Showing papers by "Nir Sochen published in 2010"

Affine-invariant diffusion geometry for the analysis of deformable 3D shapes

Abstract: We introduce an (equi-)affine invariant diffusion geometry by which surfaces that go through squeeze and shear transformations can still be properly analyzed. The definition of an affine invariant metric enables us to construct an invariant Laplacian from which local and global geometric structures are extracted. Applications of the proposed framework demonstrate its power in generalizing and enriching the existing set of tools for shape analysis.

Stereo Matching with Mumford-Shah Regularization and Occlusion Handling

Abstract: This paper addresses the problem of correspondence establishment in binocular stereo vision. We suggest a novel spatially continuous approach for stereo matching based on the variational framework. The proposed method suggests a unique regularization term based on Mumford-Shah functional for discontinuity preserving, combined with a new energy functional for occlusion handling. The evaluation process is based on concurrent minimization of two coupled energy functionals, one for domain segmentation (occluded versus visible) and the other for disparity evaluation. In addition to a dense disparity map, our method also provides an estimation for the half-occlusion domain and a discontinuity function allocating the disparity/depth boundaries. Two new constraints are introduced improving the revealed discontinuity map. The experimental tests include a wide range of real data sets from the Middlebury stereo database. The results demonstrate the capability of our method in calculating an accurate disparity function with sharp discontinuities and occlusion map recovery. Significant improvements are shown compared to a recently published variational stereo approach. A comparison on the Middlebury stereo benchmark with subpixel accuracies shows that our method is currently among the top-ranked stereo matching algorithms.

Affine-invariant geodesic geometry of deformable 3D shapes

Abstract: Natural objects can be subject to various transformations yet still preserve properties that we refer to as invariants. Here, we use definitions of affine invariant arclength for surfaces in R^3 in order to extend the set of existing non-rigid shape analysis tools. In fact, we show that by re-defining the surface metric as its equi-affine version, the surface with its modified metric tensor can be treated as a canonical Euclidean object on which most classical Euclidean processing and analysis tools can be applied. The new definition of a metric is used to extend the fast marching method technique for computing geodesic distances on surfaces, where now, the distances are defined with respect to an affine invariant arclength. Applications of the proposed framework demonstrate its invariance, efficiency, and accuracy in shape analysis.

Do Uncertainty Minimizers Attain Minimal Uncertainty

Abstract: The uncertainty principle is a fundamental concept in quantum mechanics, harmonic analysis and signal and information theory. It is rooted in the framework of quantum mechanics, where it is known as the Heisenberg uncertainty principle. In general, the uncertainty principle gives a lower bound on the product of variances for any state f with respect to two self-adjoint operators: $$v_f(A)v_f(B)\ge\frac{1}{4}|e_f([A,B])|^2.$$ The functions that attain the lower bound of the inequality have been investigated extensively, and are known as uncertainty minimizers.

Piecewise smooth affine registration of point-sets with application to DT-MRI brain fiber-data

Abstract: In this paper we present a variational probabilistic approach to the registration of brain white matter tractographies extracted from DT-MRI scans. Initially, the fibers are projected into a D-dimensional feature space based on the sequence of their spatial coordinates. The alignment of two fiber-sets is considered a probability density estimation problem, where one point-set represents Gaussian Mixture Model (GMM) centroids, and the other represents the data points. The transformation parameters are represented as spatially-dependent coefficients of the same invertible affine transformation model. The alignment term of the energy-function is minimized by maximizing the likelihood of correspondence between the data-sets while the smoothness term penalizes spatial changes in the coefficient functions. The energy-function, composed of the alignment and smoothness terms, is minimized using gradient descent optimization. Results of preliminary experiments on inter-subject full-brain data show improvement over global linear (affine) registration schemes.

Anisotropic $\alpha$-Kernels and Associated Flows

Abstract: The Laplacian raised to fractional powers can be used to generate scale spaces as was shown in recent literature by Duits, Felsberg, Florack, and Platel [$\alpha$ scale spaces on a bounded domain, in Scale Space Methods in Computer Vision, L. D. Griffin and M. Lillholm, eds., Lecture Notes in Comput. Sci. 2695, Springer, Berlin, Heidelberg, 2003, pp. 494-510] and Duits, Florack, de Graaf, and ter Haar Romeny [J. Math. Imaging Vision, 20 (2004), pp. 267-298]. In this paper, we study the anisotropic diffusion processes by defining new generators that are fractional powers of an anisotropic scale space generator. This is done in a general framework that allows us to explain the relation between a differential operator that generates the flow and the generators that are constructed from its fractional powers. We then generalize this to any other function of the operator. We discuss important issues involved in the numerical implementation of this framework and present several examples of fractional versions of the Perona-Malik and Beltrami flows along with their properties.