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

Abstract

A recently developed PLUS factorization holds great promise in image coding due to its simplicity and integer reversibility. However, existing PLUS factorizations did not consider stability and optimality. To address these problems, we propose methodologies to design stable and optimal PLUS factorization algorithms. Firstly, we propose three stable PLUS factorization algorithms, prove the stability theorem under no perturbation and analyze stability under perturbation. Furthermore, we obtain a closed-form formula for transform error, and use the formula to design an algorithm for optimal PLUS factorization. Then, we apply the PLUS factorization to image coding. The integer DCTs implemented with the optimal PLUS factorizations found by our algorithms outperform the integer DCT with expansion factors in terms of entropy. The optimal PLUS factorizations are superior to the lifting factorization in JPEG-XR. The experimental results agree with analytical results of PLUS factorization, and show superior performance of our algorithms in image coding.

Unitriangular factorizations of chevalley groups

“…We will make no attempts to systematically cover subsequent literature in the field, and limit ourselves to several typical works [20,31,35,57], where one can find further references. §5.…”

Approximating vector quantisation by transformation and scalar quantisation

“…To preserve signal energy constant, linear transforms are focused on, such as unitary transforms and volume-preserving scaling transforms, represented by matrices with unitary determinant. Such transforms can also be implemented lossless with PLUS factorisation [18].…”