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

我的许多研究可以被称为“数学成像”，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、高维数据集，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）。一个典型的HSI可以是一个区域在不同波长下拍摄的200多张图像的叠加。在代表该区域一部分的每个像素处，这200+反射率值构成了该部分的“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上扩散的许多图像的“堆栈”。每一个体素代表病人的一个小区域，这些数值给出了水如何向不同方向扩散的概念。这种图像的一个重要应用是“神经束成像”，它可以绘制出患者大脑中神经元的连接（或缺乏连接）。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，这可能会导致更好的算法来处理它们，例如，去噪，压缩。******我也一直对医学图像压缩感兴趣-减少存储医学图像所需的计算机内存量。仍然没有答案的问题是，“在诊断信息丢失之前，医学图像可以压缩到什么程度？”目前，大多数由压缩产生的扭曲评估都是主观的，由放射科医生进行，这使得它们非常昂贵和耗时。我们与麦克马斯特大学（McMaster University）的放射科医生合作，一直在研究自动化这些评估的问题。******这导致了图像处理中的另一个问题——评估图像的“视觉质量”。有一种标准的、基于数学的方法来计算两幅图像之间的“距离”。然而，在这个距离上很近的两个图像在视觉上可能并不近。我在华盛顿大学的一位合作者是“结构相似性测量”（SSIM）的合著者，该测量被认为是迄今为止视觉接近度的最佳测量之一。我们最近的研究表明，SSIM在匹配放射科医生的主观评估方面表现得更好。现在仍然需要有效地使用SSIM来确定医学图像压缩的新标准。我对SSIM的数学性质也很感兴趣。******我在数学成像方面的研究是从早期以“分形分析”为中心的密集研究项目发展而来的，在这个项目中，人们试图将一个数学对象表达为一个更小的、可能扭曲的自身副本的联盟。我继续研究这个领域，它有有趣的应用，特别是在成像方面。

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.******I have also been interested in medical image compression - reducing the amount of computer memory needed to store a medical image. The question that remains unanswered is, "To what degree can a medical image be compressed before diagnostic information is lost?" Currently, most assessments of distortions produced by compression are subjective and performed by radiologists, making them extremely expensive and time-consuming. In collaboration with radiologists at McMaster University, we have been working on the problem of automating these assessments.******This leads to another problem in image processing - assessing the "visual quality" of images. There is a standard, mathematically-based method of computing the "distance" between two images. However, two images that are close in this distance may not be close visually. One of my collaborators at UW is the co-author of the "structural similarity measure" (SSIM), recognized as one of the best measures of visual closeness to date. We have recently shown that SSIM performs much better in matching the subjective assessments of radiologists. It now remains to use SSIM effectively to determine new standards of medical image compression. I am also interested in the mathematical properties of SSIM.******My research in mathematical imaging evolved from an earlier intensive research programme centered around "fractal analysis", in which one tries to express a mathematical object as a union of smaller, possibly distorted copies of itself. I continue to pursue this area of research which has interesting applications, particularly in imaging.