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

The Kadison-Singer Problem in Mathematics and Engineering

We will show that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer

Day-Ahead Crude Oil Price Forecasting Using a Novel Morphological Component Analysis Based Model

As a typical nonlinear and dynamic system, the crude oil price movement is difficult to predict and its accurate forecasting remains the subject of intense research activity. Recent empirical evidence suggests that the multiscale data characteristics in the price movement are another important stylized fact. The incorporation of mixture of data characteristics in the time scale domain during the modelling process can lead to significant performance improvement. This paper proposes a novel morphological component analysis based hybrid methodology for modeling the multiscale heterogeneous characteristics of the price movement in the crude oil markets. Empirical studies in two representative benchmark crude oil markets reveal the existence of multiscale heterogeneous microdata structure. The significant performance improvement of the proposed algorithm incorporating the heterogeneous data characteristics, against benchmark random walk, ARMA, and SVR models, is also attributed to the innovative methodology proposed to incorporate this important stylized fact during the modelling process. Meanwhile, work in this paper offers additional insights into the heterogeneous market microstructure with economic viable interpretations

Applications of nonlinear approximation for problems in learning theory and applied mathematics

A major pillar of approximation theory in establishing the ability of one class of functions to be represented by another. Establishing such a relationship often leads to efficient numerical approximation methods. In this work, several expressibility theorems are established and several novel numerical approximation techniques are also presented. Not only are these novel methods supported by the presented theory, but also, provided numerical experiments show that these novel methods may be applied to a wide range of applications from image compression to the solutions of high-dimensional PDE