Dear Anton, 

It is most complicated and controversial how should points gives a similar or identical shape. There for need to understand the basics of morpho structure first. 

The discrete Fourier transform (DFT) is widely used in data compression, pattern recognition, and image reconstruction. The study of properties of the DFT shows that this transform can be generalized and used mere effectively for solving many problems in such areas as signal and image processing. The traditional N-point DFT is defined as the decomposition of the signal by N roots of the unit, which are on the unit circle. On different stages of the DFT, the data of the signal are multiplied by the exponential factors, or rotated around the circles. 

Try to use elliptic discrete Fourier transform (EDFT) in which basic 2×2 transformations that are not given transformations, but rather rotations around ellipses. The elliptic transformation is therefore defined by different Nth

roots of the identity matrix 2×2, matrix whose groups of motion move a point around the ellipses.  The EDFT preserves main properties of the DFT,

such as shifting and energy preservation among others. The EDFT distinguishes the carrying frequencies of the signal in both real and imaginary parts better than the DFT. It also has a simple inverse matrix. It is parameterized and includes the DFT, as a partial case. The preliminary results show that by using different parameters, the EDFT can be used effectively for solving many problems in signal and image processing field, which includes image enhancement, filtration, encryption and others. You have to go with similar way, all depend on your model

I attached  two file which me be helpful to you.

Warm regards

-Paul  

Dear Paul,

In the question I've mentioned that I've found harmonics of Elliptic FT. Now I try to use them for separating/dividing shapes (outlines) into statistically different groups. I need estimates of that differences, which allow to say that outlines in group A differs from outlines in group B by, for example, ...% (or any other estimates another than percents).

As I understand from the papers you've sent I need to use PCA?

Антон, в свое время подобными задачами занимался В.В. Меншуткин, он мне рассказывал. Посмотри у него для начало в книжке Искусство моделирования

You may use PCA to specify the important points and then goes for further analysis. 

If you want to tell whether the outlines of the taxa are different, you would do a multivariate analysis of variance.  People using Fourier Analysis commonly first calculate the principal components of the harmonics and use those (or a subset of them) in the Manova and in a Canonical Variates Analysis. These two methods, Manova and CVA can suggest greater separation among taxa than merited when the sample size is small relative to the number of variables, so the PCA is used to reduce the dimensionality of the data. 

Dear Miriam, could I apply Cluster analysis to PCA of Fourier coefficients?

PCA is just a rotation of your original variables and rotation has no effect on the distances between points. So you could use those PCs (as long as you use all of them--if not, you won't accurately reproduce the distances). Or you could just calculate the pairwise distances between the points. 

Dear Anton:

I suggest you that you look at my Book Fourier Descriptors and their applications in Biology, Cambridge University Press (1997) for background and some case studies using elliptical Fourier analysis (EFA). Also, if you do an internet search you will find a large number of studies that have now used EFA. I hope this is useful for you. If you have any questions you can reach me at [email protected].

Pete E. Lestrel, PhD