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. 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. Review of non-linear techniques • Multi-level median filter (Defee, Neuvo, 1991) • Anisotropic pyramid (You, Kaveh,1996)

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 likeaverage 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) : 5.61 • Cubic interpolation : 5.43 • Approx. opt. NL : 5.39 • MMF : 5.35 • DPCM : 5.03

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)