Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diiusion in Image Processing

In image processing, it is often desirable to remove noise, smooth or sharpen image features, or to otherwise enhance the image. Total Variation (TV) based regularization is a model case of geometry-driven diiusion for image processing. In our papers 14] and 15], we analyze the precise eeects of TV based regularization by analytically nding exact solutions to the TV regularization problem. In this paper, we used these results to develop adaptive TV regularization schemes, as well as a scheme for automatic scale recognition. We develop our results by considering the unconstrained (Tikhonov) formulation of the TV regularization problem, in which a regularization parameter must be chosen to determine the balance between the goodness of t to the original (e.g. measured) data and the amount of regularization to be done to the image. We rst discuss how to choose the regularization parameter based on the local features of objects in the image by using the theory presented in 15]. We then discuss how to choose the regularization parameter based on the noise level in the image, and we nally discuss how to develop adaptive regularization schemes which are driven by both features and noise level in the image. We discuss several heuristics and give two schemes for doing adaptive regularization based on our results, which use a priori information to manually deene a spatially varying regularization parameter. Using the same theory, we also give a simple scheme for local scale recognition. Our results are equally applicable to the constrained formulation of the TV regularization problem, as well as to other geometry-driven diiusion schemes, as these other schemes can be thought of as variations or modiications of the model TV regularization scheme.