A Multiscale Relaxation Algorithm for SNR Maximization in Nonorthogonal Subband Coding

In this paper, we develop a technique for improving the applicability of complete, nonorthogonal, multiresolution transforms to image coding. As is well known, the L2 norm of the quantization errors is not preserved by nonorthogonal transforms, so the L2 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 L2 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 intrascale 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 L2 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.