Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of displacement vectors with respect to the template surface.Comment: Accepted in Medical Image Analysi

Existence and Continuity of Minimizers for the Estimation of Growth Mapped Evolutions for Current Data Term and Couterexamples for Varifold Data Term

In the field of computational anatomy, the complexity of changes occurring during the evolution of a living shape while it is growing, aging or reacting to a disease, calls for more and more accurate models to allow subject comparison. Growth mapped evolutions have been introduced to tackle the loss of homology between two ages of an organism following a growth process that involves creation of new material. They model the evolution of longitudinal shape data with partial mappings. One viewpoint consists in a progressive embedding of the shape into an ambient space on which acts a group of diffeomorphisms. In practice, the shape evolves through a time-varying dynamic called the growth dynamic. The concept of shape space has now been widely studied and successfully applied to analyze the variability of a population of related shapes. Time-varying dynamics subsequently enlarge this framework and open the door to new optimal control problems for the assimilation of longitudinal shape data. We address in this paper an interesting problem in the field of the calculus of variations to investigate the existence and continuity of solutions for the registration of growth mapped evolutions with the growth dynamic. This theoretical question highlights the unexpected role of the data term grounded either on current or varifold representations. Indeed, in this new framework, the spatial regularity of a continuous scenario estimated from a temporal sequence of shapes with the growth dynamic depends on the temporal regularity of the deformation. Current metrics have the property to be more robust to this spatial regularity than varifold metrics. We will establish the existence and continuity of global minimizers for current data term and highlight two counterexamples for varifold data term

Nonuniformly hyperbolic systems arising from coupling of chaotic and gradient-like systems

We investigate dynamical systems obtained by coupling two maps, one of which is chaotic and is exemplified by an Anosov diffeomorphism, and the other is of gradient type and is exemplified by a N-pole-to-S-pole map of the circle. Leveraging techniques from the geometric and ergodic theories of hyperbolic systems, we analyze three different ways of coupling together the two maps above. For weak coupling, we offer an addendum to existing theory showing that almost always the attractor has fractal-like geometry when it is not normally hyperbolic. Our main results are for stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. Under these conditions, we show that the coupled systems have invariant cones and possess SRB measures even though there are genuine obstructions to uniform hyperbolicity