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A Coarse Classification Scheme Based On Clustering

and Distance Thresholds

Presented byTienwei Tsai

July, 2005

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Outline

Introduction Feature Extraction Statistical Coarse Classification Experimental Results Conclusion

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1 Introduction

Paper documents -> Computer codes

OCR(Optical Character Recognition)

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Two subproblems in the design of classification systems Feature extraction Classification

The classification process is usually divided into two stages: Coarse classification process Fine classification process

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The purpose of this paper is to design a general coarse classification scheme: low dependent on domain-specific

knowledge. To achieve this goal, we need:

reliable and general features general classification method

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2 Feature Extraction The characteristics of features

used for coarse classification: Simple Reliable Low dimension

We are motivated to apply Discrete Cosine Transform (DCT) to extract statistical features.

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Discrete Cosine Transform (DCT)

It is widely used in compression A close relative of the discrete Fourier

transform (DFT) Converting a signal into elementary

frequency components Separating the image into parts of

differing importance (with respect to the image's visual quality).

Energy compaction property

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The DCT coefficients of the character image of “ 佛” .

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Energy Compacting Property of DCT

For most images, much of the signal energy lies at low frequencies.

About 90% signal energy appears in the upper left corner of size 10×10 for the DCT coefficients of size 48×48.

That is, about 90% signal energy appears in 4.34% DCT coefficients.

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Discrete Cosine Transform (DCT)

The DCT coefficients C(u, v) of an N×N image represented by x(i, j) can be defined as

where

1

0

1

0

),()()(2

),(N

i

N

j

jixvuN

vuC ),2

)12(cos()

2

)12(cos(

N

vj

N

ui

.1

,021

)(otherwise

wforw

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3 Statistical Coarse Classification

Two philosophies of classification Statistical

the measurements that describe an object are treated only formally as statistical variables, neglecting their “meaning

Structural regards objects as compositions of

structural units, usually called primitives.

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The Classification Problem

The ultimate goal of classification is to classify an unknown pattern x to one of M possible classes (c1, c2,…, cM).

In statistical approach, each pattern is represented by a set of D features, viewed as a D-dimensional feature vector.

The statistical classification system is operated in two modes: Training (learning) Classification (testing)

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Techniques to speed up the classification process

To speed up the classification process, two techniques can be applied: A. Clustering B. pruning

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A. Clustering

Clustering is the process of grouping the data objects into clusters so that objects within a cluster have high similarity in comparison to one another, but are very dissimilar to objects in other clusters.

Two main categories of existing clustering algorithms:

hierarchical methods: agglomerative (bottom-up) divisive (top-down).

partitioning methods: k-means k-medoids

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Coarse Classification via Clustering

In the training mode: the feature vectors of learning samples are

clustered first based on a certain criterion. In the classification mode:

the distances between a test sample and every cluster is calculated, and

the clusters that are nearest to the test sample are chosen as candidate clusters.

Then the classes within those candidate clusters are selected as the candidates of the test sample.

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B. Pruning

In the training mode: The template vector of a class is generated

from the training samples of the class. In the classification mode:

When a test sample is being classified, all the distances between the feature vector of the test sample and every template vector are calculated.

Then the templates whose distance to the test sample is beyond a predefined threshold will be pruned.

Therefore, for the test sample to be classified, the size of the candidate list is reduced.

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Coarse classification scheme

Our coarse classification is accomplished by four main modules: Pre-processing Feature extraction Clustering Pruning

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In the training mode, the training samples are first normalized to a certain size.

Then the most significant D features of each training sample are extracted by DCT.

Ex. The most important 4 features of a pattern are the 4 DCT coefficients with the lowest frequencies, namely C(0,0), C(0,1), C(1,0) and C(1,1).

Then the average D features are obtained for each class.

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k-means algorithm

Chooses k templates arbitrarily as the initial centers of the k clusters.

The remaining (M-k) templates are assigned to their nearest clusters one by one, according to a certain distance measure.

Each cluster center is updated progressively from the average of the total patterns within the cluster and can be viewed as a virtual reference pattern.

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Distance Measure

Suppose Pi = [pi1, pi2,…, piD] represents the feature vector of pattern Pi.

Then the dissimilarity (distance) between patterns Pi and Pj is defined as

D

djdidji ppPPd

1

2),(

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In the classification mode, for an input pattern to be recognized, the pattern is first normalized to a certain size.

Then the features of the pattern are extracted by DCT.

After that the pattern is matched against the center of each cluster to obtain the distance to each cluster.

Finally, the clusters which do not satisfy some distance-based thresholds will be excluded from the set of candidate clusters, and the members within the remaining clusters will be served as the candidates for the input pattern.

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3.2.1. Cluster pruning rules

To observe the rate of distance per maximum distance, the relative distance dm(x, Ci) is defined as

where d(x,Ci) is the distance between the test pattern x and each cluster Ci .

denotes the maximum distance between x and any cluster.

),(

),(),(

kC

iim CxdMax

CxdCxd

k

),( kC CxdMaxk

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We also define the relative distanceda(x, Ci) as

where denotes the average distance over all clusters.

),(

),(),(

kC

iia CxdAverage

CxdCxd

k

),( kC CxdAveragek

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Pruning Rules

R1 : Pruning via absolute distance threshold d the distance d(x,Ci) is compared with a pre-

determined distance threshold d. If d(x,Ci) is larger than d, then cluster Ci is

excluded from the candidate cluster set of x. R2 : Pruning via cluster rank

threshold r to filter out the farthest clusters by eliminating

the clusters whose ranks are larger than r.

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R3 : Pruning via relative distance threshold m If dm(x,Ci) is larger than m, then cluster Ci

is excluded from the candidate cluster set of x.

R4 : Pruning via relative distance threshold a If da(x,Ci) is larger than a, then cluster Ci is

excluded from the candidate cluster set of x.

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Each threshold value is obtained from the statistics of the training samples in the training mode, based on a certain accuracy requirement.

Rule R1 and R2 are obtained straightforwardly, which cannot measure the similarity between x and each cluster precisely.

Rule R3 and R4 take advantage of the discriminating ability of the relative distances.

For instance, ‘da(x,Ci)=0.5’ means the distance between x and cluster Ci is about half those distances between x and other clusters.

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4 Experimental Results 6000 samples (about 500 categories) are

extracted from Kin-Guan ( 金剛 ) bible. Each character image was transformed into a

48×48 bitmap. 5000 of the 6000 samples are used for training and

the others are used for testing. In our experiment

The number of clusters used in k-means clustering is set to 10.

The number of DCT coefficients extracted from each sample is set to 9, i.e., C(i, j), where i,j=0, 1, 2.

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• High reduction rate always results in low accuracy rate

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• Rules R3 performs better than the other rules.

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6 Conclusions This paper presents a coarse

classification scheme based on DCT and k-means clustering.

The advantages of our approach: The DCT features are simple, reliable

and appropriate for coarse classification.

The proposed coarse classification scheme is a general approach to most of the vision-oriented applications.

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Future works Future works include the application of

another well-known vision-oriented feature extraction method: wavelet transform.

Since features of different types complement one another in classification performance, by using features of different types simultaneously, classification accuracy could be further improved.