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This paper presents an abstract analysis of bounded variation (BV) methods for ill-posed operator equations Au = z. Let T(u) def = kAu ? zk 2 + J(u); where the penalty, or \regularization", parameter > 0 and the functional J(u) is the BV norm or seminorm of u, also known as the total variation of u. Under mild restrictions on the operator A and the functional J(u), it is shown that the functional T(u) has a unique minimizer which is stable with respect to certain perturbations in the data z, the operator A, the parameter , and the functional J(u). In addition, convergence results are obtained which apply when these perturbations vanish and the regularization parameter is chosen appropriately.

Total Variation (TV) methods are very e ective for recovering \blocky", possibly discontinuous, images from noisy data. A xed point algorithm for minimizing a TV-penalized least squares functional is presented and compared with existing minimization schemes. A variant of the cell-centered nite di erence multigrid method of Ewing and Shen is implemented for solving the (large, sparse) linear subproblems. Numerical results are presented for one-and two-dimensional examples; in particular, the algorithm is applied to actual data obtained from confocal microscopy.

Tikhonov regularization with a modified total variation regularization functional is used to recover an image from noisy, blurred data. This approach is appropriate for image processing in that it does not place a priori smoothness conditions on the solution image. An efficient algorithm is presented for the discretized problem that combines a fixed point iteration to handle nonlinearity with a new, effective preconditioned conjugate gradient iteration for large linear systems. Reconstructions, convergence results, and a direct comparison with a fast linear solver are presented for a satellite image reconstruction application.

In total variation denoising, one attempts to remove noise from a signal or image by solving a nonlinear minimization problem involving a total variation criterion. Several approaches based on this idea have recently been shown to be very effective, particularly for denoising functions with discontinuities. This paper analyzes the convergence of an iterative method for solving such problems. The iterative method involves a "lagged diffusivity" approach in which a sequence of linear diffusion problems are solved. Global convergence in a finite-dimensional setting is established, and local convergence properties, including rates and their dependence on various parameters, are examined.