Publications

Topics:
  1. Y. Choukroun, A. Shtern, A. M. Bronstein, R. Kimmel, Hamiltonian operator for spectral shape analysis, arXiv:1611.01990, 2016 details

    Hamiltonian operator for spectral shape analysis

    Y. Choukroun, A. Shtern, A. M. Bronstein, R. Kimmel
    arXiv:1611.01990, 2016

    Many shape analysis methods treat the geometry of an object as a metric space that can be captured by the Laplace-Beltrami operator. In this paper, we propose to adapt the classical Hamiltonian operator from quantum me- chanics to the field of shape analysis. To this end we study the addition of a potential function to the Laplacian as a generator for dual spaces in which shape processing is performed. We present a general optimization approach for solving variational problems involving the basis defined by the Hamilto- nian using perturbation theory for its eigenvectors. The suggested operator is shown to produce better functional spaces to operate with, as demon- strated on different shape analysis tasks.

    A. M. Bronstein, Y. Choukroun, R. Kimmel, M. Sela, Consistent discretization and minimization of the L1 norm on manifolds, Proc. 3D Vision (3DV), 2016 details

    Consistent discretization and minimization of the L1 norm on manifolds

    A. M. Bronstein, Y. Choukroun, R. Kimmel, M. Sela
    Proc. 3D Vision (3DV), 2016

    The L1 norm has been tremendously popular in signal and image processing in the past two decades due to its sparsity-promoting properties. More recently, its generalization to non-Euclidean domains has been found useful in shape analysis applications. For example, in conjunction with the minimization of the Dirichlet energy, it was shown to produce a compactly supported quasi-harmonic orthonormal basis, dubbed as compressed manifold modes. The continuous L1 norm on the manifold is often replaced by the vector l1 norm applied to sampled functions. We show that such an approach is incorrect in the sense that it does not consistently discretize the continuous norm and warn against its sensitivity to the specific sampling. We propose two alternative discretizations resulting in an iteratively-reweighed l2 norm. We demonstrate the proposed strategy on the compressed modes problem, which reduces to a sequence of simple eigendecomposition problems not requiring non-convex optimization on Stiefel manifolds and producing more stable and accurate results.

    R. Litman, A. M. Bronstein, SpectroMeter: Amortized sublinear spectral approximation of distance on graphs, Proc. 3D Vision (3DV), 2016 details

    SpectroMeter: Amortized sublinear spectral approximation of distance on graphs

    R. Litman, A. M. Bronstein
    Proc. 3D Vision (3DV), 2016

    We present a method to approximate pairwise distance on a graph, having an amortized sub-linear complexity in its size. The proposed method follows the so-called heat method due to Crane et al. The only additional input is the values of the eigenfunctions of the graph Laplacian at a subset of the vertices. Using these values we estimate a random walk from the source points, and normalize the result into a unit gradient function. The eigenfunctions are then used to synthesize distance values abiding by these constraints at desired locations. We show that this method works in practice on different types of inputs ranging from triangular meshes to general graphs. We also demonstrate that the resulting approximate distance is accurate enough to be used as the input to a recent method for intrinsic shape correspondence computation.

    T. Remez, O. Litany, S. Yoseff, H. Haim, A. M. Bronstein, FPGA system for real-time computational extended depth of field imaging using phase aperture coding, arXiv:1608.01074 details

    FPGA system for real-time computational extended depth of field imaging using phase aperture coding

    T. Remez, O. Litany, S. Yoseff, H. Haim, A. M. Bronstein
    arXiv:1608.01074

    We present a proof-of-concept end-to-end system for computational extended depth of field (EDOF) imaging. The acquisition is performed through a phase-coded aperture implemented by placing a thin wavelength-dependent op- tical mask inside the pupil of a conventional camera lens, as a result of which, each color channel is focused at a different depth. The reconstruction process re- ceives the raw Bayer image as the input, and performs blind estimation of the output color image in focus at an extended range of depths using a patch-wise sparse prior. We present a fast non-iterative reconstruction algorithm operating with constant latency in fixed-point arithmetics and achieving real-time perfor- mance in a prototype FPGA implementation. The output of the system, on simu- lated and real-life scenes, is qualitatively and quantitatively better than the result of clear-aperture imaging followed by state-of-the-art blind deblurring.

    R. Giryes, G. Sapiro, A. M. Bronstein, Deep neural networks with random Gaussian weights: A universal classification strategy?, IEEE Trans. Signal Processing, Vol. 64(13), 2016 details

    Deep neural networks with random Gaussian weights: A universal classification strategy?

    R. Giryes, G. Sapiro, A. M. Bronstein
    IEEE Trans. Signal Processing, Vol. 64(13), 2016

    Three important properties of a classification machinery are: (i) the system preserves the important information of the input data; (ii) the training examples convey information for unseen data; and (iii) the system is able to treat differently points from different classes. In this work, we show that these fundamental properties are inherited by the architecture of deep neural networks. We formally prove that these networks with random Gaussian weights perform a distance-preserving embedding of the data, with a special treatment for in-class and out-of-class data. Similar points at the input of the network are likely to have the same The theoretical analysis of deep networks here presented exploits tools used in the compressed sensing and dictionary learning literature, thereby making a formal connection between these important topics. The derived results allow drawing conclusions on the metric learning properties of the network and their relation to its structure; and provide bounds on the required size of the training set such that the training examples would represent faithfully the unseen data. The results are validated with state-of-the-art trained networks.

    O. Litany, E. Rodolà, A. M. Bronstein, M. M. Bronstein, D. Cremers, Non-rigid puzzles, Computer Graphics Forum, Vol. 35(5), 2016 (SGP Best Paper Award) details

    Non-rigid puzzles

    O. Litany, E. Rodolà, A. M. Bronstein, M. M. Bronstein, D. Cremers
    Computer Graphics Forum, Vol. 35(5), 2016 (SGP Best Paper Award)

    Shape correspondence is a fundamental problem in computer graphics and vision, with applications in various problems including animation, texture mapping, robotic vision, medical imaging, archaeology and many more. In settings where the shapes are allowed to undergo non-rigid deformations and only partial views are available, the problem becomes very challenging. To this end, we present a non-rigid multi-part shape matching algorithm. We assume to be given a reference shape and its multiple parts undergoing a non-rigid deformation. Each of these query parts can be additionally contaminated by clutter, may overlap with other parts, and there might be missing parts or redundant ones. Our method simultaneously solves for the segmentation of the reference model, and for a dense correspondence to (subsets of) the parts. Experimental results on synthetic as well as real scans demonstrate the effectiveness of our method in dealing with this challenging matching scenario.

    X. Bian, H. Krim, A. M. Bronstein, L. Dai, Sparsity and nullity: paradigms for analysis dictionary learning, SIAM J. Imaging Sci., Vol. 9(3), 2016 details

    Sparsity and nullity: paradigms for analysis dictionary learning

    X. Bian, H. Krim, A. M. Bronstein, L. Dai
    SIAM J. Imaging Sci., Vol. 9(3), 2016

    Sparse models in dictionary learning have been successfully applied in a wide variety of machine learning and computer vision problems, and as a result, have recently attracted increased research interest. Another interesting related problem based on linear equality constraints, namely the sparse null space (SNS) problem, first appeared in 1986 and has since inspired results on sparse basis pursuit. In this paper, we investigate the relation between the SNS problem and the analysis dictionary learning (ADL) problem, and show that the SNS problem plays a central role, and may be utilized to solve dictionary learning problems. Moreover, we propose an efficient algorithm of sparse null space basis pursuit (SNS-BP) and extend it to a solution of ADL. Experimental results on numerical synthetic data and real-world data are further presented to validate the performance of our method.

    D. Pickup, X. Sun, P. L. Rosin, R. R. Martin, Z. Cheng, Z. Lian, M. Aono, A. Ben Hamza, A. M. Bronstein, M. M. Bronstein, S. Bu, U. Castellani, S. Cheng, V. Garro, A. Giachetti, A. Godil, J. Han, H. Johan, L. Lai, B. Li, C. Li, H. Li, R. Litman, X. Liu, Z. Liu, Y. Lu, A. Tatsuma, J. Ye, Shape retrieval of non-rigid 3D human models, Intl. Journal of Computer Vision (IJCV), 2016 details

    Shape retrieval of non-rigid 3D human models

    D. Pickup, X. Sun, P. L. Rosin, R. R. Martin, Z. Cheng, Z. Lian, M. Aono, A. Ben Hamza, A. M. Bronstein, M. M. Bronstein, S. Bu, U. Castellani, S. Cheng, V. Garro, A. Giachetti, A. Godil, J. Han, H. Johan, L. Lai, B. Li, C. Li, H. Li, R. Litman, X. Liu, Z. Liu, Y. Lu, A. Tatsuma, J. Ye
    Intl. Journal of Computer Vision (IJCV), 2016

    3D models of humans are commonly used within computer graphics and vision, and so the ability to distinguish between body shapes is an important shape retrieval problem. We extend our recent paper which provided a benchmark for testing non-rigid 3D shape retrieval algorithms on 3D human models. This benchmark provided a far stricter challenge than previous shape benchmarks.We have added 145 new models for use as a separate training set, in order to standardise the training data used and provide a fairer comparison. We have also included experiments with the FAUST dataset of human scans. All participants of the previous benchmark study have taken part in the new tests reported here, many providing updated results using the new data. In addition, further participants have also taken part, and we provide extra analysis of the retrieval results. A total of 25 different shape retrieval methods are compared.