In this project we focused on the choice of the decimation and interpolation filters for a pyramidal coder. We made several assumptions in order to assess the quality of those filters independently from other factors like the quantization method or the bit allocation. In particular, we restricted ourselves to the case of lossless pyramidal coding with no quantization and assuming optimal variable length coding.
We implemented and compared a number of different linear filters found in the literature, including Haar, Burt-Adelson [1], Optimal Linear and Cubic [2], and Linear and Cubic Spline [4]. The criteria we took into account were the entropy of the error images, the computational complexity and the visual quality of the reduced images, including the blurring and aliasing introduced by the filters.
We studied the possibility of using non-linear filters to improve the compression efficiency of the pyramid scheme. We implemented a median type filter and reviewed others from the literature. We proposed an approximation of the optimal non-linear predictor given a decimation filter. We proposed an hybrid method that tries to get the best of each of the previously mentioned filters.
The results showed that significant improvements over the initial filters proposed by Burt & Adelson were possible. In particular, the SNR of the image reconstructed without the first level error image for 'lena' increased by 5dB. However, the typical compression rates obtained with lossless pyramidal coding (computed using the weighted entropy of the pyramid levels) are still worse that those of simple causal linear predictive schemes. Indeed, the rate reduction achieved by entropy-coding the error images is altered by the 4/3 increase in the number of pixels to be coded. Thus it appears that the pyramid structure itself, if useful for multi-resolution processing or progressive transmission, has a cost in terms of compression rate.
A more thorough study of the subject should take into account the influence of other issues of a pyramid compression scheme, that is quantization with variable bit allocation and non-optimal variable length coding. The two proposed methods "approximate optimal non-linear" and "hybrid interpolation" are likely to yield further improvements especially if used together. However, a careful study of the influence of the different parameters involved in these techniques would be necessary to get the best out of them.