Improved discrete multiwavelet transform for vector-based image coding

Preconditioning is required to convert a scalar-valued signal into a vector-valued signal for the discrete multiwavelet transform (DMWT). The two most prominent approaches to achieve this are: (a) balanced multiwavelet, and (b) prefiltering. Balanced multiwavelet has the advantage of using the existing scalar filters without the addition of extra pre- and post-filters but at the expense of more complex design procedure and longer filter length which contradicts with the localization characteristic of the wavelet transform. Prefiltering on the other hand has a comparatively simple design procedure but the additional computation before and after the DMWT may increase the computational complexity that hinders its usage. Moreover, traditional prefilters are not linear phase, often non-orthogonal or non-maximally decimated. Recently, an orthogonal symmetric prefilter bank (OSPFB) was suggested to tackle this problem but it was still difficult to find a compatible filter bank structure and sometimes there was none, especially for low multiplicity multiwavelets. In this study, we propose a new DMWT structure with embedded orthogonal prefilters bank (FOSPFB) by combining the prefilters with the first level of DMWT. The advantage of this new structure is twofold. First, since the prefiltering is embedded into DMWT, the computational complexity can be greatly reduced. Our simulations show that up to 36% saving in arithmetic operations can be achieved comparing with the traditional DMWT realizations. Second, the new structure provides more freedom in designing the prefilters so that they can be maximally decimated, orthogonal and symmetric even for multiwavelets of multiplicity as low as two. Due to these nice properties, it is found that the new DMWT structure has a better energy compaction ratio (ECR) when compared with the traditional scalar wavelet transform particularly in vector-based image coding applications. In this research, the performance of the new DMWT structure is further evaluated by applying to a simple vector quantization based image coding system of which the codebook is initialized with a modified approach called cumulative absolute difference (CAD). The resulted codebook shows on average 1--2 dB gain in coding performance over the typical variance-based approach. The new DMWT structure gives an average gain of 0.5 dB and 2 dB in PSNR for coding typical gray level images over the DMWT using the traditional prefilter bank and DWT using Bior9/7 scalar wavelet respectively. These results enable the DMWT to be seriously considered as an improved replacement to the traditional DWT for industrial use.