Generative Models Applied to Inverse Problems

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My research works at the intersection of deep learning and inverse problems in imaging. Solving an inverse problem is the task of computing an unknown physical quantity given indirect measurements via a (usually) known forward model. A typical example is when imaging technologies are used in medicine, engineering, astronomy and geophysics. Interesting Inverse Problems are nearly always ill-posed and addressing this is critical in applications where decision making is based on the recovered image. In particular, sometimes the data recovered is not sufficient to solve the reconstruction problem accurately and some form of prior knowledge about the object must be incorporated in the solution process. Developing new and improving existing approaches to these cases is the aim of my project. The proposed approaches uses deep learning methods to learn a generative model. Generative models implicitly model high-dimensional distributions of data from observations. The learnt distribution can then be used as a prior when solving the inverse problem, ensuring solutions are feasible. Learnt priors could provide more specific information than that of a hand-crafted prior while remaining flexible to changes in the forward problem. For example, we could learn a generative model that can produce all feasible brain heart and lung CT images. Then receiving low radiation dose data from the CT scanner, we search through the set of feasible images to find the image that best fits the data. The resulting image will be high quality despite the small amount of provided data. There is wide scope for novel engineering and/or physical sciences research. The resulting optimisation required for finding the image produced by the generative model that best fits the data is both a non-linear and non-convex problem and will require application or development of state of the art techniques. Numerical analysis tools are needed to bound errors and convergence. Mathematical approaches working closely with computer scientists also need to be developed for checking and ensuring that the ground truth images can be produced by the generative model. This will add to a fast growing research area including variational autoencoders and generative adversarial networks.