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Radial Basis Functions and Application in Edge Detection. Project by: Chris Cacciatore, Tian Jiang, and Kerenne Paul. Abstract. - PowerPoint PPT Presentation
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Radial Basis Functions and Application in Edge Detection
Project by: Chris Cacciatore, Tian Jiang, and Kerenne Paul
Abstract
This project focuses on the use of Radial Basis Functions in Edge Detection in both one-dimensional and two-dimensional images. We will be using a 2-D iterative RBF edge detection method. We will be varying the point distribution and shape parameter. We also quantify the effects of the accuracy of the edge detection on 2-D images. Furthermore, we study a variety of Radial Basis Functions and their accuracy in Edge Detection.
Radial Basis Functions (RBF’s)
Radial Basis Function• RBF’s use the distances
between points on a given interval and epsilon( shape parameter) as variables.
Commonly Used RBF’s• Multi-quadratic • Inverse Multi-quadratic• Gaussian
Multi-quadratic
=
Gaussian
Exp()
The - adaptive method for jump discontinuity
This method changes the values of the shape parameters depending on the smoothness of f(x). Using this method allows the accuracy of the approximations to be solely determined on . The Main idea is that disappears only near the center of the discontinuity resulting in the basis functions near the discontinuity to become linear. This causes Gibbs oscillations not to appear in the approximation.
Local -adaptive method
Gibbs Phenomenon
Example graph for Gibbs phenomenon
Using the -adaptive method
Begin by finding the jump discontinuity. This can be done by finding the first derivative/slope at the centers.
Example of simple discontinuity
Multi-Quadric RBF
Multi-quadratic Derivative of Multi-quadric
=
M = zeros(N); MD = M; for ix = 1:N for iy = 1:N
M(ix,iy) = sqrt( (x(ix)-x(iy))^2 + (eps(iy))^2); if M(ix,iy) == 0 MD(ix,iy) = 0; else MD(ix,iy) = (x(ix) - x(iy))/M(ix,iy); end
Inverse Multi-Quadric RBF
Inverse Multi-quadric Derivative of Inverse Multi-quadric
M = zeros(N); MD = M; for ix = 1:N for iy = 1:NM(ix,iy) = 1/sqrt( (x(ix)-x(iy))^2 + (eps(iy))^2); if M(ix,iy) == 0 MD(ix,iy) = 0; else MD(ix,iy) = -(x(ix) - x(iy))/sqrt( ((x(ix)-x(iy))^2 + (eps(iy))^2)^3); end
Gaussian RBF
Gaussian Derivative of Gaussian
M = zeros(N); MD = M; for ix = 1:N for iy = 1:NM(ix,iy) = exp(-((eps(iy))^2)*((x(ix)-x(iy))^2)); if M(ix,iy) == 0 MD(ix,iy) = 0; else MD(ix,iy) = -2*((eps(iy))^2)*(x(ix)-x(iy))*exp(-((eps(iy))^2)*(x(ix)-x(iy))^2); end
Comparing the three
Gaussian RBF Inverse Multi-quadric RBF
Multi-quadric RBF Original Image
Comparing the three (cont.)
Kerenne as a multi-quadric RBF
Kerenne as an inverse multi-quadric RBF Kerenne as a Gaussian RBF
Kerenne as a real person
Future work
• Explore further into matrix involvement in Edge Detection
• Look into effects different parts of the code, TwoD_Example1, have on edge maps
References
Vincent Durante, Jae-Hun Jung. An iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities. Appl. Numer. Math. 57 (2007) 213-229