An Edge-Based Formulation for the Exact Computation of High-Order Zernike Moments of 2D Shapes and Images

Abstract

Zernike moments are widely used rotation-invariant descriptors for shape and image analysis, but their standard computation relies on a pixel-based quadrature that treats each pixel as a point mass located at its center. This approximation introduces spatial aliasing that increases with moment order, degrading image reconstruction and reducing the discriminative power of high-order moments. We present an edge-based formulation that eliminates this source of error by applying Green's theorem to transform the two-dimensional area integral defining a Zernike moment into a sum of one-dimensional integrals along image boundaries. The resulting framework applies equally to polygonal shapes, binary images, grayscale images, and color images. We derive recurrence relations for the required radial primitives and show that the transformed edge integrands are polynomial functions, allowing their exact evaluation using Clenshaw--Curtis quadrature. The proposed method computes Zernike moments from polygonal image representations without the spatial discretization errors inherent to conventional pixel-based approaches and remains computationally practical for high-order moments. Numerical experiments on image reconstruction, shape analysis, and character classification demonstrate that the proposed formulation matches the accuracy of classical methods at low orders while remaining stable at orders for which pixel-based moments suffer from significant aliasing and numerical degradation.