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。 在表示该区域的一部分的每个像素处，这些200+个反射率值包括该部分处的“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。**我对医学图像压缩也很感兴趣-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，对压缩产生的失真的大多数评估都是主观的，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的放射科医生合作，一直致力于解决这些评估自动化的问题。这导致了图像处理中的另一个问题-评估图像的“视觉质量”。 有一个标准的，基于几何的方法来计算两个图像之间的“距离”。 然而，在该距离上接近的两个图像可能在视觉上不接近。 我在华盛顿大学的一位合作者是“结构相似性度量”（SSIM）的合著者，该度量被认为是迄今为止最好的视觉接近度度量之一。 我们最近表明，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。 我也对SSIM的数学性质感兴趣。**我在数学成像的研究是从早期的密集研究计划围绕“分形分析”，其中一个试图表达一个数学对象作为一个更小的，可能是扭曲的副本本身的联盟。 我继续追求这个有有趣应用的研究领域，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.