MATLAB Implementation of Zernike Moments with Code Optimization Techniques

Zernike moments are image descriptors based on Zernike polynomials that exhibit excellent rotation-invariant properties in pattern recognition and image analysis applications. The MATLAB implementation of Zernike moments primarily involves three crucial steps: Coordinate System Transformation The implementation requires converting image pixel coordinates to polar coordinates and normalizing them within a unit circle. This process involves transforming Cartesian coordinates (x,y) to polar coordinates (r,θ) while ensuring all points fall within a circle of radius 1. In code implementation, this typically involves centering the coordinate system and scaling coordinates using maximum radial distance calculations. Zernike Polynomial Computation Zernike polynomials consist of radial polynomials and angular functions with orthogonal properties. The computation requires determining the order n and repetition m, with constraints that n-m must be even and m≤n. The radial polynomials are efficiently computed using recursive methods in MATLAB, where precomputation of factorial terms and recursive relationships significantly improves computational efficiency. Moment Value Calculation For each valid point in the image, the corresponding Zernike polynomial value is calculated and multiplied by the pixel's grayscale value. The real and imaginary parts of Zernike moments are obtained through integration (practically implemented as summation). For higher-order moments, numerical stability issues must be addressed using appropriate scaling factors and precision control. During implementation, performance can be optimized by precomputing radius and angle components. For large-scale images, block processing strategies can be employed to manage memory usage. The resulting Zernike moment features are suitable for subsequent applications such as shape recognition, texture analysis, and pattern classification tasks. Code implementation typically involves vectorization techniques to leverage MATLAB's matrix operations for improved computational speed.