Fast total variation wavelet inpaintingvia approximated primal-dual hybridgradient algorithm

Xiaojing Ye; Haomin Zhou

doi:10.3934/ipi.2013.7.1031

Article Contents

2013, Volume 7, Issue 3: 1031-1050. Doi: 10.3934/ipi.2013.7.1031

This issue Previous Article General convergent expectation maximization (EM)-type algorithms for image reconstruction Next Article The Gaussian beam method for the wigner equation with discontinuous potentials

Fast total variation wavelet inpainting via approximated primal-dual hybrid gradient algorithm

Xiaojing Ye^1, and
Haomin Zhou^1,

1.
School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30308

Received: May 2012

Revised: November 2012

Published: August 2013

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Abstract

The primal-dual hybrid gradient (PDHG) algorithm has been successfully applied to a number of total variation (TV) based image reconstruction problems for fast numerical solutions. We show that PDHG can also effectively solve the computational problem of image inpainting in wavelet domain, where high quality images are to be recovered from incomplete wavelet coefficients due to lossy data transmission. In particular, as the original PDHG algorithm requires the orthogonality of encoding operators for optimal performance, we propose an approximated PDHG algorithm to tackle the non-orthogonality of Daubechies 7-9 wavelet which is widely used in practice. We show that this approximated version essentially alters the gradient descent direction in the original PDHG algorithm, but eliminates its orthogonality restriction and retains low computation complexity. Moreover, we prove that the sequences generated by the approximated PDHG algorithm always converge monotonically to an exact solution of the TV based image reconstruction problem starting from any initial guess. We demonstrate that the approximated PDHG algorithm also works on more general image reconstruction problems with total variation regularizations, and analyze the condition on the step sizes that guarantees the convergence.

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Mathematics Subject Classification: Primary: 65K15, 49K35; Secondary: 90C25.

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