Directional Multiscale Representation Systems: Theory, Design, and Applications

In the era of information, everyday and everywhere, huge amount of information are acquired, processed, stored, and transmitted in the form of high-dimensional digital data through the Internet, TV, cell phones, and various other modern communication technologies. Efficient and sparse representation of high-dimensional digital data and fast extraction of valuable information from data becomes one of the main goals in nowadays' scientific research. It is well-known that high-dimensional data usually exhibit anisotropic phenomena due to data clustering of various types of structures. For example, a single-channel audio signal consisting of tonal and transient layers in musical audio analysis, two different structures, namely 'dendrites' (curve-like) and 'spines' (point-like), from neuron image in neurobiological imaging, many morphologically distinct objects concentrated near lower-dimensional structures such as points (stars), filaments, and sheets (nebulae) in cosmological data analysis, and so on. The anisotropic features thus encode a large portion of significant information of the data. The design of representation systems that are capable of capturing such anisotropic features is therefore undoubtedly the key for efficient and sparse representation of high-dimensional data.In this project, our main objective is the construction of novel representation systems with directionality selectivity--directional multiscale representation system--that are capable of capturing various anistropic information from high-dimensional data, e.g., signals, images, videos, etc.. Our goal is also to provide a unified framework for analysis, characterization and construction of directional multiscale representation systems with many desirable properties. Since efficient decomposition and reconstruction algorithms are crucial for the success of directional multiscale representation systems in many practical applications, we shall study the digitization theory associated with the directional multiscale representation systems for discrete data. Then, we shall design fast algorithms for forward and backward transforms with respect to the digital version of the directional multiscale representation systems based on (directional) filter banks in the spatial domain or FFTs (and its variants) in the frequency domain. Applications to image/signal processing such as image denoising shall be considered.