Block TERM factorization of block matrices

Abstract

Reversible integer mapping (or integer transform) is a useful way to realize lossless coding, and this technique has been used for multi-component image compression in the new international image compression standard JPEG 2000. For any nonsingular linear transform of finite dimension, its integer transform can be implemented by factorizing the transform matrix into 3 triangular elementary reversible matrices (TERMs) or a series of single-row elementary reversible matrices (SERMs). To speed up and parallelize integer transforms, we study block TERM and SERM factorizations in this paper. First, to guarantee flexible scaling manners, the classical determinant (det) is generalized to a matrix function, DET, which is shown to have many important properties analogous to those of det. Then based on DET, a generic block TERM factorization, BLUS, is presented for any nonsingular block matrix. Our conclusions can cover the early optimal point factorizations and provide an efficient way to implement integer transforms for large matrices.

Matrix factorizations for parallel integer transforms

“…, where is an elementary block matrix of which the th block is and is a block matrix with the th block zero, we studied block factorizations in [11] and [12]. In contrast to point SERM factorizations, block SERM factorizations make it possible that the factorizations and the transforms are carried out at block level and therefore boost the degree of parallelism.…”

“…In the case that all blocks are of the same size [11], we redefined the generalized determinant matrix function DET and obtained a BLUS factorization , where is a unit block SERM associated with the last block row (thus is also a unit lower TERM) and . We proposed a practical algorithm [11] as a generalization of point TERM factorization [9] and also proved that block SERM factorization exists if and only if is a diagonal matrix and all the diagonal elements are integer factors.…”

“…Interested readers may refer to [11] and [12] This property, as a counterpart of that of matrix determinant DET, is an important guarantee of the flexibility and practicability of the scaling modification.…”

“…In Section III, based on the block triangular ERM (TERM) and single-row ERM (SERM) factorizations [11], [12], we introduce two types of PERM factorizations for the parallel integer transform. Section IV is a discussion of computational complexity.…”

Stabilization and optimization of PLUS factorization and its application in image coding

“…This work is proposed to address these problems (other problems of PLUS factorization, like blocking factorization and parallelling computing, arising from solving large linear systems, could be found in [17,18,19]). Our main contributions include:…”