Scale-Coupling and Non-Locality in Large Random Fields

我们收集了大量的图像数据--卫星图像，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，例如数码相机的肖像。然而，对于科学图像，例如森林的卫星图像，混凝土裂缝的显微照片，或来自MRI的身体的医学图像，需要更专业的图像处理技术。为了从图像中获得尽可能多的信息，我们需要建立数学模型，例如地球大气层的温度或大脑的预期拓扑结构，然后将模型与测量数据联合收割机结合起来。这样的数学运算非常有价值，原因有二：第一，因为它们允许我们从数据中推断出细微的结果;第二，因为它们允许我们检验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，对于大型二维或三维问题，很难让计算机求解模型所依据的方程。我的研究旨在开发建模和算法的有效替代方案，目前专注于两个具体策略：*1。查看各种尺度上的图像建模，将查看大对象的粗尺度模型与查看细节和纹理的细尺度模型耦合：* 换句话说，我研究分层方法，将问题分解为许多层或尺度。这种想法听起来非常简单或直观，因为人类视觉系统就是这样工作的，但在实践中极具挑战性，因为大多数数学模型不允许自己被分割，并且模型行为从一个尺度到另一个尺度的外推尚未完全理解。2.允许模型不仅连接相邻的像素，而且连接距离更远的像素：* 这个目标似乎很明显，但是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，其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随机分散的像素集相连。 这样的模型更难以指定，但能够表示局部模型无法表示的图像特征和现象。该研究的重大影响是对高素质研究生的培训，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能力，并有机会与行业互动。 为了确保研究成果被采用，我与物理学和验光学的科学家合作开发可以成像视网膜和角膜的仪器，以改善眼睛健康，并与应用健康科学的科学家合作，以改善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.**