What is a Kravchuk transform and how is it related to Fourier transforms?
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:


Image analysis by Krawtchouk moments, IEEE Transactions on Image Processing 2003



  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: 


Image Analysis Using Hahn Moments, IEEE Transactions on Pattern Analysis and Machine, 2007, where Hahn moments generalize Chebyshev and Krawtchouk moments:



  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.