Transcript Two High Speed Quantization Algorithms

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: 141,000
Number of colours: 16
Quantization Algorithms

Applications
 Display
 Compression
 Classification
 Segmentation
Quantization steps

Create clusters
Quantization steps

Create clusters:
 Squared
error
SEC  
 f x x   C 
xC
 Partition
error
P  C1 ,, CK 
K
E ( P) 
 SE(C )
i
i 1
2
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
Select K “empty” clusters
For each colour c in the image colour
Aggregate c set
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
 i , j  E ( P' )  E ( P) 

Ci  C j
Ci C j
Minimize the partition error
 Select i0
and j0 such that:
 i0 , j0 
Min

i, j
2
( i , j )1,n
i j
 Ci    C j 
2
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 [Xiang97]
 Image
quality: Improved [Tremeau-96]
 Computing time  10 time faster
Our method: Results
First inverse colormap
Second inverse colormap
Original
Wu 91
Xiang 97
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
Partition Error
Number of clusters
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

Merge Step
 Find

empty clusters
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 392401. 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,
http://www.univ-reims.fr/Labos/LERI/membre/luc
1996.IS&T/SID
