David Groisser

The Riemannian geometry of the Yang-Mills moduli space

The moduli space ℳ of self-dual connections over a Riemannian 4-manifold has a natural Riemannian metric, inherited from theL2 metric on the space of connections. We give a formula for the curvature of this metric in terms of the relevant Green operators. We then examine in great detail the moduli space ℳ1 ofk=1 instantons on the 4-sphere, and obtain an explicit formula for the metric in this case. In particular, we prove that ℳ1 is rotationally symmetric and has “finite geometry:” it is an incomplete 5-manifold with finite diameter and finite volume.

Symmetric Non-rigid Registration: A Geometric Theory and Some Numerical Techniques

This paper proposes ℒ2- and information-theory-based (IT) non-rigid registration algorithms that are exactly symmetric. Such algorithms pair the same points of two images after the images are swapped. Many commonly-used ℒ2 and IT non-rigid registration algorithms are only approximately symmetric. The asymmetry is due to the objective function as well as due to the numerical techniques used in discretizing and minimizing the objective function. This paper analyzes and provides techniques to eliminate both sources of asymmetry.This paper has five parts. The first part shows that objective function asymmetry is due to the use of standard differential volume forms on the domain of the images. The second part proposes alternate volume forms that completely eliminate objective function asymmetry. These forms, called graph-based volume forms, are naturally defined on the graph of the registration diffeomorphism f, rather than on the domain of f. When pulled back to the domain of f they involve the Jacobian Jf and therefore appear “non-standard”. In the third part of the paper, graph-based volume forms are analyzed in terms of four key objective-function properties: symmetry, positive-definiteness, invariance, and lack of bias. Graph-based volume forms whose associated ℒ2 objective functions have the first three properties are completely classified. There is an infinite-dimensional space of such graph-based forms. But within this space, up to scalar multiple, there is a unique volume form whose associated ℒ2 objective function is unbiased. This volume form, which when pulled back to the domain of f is (1+det(Jf)) times the standard volume form on Euclidean space, is exactly the differential-geometrically natural volume form on the graph of f. The fourth part of the paper shows how the same volume form also makes the IT objective functions symmetric, positive semi-definite, invariant, and unbiased. The fifth part of the paper introduces a method for removing asymmetry in numerical computations and presents results of numerical experiments. The new objective functions and numerical method are tested on a coronal slice of a 3-D MRI brain image. Numerical experiments show that, even in the presence of noise, the new volume form and numerical techniques reduces asymmetry practically down to machine precision without compromising registration accuracy.

Matched G k -constructions always yield C k -continuous isogeometric elements

Gk (geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched Gk constructions for geometry and analysis automatically yield Ck isogeometric elements. This provides a formal framework for the existing and any future isogeometric elements based on geometric continuity.

Non-Rigid Shape Comparison of Plane Curves in Images

A mathematical theory for establishing correspondences between curves and for non-rigid shape comparison is developed in this paper. The proposed correspondences, called bimorphisms, are more general than those obtained from one-to-one functions. Their topology is investigated in detail.A new criterion for non-rigid shape comparison using bimorphisms is also proposed. The criterion avoids many of the mathematical problems of previous approaches by comparing shapes non-rigidly from the bimorphism.Geometric invariants are calculated for curves whose shapes can be exactly matched with a bimorphism. The invariants are related to the concave and convex segments of a curve and provide justification for parsing the curve into such segments.

Matched Gk-constructions always yield Ck-continuous isogeometric elements

Gk (geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched Gk constructions for geometry and analysis automatically yield Ck isogeometric elements. This provides a formal framework for the existing and any future isogeometric elements based on geometric continuity.

A Geometric Theory of Symmetric Registration

This paper shows that asymmetry (inverse inconsistency) in L2 and information theoretic (IT) registration objective functions is the result of using standard volume differential forms. A new space with non-standard volume differential forms is identified, and a unique symmetrizing volume differential form is found in it. This form simultaneously symmetrizes the L2 and the IT registration functions. Experimental results supporting the mathematical theory are provided.

On the convergence of some Procrustean averaging algorithms

Euclidean “(size-and-)shape spaces” are spaces of configurations of points in R N modulo certain equivalences. In many applications one seeks to average a sample of shapes, or sizes-and-shapes, thought of as points in one of these spaces. This averaging is often done using algorithms based on generalized Procrustes analysis (GPA). These algorithms have been observed in practice to converge rapidly to the Procrustean mean (size-and-)shape, but proofs of convergence have been lacking. We use a general Riemannian averaging (RA) algorithm developed in [Groisser, D. (2004) “Newtons method, zeroes of vector fields, and the Riemannian center of mass”, Adv. Appl. Math. 33, pp. 95–135] to prove convergence of the GPA algorithms for a fairly large open set of initial conditions, and estimate the convergence rate. On size-and-shape spaces the Procrustean mean coincides with the Riemannian average, but not on shape spaces; in the latter context we compare the GPA and RA algorithms and bound the distance between the averages to which they converge.

Nonparametric two-sample tests on homogeneous Riemannian manifolds, Cholesky decompositions and Diffusion Tensor Image analysis

This paper addresses much needed asymptotic and nonparametric bootstrap methodology for two-sample tests for means on Riemannian manifolds with a simply transitive group of isometries. In particular, we develop a two-sample procedure for testing the equality of the generalized Frobenius means of two independent populations on the space of symmetric positive matrices. The new method naturally leads to an analysis based on Cholesky decompositions of covariance matrices which helps to decrease computational time and does not increase dimensionality. The resulting nonparametric matrix valued statistics are used for testing if there is a difference on average at a specific voxel between corresponding signals in Diffusion Tensor Images (DTIs) in young children with dyslexia when compared to their clinically normal peers, based on data that was previously analyzed using parametric methods.

Instantons and the Information Metric

The information metric arises in statistics as a natural inner product on a space of probability distributions. In general this inner product is positive semi-definite but is potentially degenerate.By associating to an instanton its energy density, we can examine the information metric g on the moduli spaces M of self-dual connections over Riemannian four-manifolds. Compared with the more widely known L2 metric, the information metric better reflects the conformal invariance of the self-dual Yang–Mills equations, and seems to have better completeness properties. In the case of SU(2) instantons on S4 of charge one, g is known to be the hyperbolic metric on the five-ball. We show more generally that for charge-one SU(2) instantons over 1-connected, positive-definite manifolds, g is non-degenerate and complete in the collar region of M, and is “asymptotically hyperbolic” there; g vanishes at the cone points of M. We give explicit formulae for the metric on the space of instantons of charge one on CP2.

Scaling-Rotation Distance and Interpolation of Symmetric Positive-Definite Matrices

We introduce a new geometric framework for the set of symmetric positive-definite (SPD) matrices, aimed at characterizing deformations of SPD matrices by individual scaling of eigenvalues and rotation of eigenvectors of the SPD matrices. To characterize the deformation, the eigenvalue-eigenvector decomposition is used to find alternative representations of SPD matrices and to form a Riemannian manifold so that scaling and rotations of SPD matrices are captured by geodesics on this manifold. The problems of nonunique eigen-decompositions and eigenvalue multiplicities are addressed by finding minimal-length geodesics, which gives rise to a distance and an interpolation method for SPD matrices. Computational procedures for evaluating the minimal scaling-rotation deformations and distances are provided for the most useful cases of