Vision and Learning Seminar SeriesEvery Thursday at 11amVENUE: E305Coordinators: Dr.Anand Rangarajan,
Santhosh Kodipaka
Date: Thursday Oct 22, 2009
Abstract: Computational techniques adapted from classical mechanics and used in image analysis run the gamut from Lagrangian action principles to Hamilton- Jacobi field equations: witness the popularity of the fast marching and fast sweeping methods which are essentially fast Hamilton-Jacobi solvers. In sharp contrast, there are very few applications of quantum mechanics inspired computational methods. Given the fact that most of classical mechanics can be obtained as a limiting case of quantum mechanics (as Planck’s constant h tends to zero), this paucity of quantum mechanics inspired methods is surprising. In this work, we derive relationships between nonlinear Hamilton-Jacobi and linear Schrödinger equations for the Euclidean distance function problem (in 1D, 2D and 3D). We then solve the Schrödinger wave equation instead of the corresponding Hamilton- Jacobi equation. We show that the Schrödinger equation has a closed form solution and that this solution can be efficiently computed in O(N logN), N being the number of grid points. The Euclidean distance can then be recovered from the wave function. Since the wave function is computed for a small but non-zero h, the obtained Euclidean distance function is an approximation. We derive analytic bounds for the error of the approximation and experimentally compare the results of our approach with the exact Euclidean distance function on real and synthetic data.
Date: Thursday Oct 15, 2009
Abstract: This paper presents a novel and robust technique for group-wise registration of point sets with unknown correspondence. We begin by defining a Havrda-Charvát (HC) entropy valid for cumulative distribution functions (CDFs) which we dub the HC Cumulative Residual Entropy (HC-CRE). Based on this definition, we propose a new measure called the CDF-HC divergence which is used to quantify the dis-similarity between CDFs estimated from each point-set in the given population of point sets. This CDF-HC divergence generalizes the CDF based Jensen-Shannon (CDF-JS) divergence introduced earlier in the literature, but is much simpler in implementation and computationally more efficient. A closed-form formula for the analytic gradient of the cost function with respect to the nonrigid registration parameters has been derived, which is conducive for efficient quasi-Newton optimization. Our CDF-HC algorithm is especially useful for unbiased point-set atlas construction and can do so without the need to establish correspondences. Mathematical analysis and experimental results indicate that this CDF-HC registration algorithm outperforms the previous group-wise point-set registration algorithms in terms of efficiency, accuracy and robustness.
Date: Thursday Sep 17, 2009
Abstract: In this paper, we present a novel algorithm for non-rigidly registering two high angular resolution diffusion weighted MRIs (HARDI), each represented by a Gaussian mixture field (GMF). We model the non- rigid warp by a thin-plate spline and formulate the registration problem as the minimization of the L2 distance between the two given GMFs. The key mathematical contributions of this work are, (i) a closed form ex- pression for the derivatives of this objective function with respect to the parameters of the registration and (ii) a novel and simpler re-orientation scheme based on an extension to the ”Preservation of Principle Direc- tions” technique. We present results of our algorithm’s performance on several synthetic and real HARDI data sets. |
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