Estimating shortest paths and minimal distances on digitized three-dimensional surfaces

Abstract

The estimation of minimal distances and shortest paths between points on a continuous three-dimensional (3D) surface given in digitized form is considered. The suggested approach combines recently developed length estimators for digitized 3D curves with well-known algorithms for computing shortest paths in sparse graphs, to estimate the minimal distances and the corresponding shortest paths in a systematic, computationally efficient way. In comparison to differential geometry techniques for solving the continuous geodesic problem and to computational geometry algorithms for solving the discrete geodesic probelm, the suggested hybrid continuous-discrete approach is well suited to applications in which continuous surfaces are given in digitized form, computation time must be relatively short and an approximate solution is acceptable. In particular, the suggested method can be used to efficiently obtain a good initial approximation as an input to other algorithms, thus improving their speed of convergence and reducing their tendency to converge to insignificant local minima. Some confusion has recently arisen with respect to the relation between the design of length estimators and the optimization of local distances for distance transformations. Since 3D length estimation is a central concept in the suggested approach, an effort is made to resolve discrepancies and to highlight the similarities and differences between these problems. It is shown that while in two dimensions the design of length estimators and the design of local weights for distance transformations are rather similar, in three dimensions digital geometry dictates that these problems are different. The contexts in which each of the two approaches should be preferred are identified.