Abstract
It is well known that the problem of finding the shortest path amid 3-D polyhedral obstacles is a NP-Hard problem. In this paper, we propose an efficient algorithm to find the globally shortest path by solving stochastic differential equations (SDEs). The main idea is based on the simple structure of the shortest path, namely it consists of straight line segments connected by junctions on the edges of the polyhedral obstacles. Thus, finding the shortest path is equivalent to determining the junctions points. This reduces the originally infinite dimensional problem to a finite dimensional one. We use the gradient descent method in conjunction with Intermittent Diffusion (ID), a global optimization strategy, to deduce SDEs for the globally optimal solution. Compared to the existing methods, our algorithm is efficient, easier to implement, and able to obtain the solution with any desirable precisions.
Dedicated to Professor RAYMOND H. CHAN on the occasion of his 60th birthday
This work is partially supported by NSF Awards DMS-1419027, DMS-1620345, and ONR Award N000141310408
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Chow, SN., Lu, J., Zhou, HM. (2021). The Shortest Path AMID 3-D Polyhedral Obstacles. In: Tai, XC., Wei, S., Liu, H. (eds) Mathematical Methods in Image Processing and Inverse Problems. IPIP 2018. Springer Proceedings in Mathematics & Statistics, vol 360. Springer, Singapore. https://doi.org/10.1007/978-981-16-2701-9_10
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DOI: https://doi.org/10.1007/978-981-16-2701-9_10
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