The growing importance of statistical studies of Diffusion Tensor Images (DTI) requires the development of a processing framework that accounts for the non-scalar and nonlinear nature of diffusion tensors. This motivation led a number of authors to consider a Riemannian framework for DTI processing because a Riemannian structure on the data space is sufficient to redefine most processing operat...

We prove that the approximation ratio of the greedy algorithm for the metric Traveling Salesman Problem is Θ(log n). Moreover, we prove that the same result also holds for graphic, euclidean, and rectilinear instances of the Traveling Salesman Problem. Finally we show that the approximation ratio of the ClarkeWright savings heuristic for the metric Traveling Salesman Problem is Θ(log n). keywor...

We consider the problem of computing shortest paths in three-dimensions in the presence of a single-obstacle polyhedral terrain, and present a new algorithm that for any p ≥ 1, computes a (c + ε)approximation to the Lp-shortest path above a polyhedral terrain in O( ε log n log log n) time and O(n log n) space, where n is the number of vertices of the terrain, and c = 2(p−1)/p. This leads to a F...

In [Rao, SoCG 1999], it is shown that every n-point Euclidean metric with polynomial spread admits a Euclidean embedding with k-dimensional distortion bounded by O( √ log n log k), a result which is tight for constant values of k. We show that this holds without any assumption on the spread, and give an improved bound of O( √ log n(log k)). Our main result is an upper bound of O( √ log n log lo...

We investigate higher-order Voronoi diagrams in the city metric. This metric is induced by quickest paths in the L1 metric in the presence of an accelerating transportation network of axis-parallel line segments. For the structural complexity of k-order city Voronoi diagrams of n point sites, we show an upper bound of O(k(n − k) + kc) and a lower bound of Ω(n+ kc), where c is the complexity of ...

Symmetric positive definite (spd) matrices pervade numerous scientific disciplines, including machine learning and optimization. We consider the key task of measuring distances between two spd matrices; a task that is often nontrivial whenever the distance function must respect the non-Euclidean geometry of spd matrices. Typical non-Euclidean distance measures such as the Riemannian metric δR(X...

Symmetric positive definite (spd) matrices pervade numerous scientific disciplines, including machine learning and optimization. We consider the key task of measuring distances between two spd matrices; a task that is often nontrivial whenever the distance function must respect the non-Euclidean geometry of spd matrices. Typical non-Euclidean distance measures such as the Riemannian metric δR(X...

Let F be a flat vector bundle over a compact Riemannian manifold M and let f : M → R be a Morse function. Let g be a smooth Euclidean metric on F , let g t = e g and let ρ(t) be the Ray-Singer analytic torsion of F associated to the metric g t . Assuming that ∇f satisfies the Morse-Smale transversality conditions, we provide an asymptotic expansion for log ρ(t) for t → +∞ of the form a0 + a1t +...

We study the $k$ nearest neighbors problem in plane for general, convex, pairwise disjoint sites of constant description complexity such as line segments, disks, and quadrilaterals under a general family distance functions including $L_p$ norms additively weighted Euclidean distances. compose static data structure this setting with nearly optimal $O(n\log\log n)$ space, $O(\log n+k)$ query time...

We compute the k smallest spanning trees of a point set in the planar Euclidean metric in time O(n log n log k+kmin(k, n) log(k/n)), and in the rectilinear metrics in time O(n log n + n log logn log k + kmin(k, n) log(k/n)). In three or four dimensions our time bound is O(n + kmin(k, n) log(k/n)), and in higher dimensions the bound is O(n2−2/(dd/2e+1)+2 + kn log n).