It has been stated here, that the the so-called Kravchuk transform is very important in the field of image processing and possibly in signal processing in general.

I can hardly find any description about this (e.g. not mentioned in Wikipedia, etc.).

It seems to be mentioned in this paper, for example.


Transliterations of Ukrainian names have different avatars in English (and in others languages as well). You can find Kravchuk polynomials, and other papers like On Krawtchouk Transforms or Krawtchouk polynomials and Krawtchouk matrices. You can find as well Kravchuk orthogonal polynomials.

As they form an orthogonal basis of polynomials (as well as many others, listed on polpak: Bernoulli, Bernstein, Tchebychev, Hermite, Laguerre, Legendre, Zernike), they are candidates for a transform. Derived moments are used in image processing, and the following paper seems to have a wide audience:

A new set of orthogonal moments based on the discrete classical Krawtchouk polynomials is introduced. The Krawtchouk polynomials are scaled to ensure numerical stability, thus creating a set of weighted Krawtchouk polynomials. The set of proposed Krawtchouk moments is then derived from the weighted Krawtchouk polynomials. The orthogonality of the proposed moments ensures minimal information redundancy. No numerical approximation is involved in deriving the moments, since the weighted Krawtchouk polynomials are discrete. These properties make the Krawtchouk moments well suited as pattern features in the analysis of two-dimensional images. It is shown that the Krawtchouk moments can be employed to extract local features of an image, unlike other orthogonal moments, which generally capture the global features. The computational aspects of the moments using the recursive and symmetry properties are discussed. The theoretical framework is validated by an experiment on image reconstruction using Krawtchouk moments and the results are compared to that of Zernike, pseudo-Zernike, Legendre, and Tchebyscheff moments. Krawtchouk moment invariants are constructed using a linear combination of geometric moment invariants; an object recognition experiment shows Krawtchouk moment invariants perform significantly better than Hu's moment invariants in both noise-free and noisy conditions.

Later, you can read:

This paper shows how Hahn moments provide a unified understanding of the recently introduced Chebyshev and Krawtchouk moments. The two latter moments can be obtained as particular cases of Hahn moments with the appropriate parameter settings and this fact implies that Hahn moments encompass all their properties. The aim of this paper is twofold: (1) To show how Hahn moments, as a generalization of Chebyshev and Krawtchouk moments, can be used for global and local feature extraction and (2) to show how Hahn moments can be incorporated into the framework of normalized convolution to analyze local structures of irregularly sampled signals.

In Wikipedia's Discrete Fourier transform we find:

The choice of eigenvectors of the DFT matrix has become important in recent years in order to define a discrete analogue of the fractional Fourier transform—the DFT matrix can be taken to fractional powers by exponentiating the eigenvalues (e.g., Rubio and Santhanam, 2005). For the continuous Fourier transform, the natural orthogonal eigenfunctions are the Hermite functions, so various discrete analogues of these have been employed as the eigenvectors of the DFT, such as the Kravchuk polynomials (Atakishiyev and Wolf, 1997). The "best" choice of eigenvectors to define a fractional discrete Fourier transform remains an open question, however.

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  • $\begingroup$ Can we said that "Kravchouk transform" usually is a "fractional Fourier transform"? $\endgroup$ – Machupicchu Jul 20 '19 at 15:54
  • $\begingroup$ There are ambiguities on what people call "fractional Fourier transforms", and I am not a practitioner of the Fractional Fourier–Kravchuk transform, but I doubt there is a one-to-one correspondance $\endgroup$ – Laurent Duval Jul 20 '19 at 16:02
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    $\begingroup$ ah ok. It seems strange that they cite this obscure Kravchouk in phys.org/news/2019-07-quantum-technology.html as it does not seem to be known by many DSP people ? $\endgroup$ – Machupicchu Jul 20 '19 at 16:19
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    $\begingroup$ oh ok. So you would say it is "overstated" as important? $\endgroup$ – Machupicchu Jul 20 '19 at 16:32
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    $\begingroup$ This is another area where keeping the distinction and relationship between the discrete and continuous case is important. I like to use Legendre polynomials. I tend to think of them as "the orthogonal Taylor series", however, the discrete version in not orthogonal. You need to apply something like Gram-Schmidt (G-S) to rectify that for discrete usage. Then keeping the same N becomes important. $\endgroup$ – Cedron Dawg Jul 20 '19 at 16:37

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