Matrix factorizations for reversible integer mapping

Reversible integer mapping is essential for lossless source coding by transformation. A general matrix factorization theory for reversible integer mapping of invertible linear transforms is developed. Concepts of the integer factor and the elementary reversible matrix (ERM) for integer mapping are introduced, and two forms of ERM-triangular ERM (TERM) and single-row ERM (SERM)-are studied. We prove that there exist some approaches to factorize a matrix into TERMs or SERMs if the transform is invertible and in a finite-dimensional space. The advantages of the integer implementations of an invertible linear transform are (i) mapping integers to integers, (ii) perfect reconstruction, and (iii) in-place calculation. We find that besides a possible permutation matrix, the TERM factorization of an N-by-N nonsingular matrix has at most three TERMs, and its SERM factorization has at most N+1 SERMs. The elementary structure of ERM transforms is the ladder structure. An executable factorization algorithm is also presented. Then, the computational complexity is compared, and some optimization approaches are proposed. The error bounds of the integer implementations are estimated as well. Finally, three ERM factorization examples of DFT, DCT, and DWT are given.