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