Damped newtons method for image restoration

As in image enhancement, the ultimate goal of restoration techniques is to im-prove an image in some sense. For the purpose of differentiation, we consider restoration to be a process that attempts to reconstruct or recover an image that has been degraded by using some a priori knowledge of the degradation phenom-enon. Thus restoration techniques are oriented toward modeling the image. This approach usually involves formulating a criterion of goodness that will yield some optimal estimate of the desired result. By contrast, enhancement techniques ba-sically are heuristic procedures designed to manipulate an image in order to take advantage of the psychophysical aspects it might present to the viewer, whereas removal of image blur by applying a deblurring function is considered a restoration technique([6], [9]). The problem of denoising, or estimating an underlying function from error con-taminated observations, occurs in a number of important applications, particularly in probability density estimation and image reconstruction. Consider the model equation u0 = u+ n(1.1) where u represents the desired true solution, n represents error, and u0 represents the observed data. A number of approaches can be taken to estimate u. These include selective smoothing (see [1]), filtering using Fourier and wavelet transforms, level set method, and Total variation (TV) based denoising. Suffice it to say that TV denoising is extremely effective for recovering ”block”, possibly discontinuous, functions from noise data. It is the goal of this paper to present a new algorithm for TV denoising and introduce some existing TV-based methods. From 1.1 we consider the minimization of the TV-penalized least squares func-tional f(u) = α