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Block TERM factorization of block matrices. (English) Zbl 1186.15012
Summary: 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.

MSC:
15A23 Factorization of matrices
15A15 Determinants, permanents, traces, other special matrix functions
68U10 Computing methodologies for image processing
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
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