"Mathematical imaging, image multifunctions, diagnostically lossless image compression, fractal-based methods of analysis and approximation"

我的很多研究都可以被称为“数学成像”，即利用数学来开发图像处理或图像分析的新方法。其中一种图像处理方法仍然受到极大的关注，那就是“去噪”，即从图像中去除令人烦恼的噪声(例如，来自磁共振成像的仪器噪声)。 我还回到了图像压缩的研究领域--减少存储数字图像所需的计算机内存量。(数码相机中用来压缩图像的标准“JPEG”方法是基于一个众所周知的数学原理。)如何有效地存储、传输、检索和显示大型数据库中的图像，特别是医学图像，已成为一大挑战。仍然没有答案的问题是“在诊断信息丢失之前，医学图像可以被压缩到什么程度？”目前，对这种失真的大多数评估都是由放射科医生完成的，这使得评估变得主观、极其昂贵和耗时。我们正在与麦克马斯特大学的一名放射科医生和爱克发医疗保健公司的一名软件设计师合作，研究自动进行这种评估的问题，即预测诊断关键失真将在何时发生。 这就引出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”。但是，在此距离内接近的两个图像在视觉上可能并不接近。我在威斯康星州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相似性度量之一。我们一直致力于使用SSIM对图像进行去噪，现在计划将其用于上面概述的医学图像压缩问题。我对SSIM的数学特性也很感兴趣。 我在数学成像方面的研究是从早期一个以“分形分析”为中心的研究项目发展而来的。在“分形分析”中，人们试图将一个“物体”表达为其自身更小、可能扭曲的复制品的联合体。我继续从事这一领域的研究，这一领域具有有趣的应用，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 image processing or image analysis. One such image processing method, which still receives a great deal of attention, is ''denoising'', the removal of troublesome noise from images (e.g., instrument noise from an MRI). I have also returned to the research area of image compression -- reducing the amount of computer memory needed to store a digital image. (The standard "JPEG" method used to compress images in digital cameras is based on a well-known mathematical principle.) The efficient storage, transmission, retrieval and display of images - in particular medical images - in large-scale databases has become a major challenge. The question that remains unanswered is "To what degree can a medical image be compressed before diagnostic information is lost?'' Currently, most assessments of such distortions are done by radiologists, making them subjective, extremely expensive and time-consuming. In collaboration with a radiologist at McMaster University and a software designer at Agfa HealthCare, we are working on the problem of automating this assessment, i.e., predicting when diagnostically critical distortions will occur. 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 a co-author of the "structural similarity measure," (SSIM) recognized as one of the best measures of visual closeness to date. We have been working on the use of SSIM to denoise images and now plan to use it in the medical image compression problem outlined above. I am also interested in the mathematical properties of SSIM. My research in mathematical imaging evolved from an earlier research programme centered around "fractal analysis.'' In fractal analysis, one tries to express an "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.