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

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


Create a graph: Cluster Adjacency Graph


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


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


Number of clusters

Partition Error


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

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

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