Smoothing of Matrix-Valued Data

I want to thank Joachim Weickert and Christoph Schnörr for their support. I hereby certify that the work reported in this diploma thesis is my own and that work performed by others is appropriately cited. Abstract During the last decade diffusion methods became more and more popular in the fields of image processing and computer vision. They are used for smoothing and regularization in cases where discontinuity preserving properties are wanted. Since the first discontinuity preserving smoothing operator has been presented, a lot of generalizations were made in order to fit new applications. One of those generalizations was the extension from scalar-valued to vector-valued data. This diploma thesis regards a further generalization towards matrix-valued data as well as its application in the field of optic flow estimation. More specifically, it deals with the following items: • The diffusion of matrix-valued data is derived from conventional diffusion methods. • It is then applied to a special matrix: the frequently used structure tensor [FG87]. Some experiments are shown and it turns out that some modifications to the original technique make the whole process for this specific application case more robust and easier to handle. After all, an extension of the conventional linear structure tensor to a nonlinear structure tensor is obtained. • This nonlinear structure tensor again opens a set of new applications. Actually it can be applied in all cases where a linear structure tensor is used. We concentrate here on optic flow estimation. The classic method of Lucas-Kanade [LK81, Luc84] as well as its spatio-temporal counterpart of Bigün et al. [BG88, BGW91] is improved by using the new nonlin-ear structure tensor. • Since Lucas-Kanade and Bigün are only special cases of another, more general optic flow estimation technique, the CLG technique [WBS01], the nonlinear structure tensor can also be applied to improve this method. This leads to a framework unifying a whole set of differential optic flow estimation techniques. In this framework all those techniques only differ in the kind of smoothing at certain processing steps. • Experiments demonstrate the improvements that can be achieved with the new technique in the field of optic flow estimation and show also the performance in comparison to other contemporary algorithms.