This paper describes the Subgradient Marching algorithm to compute the derivative of the geodesic distance with respect to the metric. The geodesic distance being a concave function of the metric, this algorithm computes an element of the subgradient in O(N log(N)) operations on a discrete grid ofN points. It performs a front propagation that computes the subgradient of a discrete geodesic dist...

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 ...

Principal Author’s Biography Shalini Gupta received a BE degree in Electronics and Electrical Communication Engineering from Punjab Engineering College, India. She received a MS degree in Electrical and Computer Engineering from the University of Texas at Austin, where she is currently a PhD student. During her masters, she developed techniques for computer aided diagnosis of breast cancer. She...

Geodesic regression generalizes linear regression to general Riemannian manifolds. Applied to images, it allows for a compact approximation of an image time-series through an initial image and an initial momentum. Geodesic regression requires the definition of a squared residual (squared distance) between the regression geodesic and the measurement images. In principle, this squared distance sh...

We present a heuristic algorithm to compute approximate geodesic distances on a triangular manifold S containing n vertices with partially missing data. The proposed method computes an approximation of the geodesic distance between two vertices pi and pj on S and provides an upper bound of the geodesic distance that is shown to be optimal in the worst case. This yields a relative error bound of...