign reversals along occluding contours =-=[25]-=-. Annular operators at increasing scales are applied to edge maps obtained at corresponding scales, to extract a set of symmetry points at each scale. The symmetry points are grouped into types of par... |

13 |
Using a mixed wave/diffusion process to elicit the symmetry set
- Scott, Turner, et al.
- 1989
(Show Context)
Citation Context ...m [4]. Scott et al. have suggested the use of wave propagation to obtain the full symmetry set, i.e., the locus of centers of all bi-tangent circles whether or not they are contained within the shape =-=[54]-=-. Theirs is an alternating scheme of propagating waves from edge-points in a binary image, and damping them via the diffusion or heat equation. By superposition, the maximum energy to arrive at a loca... |

10 |
Remark on a flame propagation model
- Barles
- 1985
(Show Context)
Citation Context ... the concepts of entropy [34] and viscosity solutions [41, 12, 11]. For theoretical as well as numerical reasons, the original curve flow is embedded in the level set evolution of an evolving surface =-=[15, 48, 2]-=-, z = OE(x; y; t), with the correspondence that the evolving shape is represented at all times by its zero level set OE(x; y; t) = 0: It can be shown that the zero level set of surfaces evolving accor... |

9 |
Conservation Laws and a Theory of Shape
- Kimia
- 1990
(Show Context)
Citation Context ...-order shock formation. In other words, we must show that the tangent to each of the colliding boundaries exists at the second-order shock. This property holds due to the following theorem, proved in =-=[26]-=-: Theorem 1 When two distinct non-neighboring points of the boundary come together, the tangents at these points exist, and are parallel. Remark 1 Once formed, a second-order shock must give rise to t... |

9 |
The shape triangle: parts, protrusions, and bends
- Kimia, Tannenbaum, et al.
- 1992
(Show Context)
Citation Context ...represent three cooperative/competitive processes acting on shape, namely, parts, protrusions, and bends. The sides of the triangle represent continua of shapes whose extremes correspond to each node =-=[30]-=-. Definition 2 A second-order shock is formed when two distinct non-neighboring boundary points join, but none of their immediate neighbors collapse together. Definition 3 A third-order shock is forme... |