Implementation of Three-Dimensional Fast Fourier Transform

This program is developed based on the fundamental one-dimensional FFT algorithm. The enhanced implementation utilizes a multidimensional approach by performing sequential 1D FFTs along each axis (x, y, z dimensions), leveraging the separability property of Fourier transforms. The improved program not only computes three-dimensional FFTs but can be extended to handle higher-dimensional Fourier transforms through recursive dimensional processing, thereby expanding its application scope across various scientific and engineering domains. Additionally, the optimized implementation incorporates algorithmic enhancements such as butterfly operations with reduced memory access patterns and precomputed twiddle factors, resulting in significantly improved computational efficiency and numerical accuracy. These optimizations enable users to process large volumetric datasets more rapidly while maintaining precision. Consequently, this upgraded program offers enhanced functionality and convenience, facilitating broader and deeper applications across multiple fields including medical imaging, computational physics, and signal processing.