Novel integro-differential schemes for multiscale image representation

Title of dissertation: NOVEL INTEGRO-DIFFERENTIAL SCHEMES FOR MULTISCALE IMAGE REPRESENTATION Prashant Athavale, Doctor of Philosophy, 2009 Dissertation directed by: Professor Eitan Tadmor Department of Applied Mathematics & Statistics and Scientific Computation Multiscale representation of a given image is the problem of constructing a family of images, where each image in this family represents a scaled version of the given image. This finds its motivation from biological vision studies. Using the hierarchical multiscale image representation proposed by Tadmor et. al. [32], an image is decomposed into sums of simpler ‘slices’, which extract more refined information from the previous scales. This approach motivates us to propose a novel integro-differential equation (IDE), for a multiscale image representation. We examine various properties of this IDE. The advantage of formulating the IDE this way is that, although this IDE is motivated by variational approach, we no longer need to be associated with any minimization problem and can modify the IDE, suitable to our image processing needs. For example, we may need to find different scales in the image, while retaining or enhancing prominent edges, which may define boundaries of objects. We propose some edge preserving modifications to our IDE. One of the important problems in image processing is deblurring a blurred image. Images get blurred due to various reasons, such as unfocused camera lens, relative motion between the camera and the object pictured, etc. The blurring can be modeled with a continuous, linear operator. Recovering a clean image from a blurry image, is an ill-posed problem, which is solved using Tikhonov-like regularization. We propose a different IDE to solve the deblurring problem. We propose hierarchical multiscale scheme based on (BV,L) decomposition, proposed by Chan, Esedoḡlu, Nikolova and Alliney [12, 25, 3]. We finally propose another hierarchical multiscale representation based on a novel weighted (BV,L) decomposition. NOVEL INTEGRO-DIFFERENTIAL SCHEMES FOR MULTISCALE IMAGE REPRESENTATION by Prashant Athavale Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2009 Advisory Committee: Professor Eitan Tadmor, Chair/Advisor Professor Radu Balan Professor Joel Cohen Professor Kayo Ide Professor David Jacobs Professor Doron Levy c © Copyright by Prashant Athavale 2009