Radial Basis Functions and Application in Edge Detection

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Radial Basis Functions and Application in Edge Detection

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Radial Basis Functions and Application in Edge Detection

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Radial Basis Functions and Application in Edge Detection

Chris Cacciatore, Tian Jiang, and Kerenne Paul

University of Massachusetts Dartmouth Department of Mathematics

Saeja Kim and Sigal Gottlieb

RBF’s and their Results

Original Image

We have used three different radial basis functions and their results at different values of epsilon. Each function requires different values of epsilon to render a recognizable edge map.

Abstract

This project focuses on the use of Radial Basis Functions in Edge Detection in 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 while also quantifying 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.

Original image

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

Multi-quadric

Radial Basis Functions (RBF’s)

Gaussian

- Radial Basis Functions use the distance between two points on a given interval and epsilon (shape parameter) as variables. Three common types of RBF’s are Multi-quadric, Inverse Multi-quadric, and Gaussian.
- Multi-quadric
- =
- Gaussian
- Exp()

Inverse Multi-quadric

M = zeros(N); MD = M;

for ix = 1:N

for iy = 1:N

M(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

Inverse Multi-quadric

Gaussian

M = zeros(N); MD = M;

for ix = 1:N

for iy = 1:N

M(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

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.

Future Work

- Explore further into matrix involvement in edge detection
- Look into effects different parts of the code, TwoD_Example1, have on edge maps
- Test for necessary changes in epsilon for different sized images