Peter Blomgren

Color TV: total variation methods for restoration of vector-valued images

We propose a new definition of the total variation (TV) norm for vector-valued functions that can be applied to restore color and other vector-valued images. The new TV norm has the desirable properties of 1) not penalizing discontinuities (edges) in the image, 2) being rotationally invariant in the image space, and 3) reducing to the usual TV norm in the scalar case. Some numerical experiments on denoising simple color images in red-green-blue (RGB) color space are presented.

Total variation image restoration: numerical methods and extensions

We describe some numerical techniques for the total variation image restoration method, namely a primal-dual linearization for the Euler-Lagrange equations and some preconditioning issues. We also highlight extension of this technique to color images, blind deconvolution and the staircasing effect.

Extensions to Total Variation denoising

The total variation denoising method, proposed by Rudin, Osher and Fatermi, 92, is a PDE-based algorithm for edge-preserving noise removal. The images resulting from its application are usually piecewise constant, possibly with a staircase effect at smooth transitions and may contain significantly less fine details than the original non-degraded image. In this paper we present some extensions to this technique that aim to improve the above drawbacks, through redefining the total variation functional or the noise constraints.

Modular solvers for image restoration problems using the discrepancy principle

Many problems in image restoration can be formulated as either an unconstrained non-linear minimization problem, usually with a Tikhonov-like regularization, where the regularization parameter has to be determined; or as a fully constrained problem, where an estimate of the noise level, either the variance or the signal-to-noise ratio, is available. The formulations are mathematically equivalent. However, in practice, it is much easier to develop algorithms for the unconstrained problem, and not always obvious how to adapt such methods to solve the corresponding constrained problem. In this paper, we present a new method which can make use of any existing convergent method for the unconstrained problem to solve the constrained one. The new method is based on a Newton iteration applied to an extended system of non-linear equations, which couples the constraint and the regularized problem, but it does not require knowledge of the Jacobian of the irregularity functional. The existing solver is only used as a black box solver, which for a fixed regularization parameter returns an improved solution to the unconstrained minimization problem given an initial guess. The new modular solver enables us to easily solve the constrained image restoration problem; the solver automatically identifies the regularization parameter, during the iterative solution process. We present some numerical results. The results indicate that even in the worst case the constrained solver requires only about twice as much work as the unconstrained one, and in some instances the constrained solver can be even faster. Copyright