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

我的很多研究都可以被称为“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成像产生的复杂的、高维的数据集上，特别是(1)从地球表面遥感获得的“高光谱图像”(HSI)和(2)扩散磁共振图像(DMRI)。典型的HSI可以是在不同波长拍摄的一个区域的200多幅图像的堆叠。在代表该区域一部分的每个像素处，这200个反射率值构成该部分的“光谱函数”。从光谱函数可以推断出表面的物质，如金属、水、树叶等。DMRI也可以看作是代表水在不同方向扩散的许多图像的“堆叠”。在代表患者一小块区域的每个体素上，这些许多值给出了水如何向不同方向扩散的想法。这类图像的一个重要应用是“脑束成像”，它可以绘制病人大脑中神经元的连通性(或缺失)。我们正在为这些高维数据集提出一种相当新颖的数学表示，这可能导致为它们的处理带来更好的算法，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.