## Geodesic matching of triangulated surfaces (2001)

Venue: | Image Processing, IEEE Transactions on |

Citations: | 6 - 1 self |

### BibTeX

@ARTICLE{Hamza01geodesicmatching,

author = {A. Ben Hamza and Hamid Krim},

title = {Geodesic matching of triangulated surfaces},

journal = {Image Processing, IEEE Transactions on},

year = {2001},

volume = {15},

pages = {2006}

}

### OpenURL

### Abstract

Abstract — Recognition of images and shapes has long been the central theme of computer vision. Its importance is increasing rapidly in the field of computer graphics and multimedia communication because it is difficult to process information efficiently without its recognition. In this paper, we propose a new approach for object matching based on a global geodesic measure. The key idea behind our methodology is to represent an object by a probabilistic shape descriptor that measures the global geodesic distance between two arbitrary points on the surface of an object. In contrast to the Euclidean distance which is more suitable for linear spaces, the geodesic distance has the advantage to be able to capture the intrinsic geometric structure of the data. The matching task therefore becomes a one-dimensional comparison problem between probability distributions which is clearly much simpler than comparing 3D structures. Object matching can then be carried out by an information-theoretic dissimilarity measure calculations between geodesic shape distributions, and is additionally computationally efficient and inexpensive. I.

### Citations

2070 | Some methods for classification and analysis of multivariate observations
- MacQueen
- 1967
(Show Context)
Citation Context ...centroidbased methods have been used in a variety of computer vision applications including clustering, and one of the widely centroid-based technique used for cluster analysis in the Kmean algorithm =-=[20]-=-. A. Global shape measure Let M =(V, T ) be a triangle mesh. The centroid cj of a triangle Tj is the mean of its vertices, that is, the point located at the center of the triangle. Note that the cardi... |

1792 |
A global geometric framework for nonlinear dimensionality reduction
- Tenenbaum, Silva, et al.
- 2000
(Show Context)
Citation Context ... a triangle mesh M, the geodesic distance calculation is based on a similar approach used for computing the isometric feature mapping (Isomap) for multidimensional scaling [29] on nonlinear manifolds =-=[30]-=-. The algorithm has two main steps: (i) Construct a neighborhood graph by connecting a given Fig. 4. Euclidean vs. geodesic distance on a nonlinear manifold. centroid to its k-nearest neighbors, and l... |

1421 |
On information and sufficiency
- Kullback, Leibler
- 1951
(Show Context)
Citation Context ...Information-theoretic measures provide quantitative entropic divergences between two probability distributions. A common entropic dissimilarity measure is the Kulback-Liebler (or directed) divergence =-=[10]-=- which has been successfully used in many applications including indexing and image retrieval [11]. Another entropy-based measure is the Jensen-Shannon (JS) divergence which may be defined between an ... |

575 | Theory - Milnor - 1963 |

561 |
Differential Geometry of Curves and Surfaces
- CARMO, P
- 1976
(Show Context)
Citation Context ...ogy is the property that determines which parts of the shape of objects are connected to which other parts [24–26], while geometry determines where, in a given coordinate system, each part is located =-=[27]-=-. The basic principle is that the topology of a manifold is very closely related to the critical points of a smooth function on that manifold. An interesting concept related to Morse theory and very u... |

451 | Divergence measures based on the shannon entropy
- Lin
- 1991
(Show Context)
Citation Context ...ications including indexing and image retrieval [11]. Another entropy-based measure is the Jensen-Shannon (JS) divergence which may be defined between an arbitrary number of probability distributions =-=[12]-=-. Due to this generalization, the JS divergence may be used as a coherence measure between any number of distributions and may be applied to a variety of image processing and computer vision applicati... |

321 |
Topology matching for fully automatic similarity estimation of 3D shapes
- Hilaga, Shinagawa, et al.
- 2001
(Show Context)
Citation Context ...ture of 3D objects. A preliminary work on this signature was presented in [18]. The proposed method is inspired by previous works on object matching and in particular the recent works of Hilaga et al =-=[19]-=- and Osada et al [9]. The shape distribution approach [9] is based on the Euclidean distance which is not suitable for capturing the nonlinear structure of 3D objects, whereas the method presented in ... |

297 | Differential Topology - Guillemin, Pollack - 1974 |

223 | Computing geodesic paths on manifolds
- Kimmel, Sethian
- 1998
(Show Context)
Citation Context ... with zeros in the diagonal, and positive off-diagonal elements. Note that the geodesic distance on triangulated surfaces may also be effectively computed using the fast marching method introduced in =-=[31]-=-. 2) Triangle area calculation: Denote by {v1, v2, v3} the vertices of an arbitrary triangle T of a given triangle mesh M. Using Newell method, the area of the triangle T can be computed as area(T )=�... |

205 | Dobkin,Shape distributions
- Osada, Funkhouser, et al.
- 2002
(Show Context)
Citation Context ...gistration, shape analysis, motion estimation, object recognition, and surface evolution [6–8]. An alternative to feature-based representations, called shape distribution, is developed by Osada et al =-=[9]-=-. The idea here is to represent an object by a global histogram based on the Euclidean distance defined on the surface of an object. The shape matching problem is then performed by computing a dissimi... |

89 |
Surface coding based on Morse theory
- Shinagawa, Kunii, et al.
- 1991
(Show Context)
Citation Context ...pe dissimilarity problem reduces to the problem of comparing two such object representations. Feature-based methods require that features be extracted and described before two objects can be compared =-=[1, 2]-=-. Among feature-based methods, one popular approach is graph matching, where two objects are represented by their graphs composed of vertices and edges. An efficient representation that captures the t... |

86 |
Applications of entropic spanning graphs
- Hero, Ma, et al.
- 2002
(Show Context)
Citation Context ... the JS divergence may be used as a coherence measure between any number of distributions and may be applied to a variety of image processing and computer vision applications including graph matching =-=[13]-=-, image registration and segmentation [14, 15], edge detection [16], and segmentation of DNA sequences into homogenous domains [17]. In this paper, we propose a shape signature called geodesic shape d... |

67 | Critical point theory and submanifold geometry - Palais, Terng - 1988 |

59 | Curves and Singularities - Bruce, Giblin - 1992 |

58 |
Constructing a Reeb graph automatically from cross sections
- Shinagawa, Kunii
- 1991
(Show Context)
Citation Context ...epresented by their graphs composed of vertices and edges. An efficient representation that captures the topological properties of 3D objects is the Reeb graph descriptor proposed by Shinagawa et al. =-=[1, 3]-=-. The vertices of the Reeb graph are the singular points of a function defined on the underlying object [1, 3–5]. These singularities are prominent landmarks and their detection, recognition, and clas... |

44 | Local Morse theory for solutions to the heat equation and Gaussian blurring - Damon - 1995 |

35 |
Surfacing Signatures: An Orientation Independent Free-Form Surface Representation Scheme for the Purpose of Objects Registration and Matching
- Yamany, Farag
- 2002
(Show Context)
Citation Context ...pe dissimilarity problem reduces to the problem of comparing two such object representations. Feature-based methods require that features be extracted and described before two objects can be compared =-=[1, 2]-=-. Among feature-based methods, one popular approach is graph matching, where two objects are represented by their graphs composed of vertices and edges. An efficient representation that captures the t... |

35 | A generalized divergence measure for robust image registration
- He, Hamza, et al.
- 2003
(Show Context)
Citation Context ...nce measure between any number of distributions and may be applied to a variety of image processing and computer vision applications including graph matching [13], image registration and segmentation =-=[14, 15]-=-, edge detection [16], and segmentation of DNA sequences into homogenous domains [17]. In this paper, we propose a shape signature called geodesic shape distribution that captures the intrinsic geomet... |

27 | Topology of Surfaces - Kinsey - 1993 |

27 |
Information Theory and Reliable
- Gallager
- 1968
(Show Context)
Citation Context ...� , where E{·} denotes the expected value with respect to p(x). The KL dissimilarity measure, however, is non-symmetric, unbounded, and undefined if ˆp is not absolutely continuous with respect to ˆq =-=[16, 33]-=-. To overcome these limitations, we Fig. 10. Block-diagram of the proposed methodology. use the Jensen-Shannon (JS) divergence D given by D(ˆp, ˆq) = 1 � � � � �� ˆp +ˆq ˆp +ˆq K ˆp, + K ˆq, 2 2 2 � �... |

24 | H.: Geodesic object representation and recognition
- Hamza, Krim
- 2003
(Show Context)
Citation Context ...7]. In this paper, we propose a shape signature called geodesic shape distribution that captures the intrinsic geometric structure of 3D objects. A preliminary work on this signature was presented in =-=[18]-=-. The proposed method is inspired by previous works on object matching and in particular the recent works of Hilaga et al [19] and Osada et al [9]. The shape distribution approach [9] is based on the ... |

17 | Shape Spectrum Based View Grouping and Matchning of 3D Free-Form Objects
- Dorai, Jain
- 1997
(Show Context)
Citation Context ...the Euclidean distance, is its inability to capture the nonlinear structure of the data. C. Shape spectrum Shape spectrum was initially proposed for view grouping and matching of 3D free-form objects =-=[28]-=-, and it is the histogram of the shape index operator which is defined at each point p of a surface M as S(p) = 1 1 κ1(p)+κ2(p) − tan−1 2 π κ1(p) − κ2(p) , where κ1,κ2 are the surface principal curvat... |

11 | Differential Topology: First Steps - Wallace - 1968 |

10 | Image Registration and Segmentation by Maximizing the Jensen-Rényi Divergence EMMCVPR 2003 : energy minimization methods in computer vision and pattern recognition
- Hamza, Krim
(Show Context)
Citation Context ...nce measure between any number of distributions and may be applied to a variety of image processing and computer vision applications including graph matching [13], image registration and segmentation =-=[14, 15]-=-, edge detection [16], and segmentation of DNA sequences into homogenous domains [17]. In this paper, we propose a shape signature called geodesic shape distribution that captures the intrinsic geomet... |

8 |
Francos, “Image retrieval and indexing: A hierarchical approach in computing the distance between textured images
- Stoica, Zerubia, et al.
- 1998
(Show Context)
Citation Context ...istributions. A common entropic dissimilarity measure is the Kulback-Liebler (or directed) divergence [10] which has been successfully used in many applications including indexing and image retrieval =-=[11]-=-. Another entropy-based measure is the Jensen-Shannon (JS) divergence which may be defined between an arbitrary number of probability distributions [12]. Due to this generalization, the JS divergence ... |

7 |
Topological Modeling for Visualization, Springer-Verlag
- Fomenko, Kunii
- 1997
(Show Context)
Citation Context ...n defined on the underlying object [1, 3–5]. These singularities are prominent landmarks and their detection, recognition, and classification is a crucial step in image processing and computer vision =-=[4]-=-. Such singularities carry important information for further operations, such as image registration, shape analysis, motion estimation, object recognition, and surface evolution [6–8]. An alternative ... |

7 | Surface evolution under curvature flows - Lu, Cao - 2002 |

3 | A topological skeleton of illuminated manifolds - Hamza, Krim - 2003 |

3 |
An analysis of edge detection by using the Jensen-Shannon divergence
- Gomez, Martinez, et al.
- 2000
(Show Context)
Citation Context ...umber of distributions and may be applied to a variety of image processing and computer vision applications including graph matching [13], image registration and segmentation [14, 15], edge detection =-=[16]-=-, and segmentation of DNA sequences into homogenous domains [17]. In this paper, we propose a shape signature called geodesic shape distribution that captures the intrinsic geometric structure of 3D o... |

3 |
Sequence compositional complexity of DNA through an entropic segmentation method
- Roman, Bernaola, et al.
- 1998
(Show Context)
Citation Context ...processing and computer vision applications including graph matching [13], image registration and segmentation [14, 15], edge detection [16], and segmentation of DNA sequences into homogenous domains =-=[17]-=-. In this paper, we propose a shape signature called geodesic shape distribution that captures the intrinsic geometric structure of 3D objects. A preliminary work on this signature was presented in [1... |

2 |
Mulridimensional scaling, second edition
- Cox, Cox
(Show Context)
Citation Context ...f a 3D object represented by a triangle mesh M, the geodesic distance calculation is based on a similar approach used for computing the isometric feature mapping (Isomap) for multidimensional scaling =-=[29]-=- on nonlinear manifolds [30]. The algorithm has two main steps: (i) Construct a neighborhood graph by connecting a given Fig. 4. Euclidean vs. geodesic distance on a nonlinear manifold. centroid to it... |