Scale-Coupling and Non-Locality in Large Random Fields

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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.