Edge Detection
This numerical tour explores local differential operators (grad, div, laplacian) and their use to perform edge detection.
Contents
Installing toolboxes and setting up the path.
You need to download the following files: signal toolbox and general toolbox.
You need to unzip these toolboxes in your working directory, so that you have toolbox_signal and toolbox_general in your directory.
For Scilab user: you must replace the Matlab comment '%' by its Scilab counterpart '//'.
Recommandation: You should create a text file named for instance numericaltour.sce (in Scilab) or numericaltour.m (in Matlab) to write all the Scilab/Matlab command you want to execute. Then, simply run exec('numericaltour.sce'); (in Scilab) or numericaltour; (in Matlab) to run the commands.
Execute this line only if you are using Matlab.
getd = @(p)path(p,path); % scilab users must *not* execute this
Then you can add the toolboxes to the path.
getd('toolbox_signal/'); getd('toolbox_general/');
Gradient, Laplacian
Local differential operators like gradient, divergence and laplacian are the building blocks for variational image processing using PDEs.
Load an image \(f_0\) of \(N=n \times n\) pixels.
n = 256;
f0 = load_image('boat',n);
For continuous functions, the gradient reads \[ \nabla f(x) = \pa{ \pd{f(x)}{x_1}, \pd{f(x)}{x_2} } \in \RR^2. \]
We discretize this differential operator using first order finite differences. \[ (\nabla f)_i = ( f_{i_1,i_2}-f_{i_1-1,i_2}, f_{i_1,i_2}-f_{i_1,i_2-1} ) \in \RR^2. \] Note that for simplity we use periodic boundary conditions.
Compute its gradient, using finite differences.
s = [n 1:n-1]; nabla = @(f)cat(3, f-f(s,:), f-f(:,s));
One thus has \( \nabla : \RR^N \mapsto \RR^{N \times 2}. \)
v = nabla(f0);
One can display each of its components.
clf; imageplot(v(:,:,1), 'd/dx', 1,2,1); imageplot(v(:,:,2), 'd/dy', 1,2,2);
One can also display it using a color image.
clf; imageplot(v);
One can display its magnitude \(\norm{\nabla f(x)}\), which is large near edges.
clf; imageplot( sqrt( sum3(v.^2,3) ) );
The divergence operator maps vector field to images. For continuous vector fields \(v(x) \in \RR^2\), it is defined as \[ \text{div}(v)(x) = \pd{v_1(x)}{x_1} + \pd{v_2(x)}{x_2} \in \RR. \] It is minus the adjoint of the gadient, i.e. \(\text{div} = - \nabla^*\).
It is discretized, for \(v=(v^1,v^2)\) as \[ \text{div}(v)_i = v^1_{i_1+1,i_2} + v^2_{i_1,i_2+1}. \]
t = [2:n 1]; div = @(v)v(t,:,1)-v(:,:,1) + v(:,t,2)-v(:,:,2);
The Laplacian operatore is defined as \(\Delta=\text{div} \circ \nabla = -\nabla^* \circ \nabla\). It is thus a negative symmetric operator.
delta = @(f)div(nabla(f));
Display \(\Delta f_0\).
clf; imageplot(delta(f0));
Check that the relation \( \norm{\nabla f} = - \dotp{\Delta f}{f}. \)
dotp = @(a,b)sum(a(:).*b(:));
fprintf('Should be 0: %.3i\n', dotp(nabla(f0), nabla(f0)) + dotp(delta(f0),f0) );
Should be 0: -2.012e-07
Gradient Based Edge Detectiors
The simplest edge detectors only make use of the first order derivatives.
d = sqrt( sum3(nabla(f0).^2,3) );
A very simple edge detector is obtained by simply thresholding the gradient magnitude.
clf; imageplot( double(d>max(d(:))/10), 'High threshold', 1,2,1); imageplot( double(d>max(d(:))/5 ), 'Low threshold', 1,2,2);
The zero crossing of the Laplacian is a well known edge detector. This requires first blurring the image (which is equivalent to blurring the laplacian). It can be shown, however, that this operator will also return false edges corresponding to local minima of the gradient magnitude. Moreover, this operator will give poor localization at curved edges.
% compute the laplacian sigma = 6; Lh = perform_blurring(delta(f0), sigma); % display the level sets clf; plot_levelset(Lh,0,f0);
Hessian Based Edge Detectors
More advanced edge detectors make use of the second order derivatives.
A better edge detector are the zero-crossings of the second-order directional derivative in the gradient direction.
Exercice 1: (the solution is exo1.m) Compute the Hessian matrix of second derivative. It is an n x n x 3 matrix containing d^2/dx^2, d^2/dy^2, d^2/dxdy.
exo1;
Exercice 2: (the solution is exo2.m) Compute the directional derivative in the direction of the gradient, and display its zero crossing. Warning: the image needs to be blurred before the computation.
exo2;
