Euler's fluid equations: Optimal Control vs Optimization

An optimization method used in image-processing (metamorphosis) is found to imply Euler's equations for incompressible flow of an inviscid fluid, without requiring that the Lagrangian particle labels exactly follow the flow lines of the Eulerian velocity vector field. Thus, an optimal control problem and an optimization problem for incompressible ideal fluid flow both yield the \emph {same} Euler fluid equations, although their Lagrangian parcel dynamics are \emph{different}. This is a result of the \emph{gauge freedom} in the definition of the fluid pressure for an incompressible flow, in combination with the symmetry of fluid dynamics under relabeling of their Lagrangian coordinates. Similar ideas are also illustrated for SO(N) rigid body motion.Comment: 12 page

Gauge Invariant Framework for Shape Analysis of Surfaces

This paper describes a novel framework for computing geodesic paths in shape spaces of spherical surfaces under an elastic Riemannian metric. The novelty lies in defining this Riemannian metric directly on the quotient (shape) space, rather than inheriting it from pre-shape space, and using it to formulate a path energy that measures only the normal components of velocities along the path. In other words, this paper defines and solves for geodesics directly on the shape space and avoids complications resulting from the quotient operation. This comprehensive framework is invariant to arbitrary parameterizations of surfaces along paths, a phenomenon termed as gauge invariance. Additionally, this paper makes a link between different elastic metrics used in the computer science literature on one hand, and the mathematical literature on the other hand, and provides a geometrical interpretation of the terms involved. Examples using real and simulated 3D objects are provided to help illustrate the main ideas.Comment: 15 pages, 11 Figures, to appear in IEEE Transactions on Pattern Analysis and Machine Intelligence in a better resolutio

Indirect Image Registration with Large Diffeomorphic Deformations

The paper adapts the large deformation diffeomorphic metric mapping framework for image registration to the indirect setting where a template is registered against a target that is given through indirect noisy observations. The registration uses diffeomorphisms that transform the template through a (group) action. These diffeomorphisms are generated by solving a flow equation that is defined by a velocity field with certain regularity. The theoretical analysis includes a proof that indirect image registration has solutions (existence) that are stable and that converge as the data error tends so zero, so it becomes a well-defined regularization method. The paper concludes with examples of indirect image registration in 2D tomography with very sparse and/or highly noisy data.Comment: 43 pages, 4 figures, 1 table; revise

Structure-based analysis of the ultraspiracle protein and docking studies of putative ligands

The ultraspiracle protein (USP) is the insect ortholog of the mammalian retinoid X receptor (RXR). Fundamental questions concern the functional role of USP as the heterodimerization partner of insect nuclear receptors such as the ecdysone receptor. The crystallographic structures of the ligand binding domain of USPs of Heliothis virescens and Drosophila melanogaster solved recently show that helix 12 is locked in an antagonist conformation raising the question whether USPs could adopt an agonist conformation as observed in RXRα. In order to investigate this hypothesis, a homology model for USP is proposed that allows a structural analysis of the agonist conformation of helix 12 based on the sequence comparison with RXR. For USP, one of the main issues concerns its function and in particular whether its activity is ligand independent or not. The x-ray structures strongly suggest that USP can bind ligands. Putative ligands have therefore been docked in the USP homology model. Juvenile hormones and juvenile hormone analogs were chosen as target ligands for the docking study. The interaction between the ligand and the receptor are examined in terms of the pocket shape as well as in terms of the chemical nature of the residues lining the ligand binding cavity