Diffeomorphic models of image deformation are a mainstay of medical image registration due to both their advantageous mathematical properties, allowing the analysis of shape and giving rise to the field of computational anatomy, and their practical ability to accurately model the wide variety anatomical variability and physiological motion encountered in clinical practice. However, the price for these beneficial properties is a restriction on the types of motion that can be modeled, most notably the requirements of differentiability and topology preservation. In this work, we consider observed cases of nondiffeomorphic motion in medical imaging, and develop generative statistical models that accurately represent the observed motion while leveraging the useful diffeomorphic framework by representing nonsmooth motion via multiple diffeomorphic transformations. We focus on two motivating cases: nonsmooth motion in fluoroscopic imaging due to the overlapping 2D projections of 3D objects, and the nonsmooth sliding motion of the lower lungs against the thoracic wall in 4DCT imaging.

In fluoroscopic imaging, we represent the observed motion as the additive combination of smoothly deforming layers, investigate the effect of layer image priors (regularization), and show applications to denoising, frame interpolation, and digital subtraction angiography. Motivated by challenges encountered in temporal registration of contrast-enhanced vessels, a novel discrete registration technique is developed for estimating smooth, nonlinear motion of small or repetitive features. Applications of this technique on slice-to-slice microscopy registration are presented.

To model discontinuous motion in 4DCT imaging, we propose a framework for representing and estimating globally invertible and piecewise-smooth transformations. This formulation explicitly represents the location of discontinuities in the deformation field. Based on a novel representation of the invertibility constraint, our formulation allows us to automatically estimate the discontinuous motion, including the location of discontinuities, from the image data. Finally, we extend this invertible and piecewise-smooth model to represent spatially localized topological image changes as a separation between smooth segments in the composite deformation. Initial results are presented modeling the physical tearing of 2D histological sections, automatically estimating both the deformation and tearing region.

The multilinear tensor framework has emerged as a key approach in computer graphics, computer vision, and machine learning, particularly for the purposes of image synthesis, analysis, and recognition. Data ensembles such as facial images, human motion capture data, speech, among many others are the compositional consequence of multiple causal factors. For example, natural images result from the multifactor interaction between the imaging process (camera type/location), the scene illumination, and the scene geometry. These various causal factors confound each other's contributions, and challenge the performance of automated analysis, recognition and synthesis systems. Multilinear algebra, the algebra of higher-order tensors, is a principled mathematical approach for disentangling and explicitly representing these constituent causal factors, which are essential to data formation.

Recently, there has been a surge of interest in the capabilities of tensor data analysis. It has been demonstrated that the popular deep learning approach is a neural network approximation to a hierarchical multilinear (tensor) algebraic data analytic model.

I will discuss the multilinear tensor approach in in the context of facial image biometrics, where the relevant factors include different facial geometries, expressions, lighting conditions, and viewpoints. I will discuss our multilinear projection which extends several matrix algebra concepts in order to extract all the causal factor representations from a single unlabeled image, as well as our hierarchical tensor decomposition approach to data analysis. I will briefly show human motion synthesis results, brain fMRI analysis results, texture synthesis results.