Histogram Analysis to Choose the Number of Clusters for K Means

By: Matthew FawcettDept. of Computer Science and

EngineeringUniversity of South Carolina

Overview Importance and use K means cluster algorithm The changes and adaptation I used Results Conclusions and Future Work

Importance The main reason for use in Medical

Imaging is for Segmentation. Other uses outside of the realms of

Image Processing(e.g. information retrieval)

Widespread algorithm

K Means Clustering Problem is that user doesn’t know

the optimal number of clusters to pick.

This is the problem I am trying to solve by using Histogram Analysis.

Histogram of the pixel intensity to find the optimal number of clusters for a picture.

Overview Importance and use K means cluster algorithm The changes and adaptation I used Results Conclusions and Future Work

Algorithm K means clusters is a very simple

algorithm First the user picks the number of

centers that he/she would like. Next the centers are chosen

randomly.

Algorithm I have read on different ways to

choose the centers. (e.g. pick the 2 farthest points away from each other.)

After the centers have been established then we check every other point with each of the centers and find the minimum distance.

Algorithm Each point is assigned to 1 cluster

which it is closet. This makes sense that points that

are closer to each other are normally together

After each point is assigned the cluster centers are then recalculated based on these assignments

Algorithm So once the new centers have

been processed the routine starts over and continues until it converges and the centers do not move.

http://www.elet.polimi.it/upload/matteucc/Clustering/tutorial_html/AppletKM.html

Overview Importance and use K means cluster algorithm The changes and adaptation I used Results Conclusions and Future Work

The new algorithm Instead of guessing the number of

clusters to have, I have used some preprocessing information to choose the number of clusters.

The first thing to be done is to make a histogram of pixel intensity.

Histogram The histogram will probably have

many peaks and valleys so the idea is to pick the correct number.

My idea was to basically count the peaks on the histogram.

However this can cause problems Any guesses?

HistogramWhich peaks do I take?

Histogram I added a term called Threshold. The threshold term just determines the

cutoff point for a peak. For example: If the threshold is 150

then I only take peaks with 151 or more. The threshold I chose was the max color

which was 255 divided by the number of pixels which equaled to 64.

How about any other problems with a histogram?

HistogramWhat about neighboring peaks?

Histogram I know introduce another term to my

work called span. Span basically covers the number of

pixels to the left and right of the current pixel.

For example if span was set to 3 then I would check 3 pixels to the left and 3 pixels to the right and then take the maxmium one over the threshold

Histogram The span guarantees that I don’t

have 2 pixels next to each other as 2 different centers in the picture.

This seems like a reasonable idea because pixels with the same intensity or near same intensity should share the same center and are probably close together.

Find Centers Based on this information I

determine the number of peaks above the threshold and no neighbors based on the span.

This the magic number I am using for the clusters by anglicizing the histogram of the pixel intensity.

Metric Now I have the number of centers(k) Start the k means algorithm Pick k center points at random. The metric I am using is the difference

in intensity. We take the absolute value of this to make sure it positive.

Assign each pixel to one of the clusters

Resign the cluster centers Now that we have all the pixels in a

cluster we recalculate the centers. Add up each pixel in each cluster and

divide by the number of pixels in the cluster and we get the new center.

Supposed to repeat this until it converges but here I just do this 25 times.

Overview Importance and use K means cluster algorithm The changes and adaptation I used Results Conclusions and Future Work

Results Found some MRI images Used ImageMagik to change the size

of the pictures to be 120 X 120

Results Number of centers = 6

Histogram of Ankle

0

200

400

600

800

1000

1 14 27 40 53 66 79 92 105

118

131

144

157

170

183

196

209

222

235

248

Intensity

Nu

mb

er o

f P

ixel

s

Pixels

Results Number of Centers = 19

Histogram of Brain

010002000300040005000600070008000

1 14 27 40 53 66 79 92 105

118

131

144

157

170

183

196

209

222

235

248

Intensity

Nu

mb

er o

f P

ixel

s

Pixels

Results Number of Centers = 17

Histogram of Knee

0

100

200

300

400

500

1 14 27 40 53 66 79 92 105

118

131

144

157

170

183

196

209

222

235

248

Intensity

Nu

mb

er o

f P

ixel

s

Pixels

Results

Ankle

6 centers 3 centers 5 centers 7 centers

154 154 154 154

134 134 134 134

127 35.878 127 127

124 124 124

123 35.575 123

35.502 117

35.365

ResultsBrain

19 centers 3 centers 5 centers 7 centers

49.5 0.30079 49.1 76.72727

1.875 39.39565 28.94118 38.15385

84 76.82353 68.77778 6.87234

62 85.875 21

40 0.32958 102.5

45 57.6

43.5 0.052299

7.18919

79.6

0

34

39

15

21.66667

26.5

102.5

36.6

54

71.5

ResultsKnee

17 centers 3 centers 5 centers 7 centers

183.933 131.721 190.105 235

37.321 31.618 235 191.361

193.524 227.178 136.388 148.022

253.846 29.065 27.446

155.1 79.6 77.952

50.75 48.69

220.533 112.312

239.952

104.5

26.816

81.467

207.048

130.585

64.808

172.474

246.522

231.051

Results Want to compare the variance of

each cluster. The variance in each cluster should

be about the same.

Overview Importance and use K means cluster algorithm The changes and adaptation I used Results Conclusions and Future Work

Conclusions and Future Work A method to find the centers of the

clusters The parameters for threshold and

span Supersampling instead of using

just one pixel.