Shape description and matching using integral invariants on eccentricity transformed images

Matching occluded and noisy shapes is a problem frequently encountered in medical image analysis and more generally in computer vision. To keep track of changes inside the breast, for example, it is important for a computer aided detection system to establish correspondences between regions of interest. Shape transformations, computed both with integral invariants (II) and with geodesic distance, yield signatures that are invariant to isometric deformations, such as bending and articulations. Integral invariants describe the boundaries of planar shapes. However, they provide no information about where a particular feature lies on the boundary with regard to the overall shape structure. Conversely, eccentricity transforms (Ecc) can match shapes by signatures of geodesic distance histograms based on information from inside the shape; but they ignore the boundary information. We describe a method that combines the boundary signature of a shape obtained from II and structural information from the Ecc to yield results that improve on them separately

Int J Comput Vis DOI 10.1007/s11263-007-0078-4 Analysis of Two-Dimensional Non-Rigid Shapes

Abstract Analysis of deformable two-dimensional shapes is an important problem, encountered in numerous pattern recognition, computer vision and computer graphics applications. In this paper, we address three major problems in the analysis of non-rigid shapes: similarity, partial similarity, and correspondence. We present an axiomatic construction of similarity criteria for deformation-invariant shape comparison, based on intrinsic geometric properties of the shapes, and show that such criteria are related to the Gromov-Hausdorff distance. Next, we extend the problem of similarity computation to shapes which have similar parts but are dissimilar when considered as a whole, and present a construction of set-valued distances, based on the notion of Pareto optimality. Finally, we show that the correspondence between non-rigid shapes can be obtained as a byproduct of the non-rigid similarity problem. As a numerical framework, we use the generalized multidimensional scaling (GMDS) method, which is the numerical core of the three problems addressed in this paper. Keywords Non-rigid shapes · Partial similarity · Pareto optimum · Multidimensional scaling · GMDS · Gromov-Hausdorff distance · Intrinsic geometr

Multigrid multidimensional scaling

Multidimensional scaling (MDS) is a generic name for a family of algorithms that construct a conguration of points in a target metric space from information about inter-point distances measured in some other metric space. Large-scale MDS problems often occur in data analysis, representation and visualization. Solving such problems e ciently is of key importance in many applications. In this paper we present a multigrid framework for MDS problems. We demonstrate the performance of our algorithm on dimensionality reduction and isometric embedding problems, two classical problems requiring e cient large-scale MDS. Simulation results show that the proposed approach signi cantly outperforms conventional MDS algorithms. Copyright? 2006 John Wiley & Sons, Ltd. KEY WORDS: multigrid; multiresolution; multidimensional scaling; isometric embedding; SMACOF; BFGS; face recognition; bending-invariant canonical form; dimensionality reductio

Multigrid Multidimensional Scaling

... In this paper we present a multigrid framework for MDS problems. We demonstrate the performance of our algorithm on dimensionality reduction and isometric embedding problems, two classical problems requiring efficient large-scale MDS. Simulation results show that the proposed approach significantly outperforms conventional MDS algorithms