Browsing by Author "Goldfarb, Donald"

Now showing 1 - 5 of 5
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    A Comparison of Three Total Variation Based Texture Extraction Models
    (2007-01) Yin, Wotao; Goldfarb, Donald; Osher, Stanley
    This paper qualitatively compares three recently proposed models for signal/image texture extraction based on total variation minimization:the Meyer, Vese-Osher, and TV-L1 models. We formulate discrete versions of these models as second-order cone programs (SOCPs) which can be solved efficiently by interior-point methods. Our experiments with these models on 1D oscillating signals and 2D images reveal their differences: the Meyer model tends to extract oscillation patterns in the input, the TV-L1 model performs a strict multiscale decomposition, and the Vese-Osher model has properties falling in between the other two models.
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    A Curvilinear Search Method for p-Harmonic Flows on Spheres
    (2008-01) Goldfarb, Donald; Wen, Zaiwen; Yin, Wotao
    The problem of finding p-harmonic flows arises in a wide range of applications including micromagnetics, liquid crystal theory, directional diffusion, and chromaticity denoising. In this paper, we propose an innovative curvilinear search method for minimizing p-harmonic energies over spheres. Starting from a flow (map) on the unit sphere, our method searches along a curve that lies on the sphere in a manner similar to a standard inexact line search descent method. We show that our method is globally convergent if the step length satisfies the Armijo-Wolfe conditions. Computational tests are presented to demonstrate the efficiency of the proposed method and a variant of it that uses Barzilai-Borwein steps.
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    Alternating Direction Augmented Lagrangian Methods for Semidefinite Programming
    (2009-12) Wen, Zaiwen; Goldfarb, Donald; Yin, Wotao
    We present an alternating direction method based on an augmented Lagrangian framework for solving semidefinite programming (SDP) problems in standard form. At each iteration, the algorithm, also known as a two-splitting scheme, minimizes the dual augmented Lagrangian function sequentially with respect to the Lagrange multipliers corresponding to the linear constraints, then the dual slack variables and finally the primal variables, while in each minimization keeping the other variables fixed. Convergence is proved by using a fixed-point argument. A multiple-splitting algorithm is then proposed to handle SDPs with inequality constraints and positivity constraints directly without transforming them to the equality constraints in standard form. Finally, numerical results for frequency assignment, maximum stable set and binary integer quadratic programming problems are presented to demonstrate the robustness and efficiency of our algorithm.
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    On the Convergence of an Active Set Method for L1 Minimization
    (2010-07) We, Zaiwen; Yin, Wotao; Zhang, Hongchao; Goldfarb, Donald
    We analyze an abridged version of the active-set algorithm FPC_AS for solving the L1-regularized least squares problem. The active set algorithm alternatively iterates between two stages. In the first "nonmonotone line search (NMLS)" stage, an iterative first-order method based on "shrinkage" is used to estimate the support at the solution. In the second "subspace optimization"stage, a smaller smooth problem is solved to recover the magnitudes of the nonzero components of the solution x. We show that NMLS itself is globally convergent and the convergence rate is at least R-linearly. In particular, NMLS is able to identify of the zero components of a stationary point after a finite number of steps under some mild conditions. The global convergence of FPC_AS is established based on the properties of NMLS.
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    The Total Variation Regularized L1 Model for Multiscale Decomposition
    (2006-11) Yin, Wotao; Goldfarb, Donald; Osher, Stanley
    This paper studies the total variation regularization model with an L1 fidelity term (TV-L1) for decomposing an image into features of different scales. We first show that the images produced by this model can be formed from the minimizers of a sequence of decoupled geometry subproblems. Using this result we show that the TV-L1 model is able to separate image features according to their scales, where the scale is analytically defined by the G-value. A number of other properties including the geometric and morphological invariance of the TV-L1 model are also proved and their applications discussed.