This paper presents a unified image processing and analysis framework for cortical thickness in characterizing a clinical population. The emphasis is placed on the development of data smoothing and analysis framework. The human brain cortex is a highly convoluted surface. Due to the convoluted non-Euclidean surface geometry, data smoothing and analysis on the cortex are inherently difficult. Wh...

We present a method that deforms an image plane to visually meet the shape and pose of a manifold surface. The user provides constraints that couple a small number of surface points with their corresponding image pixels to initially deform the plane. Matching, based on geodesic distances, couples additional points, followed by a second deformation that brings the image plane into its final pose...

Let μ be a Borel probability measure with compact support. We consider exponential type orthonormal bases, Riesz bases and frames in L2(μ). We show that if L2(μ) admits an exponential frame, then μ must be of pure type. We also classify various μ that admits either kind of exponential bases, in particular, the discrete measures and their connection with integer tiles. By using this and convolut...

Suppose a two-member team is required to navigate through dangerous territory (a plane with polygonal obstacles) along two preselected polygonal paths. The team could consist of police officers, soldiers, or even robots on the surface of Mars. Team members are allowed to coordinate their movements, and it is imperative to keep the team close together during the mission so that if an emergency o...

Very few studies involve how to construct the efficient RBFs by means of problem features. Recently the present author presented general solution RBF (GS-RBF) methodology to create operator-dependent RBFs successfully [1]. On the other hand, the normal radial basis function (RBF) is defined via Euclidean space distance function or the geodesic distance [2]. This purpose of this note is to redef...

Two dimensional parameterizations of meshes is a dynamic field of research. Most works focus on parameterizing complete surfaces, attempting to satisfy various constraints on distances and angles and produce a 2D map with minimal errors. Except for developable surfaces no single map can be devoid of errors, and a parametrization produced for one purpose usually doesn’t suit others. This work pr...

We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann (2001). We show that the possible combinatorial types of shortest paths between two trees can be compactly represented by a partially ordered set. We calculate the shortest distance along each candidate path by converting the problem into one of ...

A geodesic in a graph G is a shortest path between two vertices of G. For a specific function e(n) of n, we define an almost geodesic cycle C in G to be a cycle in which for every two vertices u and v in C, the distance dG(u, v) is at least dC(u, v) − e(n). Let ω(n) be any function tending to infinity with n. We consider a random d-regular graph on n vertices. We show that almost all pairs of v...

The present paper discusses the problem of least-squares over the real symplectic group of matrices Sp(2n,R). The least-squares problem may be extended from flat spaces to curved spaces by the notion of geodesic distance. The resulting non-linear minimization problem on manifold may be tackled by means of a gradient-descent algorithm tailored to the geometry of the space at hand. In turn, gradi...

Within Riemannian geometry the geodesic exponential map is an essential tool for various distance related investigations and computations. Several natural questions can be formulated in terms of its preimages, usually leading to quite challenging nonlinear problems. In this context we recently proposed an approach for computing multiple geodesics connecting two arbitrary points on two-dimension...