Zernike moments of order n with repetition m for an image function f(x,y) defined on a square somain N \times N are defined as
A_{nm} = \frac{n+1}{\pi} \underset{x^2+y^2 \le 1} {\int \int} f(x,y) V^*_{nm}(x,y) dxdy
Consider a set of orthogonal functions with simple rotation properties which forms a complete orthogonal set over the interior of the unit circle. The form of these polynomials is
V_{nm} (x,y) = V_{nm} (\rho \:\text{sin} \theta, \rho \:\text{cos} \theta) = R_{nm} (\rho) e^{j m \theta}
where V^*_{nm} is the complex conjugate of the complex polynomials V_{nm}(x,y)
V_{nm}(x,y)=R_{nm}(r) e^{j m \theta}
where r=\sqrt{x^2+y^2}, 0 \leqslant r \leqslant 1, \sqrt{-1}, n \geqslant 0, |m| \leqslant n, n-m=even, and \theta = \text{arctg} \frac{y}{x}.
Zernike real valued radial polynomials R_{nm}(r) are given by
R_{nm}(r) = \underset{k=|m| \atop \: n-k=even} {\sum ^n} B_{nmk}r^k
where
B_{nmk} = \frac{ -1^{\frac{n-k}{2}}(\frac{n+k}{2})! } { (\frac{n-k}{2})! (\frac{k+m}{2})! (\frac{k-m}{2})! }
Approximating the double integration for the discrete image function on the domain of size N \times N, we get
\hat Z_{nm} = \frac {n+1}{\pi} \sum _{i=0}^{N-1} \underset{x_i^2+y_j^2 \leqslant 1}{\sum _{j=0}^{N-1}} f(x_i,y_j)V_{nm}^* (x_i,y_j) \delta a
where \delta A = dxdy is an elemental area of the normalized square image in discrete form when a square image of any size is mapped on the unit disk. If the image is square-shaped and R = \frac {N}{\sqrt{2}} is the enclosing circle radius, then \delta A = \frac{1}{R^2}.
A set of features with prefix ZERNIKE2D is calculated. In the source code, the order is controlled with class ZernikeFeature's member constant ZernikeFeature::ZERNIKE2D_ORDER (default value: 9), and the number of repetitions m is controlled via constant ZernikeFeature::NUM_FEATURE_VALS:
ZernikeFeature::NUM_FEATURE_VALS = ZernikeFeature::ZERNIKE2D_ORDER \times m.
A. Tahmasbi, F. Saki, S.B. Shokouhi. Classification of benign and malignant masses based on Zernike moments. Comput Biol Med. 2011 Aug;41(8): 726-35. doi: 10.1016/j.compbiomed.2011.06.009. Epub 2011 Jul 1. PMID: 21722886.
C. Singh, E. Walia. Algorithms for fast computation of Zernike moments and their numerical stability. Image and Vision Computing, Volume 29, Issue 4, 2011: 251-259, ISSN 0262-8856, https://doi.org/10.1016/j.imavis.2010.10.003.