Reversible Interger 2-D Discrete Fourier Transform by Control Bits

This paper describes the 2-D reversible integer discrete Fourier transform (RiDFT), which is based on the concept of the paired representation of the 2-D image, which is referred to as the unique 2-D frequency and 1-D time representation. The 2-D DFT of the image is split into a minimum set of short transforms, and the image is represented as a set of 1-D signals. The paired 2-DDFT involves a few operations of multiplication that can be approximated by integer transforms, such as one-point transforms with one control bit. 24 control bits are required to perform the 8×8-point RiDFT, and 264 control bits for the 16×16-point 2-D RiDFT of real inputs. The fast paired method of calculating the 1-D DFT is used. The computational complexity of the proposed 2-D RiDFTs is comparative with the complexity of the fast 2-D DFT.