Radial Basis Functions and Application in Edge Detection

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 twodimensional 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

Gibbs Phenomenon Example graph for Gibbs phenomenon

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 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 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

Gaussian RBF 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

Comparing the three Original Image Gaussian RBF Multi-quadric RBF Inverse Multi-quadric RBF

Comparing the three (cont. ) Kerenne as a real person Kerenne as a Gaussian RBF Kerenne as a multi-quadric RBF Kerenne as an inverse multi-quadric RBF

Future work • Explore further into matrix involvement in Edge Detection • Look into effects different parts of the code, Two. D_Example 1, 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