A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity

The richness of natural images makes the quest for optimal representations in image processing and computer vision challenging. The latter observation has not prevented the design of image representations, which trade off between efficiency and complexity, while achieving accurate rendering of smooth regions as well as reproducing faithful contours and textures. The most recent ones, proposed in the past decade, share an hybrid heritage highlighting the multiscale and oriented nature of edges and patterns in images. This paper presents a panorama of the aforementioned literature on decompositions in multiscale, multi-orientation bases or dictionaries. They typically exhibit redundancy to improve sparsity in the transformed domain and sometimes its invariance with respect to simple geometric deformations (translation, rotation). Oriented multiscale dictionaries extend traditional wavelet processing and may offer rotation invariance. Highly redundant dictionaries require specific algorithms to simplify the search for an efficient (sparse) representation. We also discuss the extension of multiscale geometric decompositions to non-Euclidean domains such as the sphere or arbitrary meshed surfaces. The etymology of panorama suggests an overview, based on a choice of partially overlapping "pictures". We hope that this paper will contribute to the appreciation and apprehension of a stream of current research directions in image understanding.Comment: 65 pages, 33 figures, 303 reference

Low Dimensional Band-Limited Framelets and Their Applications in Colour Image Restoration

Ph.DDOCTOR OF PHILOSOPH

Bayesian Methods for Metabolomics

Metabolomics, the large-scale study of small molecules, enables the underlying biochemical activity and state of cells or tissues to be directly captured. Nuclear Magnetic Resonance (NMR) Spectroscopy is one of the major data capturing tech- niques for metabolomics, as it provides highly reproducible, quantitative informa- tion on a wide variety of metabolites. This work presents possible solutions for three problems involved to aid the development of better algorithms for NMR data analy- sis. After reviewing relevant concepts and literature, we first utilise observed NMR chemical shift titration data for a range of urinary metabolites and develop a the- oretical model of chemical shift using a Bayesian statistical framework and model selection procedures to estimate the number of protonation sites, a key parameter to model the relationship between chemical shift variation and pH and usually un- known in uncatalogued metabolites. Secondly, with the aim of obtaining explicit concentration estimates for metabolites from NMR spectra, we discuss a Monte Carlo Co-ordinate Ascent Variational Inference (MC-CAVI) algorithm that com- bines Markov chain Monte Carlo (MCMC) methods with Co-ordinate Ascent VI (CAVI), demonstrate MC-CAVI’s suitability for models with hard constraints and compare MC-CAVI’s performance with that of MCMC in an important complex model used in NMR spectroscopy data analysis. The third distribution seeks to im- prove metabolite identification, one of the biggest bottlenecks in metabolomics and severely hindered by resonance overlapping in one-dimensional NMR spectroscopy. In particular, we present a novel Bayesian method for widely used two-dimensional (2D) 1H J-resolved (JRES) NMR spectroscopy, which has considerable potential to accurately identify and quantify metabolites within complex biological samples, through combining B-spline tight wavelet frames with theoretical templates. We then demonstrate the effectiveness of our approach via analyses of JRES datasets from serum and urine