A new class of regularizations is proposed for the classical Perona-Malik equation of image processing, along with a modification thereof. It differs from more standard regularization procedures in that it allows one to more finely tune the regularization degree. It can be interpreted as a slight non-local weakening of the standard nonlinear diffusion coefficient. The regularized equations obtained in this way have desirable analytical properties such as local classical well-posedness or the fact that piecewise constant functions are stationary solutions. These properties are welcome from a practical image processing point of view since they lead to discretizations which perform the task of denoising remarkably well while avoiding the blurring common to standard regularization techniques and the ``stair-casing'' phenomenon known to occur for discretizations of Perona-Malik.
Partial differential equations have a long and successful history as mathematical, quantitative models for the often nontrivial dynamics of a variety of nonlinear phenomena of practical interest. Traditional application fields include all branches of physics, chemistry and some subfields of biology. More recently they have found a whole variety of new domains to which they can contribute. Among them are econometrics, molecular biology, information technology and a vast number of industrial and/or engineering problems. This proposal deals with nonlinear diffusions and their application to image processing. In the last two or three decades many mathematical models have been successfully utilized to process raw digital data in the form of images obtained by various devices of which medical ones are maybe a prime example. Often raw data needs to be processed before it can be properly interpreted, like in medical scans, sometimes one would like to be able to develop software to automatically extract useful information from raw data without human intervention, as in surveillance devices. In both cases methods based on the use of partial differential equations have proven useful. The proposer and his collaborator take on the task of furthering the theoretical understanding of complex nonlinear systems and of enhancing state-of-the-art techniques based thereupon of relevance in information technology and image processing, in particular, but that are quite general in nature, thus opening the possibility of applications to other fields, too.
