This is a good question.

There are a number of ways to extract and classify shapes.

> Measure the similarity distance between a known and an unknown shape. A detailed presentation of similarity distance is given in my book Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces (now in press at Springer), available in December or by mid-January.

> one-dimensional function for shape representation (centroid, gradient orientation, gradient, gradient magnitude in a particular direction, shape area, shape diameter, Zernike moment, triangle area, quadrilateral area chord length, and so on).

> polynomial approximation (distance threshold method tunneling method, polygon evaluatiaon, and so on).

> spatial interrelation (bounding box, inner box, chord distribution and so on).

> Moments (boundary moments, region moments such as Zernike, invariant moments, and so on).

> scale space methods (e.g., intersection points map).

> shape transform domain (e.g., Fourier descriptors, wavelet transform, shapelets, and so on).

For more about this, see the attached paper.

Dear Abdelbar Nasri, Check the links provided below. May help you. Good Luck!

http://www.acsu.buffalo.edu/~nagi/pubs/stepfeature.pdf

http://graphics.stanford.edu/courses/cs468-03-fall/Papers/Pauly_FeatureExtraction.pdf

This is a good question.

There are a number of ways to extract and classify shapes.

> Measure the similarity distance between a known and an unknown shape. A detailed presentation of similarity distance is given in my book Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces (now in press at Springer), available in December or by mid-January.

> one-dimensional function for shape representation (centroid, gradient orientation, gradient, gradient magnitude in a particular direction, shape area, shape diameter, Zernike moment, triangle area, quadrilateral area chord length, and so on).

> polynomial approximation (distance threshold method tunneling method, polygon evaluatiaon, and so on).

> spatial interrelation (bounding box, inner box, chord distribution and so on).

> Moments (boundary moments, region moments such as Zernike, invariant moments, and so on).

> scale space methods (e.g., intersection points map).

> shape transform domain (e.g., Fourier descriptors, wavelet transform, shapelets, and so on).

For more about this, see the attached paper.

thank for this pertinent and helpful explanation.

I'll start with the first group of features:

Sir James Peters: " one-dimensional function for shape representation (centroid, gradient orientation, gradient, gradient magnitude in a particular direction, shape area, shape diameter, Zernike moment, triangle area, quadrilateral area chord length, and so on)."

I don't know how can i use this represtation to find relation between all shape of image?

in addition, i want to add same information of my question:

the image in attachment is generated with some unknown rules and i would like to find these rules with analysing here generated image.

"I'm sorry for this bad language "

Abdelbar,

You have asked another good question.

To find the unknown rules used to generate the sample image, it will be helpful to develop some shape generation rules in general for a particular class of images.

For the sample image, one can start by introducing rules that can be used to generate a parallelogram for each of the cases represented by your sample image.

Each rule would be of the following form:

Let X be a set of parallelograms.

Let x be a shape in X.

Let \Phi be a set of probe functions that represent features of each x.

Let \Phi(x) be a feature vector that descibes x in X.

After you have introduced rules to fit the cases represented by your sample image, then the features of each parallogram would be lead to a feature vector that describes a particular shape.

Let y be a randomly generated parallogram.

Let classify: \mathbb{R}^n x \mathbb{R}^n---> \mathbb{R}.

classify(\Phi(y)) = 1, if the description of \Phi(y) matches some \Phi(x).

Rule:

if \Phi(y) = \Phi(x), then classify(\Phi(y)) = 1.

Do you need to classify the shapes or to describe them? And do you have a more precise definition of the types of shapes in your images - are they always simple polygons? Do you need scale/rotation invariance? Are there possibly strong noise, complex backgrounds, or geometric distortions? These I think are all questions you should consider prior to selecting an algorithm.

Thanks to all for this helpfull discussion;

Sir Gaspar, I suppose in a first time that I have synthetic images, as my sample example, with complex polygon in different scale and rotation invariance. in addition all images of my data have a simple background.

I began with this feature V(location and orientation of each vertex; all angle between non parallel vertex; shape type; nomber of shape; ...).

And i want add geometry relation between shapes.

Dear James: have you an example of shapelets and zernik momment?

Abdelbar,

Yes, there are many beautiful examples associated with Zernike polynomials defined on the unit disk.

See the attached image for 4 orders of Zernike polynomials.

For more about shape classification using Zernike moments, see the attached 2011 paper by Michael Vorobyov.