In algorithms for processing diffusion tensor images, two common ingredients are interpolating tensors, and measuring the distance between them. We propose a new class of interpolation paths for tensors, termed geodesic-loxodromes, which explicitly preserve clinically important tensor attributes, such as mean diffusivity or fractional anisotropy, while using basic differential geometry to inter...

Clustering is a widely applied tool of data mining to detect the hidden structure of complex multivariate datasets. Hence, clustering solves two kinds of problems simultaneously, it partitions the datasets into cluster of objects that are similar to each other and describes the clusters by cluster prototypes to provide some information about the distribution of the data. In most of the cases th...

We address the Monge problem in metric spaces with a geodesic distance: (X, d) is a Polish space and dL is a geodesic Borel distance which makes (X, dL) a non branching geodesic space. We show that under the assumption that geodesics are d-continuous and locally compact, we can reduce the transport problem to 1-dimensional transport problems along geodesics. We introduce two assumptions on the ...

We discuss the enumeration of planar graphs using bijections with suitably decorated trees, which allow for keeping track of the geodesic distances between faces of the graph. The corresponding generating functions obey non-linear recursion relations on the geodesic distance. These are solved by use of stationary multi-soliton tau-functions of suitable reductions of the KP hierarchy. We obtain ...

This paper presents a boundary-based, topological shape descriptor: the distance profile. It is inspired by the LBP (= local binary pattern) scale space – a topological shape descriptor computed by a filtration with concentric circles around a reference point. For rigid objects, the distance profile is computed by the Euclidean distance of each boundary pixel to a reference point. A geodesic di...

Polygon meshes are collections of vertices, edges and faces defining surfaces in a 3D environment. Computing geometric features on a polygon mesh is of major interest for various applications. Among these features, the geodesic distance is the distance between two vertices following the surface defined by the mesh. In this paper, we propose an algorithm for fast geodesic distance approximation ...

This paper addresses the generalization of Principal Component Analysis (PCA) to Riemannian manifolds. Current methods like Principal Geodesic Analysis (PGA) and Geodesic PCA (GPCA) minimize the distance to a ”Geodesic subspace”. This allows to build sequences of nested subspaces which are consistent with a forward component analysis approach. However, these methods cannot be adapted to a backw...