Isotropic Multiresolution Analysis in Multi-Dimensions

这一建议解决了创建一种理论的根本挑战，该理论产生了对图像和多维数据结构的定向无偏、快速处理。所提出的各向同性多分辨率分析(IMRA)能够实现无方向偏好的数据结构的分解，其数值效率与一维MRA-小波构造的数值效率相当。这一新理论的一个重要组成部分是基于径向平移的创新概念的径向函数的MRA。再加上合适的角度分辨率，这允许在更高的维度上复制经典一维MRA小波的所有有益特征。因此，IMRA小波有望在分离边缘和分离纹理方面与其一维前身的成功相媲美。与以往的高维MRA构造相比，IMRA将尺度函数和小波的对称性、光滑性和紧致性结合在一起，达到了前所未有的程度。该理论有望对二维或更多维数字信号处理的所有领域产生重大影响，特别是在生物医学图像处理领域。到目前为止，数字图像处理系统通常以行和列的方式处理数据。虽然这种像素化方法对于数字机器来说是很自然的，但如果目标是从自然图像或更一般的多维数据结构中提取信息，就不太自然了。在这项建议中，我们创建了一个数学理论，模拟哺乳动物视觉系统处理视网膜的特征。众所周知，视网膜处理可以在不同的空间和时间分辨率尺度上检测边缘和纹理，而与它们的方向和边界轮廓的拓扑无关。提出的各向同性多分辨率分析(IMRA)理论提供了一种将模拟信号数字化并从数字数据中合成模拟信号的方法，其方式与我们的视网膜执行的自然图像的数字化更兼容。IMRAS的一个特殊组成部分是使用径向平移，灵感来自于当岩石落入平静的池塘时产生的不断演变的波浪图案。为了确保快速的数值处理能力，我们使用与小波框架中的概念类似的概念，这在一维信号(例如音频)的处理中被证明在计算上是有效的。这项工作的智力价值在于，它为二维或更多维数字信号处理的所有领域提供了方向无偏的快速处理能力，特别是生物医学图像处理。根据我们的理论产生的快速各向同性小波算法将应用于世界著名的德克萨斯心脏研究所(THI)向我们提供的匿名医疗患者数据，以共同努力准确和早期检测冠状动脉中易受攻击的斑块的形成。这项工作的目标是一种准确的非侵入性初步筛查测试，以使用心脏CT扫描来评估心肌梗死的风险。

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.