EE462 MLCV 1 Lecture 3-4 Clustering (1hr) Gaussian Mixture and EM (1hr) Tae-Kyun Kim

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Lecture 3-4Clustering (1hr)Gaussian Mixture and EM (1hr)

Tae-Kyun Kim

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2D data vectors (green) are grouped to two homogenous clusters (blue and red).Clustering is achieved by an iterative algorithm (left to right). The cluster centers are marked x.

Vector Clustering

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``

RGB

Pixel Clustering (Image Quantisation)Image pixels are represented by 3D vectors of R,G,B values.The vectors are grouped to K=10,3,2 clusters, and represented by the mean values of the respective clusters.

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dim

ensi

on

D………

……

or raw pixels

…

K codewords

Patch Clustering (BoW in Lecture 9-10) Image patches are harvested around feature points in a large number of images.They are represented by finite dimensional vectors, and clustered to form a visual dictionary.

SIFT

20

20D=400

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Image ClusteringWhole images are represented as finite dimensional vectors.Homogenous vectors are grouped together in Euclidean space.

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K-means vs GMM

Hard clustering: a data point is assigned only one cluster.

Soft clustering: a data point is assigned multiple Gaussians probabilistically.

Two representative techniques are k-means and Gaussian Mixture Model (GMM).K-means assigns data points to the nearest clusters, while GMM assigns data to the Gaussian densities that best represent the data.

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Matrix and Vector Derivatives

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K-means Clustering

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till converge

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K=2

μ1

μ2

rnk

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Convergence proof (yes)Global minimum (no)

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V=V=

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Statistical Pattern Recognition Toolbox for Matlab

http://cmp.felk.cvut.cz/cmp/software/stprtool/

…\stprtool\probab\cmeans.m…\stprtool\probab\cmeans_tk.m

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Mixture of Gaussians

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Maximum Likelihood

s.t.

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objective ftn. f(x)constraints g(x)

max f(x) s.t. g(x)=0 max f(x) + g(x)

http://en.wikipedia.org/wiki/Lagrange_multiplier

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till converge

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Statistical Pattern Recognition Toolbox for Matlab

http://cmp.felk.cvut.cz/cmp/software/stprtool/

…\stprtool\visual\pgmm.m…\stprtool\demos\demo_emgmm.m

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Supplementary Material

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Information Theory (for Lecture 7-8)

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Advanced topic (optional)

http://www.iis.ee.ic.ac.uk/~tkkim/mlcv/lecture_clustering_em.pdf

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EM Algorithm in General

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