Total Variation Image RestorationSOCP > SOC inequalities | Standard Forms | Group sparsity | Applications > Back | Total Variation Image Restoration | Next
The image restoration problemFunctional problem statementDigital images always contain noise. In image restoration, the problem is to filter out the noise. Early methods involved least-squares but the solutions exhibited the ‘‘ringing’ phenomenon, with spurious oscillations near edges in the restored image. To address this phenomenon, one may add to the objective of the least-squares problem a term which penalizes the variations in the image. We may represent a given (noisy) image as function from the square ![]() where the function The above is an infinite-dimensional problem, in the sense that the variable is a function, not a finite-dimensional vector. DiscretizationWe can discretize the square with a square grid, as follows: ![]() We represent the data of our problem, To approximate the gradient, we start from the first-order expansion of a function of two variables, valid for some small increment ![]() Applying this to a grid point, with the small increment set to ![]() with the convention that the terms involved are zero on the boundary (that is, if either SOCP formulationThe discretized version of our problem is thus ![]() Examples |