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

Contents Quantization algorithms Our Methods Discussion

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

Quantization Algorithms Applications Display Compression Classification Segmentation

Quantization steps Create clusters

Quantization steps Create clusters: Squared error Partition error

Quantization steps Create clusters Compute means

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

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

Top-down methods Recursive split of the image color set

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

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

Our Method: Split step Create a uniform quantization.

Our Method: Merge Step Create a graph

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

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

Our Method: Merge Step Merge clusters: Edge contraction

Our Method: Merge Step Merge clusters: Edge contraction

Our Method: Merge Step Merge clusters: Edge contraction

Our Method: Merge Step Merge clusters: Edge contraction

Our Method: Merge Step Merge clusters: Edge contraction

Our Method: Merge Step Merge clusters: Edge contraction

Our Method: Merge Step Merge clusters: Edge contraction

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

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

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

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

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

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

Discussion: Short History Number of clusters Partition Error

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

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

References [Wu 91] Xiaolin Wu and K. Zhang. A better tree structured vector quantizer. In Proceedings of the IEEE Data Compression Conference, pages IEEE Computer Society Press, [Xiang-97] Color Image quantization by minimizing the maximum inter-cluster distance. ACM Transactions on Graphics, 16(3): , July [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): , 1996.IS&T/SID