A Blend of Analytical and Numerical Methods to Compute Orthogonal Image Moments over a Unit Disk

Abstract

Accurate image moment computation is critical because they are used in a variety of fields, including image reconstruction and object recognition. Orthogonal polynomials are frequently used to compute moments due to their numerous intersecting and important theoretical properties. In polar coordinate systems, orthogonal polynomials such as the Polar Harmonic Fourier Transform (PHFT) and Polar Harmonic Transformation (PHT) are defined. However, the images are defined by a Cartesian coordinate system. To obtain image moments, a double integration over a unit circle over the product of the PHFT or PHT and the image function must be performed. The choice of double integration techniques and domain for the double integral has a significant effect on the precision of the computed moments. We have proposed a method for using the entire unit circle as an integration domain in this study, for image reconstruction. We used the PHFT and PHT to apply this technique to computing moments for the reconstruction. The proposed method outperforms other state-of-the-art methods, including Gaussian quadrature numerical integration method (GQM) and zeroth order approximation (ZOA) on variety of scenarios using three benchmarked image sets. We have demonstrated experimentally that this technique significantly improves the accuracy of image moments in higher order moments, as measured by reduced mean squared error (MSE). Additionally, the proposed method significantly improved the moment’s rotational (an improvement of 1-3% as compared to GQM and an improvement of 37-93% as compared to ZOA) and scaling invariance (an improvement of 0-500% as compared to GQM and an improvement of 3-7000% as compared to ZOA).