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Mathematical Morphology
This numerical tour explores mathematical morphology of binary images.
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);
Then you can add the toolboxes to the path.
getd('toolbox_signal/');
getd('toolbox_general/');
Binary Images and Structuring Element
Here we process binary images using local operator defined using a structuring element, which is here chosen to be a discrete
disk of varying radius.
Load an image
n = 256;
M = rescale( load_image('cortex',n) );
Display.
clf;
imageplot(M);
Make it binary.
M = double(M>.45);
Display.
clf;
imageplot(M);
Round structuring element.
wmax = 7;
[Y,X] = meshgrid(-wmax:wmax, -wmax:wmax);
normalize = @(x)x/sum(x(:));
strel = @(w)normalize( double( X.^2+Y.^2<=w^2 ) );
Exercice 1: (check the solution) Display structuring elements of increasing sizes.
exo1;
Dillation
A dilation corresponds to take the maximum value of the image aroung each pixel, in a region equal to the structuring element.
It can be implemented using a convolution with the structuring element followed by a thresholding.
dillation=@(x,w)double(perform_convolution(x,strel(w))>0);
Md = dillation(M,2);
Display.
clf;
imageplot(Md);
Exercice 2: (check the solution) Test with structing elements of increasing size.
exo2;
Errosion
An errosion corresponds to take the maximum value of the image aroung each pixel, in a region equal to the structuring element.
It can be implemented using a convolution with the structuring element followed by a thresholding.
errosion=@(x,w)double( perform_convolution(x,strel(w))>=.999 );
Me = errosion(M,2);
Display.
clf;
imageplot(Me);
Exercice 3: (check the solution) Test with structing elements of increasing size.
exo3;
Opening
An opening smooth the boundary of object (and remove small object) by performing an errosion and then a dillation.
Define a shortcut.
opening = @(x,w)dillation(errosion(x,w),w);
Perform the opening, here using a very small disk.
w = 1;
Mo = opening(M,w);
Display.
clf;
imageplot(Mo);
Exercice 4: (check the solution) Test with structing elements of increasing size.
exo4;
