Multiplierless lifting-based fast X transforms derived from fast Hartley transform factorization

Abstract

This paper presents M-channel ( $$M=2^{N}$$ , $$N\in \mathbb {N}$$ , $$N\ge 1$$ ) multiplierless lifting-based (ML-) fast X transforms (FXTs), where X $$=$$ F (Fourier), C (cosine), S (sine), and H (Hartley), i.e., FFT, FCT, FST, and FHT, derived from FHT factorization as way of lowering the cost of signal (image) processing. The basic forms of ML-FXTs are described. Then, they are customized for efficient image processing. The customized ML-FFT has a real-valued calculation followed by a complex-valued one. The ML-FCT customization for a block size of 8, which is a typical size for image coding, further reduces computational costs. We produce two customized ML-FCTs for lossy and lossless image coding. Numerical simulations show that ML-FFT and ML-FCTs perform comparably to the conventional methods in spite of having fewer operations.