## Shape Matching and Object Recognition Using Shape Contexts (2001)

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Venue: | IEEE Transactions on Pattern Analysis and Machine Intelligence |

Citations: | 1332 - 19 self |

### BibTeX

@ARTICLE{Belongie01shapematching,

author = {Serge Belongie and Jitendra Malik and Jan Puzicha},

title = {Shape Matching and Object Recognition Using Shape Contexts},

journal = {IEEE Transactions on Pattern Analysis and Machine Intelligence},

year = {2001},

volume = {24},

pages = {509--522}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a novel approach to measuring similarity between shapes and exploit it for object recognition. In our framework, the measurement of similarity is preceded by (1) solv- ing for correspondences between points on the two shapes, (2) using the correspondences to estimate an aligning transform. In order to solve the correspondence problem, we attach a descriptor, the shape context, to each point. The shape context at a reference point captures the distribution of the remaining points relative to it, thus offering a globally discriminative characterization. Corresponding points on two similar shapes will have similar shape con- texts, enabling us to solve for correspondences as an optimal assignment problem. Given the point correspondences, we estimate the transformation that best aligns the two shapes; reg- ularized thin plate splines provide a flexible class of transformation maps for this purpose. The dissimilarity between the two shapes is computed as a sum of matching errors between corresponding points, together with a term measuring the magnitude of the aligning trans- form. We treat recognition in a nearest-neighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image. Results are presented for silhouettes, trademarks, handwritten digits and the COIL dataset.

### Citations

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Citation Context ... one relies on a learning algorithm having enough examples to acquire the appropriate invariances. In the area of face recognition good results were obtained using principal components analysis (PCA) =-=[53, 55]-=- particularly when used in a probabilistic framework [37]. Murase and Nayar applied these ideas to 3D object recognition [39]. Several authors have applied discriminative classification methods in the... |

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Citation Context ...neural network for handwritten digit recognition, and the Support Vector Machine (SVM)-based methods of [40] (for discriminating between templates of pedestrians based on 2D wavelet coefficients) and =-=[10, 7]-=- (for handwritten digit recognition). The MNIST database of handwritten digits is a particularly important dataset as many different pattern recognition algorithms have been tested on it. We will show... |

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Citation Context ...tly solving for correspondences. Amit et al. [1] train decision trees for recognition by learning discriminative spatial configurations of keypoints. Leung et al. [34], Schmid and Mohr [48], and Lowe =-=[35]-=- additionally use gray level information at the keypoints to provide greater discriminative power. It should be noted that not all objects have distinguished key points (think of a circle for instance... |

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Citation Context ... matching be one-to-one, i.e. 7r is a permutation. This is an instance of the square assignment (or weighted bipartite matching) problem, which can be solved in O(N a) time using the Hungarian method =-=[41]-=-. In our experiments, we use the aAlternatives include Bickel's generalization of the Kolmogorov-Smirnov test for 2D distributions [4], which does not require binning. 11 more efficient algorithm of [... |

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Citation Context ...n indefinitely. Prototype-based recognition translates readily into the computational framework of nearest neighbor methods using multiple stored views. Nearest neighbor classifiers have the property =-=[45]-=- that as the number of examples n in the training set goes to infinity, the 1-NN error converges to a value _ 2E*, where E* is the Bayes Risk (for K-NN, K -- c and Kin -- 0, the error -- E*). This is ... |

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Citation Context ...n good results were obtained using principal components analysis (PCA) [53, 55] particularly when used in a probabilistic framework [37]. Murase and Nayar applied these ideas to 3D object recognition =-=[39]-=-. Several authors have applied discriminative classification methods in the appearance-based shape matching framework. Some examples are the LeNet classifier [33], a convolutional neural network for h... |

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Citation Context ... these ideas to 3D object recognition [39]. Several authors have applied discriminative classification methods in the appearance-based shape matching framework. Some examples are the LeNet classifier =-=[33]-=-, a convolutional neural network for handwritten digit recognition, and the Support Vector Machine (SVM)-based methods of [40] (for discriminating between templates of pedestrians based on 2D wavelet ... |

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Citation Context ...rp the images into alignment. The third term Dbe (7 >, Q) corresponds to the 'amount' of transformation necessary to align the shapes. In the TPS case the bending energy (9) is a natural measure (see =-=[5]-=-). 5.2 Choosing Prototypes In a prototype based approach, the key question is: what examples shall we store? Different categories need different numbers of views. For example, certain handwritten digi... |

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Citation Context ...sure, we can use nearest neighbor techniques for object recognition. Philosophically, nearest neighbor techniques can be related to prototype-based recognition as developed by Rosch and collaborators =-=[46, 47]-=-. They have the advantage that Figure 1. Examples of two handwritten digits. In terms of pixel-to-pixel comparisons, these two images are quite different, but to the human observer, the shapes appear ... |

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Citation Context ...hand-crafted parametrized models, e.g. for eyes, in the image domain using gradient descent. Another well-known computational approach in this vein was developed by von der Malsburg and collaborators =-=[30]-=- using elastic graph matching. Our primary contribution in this work is a robust and simple algorithm for finding correspondences between shapes. Shapes are represented by a set of points sampled from... |

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Citation Context ...nts in the 2D image. Extracting these from an image is less of a problem e.g. one can just use an edge detector. Huttenlocher et al. developed methods in this category based on the Hausdorff distance =-=[22]-=-; this can be extended to deal with partial matching and clutter. A drawback for our purposes is that the method does not return correspondences. Methods based on Distance Transforms, such as [15], ar... |

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Citation Context ...without explicitly solving for correspondences. Amit et al. [1] train decision trees for recognition by learning discriminative spatial configurations of keypoints. Leung et al. [34], Schmid and Mohr =-=[48]-=-, and Lowe [35] additionally use gray level information at the keypoints to provide greater discriminative power. It should be noted that not all objects have distinguished key points (think of a circ... |

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Citation Context ... one relies on a learning algorithm having enough examples to acquire the appropriate invariances. In the area of face recognition good results were obtained using principal components analysis (PCA) =-=[53, 55]-=- particularly when used in a probabilistic framework [37]. Murase and Nayar applied these ideas to 3D object recognition [39]. Several authors have applied discriminative classification methods in the... |

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Citation Context ...gning transformation. At the heart of our approach is a tradition of matching shapes by deformation that can be traced at least as far back as D'Arcy Thompson. In his classic work, On Growth and Form =-=[55]-=-, Thompson observed that related but not identical shapes can often be deformed into alignment using simple . S. Belongie is with the Department of Computer Science and Engineering, AP&M Building, Roo... |

329 | Regularization Theory and Neural Networks Architectures
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Citation Context ...i- f(xi, Yi)) 2 + Air . (10) i----1 The regularization parameter A, a positive scalar, controls the amount of smoothing; the limiting case of A -- 0 reduces to exact interpolation. As demonstrated in =-=[59, 17]-=-, we can solve for the TPS coefficients in the regularized case by replacing the matrix K by K + I, where I is the n x n identity matrix. It is interesting to note that the highly regularized TPS mode... |

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Citation Context ...hat related but not identical shapes can often be deformed into alignment using simple coordinate transformations, as illustrated in Fig. 2. In the computer vision literature, Fischler and Elschlager =-=[14]-=- operationalized such an idea by means of energy minimization in a mass-spring model. Grenander et al. [20] developed these ideas in a probabilistic setting. Yuille [60] developed another variant of t... |

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Citation Context ...n. In the nearest neighbor classifier literature, the problem of selecting exemplars is called editing. Extensive reviews of nearest neighbor editing methods can be found in Ripley [45] and Dasarathy =-=[11]-=-. We have developed a novel editing algorithm based on shape distance and Kmedoid clustering. K-medoids can be seen as a variant of K-means that restricts prototype positions to data points. First a m... |

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Citation Context ...ontain the homogeneous coordinates of ' and Q, respectively, i.e. I p pm P = ' ' ' (6) 1 pn pn2 Here, Q+ denotes the pseudo inverse of Q. In this work, we mostly use the thin plate spline (TPS) model =-=[13, 36]-=-, which is commonly used for representing flexible coordinate transformations. Bookstein [6] found it to be highly effective for modeling changes in biological forms. Powell applied the TPS model to r... |

257 | Fourier descriptors for plane closed curves
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Citation Context ... not have holes or internal markings, the associated boundaries are conveniently represented by a single closed curve which can be parametrized by arclength. Early work used Fourier descriptors, e.g. =-=[61, 42]-=-. Blum's medial axis transform has led to attempts to capture the part structure of the shape in the graph structure of the skeleton by Kimia, Zucker and collaborators, e.g. Sharvit et al. [52]. The 1... |

240 |
Morphometric Tools for Landmark Data: Geometry and Biology
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Citation Context ...al analysis. This only tells us when two shapes are exactly the same; we need more than that for a theory of shape similarity or shape distance. The statistician's definition of shape, e.g. Bookstein =-=[6]-=- or Kendall [28], addresses the problem of shape distance, but assumes that correspondences are known. Other statistical approaches to shape comparison do not require correspondences e.g. one could co... |

236 | Pedestrian Detection Using Wavelet Templates
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(Show Context)
Citation Context ...ce-based shape matching framework. Some examples are the LeNet classifier [33], a convolutional neural network for handwritten digit recognition, and the Support Vector Machine (SVM)-based methods of =-=[40]-=- (for discriminating between templates of pedestrians based on 2D wavelet coefficients) and [10, 7] (for handwritten digit recognition). The MNIST database of handwritten digits is a particularly impo... |

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The internal representation of solid shape with respect to vision
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Citation Context ... 3D objects, a sphere needs only one view, for example, while a telephone needs several views to capture the variety of visual appearance. This idea is related to the "aspect" concept as dis=-=cussed in [29]-=-. We will now discuss how we approach the problem of prototype selection. In the nearest neighbor classifier literature, the problem of selecting exemplars is called editing. Extensive reviews of near... |

203 | Real-time object detection for smart vehicles
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(Show Context)
Citation Context ...nce [22]; this can be extended to deal with partial matching and clutter. A drawback for our purposes is that the method does not return correspondences. Methods based on Distance Transforms, such as =-=[15]-=-, are similar in spirit and behavior in practice. The work of Sclaroff and Pentland [49] is representative of the eigenvector- or modalmatching based approaches; see also [51, 50, 56]. In this approac... |

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Citation Context ...sure, we can use nearest neighbor techniques for object recognition. Philosophically, nearest neighbor techniques can be related to prototype-based recognition as developed by Rosch and collaborators =-=[46, 47]-=-. They have the advantage that Figure 1. Examples of two handwritten digits. In terms of pixel-to-pixel comparisons, these two images are quite different, but to the human observer, the shapes appear ... |

200 |
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Citation Context ... not have holes or internal markings, the associated boundaries are conveniently represented by a single closed curve which can be parametrized by arclength. Early work used Fourier descriptors, e.g. =-=[61, 42]-=-. Blum's medial axis transform has led to attempts to capture the part structure of the shape in the graph structure of the skeleton by Kimia, Zucker and collaborators, e.g. Sharvit et al. [52]. The 1... |

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(Show Context)
Citation Context ... our purposes is that the method does not return correspondences. Methods based on Distance Transforms, such as [15], are similar in spirit and behavior in practice. The work of Sclaroff and Pentland =-=[49]-=- is representative of the eigenvector- or modalmatching based approaches; see also [51, 50, 56]. In this approach, sample points in the image are cast into a finite element spring-mass model and corre... |

184 | Shape Manifolds, Procrustean Metrics, and Complex Projective Spaces
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Citation Context ...is only tells us when two shapes are exactly the same; we need more than that for a theory of shape similarity or shape distance. The statistician's definition of shape, e.g. Bookstein [6] or Kendall =-=[28]-=-, addresses the problem of shape distance, but assumes that correspondences are known. Other statistical approaches to shape comparison do not require correspondences e.g. one could compare feature ve... |

167 | A New Algorithm for Non-Rigid Point Matching
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Citation Context ... image are cast into a finite element spring-mass model and correspondences are found by comparing modes of vibration. Most closely related to our approach is the work of Rangarajan and collaborators =-=[18, 8]-=-, which is discussed in Section 3.4. There have been several approaches to shape recognition based on spatial configurations of a small number of keypoints or landmarks. In geometric hashing [31], the... |

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Citation Context ...]. In our experiments, we use the aAlternatives include Bickel's generalization of the Kolmogorov-Smirnov test for 2D distributions [4], which does not require binning. 11 more efficient algorithm of =-=[27]. The-=- input to the assignment problem is a square cost matrix with entries Cij. The result is a permutation r(i) such that (2) is minimized. In order to have robust handling of outliers, one can add "... |

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An eigendecomposition approach to weighted graph matching problems
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Citation Context ...ance Transforms, such as [15], are similar in spirit and behavior in practice. The work of Sclaroff and Pentland [49] is representative of the eigenvector- or modalmatching based approaches; see also =-=[51, 50, 56]-=-. In this approach, sample points in the image are cast into a finite element spring-mass model and correspondences are found by comparing modes of vibration. Most closely related to our approach is t... |

153 | Improving the accuracy and speed of support vector machines
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Citation Context ...neural network for handwritten digit recognition, and the Support Vector Machine (SVM)-based methods of [40] (for discriminating between templates of pedestrians based on 2D wavelet coefficients) and =-=[10, 7]-=- (for handwritten digit recognition). The MNIST database of handwritten digits is a particularly important dataset as many different pattern recognition algorithms have been tested on it. We will show... |

152 | Matching shapes
- Belongie, Malik, et al.
(Show Context)
Citation Context ...ed by (ARO) DAAH04-96-1-0341, the Digital Library Grant IRI-9411334, an NSF graduate Fellowship for S.B., and the German Research Foundation by DFG grant PU-165/1. Parts of this work have appeared in =-=[3, 2]-=- We wish 32 to thank H. Chui and A. Rangarajan for providing the synthetic testing data used in 4.2. We would also like to thank them and various members of the Berkeley computer vision group, particu... |

147 | Training invariant support vector machines
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(Show Context)
Citation Context ...left) our shape distance is compared to SSD (sum of squared differences between pixel brightness values). In Fig. 7 (right) we compare the classification rates for different K. 4DeCoste and Sch51kopf =-=[12]-=- report an error rate of 0.56% on the same database using Virtual Support Vectors (VSV) with the full training set of 60,000. VSVs are found as follows: (1) obtain SVs from the original training set u... |

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Hands: A Pattern Theoretic Study of Biological Shapes
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Citation Context ...ations, as illustrated in Fig. 2. In the computer vision literature, Fischler and Elschlager [14] operationalized such an idea by means of energy minimization in a mass-spring model. Grenander et al. =-=[20]-=- developed these ideas in a probabilistic setting. Yuille [60] developed another variant of the deformable template concept by means of fitting hand-crafted parametrized models, e.g. for eyes, in the ... |

136 | Shape context: A new descriptor for shape matching and object recognition
- Belongie, Malik, et al.
(Show Context)
Citation Context ...ed by (ARO) DAAH04-96-1-0341, the Digital Library Grant IRI-9411334, an NSF graduate Fellowship for S.B., and the German Research Foundation by DFG grant PU-165/1. Parts of this work have appeared in =-=[3, 2]-=- We wish 32 to thank H. Chui and A. Rangarajan for providing the synthetic testing data used in 4.2. We would also like to thank them and various members of the Berkeley computer vision group, particu... |

132 | Shape descriptors for non-rigid shapes with a single closed contour
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(Show Context)
Citation Context ...kinds of transformation including some articulation and occlusion. A comprehensive comparison of different shape descriptors for comparing silhouettes was done as part of the MPEG-7 standard activity =-=[32]-=-, with the leading approaches being those due to Latecki et al. [32] and Mokhtarian et al. [38]. Silhouettes are fundamentally limited as shape descriptors for general objects; they ignore internal co... |

132 | Efficient and Robust Retrieval by Shape Content through Curvature Scale Space
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(Show Context)
Citation Context ...f different shape descriptors for comparing silhouettes was done as part of the MPEG-7 standard activity [32], with the leading approaches being those due to Latecki et al. [32] and Mokhtarian et al. =-=[38]-=-. Silhouettes are fundamentally limited as shape descriptors for general objects; they ignore internal contours and are difficult to extract from real images. More promising are approaches that treat ... |

128 |
Bayesian face recognition
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(Show Context)
Citation Context ... acquire the appropriate invariances. In the area of face recognition good results were obtained using principal components analysis (PCA) [53, 55] particularly when used in a probabilistic framework =-=[37]-=-. Murase and Nayar applied these ideas to 3D object recognition [39]. Several authors have applied discriminative classification methods in the appearance-based shape matching framework. Some examples... |

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Feature-based correspondence - an eigenvector approach
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(Show Context)
Citation Context ...ance Transforms, such as [15], are similar in spirit and behavior in practice. The work of Sclaroff and Pentland [49] is representative of the eigenvector- or modalmatching based approaches; see also =-=[51, 50, 56]-=-. In this approach, sample points in the image are cast into a finite element spring-mass model and correspondences are found by comparing modes of vibration. Most closely related to our approach is t... |

123 |
Affine invariant model-based object recognition
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(Show Context)
Citation Context ...s [18, 8], which is discussed in Section 3.4. There have been several approaches to shape recognition based on spatial configurations of a small number of keypoints or landmarks. In geometric hashing =-=[31]-=-, these configurations are used to vote for a model without explicitly solving for correspondences. Amit et al. [1] train decision trees for recognition by learning discriminative spatial configuratio... |

122 | A computational framework for determining stereo correspondence from a set of linear spatial filters
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(Show Context)
Citation Context ...s a correspondence problem similar to that in stereopsis. Experience there suggests that matching is easier if one uses a rich local descriptor, e.g. a gray scale window or a vector of filter outputs =-=[26]-=-, instead of just the brightness at a single pixel or edge location. Rich descriptors reduce the ambiguity in matching. As a key contribution we propose a novel descriptor, the shape context, that cou... |

121 |
Deformable templates for face recognition
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(Show Context)
Citation Context ...ature, Fischler and Elschlager [14] operationalized such an idea by means of energy minimization in a mass-spring model. Grenander et al. [20] developed these ideas in a probabilistic setting. Yuille =-=[60]-=- developed another variant of the deformable template concept by means of fitting hand-crafted parametrized models, e.g. for eyes, in the image domain using gradient descent. Another well-known comput... |

118 | Flexible syntactic matching of curves and its application to automatic hierarchical classification of silhouettes
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(Show Context)
Citation Context ...e graph structure of the skeleton by Kimia, Zucker and collaborators, e.g. Sharvit et al. [52]. The 1D nature of silhouette curves leads naturally to dynamic programming approaches for matching, e.g. =-=[16]-=-, which uses the edit distance between curves. This algorithm is fast and invariant to several kinds of transformation including some articulation and occlusion. A comprehensive comparison of differen... |

108 | Finding faces in cluttered scenes using random labeled graph matching
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(Show Context)
Citation Context ...d to vote for a model without explicitly solving for correspondences. Amit et al. [1] train decision trees for recognition by learning discriminative spatial configurations of keypoints. Leung et al. =-=[34]-=-, Schmid and Mohr [48], and Lowe [35] additionally use gray level information at the keypoints to provide greater discriminative power. It should be noted that not all objects have distinguished key p... |

106 |
An Algorithm for Associating the Features of Two Images
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(Show Context)
Citation Context ...ance Transforms, such as [15], are similar in spirit and behavior in practice. The work of Sclaroff and Pentland [49] is representative of the eigenvector- or modalmatching based approaches; see also =-=[51, 50, 56]-=-. In this approach, sample points in the image are cast into a finite element spring-mass model and correspondences are found by comparing modes of vibration. Most closely related to our approach is t... |

98 |
Modelling spatial patterns (with discussion
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(Show Context)
Citation Context ...have a certain minimum distance between them as this makes sure that the sampling along the contours is somewhat uniform. (This corresponds to sampling from a point process which is a hard-core model =-=[44]-=-.) Since the sample points are drawn randomly and independently from the two shapes, there is inevitably jitter noise in the output of the matching algorithm which finds correspondences between these ... |

95 |
Multivariate interpolation at arbitrary points made simple
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(Show Context)
Citation Context ...ontain the homogeneous coordinates of ' and Q, respectively, i.e. I p pm P = ' ' ' (6) 1 pn pn2 Here, Q+ denotes the pseudo inverse of Q. In this work, we mostly use the thin plate spline (TPS) model =-=[13, 36]-=-, which is commonly used for representing flexible coordinate transformations. Bookstein [6] found it to be highly effective for modeling changes in biological forms. Powell applied the TPS model to r... |

86 | New algorithms for 2d and 3d point matching: Pose estimation and correspondence
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(Show Context)
Citation Context ... image are cast into a finite element spring-mass model and correspondences are found by comparing modes of vibration. Most closely related to our approach is the work of Rangarajan and collaborators =-=[18, 8]-=-, which is discussed in Section 3.4. There have been several approaches to shape recognition based on spatial configurations of a small number of keypoints or landmarks. In geometric hashing [31], the... |