two high speed quantization algorithms luc brun myriam mokhtari l.e.r.i. reims university (i.u.t.)
TRANSCRIPT
Two High Speed Quantization Algorithms
Luc BrunMyriam 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: 141,000 Number of colours: 16
Quantization Algorithms
Applications
Display
Compression
Classification
Segmentation
Quantization steps
Create clusters
Quantization steps
Create clusters: Squared error
Partition error
K
iiCSEPE
1
)()(
Cx
CxxfCSE2
KCCP ,,1
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: Ci and Cj
Minimize the partition error Select i0 and j0 such that:
2
, )()'( ji
ji
ji
ji CCCC
CCPEPE
jinji
ji
ji
Min ,,1),(
, 200
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 clusterFind its enclosing meta-clusterMap c to its mean
Our Method: Second Inverse colormap
Given a color cFind its enclosing clusterFind the adjacent meta-clustersMap c to the closest mean
Our Method: Results
Compared to the Top-down method [Wu-91] Image quality:
First inverse colormap: slightly lowerSecond 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 colormap Second inverse colormap
Wu 91 Xiang 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(n2) (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 methodsFirst 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 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