Geodesics in heat

Abstract

We introduce the heat method for computing the shortest geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it is based on solving a pair of standard linear elliptic problems. The method represents a significant breakthrough in the practical computation of distance on a wide variety of geometric domains, since the resulting linear systems can be prefactored once and subsequently solved in near-linear time. In practice, distance can be updated via the heat method an order of magnitude faster than with state-of-the-art methods while maintaining a comparable level of accuracy. We provide numerical evidence that the method converges to the exact geodesic distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where more regularity is required.

Earth mover's distances on discrete surfaces

“…Like the biharmonic distance, our distances are smooth and follow the natural cross-sections of the shape even in more distant areas. Similarly to geodesic distance, we find that even d 0 W has isotropic and evenly-spaced level [Crane et al 2013] as d h . sets even though it is the lowest-order approximation of dg; this is in contrast to biharmonic distance that may have unevenly-spaced isocontours at different parts of a mesh.…”

“…Finally, Figure 8(c) shows insensitivity to tessellation; the distance remains almost unchanged as the mesh is refined considerably. Figure 9 compares our technique to [Crane et al 2013]. Metric properties hold for our distances at all levels of spectral truncation even after discretization, while their smoothed geodesics at larger and larger diffusion times no longer benefit from an infinitesimal relationship with geodesic distances.…”

“…In [Crane et al 2013] an approximation of geodesic distance was formulated by integrating the normalized gradient field of the heat kernel. This approximation gives smoothed versions of the geodesic distance parameterized by the time parameter of the heat kernel.…”

“…The drawbacks of geodesic and spectral distances indicate that a hybrid approach is needed. The approximation of geodesic distance formulated in [Crane et al 2013] is an important step in this direction. While these approximations are smoothed versions of the geodesic distance that are robust and globally shape-aware, they may not be symmetric or satisfy the triangle inequality.…”

Shape conforming volumetric interpolation with interior distances

“…Also, the inner distance can be defined by using other ways, for example, in terms of Eikonal equation [9] and heat flow [10]. In general, as noted in [7], the interior distance can be expressed in a continuous setting, however in practical applications usually approximations are used.…”