On Constructing the Value Function for Optimal Trajectory Problem and its Application to Image Processing

TL;DR

This paper introduces a generalized numerical algorithm for solving Hamilton-Jacobi equations related to optimal trajectory problems, with applications demonstrated in image processing tasks like fingerprint verification.

Contribution

It presents a novel, generalized algorithm building on Fast Marching and Ordered Upwind Methods for trajectory optimization and image processing applications.

Findings

Algorithm effectively computes optimal trajectories minimizing combined costs.

Successful application to fingerprint verification demonstrates practical utility.

Generalizes existing methods for broader trajectory and image processing problems.

Abstract

We proposed an algorithm for solving Hamilton-Jacobi equation associated to an optimal trajectory problem for a vehicle moving inside the pre-specified domain with the speed depending upon the direction of the motion and current position of the vehicle. The dynamics of the vehicle is defined by an ordinary differential equation, the right hand of which is given by product of control(a time dependent fuction) and a function dependent on trajectory and control. At some unspecified terminal time, the vehicle reaches the boundary of the pre-specified domain and incurs a terminal cost. We also associate the traveling cost with a type of integral to the trajectory followed by vehicle. We are interested in a numerical method for finding a trajectory that minimizes the sum of the traveling cost and terminal cost. We developed an algorithm solving the value function for general trajectory optimization problem. Our algorithm is closely related to the Tsitsiklis's Fast Marching Method and J. A. Sethian's OUM and SLF-LLL[1-4] and is a generalization of them. On the basis of these results, We applied our algorithm to the image processing such as fingerprint verification.