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This course covers essential clustering methods, including graph-theoretic clustering and the Hough transform. Students will learn about the basics of clustering, focusing on k-means; the process of assigning points to clusters and recalculating means. Additional topics include agglomerative and divisive clustering, and the importance of feature space in determining token similarity. Course materials include chapters from Forsyth & Ponce, and practical examples will enhance understanding of both the theoretical and practical aspects of clustering in computer vision.
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Segmentation Course web page: vision.cis.udel.edu/~cv May 5, 2003 Lecture 30
Announcements • Read Forsyth & Ponce Chapters 14.5 and 15.1 on graph-theoretic clustering and the Hough transform, respectively, for Wednesday • HW 5 due Friday • Class is at 4 pm on Friday due to Honors Day activities
Outline • Clustering basics • Methods • k-means clustering • Graph-theoretic clustering
Basic Approaches to Clustering • Unknown number of clusters • Agglomerative clustering • Start with as many clusters as tokens and selectively merge • Divisive clustering • Start with one cluster for all tokens and selectively split • Known number of clusters • Selectively change cluster memberships of tokens • Merging/splitting/rearranging stops when threshold on token similarity is reached • Within cluster: As similar as possible • Between clusters: As dissimilar as possible
Feature Space • Every token is identified by a set of salient visual characteristics called features (akin to gestalt grouping factors). For example: • Position • Color • Texture • Optical flow vector • Size, orientation (if token is larger than a pixel) • The choice of features and how they are quantified implies a feature space in which each token is represented by a point • Token similarity is thus measured by distance between points (aka “feature vectors”) in feature space
k-means Clustering • Initialization: Given k categories, N points (in feature space). Pick k points randomly; these are initial cluster centers (means) ¹1, …,¹k. Repeat the following: • Assign all N points to clusters by nearest ¹i (make sure no cluster is empty) • Recompute mean ¹i of each cluster from Ci member points • If no mean has changed more than some ¢, stop • Effectively carries out gradient descent on
Example: 3-means Clustering from Duda et al. Convergence in 3 steps
Example: k-means Clustering from Forsyth & Ponce 4 of 11 clusters using color alone
Clustering vs. Connected Components • Connected components: • Agglomerative • Feature space is position (adjacency) • Tokens are pixels that have passed a similarity test with respect to an absolute standard (e.g., brightness, redness, motion magnitude) • k-means • Known number of clusters (but can be estimated adaptively) • Feature space is general • Tokens are general. The similarity test guiding cluster membership is relative—i.e., only between the points themselves courtesy of HIPR Connected components example
Example: k-means Clustering from Forsyth & Ponce 4 of 20 clusters using color and position
k-means Clustering: Axis Scaling • Features of different types may have different scales • E.g., pixel position on a 640 x 480 image vs. RGB color in range [0, 1] for each channel • Problem: Features with bigger scales dominate Euclidean metric • Solutions: Manually weight features or use Mahalanobis distance in objective function • Requires guarantee of more points in each cluster
Example: k-means Clustering for Tracking from B. Heisele et al.
