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.
𝜑= (𝑥 − 𝑥𝑖 )2 +𝜖𝑖 2
Commonly Used RBF’s
• Multi-quadratic
• Inverse Multi-quadratic
• Gaussian
Gaussian
Exp(−𝜖𝑖 2 (𝑥 − 𝑥𝑖 )2 )
The 𝜖- 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 𝜖. The Main idea is that
𝜖 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 𝜖-adaptive method
𝜖𝑖 ≠ 0, 𝑥𝑖 ∈ 𝑋 𝑆
0,
𝑥𝑖 ∈ 𝑆
Gibbs Phenomenon
Example graph for Gibbs phenomenon
Using the 𝜖-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
𝜑= (𝑥 −
Derivative of Multi-quadric
𝑥𝑖 )2 +𝜖𝑖 2
(𝑥−𝑥𝑖 )
(𝑥−𝑥𝑖 )2 +𝜖𝑖 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
(𝑥−𝑥𝑖 )2 +𝜖𝑖 2
Derivative of Inverse Multi-quadric
−(𝑥−𝑥𝑖 )
((𝑥−𝑥𝑖 )2 +𝜖𝑖 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( −𝜖𝑖 2 (𝑥 − 𝑥𝑖 )2 )
Derivative of Gaussian
−2𝜖𝑖 2 𝑥 − 𝑥𝑖 ∗ exp(− 𝜖𝑖 2
∗ 𝑥 − 𝑥𝑖 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