Characterizing the maximum parameter of the total-variation denoising through the pseudo-inverse of the divergence
This provides a practical tool for tuning regularization parameters in image processing, though it is incremental as it extends known concepts from Lasso to total-variation denoising.
The paper tackles the problem of determining the maximum regularization parameter for anisotropic total-variation denoising, which indicates when the solution becomes constant, and establishes a closed-form expression for the 1D case and a tight upper bound for the 2D case.
We focus on the maximum regularization parameter for anisotropic total-variation denoising. It corresponds to the minimum value of the regularization parameter above which the solution remains constant. While this value is well know for the Lasso, such a critical value has not been investigated in details for the total-variation. Though, it is of importance when tuning the regularization parameter as it allows fixing an upper-bound on the grid for which the optimal parameter is sought. We establish a closed form expression for the one-dimensional case, as well as an upper-bound for the two-dimensional case, that appears reasonably tight in practice. This problem is directly linked to the computation of the pseudo-inverse of the divergence, which can be quickly obtained by performing convolutions in the Fourier domain.