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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 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 Multi-quadratic • RBF’s use the distances between points on a given interval and epsilon( shape parameter) as variables. = ( − )2 + 2 Commonly Used RBF’s • Multi-quadratic • Inverse Multi-quadratic • Gaussian Gaussian Exp(− 2 ( − )2 ) 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 ≠ 0, ∈ 0, ∈ 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 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-quadratic = ( − Derivative of Multi-quadric )2 + 2 (− ) (− )2 + 2 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 Inverse Multi-quadric 1 (− )2 + 2 Derivative of Inverse Multi-quadric −(− ) ((− )2 + 2 )3 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 Gaussian Exp( − 2 ( − )2 ) Derivative of Gaussian −2 2 − ∗ exp(− 2 ∗ − 2 ) 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, 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