Estimation and regularization of probability density functions in image processing
Christian Heinemann, Jens-Rainer Ohm · RWTH Publications (RWTH Aachen) · 2014
Probability density functions (PDFs) are fundamental in a considerable part of computer vision problems. Therefore, it is important to further understand and analyse the role of probabilities for different applications in image processing. One such application analysed in this thesis is a framework showing how PDFs can represent Parameters in optical flow estimation. A novel likelihood function is developed allowing to estimate parameters in a statistically optimal way. Comparisons of the novel estimator show favourable results compared to other known methods in parameter estimation for optical flow. The advantages of using PDFs defined on a sphere are shown explicitly. A second application is the spatial regularization of PDFs in medical image denoising for so called orientation distribution function (ODF) images. A consistently derived diffusion filtering framework is presented for these ODF images which are defined on a Riemannian manifold within a representation called square root representation. Experimentally, it turned out that the new derived Riemannian diffusion methods give results very close to methods based on Euclidean metrics. Further investigations revealed that this is due to the square root representation leading to vanishing Christoffel symbols which is important for practical regularization of ODF images. Different synthetic and real data experiments verify this result and investigate the general behaviour of the novel methods. A third focus of the thesis is the modification of single PDFs, showing how to obtain estimates of distributions from noisy measurements. By this, a process is derived for sharpening single density distributions according to the Gauss Markov Theorem. This process targets reduced variance of modes, while generally retaining them in a multi modal distribution. The proposed framework is incorporated into the so called channel representation, which can be interpreted as discrete PDFs under certain conditions. Several experiments are provided on single distributions as well as synthetic images revealing improved results in reducing the variance in a desired way. Finally, it is shown how channel representations can be utilized for diffusion based denoising of gray scale images. To derive the diffusion update scheme, a novel energy functional is formulated including the channel representations. Minimizing the energy leads to a robust filtering using the reconstruction from the result of separate diffusion based smoothing in channels. Experiments are performed which show a better Performance of the channel based diffusion compared to other established diffusion based methods applied directly to gray scale images, as well as channel smoothing without diffusion.