Topologically invariant manifold learning for medical imaging

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An important goal of scientific data analysis in engineering problems is to understand the behaviour of a complex system, biological process or physical-state alterations given some observation samples. This introduces the need to synthesize large amounts of multivariate data and raises the fundamental question of data reduction: how to discover compact and meaningful representations of high-dimensional data? While significant progress has been made in the field of manifold learning, these approaches are still hamstrung by fundamental challenges such as lack of isometry, highly variable structural topologies demonstrated by real-world samples and handling very large amounts of data to efficiently learn the associated representation. We propose to take the required steps to bridge the existing chasm between theory and application. The overall objective of the research program is to develop a new paradigm of manifold learning techniques, which data representation is invariant to the underlying topology. This will be geared towards applied problems encountered in medical imaging and in computer vision. The approach will be able to reduce large amounts of complex, high-dimensional medical images by decomposing structures into joint embeddings and preserving smoothness properties. The proposed computational framework would incorporate highly innovative theoretical developments, namely: (1) an algorithm to reconstruct both manifold and non-manifold surfaces from high-dimensional training sets independently of the inherent data structure using joint Grassmannian embeddings, (2) a parameterization model to perform mappings between ambient and manifold spaces and (3) a discrete optimization framework using higher-order Markov Random Fields to infer new models from the underlying space. This innovative platform can be of great value for a number of applications, such as in medical imaging and in computer vision, by providing the required tools to process high volume and multi- parametric data. It has the potential to contribute new insights and indications for possible early predictors of neurological disorders and open new leads in computational anatomy, tumor growth and organ modeling.