1. Pyramid coder with nonlinear prediction
Laurent Meunier
Antoine Manens

2. Framework No quantization : lossless coding Open-loop = Closed-loop
Ideal VLC coder for each level of the pyramid

3. Criteria Global compression rate of the pyramid
SNR and visual quality of the partially reconstructed pictures
Cost of the decoding process

4. Review of linear techniques
Haar
Gaussian filters (Burt & Adelson, 1983)
Ideal filters
Optimal filters for piecewise polynomial fitting (Chin, Choi, Luo, 1992)
Splines (Unser, Aldroubi, Eden, 1993)
Efficient, but introduces blurring and aliasing

5. Review of non-linear techniques
Multi-level median filter (Defee, Neuvo, 1991)
Anisotropic pyramid (You, Kaveh,1996)
Improvement can be obtained on specific visual patterns like edges
More complicated to analyse.
Reduce and Expand Filters chosen from intuition/experiments, no guarantee of optimality.

6. Optimal NL interpolation
Hyp: Decimation filter is given
Problem : find 4 predictors for the even-even, odd-even, even-odd and odd-odd pixels.
Optimal solution : conditional expected value of the pixel given its neighbourhood for each predictor.
The implementation requires to reduce the number of possible neighbourhoods
=> Partition the image using features like average intensity, gradient, presence of edges, texture.

7. Implementation of the optimal NL filter
Example : image obtained with  3 features (avg intensity, grad/x, grad/y)  8 levels of quantization  8x8x8 = 512 cells
Pretty coarse because only one intensity per cell.
Solution : Use an optimal linear predictor that takes the local best fitting plane instead of the expected value.
Train the predictor using a set of images.

8. Hybrid Method
Motivation : some methods do a better job than the others in some kind of neighborhoods
Implementation : the algorithm switches technique depending on the type of neighborhood. Use a training set to learn decision table.

9. Method mapping

10. Visual comparison Original Burt&Adelson with a = 0.6
Cubic interpolation
Optimal non-linear

11. Numerical results Entropies : Lena : 7.44 Burt(0.6) : 5.69
Spline(3) :
Cubic interpolation : 5.43
Approx. opt. NL : 5.39
MMF :
DPCM :

12. Conclusion
Significant improvements over the Burt&Adelson pyramid were achieved both in terms of compression rate and of SNR of the partially reconstructed images
Rate reduction is lower than with DPCM. The lossless algorithm should therefore be used only where progressive transmission is necessary.
More thorough study of the feature choice and of the number of bins for the proposed NL technique is necessary.
Further study should include the issue of quantization (variable bit- allocation and non-optimal VLC)