2. Two High Speed Quantization Algorithms Luc Brun Myriam Mokhtari L.E.R.I. Reims University (I.U.T.)

3. Contents Quantization algorithms Our Methods Discussion

4. Quantization algorithms Reduce the number of colours Number of colours: 141,000Number of colours: 16

5. Quantization Algorithms Applications Display Compression Classification Segmentation

6. Quantization steps Create clusters

7. Quantization steps Create clusters: Squared error Partition error

8. Quantization steps Create clusters Compute means

9. Quantization steps Create clusters Compute means Create output image (inverse colormap) Quantization Inverse colormap dithtering

10. Type of quantization methods Three kind of Methods Top-down Bottom-up Split & Merge

11. Top-down methods Recursive split of the image color set

12. Bottom-up methods For each colour c in the image colour set Select K “empty” clusters Aggregate c to its closest cluster

13. Split and Merge methods Select N>K clusters (split step) Merge these clusters to obtain the K final clusters (merge step)

14. Our Method: Split step Create a uniform quantization.

15. Our Method: Merge Step Create a graph

16. Our Method: Merge Step Create a graph: Cluster Adjacency Graph

17. Our Method: Merge Step Merge of clusters: C i and C j Minimize the partition error Select i 0 and j 0 such that:

18. Our Method: Merge Step Merge clusters: Edge contraction

19. Our Method: Merge Step Merge clusters: Edge contraction

20. Our Method: Merge Step Merge clusters: Edge contraction

21. Our Method: Merge Step Merge clusters: Edge contraction

22. Our Method: Merge Step Merge clusters: Edge contraction

23. Our Method: Merge Step Merge clusters: Edge contraction

24. Our Method: Merge Step Merge clusters: Edge contraction

25. Our Method: First Inverse colormap Given a colour c Find its enclosing cluster Find its enclosing meta-cluster Map c to its mean

26. Our Method: Second Inverse colormap Given a color c Find its enclosing cluster Find the adjacent meta-clusters Map c to the closest mean

27. Our Method: Results Compared to the Top-down method [Wu-91] Image quality: First inverse colormap: slightly lower Second Inverse colormap: Improved Computing time  15 time faster Compared to the Bottom-up method [Xiang- 97] Image quality: Improved [Tremeau-96] Computing time  10 time faster

28. Our method: Results First inverse colormapSecond inverse colormap Wu 91Xiang 97 Original

29. Discussion: The idea Merge at each step the two closest clusters. Reduce the amount of data (uniform quantization) Apply an expansive heuristic: O(n 2 ) (merge step)  Split & Merge strategy

30. Discussion: Short History Top down methods Intensively explored since 1982 [Heckbert 82] Bottom-up methods Restricted to simple Heuristics

31. Discussion: Short History Number of clusters Partition Error

32. Discussion: Short History Top down methods Bottom-up methods Split & Merge methods First attempts based on top-down algorithms.

33. Conclusion Possible improvements Uniform quantization Avoid empty clusters Merge Step Find a better heuristic Inverse colormap No improvement needed. Combinatorial optimisation ?

34. References [Wu 91] Xiaolin Wu and K. Zhang. A better tree structured vector quantizer. In Proceedings of the IEEE Data Compression Conference, pages 392- 401. IEEE Computer Society Press, 1991. [Xiang-97] Color Image quantization by minimizing the maximum inter-cluster distance. ACM Transactions on Graphics, 16(3):260-276, July 1997. [Tremeau-96] A. Tremeau, E. Dinet and E. Favier. Measurement and display of color image differences based on visual attention. Journal of Imaging Science and Technology, 40(6):522-534, 1996.IS&T/SID http://www.univ-reims.fr/Labos/LERI/membre/luc