Covariant Image Representation with Applications to Classification Problems in Medical Imaging

Images are often considered as functions defined on the image domains, and as functions, their (intensity) values are usually considered to be invariant under the image domaintransforms. This functionalviewpoint is both influential and prevalent, and it provides the justification for comparing images using functional $$\mathbf {L}^p$$Lp-norms. However, with the advent of more advanced sensing technologies and data processing methods, the definition and the variety of images has been broadened considerably, and the long-cherished functional paradigm for images is becoming inadequate and insufficient. In this paper, we introduce the formal notion of covariant images and study two types of covariant images that are important in medical image analysis, symmetric positive-definite tensor fieldsand Gaussian mixture fields, images whose sample values covary i.e., jointly vary with image domain transforms rather than being invariant to them. We propose a novel similarity measurebetween a pair of covariant images considered as embedded shapes (manifolds) in the ambient space, a Cartesian productof the image and its sample-value domains. The similarity measureis based on matching the two embedded low-dimensional shapes, and both the extrinsic geometry of the ambient spaceand the intrinsic geometry of the shapes are incorporated in computing the similarity measure. Using this similarity as an affinity measure in a supervised learning framework, we demonstrate its effectiveness on two challenging classification problems: classification of brain MR images based on patients' age and (Alzheimer's) disease status and seizure detection from high angular resolution diffusion magnetic resonance scans of rat brains.