A Multiscale Relaxation Algorithm for
$SNR$ Maximization in Nonorthogonal Subband Coding

by Pierre Moulin

In this paper, we develop a technique for improving the applicability
of complete, nonorthogonal, multiresolution transforms to image coding.
As is well known, the $L^2$ norm of the quantization errors is not preserved
by nonorthogonal transforms, so the $L^2$ reconstruction error
may be unacceptably large.
However, given the quantizers and synthesis filters, we show that
this artifact can be eliminated by formulating the coding problem
as that of minimizing the $L^2$ reconstruction error
over the set of possible encoded images.
With this new formulation, the coding problem becomes a high--dimensional,
discrete optimization problem and features a coupling
between the redundancy--removing and quantization operations.
A practical solution to the optimization problem is presented in the form
of a multiscale relaxation algorithm,
using inter-- and intra--scale quantization noise feedback filters.
Bounds on the coding gain over the standard coding technique are derived.
A simple extension of the algorithm allows for the use of a weighted $L^2$
error criterion and deadband (non--MMSE) quantizers.
Experiments using biorthogonal spline filter banks demonstrate appreciable SNR
gains over the standard coding technique, and comparable visual improvements.