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

Abstract

The primal-dual hybrid gradient (PDHG) algorithm has beensuccessfully applied to a number of total variation (TV) basedimage reconstruction problems for fast numerical solutions.We show that PDHG can also effectively solvethe computational problem of image inpainting in waveletdomain, where high quality images are to be recoveredfrom incomplete wavelet coefficients due to lossydata transmission. In particular, as the original PDHG algorithmrequires the orthogonality of encoding operators for optimal performance,we propose an approximated PDHG algorithm to tacklethe non-orthogonality of Daubechies 7-9 wavelet which iswidely used in practice.We show that this approximated version essentially alters the gradientdescent direction in the original PDHG algorithm, but eliminatesits orthogonality restriction and retains low computation complexity.Moreover, we prove that the sequences generated by the approximatedPDHG algorithm always converge monotonically to an exactsolution of the TV based image reconstruction problemstarting from any initial guess. We demonstrate thatthe approximated PDHG algorithm also works on more general imagereconstruction problems with total variation regularizations,and analyze the condition on the step sizes thatguarantees the convergence.