New 3D Fractional-Order Zernike Moments for Shape Analysis

We introduce a new class of three-dimensional (3D) fractional-order Zernike moments by extending the radial component of classical 3D Zernike polynomials to admit fractional exponents. We derive explicit formulas for the fractional radial coefficients to establish orthogonality of the new Zernike functions in the unit ball and prove that they form a complete basis for square-integrable functions. We also present a three-way recurrence relation that enables efficient computation of high-order fractional moments. From these moments, we further construct rotationally invariant fractional-order Zernike descriptors for orientation-independent shape analysis. Numerical experiments demonstrate that fractional-order moments improve reconstruction accuracy and enhance discriminative performance compared with classical 3D Zernike moments. These results highlight the effectiveness of fractional Zernike functions as a robust basis for 3D shape representation and retrieval.