Directional Multiresolution Image Representations

Efficient representation of visual information lies at the foundation of many image processing tasks, including compression, filtering, and feature extraction. Efficiency of a representation refers to the ability to capture significant information of an object of interest in a small description. For practical applications, this representation has to be realized by structured transforms and fast algorithms. Recently, it has become evident that commonly used separable transforms (such as wavelets) are not necessarily best suited for images. Thus, there is a strong motivation to search for more powerful schemes that can capture the intrinsic geometrical structure of pictorial information. This thesis focuses on the development of new “true” two-dimensional representations for images. The emphasis is on the discrete framework that can lead to algorithmic implementations. The first method constructs multiresolution, local and directional image expansions by using non-separable filter banks. This discrete transform is developed in connection with the continuous-space curvelet construction in harmonic analysis. As a result, the proposed transform provides an efficient representation for two-dimensional piecewise smooth signals that resemble images. The link between the developed filter banks and the continuous-space constructions is set up in a newly defined directional multiresolution analysis. The second method constructs a new family of block directional and orthonormal transforms based on the ridgelet idea, and thus offers an efficient representation for images that are smooth away from straight edges. Finally, directional multiresolution image representations are employed together with statistical modeling, leading to powerful texture models and successful image retrieval systems.