Computational Methods for 2D/Multi-Dimensional Cyclic Autocorrelation Function and Autocorrelation Density Function

To compute the two-dimensional or multi-dimensional cyclic autocorrelation function and autocorrelation density function, implement the following algorithm with appropriate signal processing libraries: 1. Define the dimensional parameters of the dataset using shape/size functions (e.g., numpy.shape() in Python or size() in MATLAB) to establish the grid structure for analysis. 2. Calculate the dataset mean using statistical functions (mean() or average()) to establish the baseline reference for deviation calculations. 3. Subtract the mean from each data point using element-wise array operations to obtain mean-centered deviations, ensuring proper broadcasting for multi-dimensional arrays. 4. Compute the autocovariance function through vectorized operations that calculate the product of deviations at point pairs separated by distance vectors. This can be implemented using sliding window algorithms or convolution operations (e.g., convn() in MATLAB). 5. Normalize the autocovariance function by dividing it by the dataset variance (calculated using var() function), ensuring dimensional consistency for multi-dimensional cases. 6. Derive the autocorrelation function by normalizing the autocovariance function with the product of corresponding variances, implementing proper indexing for cyclic boundary conditions. 7. Calculate the autocorrelation density function using Fast Fourier Transform (FFT) algorithms (fft2() for 2D, fftn() for n-dimensional) applied to the autocorrelation function, followed by appropriate spectral density normalization. This computational pipeline enables comprehensive spatial analysis of datasets through autocorrelation properties, with implementations requiring careful handling of cyclic boundaries and multi-dimensional array operations.