FFTlles

FFTlles is a theoretical framework in signal processing and numerical analysis that describes a family of Fourier-transform inspired algorithms designed to operate efficiently on data arranged on or near regular lattices. The concept combines the speed of classic fast Fourier transform techniques with lattice-based decomposition methods to handle boundary conditions and nonuniform sampling more gracefully than standard FFTs.

Method: The approach partitions the transform domain into lattice tiles and uses precomputed, orientation-specific kernels to interpolate or approximate the spectrum from lattice samples. By exploiting convolution theorems on the lattice and limiting the number of corrective terms, FFTlles aims to achieve near-FFT computational complexity in favorable cases. For irregular grids, local remapping and sparse representations are employed to maintain accuracy while keeping runtimes practical.

Performance and scope: In ideal cases with data aligned to a regular lattice, computational cost scales similarly to O(N log N) as with the FFT; in more general situations, accuracy is traded for speed, yielding approximate spectra. The method is compatible with parallel hardware such as CPUs and GPUs and can be extended to multiple dimensions.

Applications: The framework is discussed in theoretical contexts for real-time spectrum estimation, high-dimensional image and volume processing, seismic imaging, and acoustics. It serves as a conceptual bridge between FFT-based methods and lattice or grid-based spectral techniques.

Status: FFTlles remains a niche, largely theoretical construct without a standardized definition or widespread implementation. It is used mainly in discussions of hybrid spectral methods and remains an area of ongoing research, with variations appearing under related names in the literature.