A Mathematical Framework for Tensor Image Processing

研究者开发了一个数学框架，用于表示离散张量图像，适合后处理应用，如模式识别，注册和几何变换。特别地，他构造了由连续张量场组成的原子Wiener amalgam张量空间，并在生成张量上建立了保证连续空间的任意正则抽样得到的离散张量场空间是原子的且与连续空间同构的条件。利用原子空间中的近似问题与信号和图像处理中的滤波范式之间的联系，他开发并实现了各种基本处理算子的快速滤波算法，如降噪、旋转、平移和一般仿射变换。由于张量图像数据具有关于材料、组织或器官的纤维结构和几何的空间信息，他还使用微分几何的结果来提取有序介质的不同建筑特征。项目的一部分用于测试这些算法的准确性、精密度和速度。为此，他生成了合成数据集，并使用从体内临床弥散张量MRI研究和其他成像方式获得的真实数据来评估这些算法的性能。这个项目的动机主要是需要处理和分析从扩散张量MRI获得的临床数据，扩散张量MRI是一种新的无创成像方式，允许医生可视化体内的神经和肌肉纤维束。然而，这里发展的数学适用于处理和分析从不同应用领域使用的大量成像设备和模式获得的数据，包括医学、材料科学、海洋学、气象学、流体力学、卫星侦察和天文学。许多新的扫描系统在图像的每个点或位置测量多个量，而不是单个量。这些数字列表可以表示重要的物理量，例如速度或位移，或者在特定位置以不同波长吸收、反射或发射的光量。这里正在发展的理论提供了一种合理的方法来表示、处理、分析和压缩这些数据——这是目前没有理论解决的问题。此外，这项工作旨在改善这些新型数据集测量中固有的许多问题。特别是，成像数据通常被噪声破坏，是离散的而不是连续的，并且是空间平均的。最后，由于这些新的成像模式可以产生大量的数据，研究人员用于表示、处理、分析和压缩数据的算法也必须快速有效。

Aldroubi9805483The investigator develops a mathematical framework for representing discrete tensor images that is suited to post-processing applications, such as pattern recognition, registration and geometric transformations. In particular, he constructs atomic Wiener amalgam tensor spaces that consist of continuous tensor fields, and establishes conditions on the generating tensors that guarantee that the discrete tensor field spaces obtained by any regular sampling of the continuous spaces are atomic and are isomorphic to the continuous spaces. Using the connection between the approximation problems in atomic spaces and the filtering paradigm in signal and image processing, he develops and implements fast filtering algorithms for various fundamental processing operators, such as noise reduction, rotation, translation, and general affine transformations. Because tensor image data possess spatial information about the fiber structure and geometry of materials, tissues, or organs, he also uses results from differential geometry to extract different architectural features of ordered media. Part of the project is devoted to testing the accuracy, precision and speed of these algorithms. For this purpose, he generates synthetic data sets and also uses real data acquired from in vivo clinical diffusion tensor MRI studies, and other imaging modalities, to evaluate the performance of these algorithms.This project is motivated primarily by the need to process and analyze clinical data obtained from diffusion tensor MRI, a new noninvasive imaging modality that allows physicians to visualize nerve and muscle fiber tracts in the body. However, the mathematics developed here is applicable to processing and analyzing data acquired from a much larger number of imaging devices and modalities used in diverse application areas including medicine, material sciences, oceanography, meteorology, fluid mechanics, satellite reconnaissance, and astronomy. Many new scanning systems measure several quantities at each point or position within an image rather than a single quantity. These lists of numbers may represent important physical quantities, such as velocities or displacements, or the amount of light absorbed, reflected, or emitted at different wavelengths at a particular location. The theory being developed here provides a rational means to represent, process, analyze and compress this data -- a problem which no theory currently treats. In addition, this work is intended to ameliorate many problems inherent in the measurement of these new types of data sets. In particular, imaging data are usually corrupted by noise, are discrete rather than continuous, and are spatially averaged. Finally, because these new imaging modalities can generate vast amounts of data, the algorithms that the investigator implements for representing, processing, analyzing, and compressing the data must be fast and efficient, as well.