Factorization of the Coefficient Variance Matrix in Orthogonal Transforms

Orthogonal transforms are used in speech and image coding in order to redistribute the total signal energy compactly among the transform spectral coefficients. The key point is the allocation of available bits based on the variances of the coefficients. The calculation of these variances based on image correlation models has been studied by several authors. This correspondence presents a new matrix factorization of the coefficient variances which effectively separates the orthogonal transformation from the correlation model for the signal source. A transform-specific matrix W is derived which links the transform to the signal correlation model. This factorization provides a conceptual framework for understanding the behavior of the transform for different source models. It also significantly simplifies the calculation of the variances of each transform coefficient from a quadruple sum over four indices to a double sum over two indices. This simplification can be used in adaptive transform coding which requires an estimate of the variances. Test results demonstrate that this model based variance calculation provides an accurate basis for adaptive transform coding, particularly at low bit rates.