Geometry, Imaging and Computing

In this paper, we formulate new algorithms to obtain the differential geometric properties of surface-surface intersection curves in $\mathbb{R}^3$. We also show its applicability to obtain geodesic curvature and geodesic torsion of two regular implicit surfaces in $\mathbb{R}^3$. Our new method will be distinguished from the older ones in the sense that we will use Rodrigues’ rotation formula and using a new operator $T$.

Thurston’s circle packing approximation of the Riemann Mapping (proven to give the Riemann Mapping in the limit by Rodin–Sullivan) is largely based on the theorem that any topological disk with a circle packing metric can be deformed into a circle packing metric in the disk with boundary circles internally tangent to the circle. The main proofs of the uniformization use hyperbolic volumes (Andreev) or hyperbolic circle packings (by Beardon and Stephenson). We reformulate these problems into a Euclidean context, which allows more general discrete conformal structures and boundary conditions. The main idea is to replace the disk with a double covered disk with one side forced to be a circle and the other forced to have interior curvature zero. The entire problem is reduced to finding a zero curvature structure. We also show that these curvatures arise naturally as curvature measures on generalized manifolds (manifolds with multiplicity) that extend the usual discrete Lipschitz–Killing curvatures on surfaces.

Surface parameterization plays a fundamental role in geometric modeling and processing. Surface Ricci flow deforms the Riemannian metric proportional to the curvature, such that the curvature evolves according to a diffusion-reaction process, and converges to the target curvature. Surface Ricci flow is a powerful tool to design Riemannian metrics from user-prescribed curvatures. In discrete setting, there are several schemes, which can be unified to a coherent framework.

In this work, we are concerned with the spherical quasiconformal parameterization of genus-$0$ closed surfaces. Given a genus-$0$ closed triangulated surface and an arbitrary user-defined quasiconformal distortion, we propose a fast algorithm for computing a spherical parameterization of the surface that satisfies the prescribed distortion. The proposed algorithm can be effectively applied to adaptive surface remeshing for improving the visualization in computer graphics and animations. Experimental results are presented to demonstrate the effectiveness of our algorithm.

We present, in a natural, developmental manner, the main types of metric curvatures and investigate their relationship with the notions of Hausdorff and Gromov–Hausdorff distances, which by now have been widely adopted in the fields of Imaging, Vision and Graphics. In addition we present a number of applications to the fields above, as well as to Communication Networks and Regge Calculus. Further connections with such established notions as excess and fatness are also investigated. While the present paper represents essentially a survey, a number of possible applications are presented here for the first time, for instance to a numerically feasible quantification of a notion of quasi-flatness of manifolds, with applications in Imaging; and to the introduction of curvatures, in particular the Lipschitz–Killing curvature measures, for almost Riemannian manifolds, with a view to their usage in Regge Calculus, as well as in Graphics. Further directions of study are also suggested.

Surface based shape analysis plays a fundamental role in computer vision and medical imaging. In this work, we proposes a novel method for shape classification of brain’s hippocampus using Wasserstein distance based on optimal mass transport theory. In comparison with the conventional method based on Monge–Kantorovich theory, our proposed method employs Monge–Brenier theory for the computation of the optimal mass transport map, which remarkably ameliorates the efficiency by reducing computational complexity from $O(n^2)$ to $O(n)$. Using the conformal mapping, our method maps the metric surface with disk topology to the unit planar disk, which pushes the area element on the surface to the disk and incurs the area distortion. A probability measure is then determined by this area distortion. Given any two probability measures on two surfaces, our method is capable of obtaining a unique optimal mass transport map between them. The transportation cost of this optimal mass transport defines the Wasserstein distance between two surfaces, which intrinsically measures the dissimilarities between surface based shapes and thus can be used for shape classification. Experimental results on surface based hippocampal shape analysis demonstrates the efficiency and efficacy of our proposed method.

We extend two existing variational models from the Euclidean space to a vector bundle over a Riemannian manifold. The Euclidean models, dedicated to regularize or enhance some color image features, are based on the concept of nonlocal gradient operator acting on a function of the Euclidean space. We then extend these models by generalizing this operator to a vector bundle over a Riemannian manifold with the help of the parallel transport map associated to some class of covariant derivatives. Through the dual formulations of the proposed models, we obtain the expressions of their solutions, which exhibit the functional spaces that describe the image features. Finally, for a well-chosen covariant derivative and its nonlocal extension, the proposed models perform local contrast modification (reduction or enhancement) and experiments show that they preserve more the aspect of the original image than the Euclidean models do while modifying equally its contrast.

The representation of curves by their square root velocity functions (SRVF) provides a useful and computationally effective way to make a metric space out of the set of all absolutely continuous curves modulo reparametrization. The first part of this paper establishes some important theoretical properties of this method, proving the completeness of this metric space, characterizing the exact nature of the closed orbits under reparametrization, and proving the existence of optimal matchings between pairs of curves, provided that at least one of the two curves is piecewise linear.

A new framework for mesh optimization, the Filtered Hooke’s Optimization, is proposed. With the notion of the elasticity theory, the Hooke’s Optimization is developed by modifying the Hooke’s law, in which an elastic force is simulated on the edges of a mesh so that adjacent vertices are either attracted to each other or repelled from each other, so as to regularize the mesh in terms of triangulation. A normal torque force is acted on vertices to guaranteed smoothness of the surface. In addition, a filtering scheme, called the Newtonian Filtering, is proposed as a supplementary tool for the proposed Hooke’s Optimization to preserve the geometry of the mesh. Numerical simulations on meshes with different geometry indicate an impressive performance of our proposed framework to significantly improves the mesh triangulation without noteworthy distortions of the mesh geometry.

We obtain new invariant Einstein metrics on the compact Lie groups $\mathrm{SO}(n) \; (n \geq 7)$ which are not naturally reductive. This is achieved by imposing certain symmetry assumptions in the set of all left-invariant metrics on $\mathrm{SO}(n)$ and by computing the Ricci tensor for such metrics. The Einstein metrics are obtained as solutions of systems polynomial equations, which we manipulate by symbolic computations using Gröbner bases.