Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems

Minas Benyamin, Jeff Calder, Ganesh Sundaramoorthi, Anthony Yezzi

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

We further develop a new framework, called PDE acceleration, by applying it to calculus of variation problems defined for general functions on Rn, obtaining efficient numerical algorithms to solve the resulting class of optimization problems based on simple discretizations of their corresponding accelerated PDEs. While the resulting family of PDEs and numerical schemes are quite general, we give special attention to their application for regularized inversion problems, with particular illustrative examples on some popular image processing applications. The method is a generalization of momentum, or accelerated, gradient descent to the PDE setting. For elliptic problems, the descent equations are a nonlinear damped wave equation, instead of a diffusion equation, and the acceleration is realized as an improvement in the CFL condition from Δt∼ Δx2 (for diffusion) to Δt∼ Δx (for wave equations). We work out several explicit as well as a semi-implicit numerical scheme, together with their necessary stability constraints, and include recursive update formulations which allow minimal-effort adaptation of existing gradient descent PDE codes into the accelerated PDE framework. We explore these schemes more carefully for a broad class of regularized inversion applications, with special attention to quadratic, Beltrami, and total variation regularization, where the accelerated PDE takes the form of a nonlinear wave equation. Experimental examples demonstrate the application of these schemes for image denoising, deblurring, and inpainting, including comparisons against primal–dual, split Bregman, and ADMM algorithms.

Original languageEnglish (US)
Pages (from-to)10-36
Number of pages27
JournalJournal of Mathematical Imaging and Vision
Volume62
Issue number1
DOIs
StatePublished - Jan 1 2020

Keywords

  • Accelerated gradient descent
  • Beltrami regularization
  • Image deblurring
  • Image denoising
  • Image restoration
  • Nesterov acceleration
  • Nonlinear wave equations
  • PDE acceleration
  • Total variation

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