Grant: Discovery Grants Program - Individual

This proposal has two parts: (i) Methods for image analysis and (ii) summarization of high-dimensional dynamics. In the first part, I will investigate mathematical and computational methods for image analysis, with a focus on medical images such as X-Rays and Magnetic Resonance Images (MRIs). Together with my trainees, we will develop new methods for registering, i.e. aligning, images, and also for finding lesions and other abnormalities. These tasks are central to medical image analysis and many computer vision applications such as tracking and change detection. In medical imaging, registration allows us to compare images from different people, or from the same person at different times, and do statistics on groups of images. It turns out that registering images first helps with finding lesions, and finding lesions first helps with registering images. We will attempt to resolve this "chicken-and-egg" problem by developing integrated analysis methods that do both at the same time. The second part of the proposal is about dynamical systems, a term that includes both differential equations and real-world systems that change according to physical rules, such as weather and climate, or planetary motion, or a beating heart. Many dynamical systems are so big and complex that the task of understanding them can seem hopeless at first. Even if we know all the underlying physical rules, it is hard to be sure we've considered all the possible ramifications of those rules. Together with trainees and colleagues, we will develop automatic computational methods for generating summaries of the possible behaviours of a dynamical system. develop an automatic computational method for generating summaries of the possible behaviours of a dynamical system.