Deep learning of diffeomorphisms for optimal reparametrizations of shapes

In shape analysis, one of the fundamental problems is to align curves or surfaces before computing a (geodesic) distance between these shapes. To find the optimal reparametrization realizing this alignment is a computationally demanding task which leads to an optimization problem on the diffeomorphism group. In this paper, we construct approximations of orientation-preserving diffeomorphisms by composition of elementary diffeomorphisms to solve the approximation problem. We propose a practical algorithm implemented in PyTorch which is applicable both to unparametrized curves and surfaces. We derive universal approximation results and obtain bounds for the Lipschitz constant of the obtained compositions of diffeomorphisms.Comment: 26 pages, 11 figures. Submitted to SIAM Journal of Scientific Computin

Pastoral Herding Strategies and Governmental Management Objectives: Predation Compensation as a Risk Buffering Strategy in the Saami Reindeer Husbandry

Previously it has been found that an important risk buffering strategy in the Saami reindeer husbandry in Norway is the accumulation of large herds of reindeer as this increases long-term household viability. Nevertheless, few studies have investigated how official policies, such as economic compensation for livestock losses, can influence pastoral strategies. This study investigated the effect of received predation compensation on individual husbandry units’ future herd size. The main finding in this study is that predation compensation had a positive effect on husbandry units’ future herd size. The effect of predation compensation, however, was nonlinear in some years, indicating that predation compensation had a positive effect on future herd size only up to a certain threshold whereby adding additional predation compensation had little effect on future herd size. More importantly, the effect of predation compensation was positive after controlling for reindeer density, indicating that for a given reindeer density husbandry units receiving more predation compensation performed better (measured as the size of future herds) compared to husbandry units receiving less compensation

Gradient-Based Optimization in Shape Analysis for Reparametrization of Parametric Curves and Surfaces

I denne oppgaven studerer vi to gradientbaserte optimeringsalgoritmer i formanalyse, for reparametrisering av parametriske kurver og overflater. Den ene algoritmen er en tidligere kjent “gradient descent”-algoritme på gruppen av orienteringsbevarende diffeomorfier. Den andre er en ny tilnærming til parametriseringsproblemet, der det å finne en optimal reparametrisering, tilsvarer treningen av et restnevralt nettverk. Vi sammenligner ytelsen til de to algoritmene ved hjelp av noen få eksempler for både kurver og overflater. I begge tilfeller presterer det restnevrale nettverket bedre enn “gradient descent”-algoritmen

Multi-compartmental model of glymphatic clearance of solutes in brain tissue.

The glymphatic system is the subject of numerous pieces of research in biology. Mathematical modelling plays a considerable role in this field since it can indicate the possible physical effects of this system and validate the biologists' hypotheses. The available mathematical models that describe the system at the scale of the brain (i.e. the macroscopic scale) are often solely based on the diffusion equation and do not consider the fine structures formed by the perivascular spaces. We therefore propose a mathematical model representing the time and space evolution of a mixture flowing through multiple compartments of the brain. We adopt a macroscopic point of view in which the compartments are all present at any point in space. The equations system is composed of two coupled equations for each compartment: One equation for the pressure of a fluid and one for the mass concentration of a solute. The fluid and solute can move from one compartment to another according to certain membrane conditions modelled by transfer functions. We propose to apply this new modelling framework to the clearance of 14C-inulin from the rat brain

Multi-compartmental model of glymphatic clearance of solutes in brain tissue

The Glymphatic system is the subject of numerous pieces of research in Biology. Mathematical modeling plays a considerable role in this field since it can indicate the possible physical effects in this system and validate the biologists' hypotheses. The available mathematical models that describe the system at the scale of the brain (i.e. the macroscopic scale) are often solely based on the diffusion equation and do not consider the fine structures formed by the perivascular spaces. We therefore propose a mathematical model representing the time and space evolution of a mixture flowing through multiple compartments of the brain. We adopt a macroscopic point of view in which the compartments are all present at any point in space. The equations system is composed of two coupled equations for each compartment: One equation for the pressure of a fluid and one for the mass concentration of a molecule. The fluid and solute can move from one compartment to another according to certain membrane conditions modeled by transfer functions. We propose to apply this new modeling framework to the clearance of 14 C-inulin from the rat brain