2,542 research outputs found
A Rotating-Grid Upwind Fast Sweeping Scheme for a Class of Hamilton-Jacobi Equations
We present a fast sweeping method for a class of Hamilton-Jacobi equations
that arise from time-independent problems in optimal control theory. The basic
method in two dimensions uses a four point stencil and is extremely simple to
implement. We test our basic method against Eikonal equations in different
norms, and then suggest a general method for rotating the grid and using
additional approximations to the derivatives in different directions in order
to more accurately capture characteristic flow. We display the utility of our
method by applying it to relevant problems from engineering
Sub-Riemannian Fast Marching in SE(2)
We propose a Fast Marching based implementation for computing sub-Riemanninan
(SR) geodesics in the roto-translation group SE(2), with a metric depending on
a cost induced by the image data. The key ingredient is a Riemannian
approximation of the SR-metric. Then, a state of the art Fast Marching solver
that is able to deal with extreme anisotropies is used to compute a SR-distance
map as the solution of a corresponding eikonal equation. Subsequent
backtracking on the distance map gives the geodesics. To validate the method,
we consider the uniform cost case in which exact formulas for SR-geodesics are
known and we show remarkable accuracy of the numerically computed SR-spheres.
We also show a dramatic decrease in computational time with respect to a
previous PDE-based iterative approach. Regarding image analysis applications,
we show the potential of considering these data adaptive geodesics for a fully
automated retinal vessel tree segmentation.Comment: CIARP 201
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