Noise Reduction in Images: Some Recent Edge-Preserving Methods

We introduce some recent and very recent smoothing methods which focus on the preservation of boundaries, spikes and canyons in presence of noise. We try to point out basic principles they have in common; the most important one is the robustness aspect. It is reflected by the use of `cup functions' in the statistical loss functions instead of squares; such cup functions were introduced early in robust statistics to down weight outliers. Basically, they are variants of truncated squares. We discuss all the methods in the common framework of `energy functions', i.e we associate to (most of) the algorithms a `loss function' in such a fashion that the output of the algorithm or the `estimate' is a global or local minimum of this loss function. The third aspect we pursue is the correspondence between loss functions and their local minima and nonlinear filters. We shall argue that the nonlinear filters can be interpreted as variants of gradient descent on the loss functions. This way we can show that some (robust) M-estimators and some nonlinear filters produce almost the same result

Image compression with anisotropic diffusion

Compression is an important field of digital image processing where well-engineered methods with high performance exist. Partial differential equations (PDEs), however, have not much been explored in this context so far. In our paper we introduce a novel framework for image compression that makes use of the interpolation qualities of edge-enhancing diffusion. Although this anisotropic diffusion equation with a diffusion tensor was originally proposed for image denoising, we show that it outperforms many other PDEs when sparse scattered data must be interpolated. To exploit this property for image compression, we consider an adaptive triangulation method for removing less significant pixels from the image. The remaining points serve as scattered interpolation data for the diffusion process. They can be coded in a compact way that reflects the B-tree structure of the triangulation. We supplement the coding step with a number of amendments such as error threshold adaptation, diffusion-based point selection, and specific quantisation strategies. Our experiments illustrate the usefulness of each of these modifications. They demonstrate that for high compression rates, our PDE-based approach does not only give far better results than the widely-used JPEG standard, but can even come close to the quality of the highly optimised JPEG2000 codec

Understanding and advancing PDE-based image compression

This thesis is dedicated to image compression with partial differential equations (PDEs). PDE-based codecs store only a small amount of image points and propagate their information into the unknown image areas during the decompression step. For certain classes of images, PDE-based compression can already outperform the current quasi-standard, JPEG2000. However, the reasons for this success are not yet fully understood, and PDE-based compression is still in a proof-of-concept stage. With a probabilistic justification for anisotropic diffusion, we contribute to a deeper insight into design principles for PDE-based codecs. Moreover, by analysing the interaction between efficient storage methods and image reconstruction with diffusion, we can rank PDEs according to their practical value in compression. Based on these observations, we advance PDE-based compression towards practical viability: First, we present a new hybrid codec that combines PDE- and patch-based interpolation to deal with highly textured images. Furthermore, a new video player demonstrates the real-time capacities of PDE-based image interpolation and a new region of interest coding algorithm represents important image areas with high accuracy. Finally, we propose a new framework for diffusion-based image colourisation that we use to build an efficient codec for colour images. Experiments on real world image databases show that our new method is qualitatively competitive to current state-of-the-art codecs.Diese Dissertation ist der Bildkompression mit partiellen Differentialgleichungen (PDEs, partial differential equations) gewidmet. PDE-Codecs speichern nur einen geringen Anteil aller Bildpunkte und transportieren deren Information in fehlende Bildregionen. In einigen Fällen kann PDE-basierte Kompression den aktuellen Quasi-Standard, JPEG2000, bereits schlagen. Allerdings sind die Gründe für diesen Erfolg noch nicht vollständig erforscht, und PDE-basierte Kompression befindet sich derzeit noch im Anfangsstadium. Wir tragen durch eine probabilistische Rechtfertigung anisotroper Diffusion zu einem tieferen Verständnis PDE-basierten Codec-Designs bei. Eine Analyse der Interaktion zwischen effizienten Speicherverfahren und Bildrekonstruktion erlaubt es uns, PDEs nach ihrem Nutzen für die Kompression zu beurteilen. Anhand dieser Einsichten entwickeln wir PDE-basierte Kompression hinsichtlich ihrer praktischen Nutzbarkeit weiter: Wir stellen einen Hybrid-Codec für hochtexturierte Bilder vor, der umgebungsbasierte Interpolation mit PDEs kombiniert. Ein neuer Video-Dekodierer demonstriert die Echtzeitfähigkeit PDE-basierter Interpolation und eine Region-of-Interest-Methode erlaubt es, wichtige Bildbereiche mit hoher Genauigkeit zu speichern. Schlussendlich stellen wir ein neues diffusionsbasiertes Kolorierungsverfahren vor, welches uns effiziente Kompression von Farbbildern ermöglicht. Experimente auf Realwelt-Bilddatenbanken zeigen die Konkurrenzfähigkeit dieses Verfahrens auf