MATLAB Implementation of Zernike Moments with Code Description

Zernike moments are image feature descriptors based on Zernike polynomials, playing a significant role in image processing and pattern recognition. They achieve rotation-invariant moment features by projecting images onto a set of orthogonal Zernike polynomial bases. The MATLAB implementation of Zernike moments typically involves these key steps: First, constructing the Zernike polynomial basis is essential. Zernike polynomials are a set of orthogonal polynomials defined within the unit circle, consisting of radial polynomials and angular functions. Implementation requires handling polar coordinate transformations and ensuring computations occur within the unit circle boundary. In MATLAB, this can be implemented using meshgrid functions to create coordinate systems and calculating radial distances with sqrt(x.^2 + y.^2). The image preprocessing phase follows. Images must be normalized to fit within the unit circle, with special treatment for pixels outside this boundary. This step is critical for maintaining Zernike moment calculation accuracy. MATLAB's imresize function can be used for normalization, while logical indexing helps handle out-of-bound pixels efficiently. The core calculation involves numerical integration. Since Zernike moments essentially represent the inner product between the image function and Zernike polynomials, appropriate numerical integration methods must be designed. Discrete summation is typically used to approximate continuous integration. MATLAB's array operations and element-wise multiplication (.* operator) facilitate efficient computation of these sums. To enhance computational efficiency, precomputation and storage of Zernike polynomial values can avoid redundant calculations. For higher-order Zernike moments, numerical stability considerations become important. MATLAB's persistent variables or memoization techniques can optimize polynomial value storage. A key characteristic of Zernike moments is their rotation invariance, making them particularly valuable for image recognition. By taking the magnitude of Zernike moments, features completely independent of image rotation can be obtained. This is implemented in MATLAB using the abs() function on complex moment values. In practical applications, Zernike moments are commonly used for image reconstruction, shape analysis, and pattern recognition tasks. By selecting appropriate orders of Zernike moments, a balance between feature dimensionality and descriptive capability can be achieved. MATLAB's matrix computation capabilities are particularly suitable for Zernike moment calculations, where vectorized operations can significantly improve computation speed through built-in functions like sum() and cumsum(). For real-time processing applications, MATLAB Coder can be considered to convert the algorithm into optimized C code.