Radial Basis Functions and Application in Edge

Detection

Chris Catiatore, 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

Multi-Quadric RBF: Inverse Multi-Quadric RBF:

Gaussian RBF: ()

Shape Parameter: ε Uses distance between points on

a given interval

Is used as a variable in RBF representation

Epsilon Variable

epsilon = 0 epsilon = 0.05 epsilon = 0.1

epsilon = 0.2 epsilon = 0.3 epsilon = 0

Epsilon Variable

epsilon = 0

epsilon = 2

epsilon = 0.05epsilon = 0.01

epsilon = 0.1 epsilon = 1

Edge map from x-direction

Edge map from y-direction

Edge Map at 0.1 (x-direction and y-direction)

Edge map from x-direction, y-direction and total

Example of Gibbs Phenomenom

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

Future works:• Start using other types of Radial Basis Functions• Research more about matrix involvement in the code• Try to understand more about the code itself and what’s causing the variation in edge maps