Publications

Topics:
  1. A. M. Bronstein, M. M. Bronstein, R. Kimmel, Calculus of non-rigid surfaces for geometry and texture manipulation, IEEE Trans. Visualization and Computer Graphics, Vol 13(5), 2007 details

    Calculus of non-rigid surfaces for geometry and texture manipulation

    A. M. Bronstein, M. M. Bronstein, R. Kimmel
    IEEE Trans. Visualization and Computer Graphics, Vol 13(5), 2007

    We present a geometric framework for automatically finding intrinsic correspondence between three-dimensional nonrigid objects. We model object deformation as near isometries and find the correspondence as the minimum-distortion mapping. A generalization of multidimensional scaling is used as the numerical core of our approach. As a result, we obtain the possibility to manipulate the extrinsic geometry and the texture of the objects as vectors in a linear space. We demonstrate our method on the problems of expression-invariant texture mapping onto an animated three-dimensional face, expression exaggeration, morphing between faces, and virtual body painting.

    A. M. Bronstein, M. M. Bronstein, R. Kimmel, Rock, Paper, and Scissors: extrinsic vs. intrinsic similarity of non-rigid shapes, Proc. Int'l Conf. Computer Vision (ICCV), 2007 details

    Rock, Paper, and Scissors: extrinsic vs. intrinsic similarity of non-rigid shapes

    A. M. Bronstein, M. M. Bronstein, R. Kimmel
    Proc. Int'l Conf. Computer Vision (ICCV), 2007

    This paper explores similarity criteria between non-rigid shapes. Broadly speaking, such criteria are divided into intrinsic and extrinsic, the first referring to the metric structure of the objects and the latter to the geometry of the shapes in the Euclidean space. Both criteria have their advantages and disadvantages; extrinsic similarity is sensitive to non-rigid deformations of the shapes, while intrinsic similarity is sensitive to topological noise. Here, we present an approach unifying both criteria in a single distance. Numerical results demonstrate the robustness of our approach in cases where using only extrinsic or intrinsic criteria fail.

    A. M. Bronstein, M. M. Bronstein, R. Kimmel, Weighted distance maps computation on parametric three-dimensional manifolds, Journal of Computational Physics, Vol. 255(1), 2007 details

    Weighted distance maps computation on parametric three-dimensional manifolds

    A. M. Bronstein, M. M. Bronstein, R. Kimmel
    Journal of Computational Physics, Vol. 255(1), 2007

    We propose an effcient computational solver for the eikonal equations on parametric three-dimensional manifolds. Our approach is based on the fast marching method for solving the eikonal equation in O(n log n) steps by numerically simulating wavefront propagation. The obtuse angle splitting problem is reformulated as a set of small integer linear programs, that can be solved in O(n). Numerical simulations demonstrate the accuracy of the proposed algorithm.

    A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel, Paretian similarity for partial comparison of non-rigid objects, Proc. Scale Space and Variational Methods in Computer Vision (SSVM), 2007 details

    Paretian similarity for partial comparison of non-rigid objects

    A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel
    Proc. Scale Space and Variational Methods in Computer Vision (SSVM), 2007

    In this paper, we address the problem of partial comparison of non-rigid objects. We introduce a new class of set-valued distances, related to the concept of Pareto optimality in economics. Such distances allow to capture intrinsic geometric similarity between parts of non-rigid objects, obtaining semantically meaningful comparison results. The numerical implementation of our method is computationally efficient and is similar to GMDS, a multidimensional scaling-like continuous optimization problem.

    A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel, Partial similarity of objects and text sequences, Proc. Information Theory and Applications Workshop, 2007 details

    Partial similarity of objects and text sequences

    A. M. Bronstein, M. M. Bronstein, A. M. Bruckstein, R. Kimmel
    Proc. Information Theory and Applications Workshop, 2007

    Similarity is one of the most important abstract concepts in the human perception of the world. In computer vision, numerous applications deal with comparing objects observed in a scene with some a priori known patterns. Often, it happens that while two objects are not similar, they have large similar parts, that is, they are partially similar. Here, we present a novel approach to quantify this semantic definition of partial similarity using the notion of Pareto optimality. We exemplify our approach on the problems of recognizing non-rigid objects and analyzing text sequences.

    A. M. Bronstein, M. M. Bronstein, R. Kimmel, Expression-invariant representation of faces, IEEE Trans. Image Processing, Vol. 16(1), 2007 details

    Expression-invariant representation of faces

    A. M. Bronstein, M. M. Bronstein, R. Kimmel
    IEEE Trans. Image Processing, Vol. 16(1), 2007

    We present an efficient computational framework for isometry-invariant comparison of smooth surfaces. We formulate the Gromov-Hausdorff distance as a multidimensional scaling (MDS)-like continuous optimization problem. In order to construct an efficient optimization scheme, we develop a numerical tool for interpolating geodesic distances on a sampled surface from precomputed geodesic distances between the samples. For isometry-invariant comparison of surfaces in the case of partially missing data, we present the partial embedding distance, which is computed using a similar scheme. The main idea is finding a minimum-distortion mapping from one surface to another while considering only relevant geodesic distances. We discuss numerical implementation issues and present experimental results that demonstrate its accuracy and efficiency.

    A. M. Bronstein, M. M. Bronstein, R. Kimmel, Story of Cinderella: biometrics and isometry-invariant distances, Chapter in 3D Imaging for Safety and Security (A. Koschan, M. Pollefeys, M. Abidi Eds.), Springer, 2007 details

    Story of Cinderella: biometrics and isometry-invariant distances

    A. M. Bronstein, M. M. Bronstein, R. Kimmel
    Chapter in 3D Imaging for Safety and Security (A. Koschan, M. Pollefeys, M. Abidi Eds.), Springer, 2007

    In this chapter, we address the question of what are the facial measures one could use in order to distinguish between people. Our starting point is the fact that the expressions of our face can, in most cases, be modeled as isometries, which we validate empirically. Then, based on this observation, we introduce a technique that enables us to distinguish between people based on the intrinsic geometry of their faces. We provide empirical evidence that the proposed geometric measures are invariant to facial expressions and relate our findings to the broad context of biometric methods, ranging from modern face recognition technologies to fairy tales and biblical stories.

    D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel, Symmetries of non-rigid shapes, Proc. Workshop on Non-rigid Registration and Tracking through Learning (NRTL), 2007 details

    Symmetries of non-rigid shapes

    D. Raviv, A. M. Bronstein, M. M. Bronstein, R. Kimmel
    Proc. Workshop on Non-rigid Registration and Tracking through Learning (NRTL), 2007

    Symmetry and self-similarity is the cornerstone of Nature, exhibiting itself through the shapes of natural creations and ubiquitous laws of physics. Since many natural objects are symmetric, the absence of symmetry can often be an indication of some anomaly or abnormal behavior. Therefore, detection of asymmetries is important in numerous practical applications, including crystallography, medical imaging, and face recognition, to mention a few. Conversely, the assumption of underlying shape symmetry can facilitate solutions to many problems in shape reconstruction and analysis. Traditionally, symmetries are described as extrinsic geometric properties of the shape. While being adequate for rigid shapes, such a description is inappropriate for non-rigid ones. Extrinsic symmetry can be broken as a result of shape deformations, while its intrinsic symmetry is preserved. In this paper, we pose the problem of finding intrinsic symmetries of non-rigid shapes and propose an efficient method for their computation.