Efficient Threshold Selection for Multivariate Total Variation Denoising
Sylvain Sardy, Hatef Monajemi · Journal of Computational and Graphical Statistics · 2018
Total variation (TV) denoising is still attracting attention with theoretical and computational motivations, for its conceptual simplicity of solving a lasso-like convex problem and its good properties for preserving sharp edges and contours in objects with spatial structures like natural images, although more modern and recent techniques specifically tailored to image processing have been developed. TV induces variation-sparsity in the sense that the reconstruction is piecewise constant with a small number of jumps. A threshold parameter λ controls the number of jumps and the quality of the estimation. Since calculation of the TV estimate in high dimension is computationally intensive for a given λ, we propose to calculate the TV estimate for only two sequential λ’s. Our adaptive procedure is based on large deviation of stochastic processes and extreme value theory. We also show that TV can perform exact segmentation in dimension one, under an alternating sign condition for some prescribed threshold. We apply our procedure to denoise a collection of 1D and 2D test signals verifying empirically the effectiveness of our approach. Codes are given to reproduce our results in a provided PURL.