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93
Shape Matching and Object Recognition Using Shape Contexts
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2001
"... 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 ..."
Abstract

Cited by 1274 (19 self)
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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 nearestneighbor 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.
Regularization Theory and Neural Networks Architectures
 Neural Computation
, 1995
"... We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Ba ..."
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Cited by 317 (31 self)
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We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...
On the mathematical foundations of learning
 Bulletin of the American Mathematical Society
, 2002
"... The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton ..."
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Cited by 227 (12 self)
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The problem of learning is arguably at the very core of the problem of intelligence, both biological and arti cial. T. Poggio and C.R. Shelton
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 212 (22 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
A Theory of Networks for Approximation and Learning
 Laboratory, Massachusetts Institute of Technology
, 1989
"... Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, t ..."
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Cited by 200 (24 self)
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Learning an inputoutput mapping from a set of examples, of the type that many neural networks have been constructed to perform, can be regarded as synthesizing an approximation of a multidimensional function, that is solving the problem of hypersurface reconstruction. From this point of view, this form of learning is closely related to classical approximation techniques, such as generalized splines and regularization theory. This paper considers the problems of an exact representation and, in more detail, of the approximation of linear and nonlinear mappings in terms of simpler functions of fewer variables. Kolmogorov's theorem concerning the representation of functions of several variables in terms of functions of one variable turns out to be almost irrelevant in the context of networks for learning. Wedevelop a theoretical framework for approximation based on regularization techniques that leads to a class of threelayer networks that we call Generalized Radial Basis Functions (GRBF), since they are mathematically related to the wellknown Radial Basis Functions, mainly used for strict interpolation tasks. GRBF networks are not only equivalent to generalized splines, but are also closely related to pattern recognition methods suchasParzen windows and potential functions and to several neural network algorithms, suchas Kanerva's associative memory,backpropagation and Kohonen's topology preserving map. They also haveaninteresting interpretation in terms of prototypes that are synthesized and optimally combined during the learning stage. The paper introduces several extensions and applications of the technique and discusses intriguing analogies with neurobiological data.
Hierarchical and Variational Geometric Modeling with Wavelets
 IN PROCEEDINGS SYMPOSIUM ON INTERACTIVE 3D GRAPHICS
, 1995
"... This paper discusses how wavelet techniques may be applied to a variety of geometric modeling tools. In particular, wavelet decompositions are shown to be useful for hierarchical control point or least squares editing. In addition, direct curve and surface manipulation methods using an underlying ge ..."
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Cited by 63 (1 self)
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This paper discusses how wavelet techniques may be applied to a variety of geometric modeling tools. In particular, wavelet decompositions are shown to be useful for hierarchical control point or least squares editing. In addition, direct curve and surface manipulation methods using an underlying geometric variational principle can be solved more efficiently by using a wavelet basis. Because the wavelet basis is hierarchical, iterative solution methods converge rapidly. Also, since the wavelet coefficients indicate the degree of detail in the solution, the number of basis functions needed to express the variational minimum can be reduced, avoiding unnecessary computation. An implementation of a curve and surface modeler based on these ideas is discussed and experimental results are reported.
Graphical Templates For Model Registration
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 1996
"... A new method of model registration is proposed using graphical templates. A graph of landmarks is chosen in the template image. All possible candidates for these landmarks are found in the data image using local operators. A dynamic programming algorithm on decomposable subgraphs of the template gra ..."
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Cited by 54 (2 self)
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A new method of model registration is proposed using graphical templates. A graph of landmarks is chosen in the template image. All possible candidates for these landmarks are found in the data image using local operators. A dynamic programming algorithm on decomposable subgraphs of the template graph finds the optimal match to a subset of the candidate points in polynomial time. This combination of local operators to describe points of interest/landmarks and a graph to describe their geometric orientation in the plane, yields fast and precise matches of the model to the data, with no initialization required. Key words: Graphical templates, decomposable graphs, model registration, dynamic programming, image matching. Research supported in part by The University of Chicago Block Fund, and ARO DAAL0392 G0322. y Research supported in part by National Institutes of Health grant no. R01GM46800 x1 Introduction In recent years there has been a growing interest in deformable models for ...
Feature matching and deformation for texture synthesis
 SIGGRAPH ’04, Computer Graphics Proceedings
"... One significant problem in patchbased texture synthesis is the presence of broken features at the boundary of adjacent patches. The reason is that optimization schemes for patch merging may fail when neighborhood search cannot find satisfactory candidates in the sample texture because of an inaccur ..."
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Cited by 51 (3 self)
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One significant problem in patchbased texture synthesis is the presence of broken features at the boundary of adjacent patches. The reason is that optimization schemes for patch merging may fail when neighborhood search cannot find satisfactory candidates in the sample texture because of an inaccurate similarity measure. In this paper, we consider both curvilinear features and their deformation. We develop a novel algorithm to perform feature matching and alignment by measuring structural similarity. Our technique extracts a feature map from the sample texture, and produces both a new feature map and texture map. Texture synthesis guided by feature maps can significantly reduce the number of feature discontinuities and related artifacts, and gives rise to satisfactory results.
Geodesic Interpolating Splines
 IN ENERGY MINIMIZATION METHODS FOR COMPUTER VISION AND PATTERN RECOGNITION
, 2001
"... We propose a simple and efficient method to interpolate landmark matching by a nonambiguous mapping (a diffeomorphism). This method is based ..."
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Cited by 46 (12 self)
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We propose a simple and efficient method to interpolate landmark matching by a nonambiguous mapping (a diffeomorphism). This method is based
Matching with shape contexts
 IEEE Workshop on Contentbased access of Image and VideoLibraries
, 2000
"... Summary. 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) solving for correspondences between points on the two shapes, (2) using the correspondences to estimate an aligning tr ..."
Abstract

Cited by 44 (2 self)
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Summary. 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) solving 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 contexts, 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; regularized 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 transform. We treat recognition in a nearestneighbor classification framework as the problem of finding the stored prototype shape that is maximally similar to that in the image. We also demonstrate that shape contexts can be used to quickly prune a search for similar shapes. We present two algorithms for rapid shape retrieval: representative shape contexts, performing comparisons based on a small number of shape contexts, and shapemes, using vector quantization in the space of shape contexts to obtain prototypical shape pieces. Results are presented for silhouettes, handwritten digits and visual CAPTCHAs. 1