This proposal addresses the fundamental challenge of creating a theory that gives rise to directionally unbiased, fast processing of images and multidimensional data structures. The proposed Isotropic Multiresolution Analysis (IMRA) enables decompositions of data structures without orientational preference that parallel the numerical efficiency of one dimensional MRA-wavelet constructions. A crucial ingredient of this new theory is an MRA of radial functions based on an innovative concept of radial translations. Together with a suitable angular resolution, this allows to replicate all the beneficial characteristics of classical one-dimensional MRA-wavelets in higher dimensions. Therefore, IMRA-wavelets are expected to parallel the success of their one-dimensional predecessors in isolating edges and separating textures. Compared to previous attempts of MRA-construtions in higher dimensions, IMRAs combine symmetry properties, smoothness, and compactness of support of scaling functions and wavelets to an unprecedented degree. The proposed theory is anticipated to have a significant impact on all areas of digital signal processing in two or more dimensions, especially in biomedical image processing.
To date, digital image processing systems commonly handle data in a row and column fashion. Although this pixelized approach is natural for digital machines, it is much less natural if the objective is to extract information from natural images or more general multidimensional data structures. In this proposal, we create a mathematical theory that mimicks features of retinal processing by mammalian visual systems. Retinal processing is known to detect edges and textures at different scales of spatial and temporal resolution, regardless of their orientation and of the topology of their boundary contours. The proposed theory of Isotropic Multiresolution Analysis (IMRA) offers a way of digitizing analog signals and of synthesizing analog signals from digital data in a manner that is more compatible with the "digitization" of natural images performed by our retina. One particular component of IMRAs is the use of radial translations, inspired by the evolving wave pattern created when a rock falls in a pond of calm water. To ensure fast numerical processing capabilities, we use concepts similar to those in the framework of wavelets, which have proved computationally efficient in the processing of one- dimensional signals, e.g. audio. The intellectual merit of this work is that it delivers directionally unbiased, fast processing capabilities to all areas of digital signal processing of two or more dimensions, in particular to biomedical image processing. The Fast Isotropic Wavelet Algorithms resulting from our theory will be applied to anonymized medical patient data provided to us by the world renowned Texas Heart Institute (THI) in a joint effort for the accurate and early detection of the formation of vulnerable plaque in coronary arteries. The goal of this effort is an accurate non-invasive initial screening test to assess the risk of mycardial infarcts using CT-scans of the heart.
