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Perceptual Vector Quantization of Binary Image

Blocks for Pattern Analysis

- Aiyesha Ma, Rishi Mukhopadhyay, and Ishwar K.

Sethi Undergraduate Computer Research Program,

REU - Intelligent Information Engineering

Laboratory Department of Computer Science and

Engineering Oakland University

Introduction

- Most Vector Quantization techniques have been

developed for compression. - Previous VQ methods employ Euclidean metrics for

distance measures and averaging. - These measures are unsuitable for the extraction

of descriptors where invariance to translation,

rotation, and scale is important. - We focused on developing local descriptors

invariant to translation.

Introduction

- We present a vector quantization method that is

based on the Hausdorff metric of distance between

sets of points. - The goal of our method is to cluster image blocks

containing line segments with similar shapes

together.

Data Sets

Clustering

- We used the K-means approach to clustering.
- K-means is of order n complexity, where n is the

size of the set of data being clustered. - K-means algorithm
- Begin with an initial set of cluster means.
- Every element in the set is assigned to the

nearest cluster center. - The center of each cluster is re-averaged based

on its constituents. - The process of reassignment and re-averaging

repeats for some predefined number of iterations

or until the number of reassignments falls below

some threshold.

Distance Measures

- In VQ, the Euclidean or Mean Squared Error

distance is a popular distance metric. - The Euclidean metric is inappropriate for

categorizing descriptors. - Consider the following 7-by-7 image blocks with

their perceptual classifications

Distance Measures

- To group these images perceptually, the distance

between the two diagonal lines should be less

than the distance from either diagonal line to

either vertical line. - The Mean Square Error distance measure
- In the case of binary images is equivalent to the

Hamming distance - Results

Distance Measures

- Hausdorff metric of distance between sets of

points - The Manhattan distance was selected for .

Distance Measures

- The Hausdorff metric results in
- a distance of 5 from image D1 to image V1
- a distance of 4 from image D1 to image D2
- a distance of 4 from image V1 to image V2 because

of translation - a distance of 4 from image V2 to image D2.

Distance Measures

- Since, in cases of translation, all the nearest

neighbor distances are increased by the same

amount, we modify the Hausdorff distance to - Results

Distance Measures

- Now consider the following noisy images
- Which results in distances of

Distance Measures

- So instead of taking the maximum and subtracting

the minimum, we take a percentile - Then to mitigate the effects of asymmetry

inherent in the Hausdorff metric, we sum the

distance from A to B with the distance from B to

A

Distance Measures

- Yields the following result for the noisy images
- This modified Hausdorff measure yields a distance

measure invariant to translation. - Although not impervious to noise, this measure is

still moderately robust.

Averaging Methods

- Previous methods for averaging binary images

include soft-centroids and hard centroids. - These methods produce codewords that are an

accurate reflection of pixel distribution. - They lack the ability to produce codewords that

represent the shape a set of pixels form. - Consider the following images

Averaging Methods

- The ideal codeword
- Soft and hard centroid codewords

Averaging Methods

- Another averaging method is to take the Clustroid

as the codeword. - Using our modified Hausdorff measure, the

Clustroid method results in the following

codeword - We present a new averaging method based on the

Hausdorff mapping concept of the nearest neighbor

point.

Averaging Methods

- Given a set, , of binary images (where

each is a binary image and i ranges from 1 to

m, the number of images in the cluster) and the

set of points, , in the key-block (where

each is a coordinate pair representing one

of the black points in the key-block), then the

new average is defined as , where each new

coordinate pair - and where is a function that

returns the coordinate of the nearest neighbor

point in image C relative to point P.

Averaging Methods

Averaging Methods

- Significant improvement over the Euclidean based

methods. - Performs at least as well as the Clustroid method.

Data Sets

- 5-by-5, 7-by-7, 9-by-9 blocks
- Codebook size of 16 and 8, with one codeword

designated as a blank block. - Clustroid method and HBA methods one and two with

the modified Hausdorff distance measure. - Hard and Soft centroids, with Euclidean distance.
- 5-by-5 blocks with preset initial clusters,

instead of randomly chosen. - 5-by-5 and 7-by-7 blocks enlarged by a factor of

two.

Cluster Separation

- The ratio between the average inter-cluster

distance (codeword to codeword) and the average

intra-cluster distance (cluster member to

codeword) for each data set was calculated. - This ratio is an indicator of the degree of

separation of the clusters from each other. - A ratio greater than one indicates that the

codewords are more separated from each other than

they are from the blocks they represent.

Cluster Separation

16 Codewords

Cluster Separation

8 Codewords

Cluster Separation

5-by-5 blocks, preset initial codewords

Visual Comparison of Methods

Two clusters from 5-by-5 blocks with 8 means,

HBA method 1, without enlargement.

Two clusters from 5-by-5 blocks with 8 means,

HBA method 1, with enlargement.

Visual Comparison of Methods

Two clusters from 5-by-5 blocks with 8 means,

Clustroid method, with enlargement.

Visual Comparison of Methods

Four clusters from 5-by-5 blocks with 8 means,

Hard centroid method.

Three clusters from 5-by-5 blocks with 8 means,

Soft centroid method.

Visual Comparison of Methods

5-by-5 blocks with 8 means, Hard centroid

method.

5-by-5 blocks with 8 means, Soft centroid

method.

Visual Comparison of Methods

5-by-5 blocks without enlargement, 8 means, HBA

1 method.

5-by-5 blocks with enlargement, 8 means, HBA 1

method.

Visual Comparison of Methods

5-by-5 blocks without enlargement, 8 means,

Clustroid method.

5-by-5 blocks with enlargement, 8 means,

Clustroid method.

Computational Complexity

- The assignment portion of each cycle in the

k-means algorithm is of order - where b is the total number of blocks, k is the

number of clusters and where each block is n

pixels by n pixels. - The averaging portion of each cycle for both HBA

methods is of order - For the Clustroid algorithm, the averaging

portion of the cycle takes order - where is the number of members in the ith

cluster.

Using Perceptual Descriptors

- Represent the original images as matrices of

indices to the generated codebook. - Two approaches for grouping the representative

images - Co-occurrence matrices
- Frequency histograms
- The co-occurrence matrix approach successfully

separated 6 images into two groups.

Using Perceptual Descriptors

- Taking blocks from the representative image

matrix and clustering them can be used to build a

hierarchy of relationships. - Using the frequency histogram approach on a

second level representation of 6 images the

images were accurately separated into two groups.

Conclusion

- Presented a method for obtaining perceptually

meaningful descriptors for use in pattern

analysis. - These methods appear to be better suited than

previously established methods for obtaining

descriptors. - Initial progress using these descriptors seems

promising.

References

- 1 N.M. Nasrabadi and R.A. King, Image Coding

Using Vector Quantization A Review, IEEE Trans.

Commun., pp. 957-971, Vol. 36, No. 8, Aug. 1988. - 2 A.K. Jain, M.N. Murty, and P.J. Flynn, Data

Clustering A Review, ACM Computing Surveys,

Vol. 31, No. 3, Sept. 1999. - 3 Y. Linde, A. Buzo, and R.M. Gray, An

Algorithm for Vector Quantizer Design, IEEE

Trans. Communications, pp. 84-95, Vol. COM-28,

No. 1, Jan. 1980 - 4 S.P. Lloyd, Least Squares Quantization in

PCM, IEEE Trans. On Information Theory, pp.

129-137, Vol. IT-28, No. 2, Mar. 1982. - 5 A. Gersho, On The Structure of Vector

Quantizers, IEEE Trans. On Information Theory,

pp. 157-166, Vol. IT-28, No. 2, Mar. 1982. - 6 P. Franti and T. Kaukoranta, Binary Vector

Quantizer Design Using Soft Centroids, Signal

Processing Image Communication, 14(1999),

677-681.

References

- 7 D.P. Huttenlocher, G.A. Klanderman, and W.J.

Rucklidge, Comparing Images Using the Hausdorff

Distance, IEEE Trans. On Pattern Analysis and

Machine Intelligence, pp 850-863, Vol. 15, No. 9,

Sept. 1993. - 8 Q. Iqbal and J.K. Aggarwal, Applying

Perceptual Grouping to Content-Based Image

Retrieval Building Images, Proceedings of the

IEEE International Conference on Computer Vision

and Pattern Recognition, June 23-26, 1999, pp.

42-48. - 9 L. Zhu, A. Rao, and A. Zhang,Advanced

Feature Extraction for Keyblock-Based Image

Retrieval, pp. 179-182, ACM Multimedia Workshop,

2000. - 10 D. Stan and I. K. Sethi, Image Retrieval

Using a Hierarchy of Clusters, Lecture Notes in

Computer Science Advances in Pattern

Recognition, ICAPR, 2001, Springer-Verlag Ltd.

(ed.), pg. 377-388, 2001.