Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis(PCA for short)approach. When the information about the signa...Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis(PCA for short)approach. When the information about the signals one would like to represent is a more general property, like smoothness, a different basis should be considered. One example is the Fourier basis which is optimal for representation smooth functions sampled on regular grid. It is derived as the eigenfunctions of the circulant Laplacian operator. In this paper,based on the optimality of the eigenfunctions of the Laplace-Beltrami operator(LBO for short), the construction of PCA for geometric structures is regularized. By assuming smoothness of a given data, one could exploit the intrinsic geometric structure to regularize the construction of a basis by which the observed data is represented. The LBO can be decomposed to provide a representation space optimized for both internal structure and external observations. The proposed model takes the best from both the intrinsic and the extrinsic structures of the data and provides an optimal smooth representation of shapes and forms.展开更多
In this paper,we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local and global shape descriptors.Our construction is based on the definition of a diffu...In this paper,we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local and global shape descriptors.Our construction is based on the definition of a diffusion process on the shape manifold embedded into a high-dimensional space where the embedding coordinates represent the photometric information.Experimental results show that such data fusion is useful in coping with different challenges of shape analysis where pure geometric and pure photometric methods fail.展开更多
Detecting similarity between non-rigid shapes is one of the fundamental problems in computer vision.In order to measure the similarity the shapes must first be aligned.As opposite to rigid alignment that can be parame...Detecting similarity between non-rigid shapes is one of the fundamental problems in computer vision.In order to measure the similarity the shapes must first be aligned.As opposite to rigid alignment that can be parameterized using a small number of unknowns representing rotations,reflections and translations,non-rigid alignment is not easily parameterized.Majority of the methods addressing this problem boil down to a minimization of a certain distortion measure.The complexity of a matching process is exponential by nature,but it can be heuristically reduced to a quadratic or even linear for shapes which are smooth two-manifolds.Here we model the shapes using both local and global structures,employ these to construct a quadratic dissimilarity measure,and provide a hierarchical framework for minimizing it to obtain sparse set of corresponding points.These correspondences may serve as an initialization for dense linear correspondence search.展开更多
文摘Given a set of signals, a classical construction of an optimal truncatable basis for optimally representing the signals, is the principal component analysis(PCA for short)approach. When the information about the signals one would like to represent is a more general property, like smoothness, a different basis should be considered. One example is the Fourier basis which is optimal for representation smooth functions sampled on regular grid. It is derived as the eigenfunctions of the circulant Laplacian operator. In this paper,based on the optimality of the eigenfunctions of the Laplace-Beltrami operator(LBO for short), the construction of PCA for geometric structures is regularized. By assuming smoothness of a given data, one could exploit the intrinsic geometric structure to regularize the construction of a basis by which the observed data is represented. The LBO can be decomposed to provide a representation space optimized for both internal structure and external observations. The proposed model takes the best from both the intrinsic and the extrinsic structures of the data and provides an optimal smooth representation of shapes and forms.
基金The author would like to thank the referees for the helpful suggestionsThis work has been supported in part by the Israeli Science Foundation grant 615/11,the German-Israeli Foundation grant 2269/2010the Swiss High Performance and High Productivity Computing(HP2C)grant and grant agreement No.267414 of European Community’s FP7-ERC program.
文摘In this paper,we explore the use of the diffusion geometry framework for the fusion of geometric and photometric information in local and global shape descriptors.Our construction is based on the definition of a diffusion process on the shape manifold embedded into a high-dimensional space where the embedding coordinates represent the photometric information.Experimental results show that such data fusion is useful in coping with different challenges of shape analysis where pure geometric and pure photometric methods fail.
基金This research was supported by European Community’s FP7-ERC program,grant agreement no.267414.
文摘Detecting similarity between non-rigid shapes is one of the fundamental problems in computer vision.In order to measure the similarity the shapes must first be aligned.As opposite to rigid alignment that can be parameterized using a small number of unknowns representing rotations,reflections and translations,non-rigid alignment is not easily parameterized.Majority of the methods addressing this problem boil down to a minimization of a certain distortion measure.The complexity of a matching process is exponential by nature,but it can be heuristically reduced to a quadratic or even linear for shapes which are smooth two-manifolds.Here we model the shapes using both local and global structures,employ these to construct a quadratic dissimilarity measure,and provide a hierarchical framework for minimizing it to obtain sparse set of corresponding points.These correspondences may serve as an initialization for dense linear correspondence search.
