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Applying Permanent, Robust Stenciled Patterns of Fine Particles to Elastomeric Surfaces

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Multi-stencils fast marching methods: a highly accurate solution to the eikonal equation on cartesian domains.

M Sabry Hassouna¹, A A Farag

¹Department of Electrical and Computer Engineering, University of Louisville, Louisville, KY 40292, USA. msabry@cvip.uofl.edu

IEEE Transactions on Pattern Analysis and Machine Intelligence

|July 14, 2007

Summary

We introduce the multi-stencils fast marching (MSFM) method for accurately solving the Eikonal equation in 2D and 3D. This enhanced fast marching method (FMM) improves upon existing techniques for computer vision applications.

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Area of Science:

Computational mathematics
Computer vision algorithms
Numerical analysis

Background:

Hamilton-Jacobi (HJ) equations, specifically the Eikonal equation, are crucial for various computer vision tasks.
Existing fast marching methods (FMM) offer solutions but can be improved for accuracy in 2D and 3D Cartesian domains.

Purpose of the Study:

To propose an improved fast marching method (FMM) for accurate Eikonal equation solutions.
Introduce the multi-stencils fast marching (MSFM) method for enhanced accuracy in 2D and 3D.
Validate the superiority of MSFM over current state-of-the-art FMM techniques.

Main Methods:

Developed the multi-stencils fast marching (MSFM) method, an advancement of the fast marching method (FMM).
MSFM computes solutions by solving the Eikonal equation along multiple stencils at each grid point, selecting the upwind-satisfying solution.
Employs directional derivatives and higher-order finite difference schemes for non-aligned stencils in 2D and 3D.

Main Results:

The multi-stencils fast marching (MSFM) method demonstrates high accuracy for the Eikonal equation in both 2D and 3D Cartesian domains.
MSFM utilizes multiple stencils covering nearest neighbors (8 in 2D, 26 in 3D) for comprehensive solution computation.
Numerical experiments confirm the enhanced accuracy of MSFM compared to existing FMM-based approaches.

Conclusions:

The proposed multi-stencils fast marching (MSFM) method offers a significant accuracy improvement for solving the Eikonal equation.
MSFM provides a robust and accurate computational tool for computer vision applications relying on Hamilton-Jacobi equations.
The method's effectiveness is validated through extensive numerical experiments, highlighting its advantage over traditional FMM techniques.