Stable, fast computation of high-order Zernike moments using a recursive method

• This study has proposed a recursive method for fast computation of Zernike moments.
• The idea of Pascal׳s triangle is introduced to pre-calculate the binomial coefficients.
• Symmetry property and Farey sequence are applied to speed up the computation.
• The proposed method can yield accurate values of the high-order Zernike moments.
• The proposed took 5.398 seconds to compute the top 500-order Zernike moments.

Zernike moments and Zernike polynomials have been widely applied in the fields of image processing and pattern recognition. When high-order Zernike moments are computed, both computing speed and numerical accuracy become inferior. The main purpose of this study is to propose a stable, fast method for computing high-order Zernike moments. Based on the recursive formulas for computing Zernike radial polynomials, this study develops stable, fast algorithms to compute Zernike moments. Symmetry under group action and Farey sequence are both applied to shorten the computing time. The experimental results show that the proposed method took 5.292 seconds to compute the top 500-order Zernike moments of an image with 512×512 pixels. The normalized mean square error is 0.00124846 if 450-order moments are used to reconstruct the image. When computing the high-order Zernike moments, the proposed method outperformed other compared methods in terms of speed and accuracy.