Scale-Coupling and Non-Locality in Large Random Fields

收集了大量的图像数据--卫星图片、显微图像、谷歌街景--我们如何处理所有这些图像？*有许多图像处理算法可用于常规图像，例如数码相机拍摄的肖像。然而，对于科学图像，如森林的卫星图像，混凝土裂缝的显微图像，或来自核磁共振的身体医学图像，需要更多专门的图像处理技术。*为了从图像中获得尽可能多的信息，我们需要建立一个数学模型，如地球大气的最高温度或预期的大脑拓扑结构，然后将模型与测量数据结合起来。这样的数学运算非常有价值，原因有两个：*第一，因为它们允许我们从数据中推断出微妙的结果；*第二，因为它们允许我们测试给定的数学模型是否有意义，这是推进我们理解的关键一步。*然而，问题是，让计算机求解大型二维或三维问题模型所依据的方程是非常困难的。我的研究试图开发高效的建模和算法替代方案，目前专注于两个具体的策略：*1.查看各种尺度上的图像建模，将关注大型对象的粗略模型与关注细节和纹理的精细模型结合起来：*换句话说，我研究分层方法，将问题分解为多个层或尺度。这样的想法听起来非常简单或直观，因为人类的视觉系统在很大程度上是以这种方式工作的，但在实践中却极具挑战性，因为大多数数学模型不允许自身分裂，并且没有完全理解模型行为从一个尺度到另一个尺度的外推。*2.允许模型不仅连接相邻的像素，而且连接更远的像素：*这个目标看起来很明显，然而几乎所有的空间统计模型都专注于高度局部的相互作用。在我的研究中，我希望探索这样的模型，在这种模型中，每个像素都与所有其他像素相连，或者可能与基于底层图像的一组随机分散的像素相连。*这类模型更难具体说明，但能够代表本地模型无法代表的图像特征和现象。*这项研究的重大影响是培养高素质的研究生，他们将获得图像处理和计算机视觉方面的专业知识，同时具有沟通和领导技能，以及与行业互动的机会。*为了确保研究成果被采纳，我与物理学和验光学的科学家合作，开发可以对视网膜和角膜成像的仪器，以改善眼睛健康，并与应用健康科学的科学家合作，改进分析医学图像和临床数据的方法。**

There is an awful lot of image data being collected -- satellite pictures, microscopic images, Google Streetview -- how can we process all of this imagery? ***There are many image processing algorithms available for regular images, such as portraits from digital cameras. However for scientific imagery, such as satellite images of a forest, microscopic pictures of a cracks in concrete, or medical images of the body from an MRI, more specialized techniques of image processing are required.***To get as much information out of an image as possible, we need to take a mathematical model, such as the temperature of the earth's atmosphere or the expected topology of the brain, and then to combine the model with measured data. Such mathematical operations are tremendously valuable, for two reasons: *** First, because they allow us to infer subtle results from the data, and*** Second, because they allow us to test whether a given mathematical model makes sense or not, a crucial step in advancing our understanding.***The problem, however, is that it is very difficult to have a computer solve the equations underlying the models for large two- or three-dimensional problems. My research seeks to develop efficient alternatives to modeling and algorithms, currently focusing on two specific strategies:***1. Look at image modeling over a variety of scales, coupling a coarse-scale model looking at large objects with a fine-scale model looking at details and textures:***In other words, I research hierarchical approaches which break a problem into a number of layers or scales. Such an idea sounds very simple or intuitive, since the human visual system very much works this way, but is extremely challenging in practice, because most mathematical models do not allow themselves to be split up, and the extrapolation of model behaviour from one scale to another is not fully understood.*** 2. Allow the model to connect not just neighbouring pixels, but also pixels further apart:***This goal seems obvious, however virtually all spatial statistical models have focused on highly local interactions.  In my research I wish to explore models in which every pixel is connected to all other pixels, or possibly to a random scattered set of pixels based on the underlying image.  Such models are much more difficult to specify, but are able to represent image features and phenomena that local models cannot.***A significant impact of the research is on training of highly qualified graduate students, who will receive expertise in image processing and computer vision, at the same time with communication and leadership skills, and with interaction opportunities with industry.  To ensure that the research results become adopted, I collaborate with scientists in Physics and Optometry to develop instruments that can image the retina and the cornea, to improve eye health, and with scientists in Applied Health Sciences to improve ways of analyzing medical images and clinical data.**