Clustering Algorithms

1. Clustering Algorithms Padhraic Smyth Department of Computer Science CS 175, Fall 2007

2. Timeline • Today • Discussion of project presentations and final report • Overview of clustering algorithms and how they can be used with image data • Tuesday December 4th • No lecture (out of town) • Thursday Dec 6th: Student Presentations: • About 4 minutes per student, + questions • Wednesday Dec 12th: Final project reports due 12 noon to EEE • Instructions on format provided on the class Web site

3. Project Presentations • Thursday next week • Each student will make a 4 minute presentation + 1 minute for questions from the professor and/or students • 13 students * 5 minutes = 65 minutes + setup time • IMPORTANT: • We will go in a fixed order (alphabetical by last name – next slide) • Be here on time! • Your slides must be uploaded BEFORE 9am THURSDAY (day of presentations) • Powerpoint or PDF is acceptable

4. Order of Student Presentations • Austgen • Duran • Hall • Hooper • Kong • Lipeles • Newton • Nguyen (Nam) • Nguyen (Son) • Nilsen • Salanga • Schmitt • Sheldon • Rodriguez

5. Guidelines for Presentations • Your slides should at least contain the following elements: • clear statement of what task/problem you are addressing • Outline of the technical approach you are taking • You will not have time to go into details • Provide a high-level description of your methods • e.g., a figure or flow chart • Show an example (e.g., of template matching, edge map, etc) • Describe your results so far • Visual examples, tables of accuracy numbers, etc • Its ok if your project is not yet finished: describe what you have • General tips • Speak clearly and loudly – face the audience • Practice beforehand – know what is in your slides • Be creative – use figures rather than text where possible

6. Presentation Grading • 5% of your grade • You will get at least 2.5% for just showing up  • Remainder of the grade will depend on • How much work/effort did you put into your slides? • How clear are your slides and your presentation? • Creativity, e.g., a clever way to illustrate visually how your feature-extractor/detector/classifier is working • Questions?

7. Final Project Reports • Due noon Wednesday December 12th (to EEE) • See class Web page for detailed instructions • You will submit a 5 to 10 page report and your code • Worth 35% of your total grade • Make sure to spend time on the report • Much of your grade will depend on how well and how clearly your report is written • Lower grades will go to • Poorly written reports and/or poorly executed project • High grade will need • Well-written report AND well-executed project • If your system is not performing accurately, don’t panic! Carefully describe what you did, and try to identify why your system is not performing accurately (look at errors, etc). If you write a good report and document what you did, you can still get a high grade.

8. Normalized Template Matching

9. Putting 2 Vectors on the same Scale • Two vectors x and y • mx = mean(x), sx = standard_deviation(x) • Let x’ = (x – mx)/sx • x’ values now have mean 0 and standard deviation 1 • Why? • Mean(x’) = mean( (x-mx)/sx ) = (mean(x) – mx)/sx = 0 • Same type of argument for standard deviation • Can apply the same normalization to y to get y’ • y ‘ = (y – my)/sy

10. Applying this Idea to Template Matching • x = template - > x’ = normalized template • y = patch of image being matched to template y’ = normalized patch • Normalized template matching • Replace template x by normalized template x’ • Has mean pixel intensity 0 and standard deviation 1 • Only needs to be done once (at start of function) • Replace each image patch y by normalized image patch y’ • Has mean pixel intensity 0 and standard deviation 1 • Likely to lead to better matching • However: the patch normalization has to be done at every patch in the image (so will slow down the template-matching code)

11. Normalized Template Matching

12. Normalized Template Matching

13. Normalized Template Matching

14. Normalized Template Matching

15. Modified Template-Matching Code % Reshape template reshtemp = reshape(template,1,tmrows*tmcols); % Remove mean of template and divide by standard deviation: reshtemp = (reshtemp - mean(reshtemp))./std(reshtemp); % Template now has mean 0 and standard deviation 1; ……. for x=1:xspan for y=1:yspan % Take a piece of the image where the template is bite = image(y:y+tmrows-1,x:x+tmcols-1); % Reshape reshbite = reshape(bite,1,tmrows*tmcols); % Remove mean of “bite” and divide by standard deviation: sbite = std(reshbite); if sbite>0 reshbite = (reshbite - mean(reshbite))./(std(reshbite)); end % “patch” now has mean 0 and standard deviation 1;

16. Clustering Algorithms

17. Unsupervised Learning or Clustering • In “supervised learning” each data point had a class label • in many problems there are no class labels • this is “unsupervised learning” • human learning: how do we form categories of objects? • Humans are good at creating groups/categories/clusters from data • in image analysis finding groups in data is very useful • e.g., can find pixels with similar intensities • > automatically finds regions in images • e.g., can find images that are similar -> can automatically find classes/clusters of images

18. Example: Data in 2 Clusters Feature 2 Feature 1

19. The Clustering Problem • Let x = (x1, x2,…, xd,) be a d-dimensional feature vector • Let D be a set of x vectors, • D = { x(1), x(2), ….. x(N) } • Given data D, group the N vectors into K groups such that the grouping is “optimal” • One definition of “optimal”: • Let mean_k be the mean (centroid) of the Kth group • Let d_i be the distance from vector x(i) to the closest mean • so each data point x(i) is assigned to one of the K means

20. Optimal Clustering • Let mean_k be the mean (centroid) of the kth cluster • mean_k is the average vector of all vectors x “assigned to” cluster k • mean_k = (1/n) Sx(i), • where the sum is over x(i) assigned to cluster k • One definition of “optimal”: • Let d_i be the distance from vector x(i) to the closest mean • so each data point x(i) is assigned to one of the K means • Q_k = quality of cluster k = S d_i , • where the sum is over x(i) assigned to cluster k • the Q_k’s measure how “compact” each cluster is • We want to minimize the total sum of the Q_k’s

21. The Total Squared Error Objective function • Let d_i = distance from feature vector x(i) to the closest mean = squared Euclidean distance between x(i) and mean_k • Now Q_k = sum of squared distances for points in cluster k • Total Squared Error (TSE) • TSE = Total Squared_Error = S Q_k • where sum is over all K clusters (and each Q_k is itself a sum) • TSE measures how “compact” a clustering is

22. Example: Data in 2 Clusters Feature 2 Feature 1

23. “Compact” Clustering: Low TSE Feature 2 Cluster Center 2 Cluster Center 1 Feature 1

24. “Compact” Clustering: Low TSE Feature 2 Cluster Center 2 Cluster Center 1 Feature 1 Here we have 2 clusters, and TSE = Q1 + Q2

25. “Non-Compact” Clustering: High TSE Feature 2 Cluster Center 2 Cluster Center 1 Feature 1 TSE = Q1 + Q2 would be much higher now: so we want to find the cluster centers that minimize TSE

26. The Clustering Problem • Let D be a set of x vectors, • D = { x(1), x(2), ….. x(N) } • Fix a value for K, e.g., K = 2 • Find the locations of the K means that minimize the TSE • no direct solution • Exhaustive search: how many possible clusterings of N objects into K subsets? • O(KN) -> way too many to search directly • can use an iterative greedy search algorithm to minimize TSE

27. The K-means Algorithm for Clustering Inputs: data D, with N feature vectors K = number of clusters Outputs: K mean vectors (centers of K clusters) memberships for each of the N feature vectors

28. The K-means Algorithm for Clustering kmeans(D, k) choose K initial means randomly (e.g., pick K points randomly from D) while means_are_changing % assign each point to a cluster for i = 1: N membership[x(i)] = cluster with mean closest to x(i) end % update the means for k = 1:K mean_k = average of vectors x(i) assigned to cluster k end % check for convergence if (new means are the same as old means) then halt else means_are_changing = 1 end

29. Original Data (2 dimensions)

30. Initial Cluster Centers for K-means (K=2)

31. Update Memberships (Iteration 1)

32. Update Cluster Centers at Iteration 2

33. Update Memberships (Iteration 2)

34. Update Cluster Centers at Iteration 3

35. Update Memberships (Iteration 3)

36. Update Cluster Centers at Iteration 4

37. Updated Memberships (Iteration 4)

38. Comments on the K-means algorithm • Time Complexity • per iteration = O(KNd) • Can prove that TSE decreases (or converges) at each iteration • Does it find the global minimum of TSE? • No, not necessarily • in a sense it is doing “steepest descent” from a random initial starting point • thus, results will be sensitive to the starting point • in practice, we can run it from multiple starting points and pick the solution with the lowest TSE (the most “compact” solution)

39. Clustering Pixels in an Image • We can use K-means to cluster pixel intensities in an image into K clusters • this provides a simple way to “segment” an image into K regions of similar “compact” image intensities • more automated than manual thresholding of an image • How to do this? • Size(image pixel matrix) = m x n • convert to a vector with (m x n) rows and 1 column • this is a 1-dimensional feature vector of pixel intensities • run the k-means algorithm with input = vector of intensities • assign each pixel the “grayscale” of the cluster it is assigned to • Note: with color images we can use a 3-dimensional feature vector per pixel, i.e, R, G, B values at each pixel

40. Clustering in RGB (color) space Image Clusters on color K-means clustering of RGB (3 value) pixel color intensities, K = 11 segments (courtesy of David Forsyth, UC Berkeley)

41. Example: Original Image

42. Segmentation with K-means: K = 2

43. Segmentation with K=3

44. Segmentation with K=5 Note: what K-means is doing in effect is finding 4 threshold intensities (based on the data) and assigning each intensity to 1 of 5 “bins” (clusters) based on these thresholds

45. Another Image Example

46. Segmentation with K=2

47. Segmentation with K=3

48. Segmentation with K=8 (with pseudocolor display) colormap(‘hsv’)

49. Using pixel clustering for region finding • How could you use K-means in your project? • K-means puts pixels into K groups based on intensity similarities • The result is a set of regions in an image, where each region is relatively homogeneous in terms of pixel intensity • => K-means can be used as a simple technique for region-finding • Note that K-means clustering knows nothing about the spatial aspects of the image • Other region-finding algorithms can operate spatially (more on this in a later lecture)

50. Using pixel clustering for region finding • How could you use K-means in your project? • K-means puts pixels into K groups based on intensity similarities • The result is a set of regions in an image, where each region is relatively homogeneous in terms of pixel intensity • => K-means can be used as a simple technique for region-finding • Note that K-means clustering knows nothing about the spatial aspects of the image • Other region-finding algorithms can operate spatially • Note that regions and edges are “duals” edges  images • So one could find regions give edges • Or one could find edges given regions (e.g., boundaries between clusters of pixels produced by K-means)