Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems.
Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems.
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DOI：
10.1007/s10851-019-00910-2
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发表时间：
2020-01
影响因子：
2
通讯作者：
Yezzi A
中科院分区：
文献类型：
--
作者：
Benyamin M;Calder J;Sundaramoorthi G;Yezzi A
我们进一步开发了一个新的框架，称为PDE加速，通过将其应用于为一般函数定义的变分问题，获得有效的数值算法来解决基于相应加速PDE的简单离散化的优化问题。虽然所得到的偏微分方程和数值格式是相当通用的，但我们特别关注它们在正则化反演问题中的应用，并在一些流行的图像处理应用中提供了特别的示例。该方法是动量的推广，或加速，梯度下降到PDE设置。对于椭圆型问题，下降方程是一个非线性阻尼波动方程，而不是扩散方程，加速是通过将CFL条件从Δt ~ Δx2（扩散方程）改进到Δt ~ Δx（波动方程）来实现的。我们制定了几个显式和半隐式的数值格式，连同它们必要的稳定性约束，并包括递归更新公式，允许最小的努力适应现有的梯度下降PDE代码到加速的PDE框架。我们更仔细地探讨了这些方案，用于广泛的正则化反演应用，特别关注二次型、贝尔特拉米型和总变分正则化，其中加速偏微分方程采用非线性波动方程的形式。实验示例演示了这些方案在图像去噪、去模糊和上色方面的应用，包括与原始对偶、分裂Bregman和ADMM算法的比较。
We further develop a new framework, called PDE acceleration, by applying it to calculus of variation problems defined for general functions on , 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.
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影响因子：
10.6
作者：
Chan, TF;Vese, LA
通讯作者：
Vese, LA
影响因子：
4
作者：
RUDIN, LI;OSHER, S;FATEMI, E
通讯作者：
FATEMI, E
影响因子：
10.6
作者：
Sochen, N;Kimmel, R;Malladi, R
通讯作者：
Malladi, R
影响因子：
2.7
作者：
Goudou, X.;Munier, J.
通讯作者：
Munier, J.
DOI：
10.1109/34.56205
发表时间：
1990-07-01
影响因子：
23.6
作者：
PERONA, P;MALIK, J
通讯作者：
MALIK, J