| # Total Variation Image Restoration SOCP > SOC inequalities | Standard Forms | Group sparsity | Applications > Back | Total Variation Image Restoration | Next The image restoration problem Discretization SOCP formulation ## The image restoration problem ### Functional problem statement Digital 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 to . We define the image restoration problem as minimizing, over functions , the objective where the function is our estimate. The first term penalizes functions which exhibit large variations, while the second term accounts for the distance from the estimate to the noisy image, . The above is an infinite-dimensional problem, in the sense that the variable is a function, not a finite-dimensional vector. ## Discretization We can discretize the square with a square grid, as follows: We represent the data of our problem, , as a matrix , with elements . Similarly, we represent the variable of our problem with a matrix , which contains the values of at the grid points . 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 , we obtain that the gradient of at a grid point can be approximated as with the convention that the terms involved are zero on the boundary (that is, if either or is ). ## SOCP formulation The discretized version of our problem is thus ## Examples