Applied analysis: Mathematical imaging, image multifunctions, fractal-based methods in analysis

我的大部分研究可以被称为“数学成像”，即利用数学来开发分析或处理图像的新方法。 我目前的注意力集中在复杂的，高维的，从成像中产生的数据集，特别是，（1）“高光谱图像”（HSI）从地球表面的遥感和（2）扩散磁共振图像（DMRI）。典型的HSI可以是在不同波长下拍摄的区域的超过200个图像的堆叠。在代表该区域一部分的每个像素处，这些200多个反射率值构成了该部分的“光谱函数”。从光谱函数，可以推断表面上的材料，例如，金属水树叶DMRI也可以被看作是许多图像的“堆栈”，这些图像表示水在不同方向上的扩散。在代表患者的一个小区域的每个体素处，这些多个值给出了水如何在不同方向上扩散的概念。这种图像的一个重要应用是“纤维束成像”，其中可以映射患者大脑中神经元的连接性（或缺乏连接性）。我们提出了一个相当新颖的数学表示这些高维数据集，这可能会导致更好的算法，他们的处理，例如，去噪、压缩。

Much of my research can be called "mathematical imaging," the use of mathematics to develop new methods of analyzing or processing images. My current attention has focussed on complex, high-dimensional, data sets arising from imaging, in particular, (1) "hyperspectral images" (HSI) obtained from remote sensing of the earth's surface and (2) diffusion magnetic resonance images (DMRI). A typical HSI can be a stack of well over 200 images of a region taken at different wavelengths. At each pixel representing a portion of this region, these 200+ reflectance values comprise the "spectral function" at that portion. From the spectral function, one can infer the materials on the surface, e.g., metals, water, foliage. A DMRI can also be viewed as a "stack" of many images representing the diffusion of water in different directions. At each voxel representing a small region of a patient, these many values give an idea of how water can diffuse in different directions. One important application of such images is "tractography", where the connectivity (or lack thereof) of neurons in the brain of a patient can be mapped. We are proposing a rather novel mathematical representation of these high-dimensional data sets which could lead to better algorithms for their processing, e.g., denoising, compression.