Adaptive total variation minimizing image restoration

We analyze the exact effects of total variation (TV) minimizing function regularization in {dollar}Rsp1, Rsp2{dollar} and {dollar}Rsp3.{dollar} Our more precise understanding of TV regularization enables us to construct more effective TV minimizing image restoration schemes, as well as to better understand what types of images (and image degradation) are most effectively improved by TV restoration. We analytically find exact solutions to the nonlinear TV minimizing function regularization problem for simple but important cases, which can be used to better understand the effects of TV regularization for more general cases.; We give formulae that describe qualitatively and quantitatively the effects of TV regularization. Four important results which we prove are: (1) TV regularization of piecewise constant (noise-free or noisy) radially symmetric functions results in piecewise constant functions, with edge location being preserved exactly; (2) function intensity change is inversely proportional to local feature scale, is independent of original intensity, and is directly proportional to the regularization parameter; (3) for smooth function features, function intensity change is inversely proportional to radial position and directly proportional to the regularization parameter; and (4) TV regularization is local in a certain sense.; We develop two adaptive TV image restoration schemes. Both schemes are motivated by and constructed using our theoretical results. In the first scheme we accomplish adaptivity by locally weighting the measure of the total variation of the image. The weighting factor decreases as the relatively likelihood of the presence of an edge in the image increases. The second adaptive image restoration scheme is a multi-step scheme driven by the scale of individual image features. Each step involves selectively applying restoration, only where the scale of the image features is smaller than a user-controlled threshhold, {dollar}scalesb{lcub}thresh{rcub}.{dollar} The end result is an image comprised of features with scale greater than {dollar}scalesb{lcub}thresh{rcub}.{dollar} The process can be characterized as a scale-sensitive, anisotropic diffusion process. These two adaptive schemes can be viewed as prototypes of an array of adaptive TV minimizing image restoration schemes which may be developed in the future as a result of our theoretical results.