Outline. EM Algorithm and its Applications. K-Means Classifier. K-Means Classifier (Cont.) Introduction of EM K-Means EM EM Applications.

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1 EM Algorthm and ts Alcatons Y L Deartment of omuter Scence and Engneerng Unversty of Washngton utlne Introducton of EM K-Means EM EM Alcatons Image Segmentaton usng EM bect lass Recognton n BIR olor lusterng by K-means Algorthm K-Means lassfer Form K-means clusters from a set of n-dmensonal vectors. Set c teraton count to 2. hoose randomly a set of K means m,, m K. 3. For each vector, comute D,m k c, k,k and assgn to the cluster wth nearest mean. 4. Increment c by, udate the means to get m c,,m K c. 5. Reeat stes 3 and 4 untl k c k c for all k. {r, g, b } 2 {r 2, g 2, b 2 } {r, g, b } lassfer K-Means lassfcaton Results 2 2 luster Parameters m for m 2 for 2 m k for k K-Means lassfer ont. {r, g, b } Inut Known {r, g, b } 2 {r 2, g 2, b 2 } {r, g, b } luster Parameters m for m 2 for 2 m k for k utut Unknown lassfcaton Results 2 2 Inut Known 2 {r 2, g 2, b 2 } {r, g, b } Intal Guess of luster Parameters m, m 2,, m k utut Unknown lassfcaton Results, 2,, luster Parameters m, m 2,, m k lassfcaton Results 2 luster Parameters 2 m, m 2,, m k, 2,, luster Parameters c m, m 2,, m k lassfcaton Results c, 2,,

2 K-Means ont. K-Means Eamle Boot Ste: Intalze K clusters:,, K Each luster s reresented by ts mean m Iteraton Ste: Estmate the cluster of each data Re-estmate the cluster arameters m mean } { K-Means Eamle K-Means EM Boot Ste: Intalze K clusters:,, K, Σ and P for each cluster. Iteraton Ste: Estmate the cluster of each data Re-estmate the cluster arameters, Σ, For each cluster Eectaton Mamzaton EM lassfer EM lassfer ont. {r, g, b } 2 {r 2, g 2, b 2 } {r, g, b } lassfer EM lassfcaton Results 2 luster Parameters,Σ, for 2,Σ 2, 2 for 2 Inut Known {r, g, b } 2 {r 2, g 2, b 2 } {r, g, b } luster Parameters,Σ, for 2,Σ 2, 2 for 2 k,σ k, k for k utut Unknown lassfcaton Results 2 k,σ k, k for k 2

3 3 {r, g, b } 2 {r 2, g 2, b 2 } {r, g, b } luster Parameters,Σ, for 2,Σ 2, 2 for 2 k,σ k, k for k Eectaton Ste Inut Known Inut Estmaton utut lassfcaton Results 2 {r, g, b } 2 {r 2, g 2, b 2 } {r, g, b } luster Parameters,Σ, for 2,Σ 2, 2 for 2 k,σ k, k for k Mamzaton Ste Σ T Inut Known Inut Estmaton utut lassfcaton Results 2 EM Algorthm Boot Ste: Intalze K clusters:,, K Iteraton Ste: Eectaton Ste Mamzaton Ste, Σ and P for each cluster. Σ T EM Demo Demo htt:// Eamle htt://www-2.cs.cmu.edu/~awm/tutorals/gmm3.df EM Alcatons Blobworld: Image Segmentaton Usng Eectaton-Mamzaton and ts Alcaton to Image Queryng Image Segmentaton usng EM Ste : Feature Etracton Ste 2: Image Segmentaton usng EM

4 Symbols The feature vector for el s called. There are gong to be K segments; K s gven. The -th segment has a Gaussan dstrbuton wth arameters θ,σ. α 's are the weghts whch sum to of Gaussans. Θ s the collecton of arameters: Θ α,, α k, θ,, θ k Intalzaton Each of the K Gaussans wll have arameters θ,σ, where s the mean of the -th Gaussan. Σ s the covarance matr of the -th Gaussan. The covarance matrces are ntaled to be the dentty matr. The means can be ntalzed by fndng the average feature vectors n each of K wndows n the mage; ths s data-drven ntalzaton. E-Ste, Θ α f θ K k k f θ d / 2 2π Σ k α f θ T Σ 2 e / 2 k M-Ste new Σ new, Θ, Θ, Θ, Θ new new T α new, Θ Samle Results bect lass Recognton n BIR The Goal: Automatc mage labelng annotaton to enable obect-based mage retreval 4

5 Problem Statement Known: Some mages and ther corresondng descrtons Abstract Regons rgnal Images olor Regons Teture Regons Lne lusters {trees, grass, cherry trees} {cheetah, trunk} {mountans, sky} {beach, sky, trees, water} To solve: What obect classes are resent n new mages bect Model Learnng Ideal Boat, Water, Sky,! Sky Tree Sky boat buldng sky tree Water Tree Water Boat water Boat Learned Models boat regon attrbutes obect bect Model Learnng Real Model Intal Estmaton Sky Estmate the ntal model of an obect usng all the regon features from all mages that contan the obect {sky, tree, water, boat} Tree Water Boat Tree Learned Models regon attrbutes obect Sky 5

6 EM Varant bect Model Learnng Intal Model for trees Intal Model for sky EM Fnal Model for trees Fnal Model for sky Assumtons The feature dstrbuton of each obect wthn a regon s a Gaussan; Each mage s a set of regons, each of whch can be modeled as a mture of multvarate Gaussan dstrbutons.. Intalzaton Ste Eamle Image & descrton I 2 I 2 3 I Iteraton Ste Eamle I I 2 I E-Ste q q 2 q 3 0 q 0 q 2 0 q 3 W0.8 W0.2 W0.2 W0.8 W0.2 W0.8 M-Ste W0.8 W0.2 W0.8 W0.2 W0.2 W0.8 q q q 3 2 Image Labelng Test Image To calculate tree mage olor Regons tree a a tree mage ma tree o FI ma o r a r a F I tree tree comare bect Model Database Tree Sky Eerments 860 mages 8 keywords: mountans 30, orangutan 37, track 40, tree trunk 43, football feld 43, beach 45, rare grass 53, cherry tree 53, snow 54, zebra 56, olar bear 56, lon 7, water 76, chmanzee 79, cheetah 2, sky 259, grass 272, tree 36. A set of cross-valdaton eerments 80% as the tranng set and the other 20% as the test set 6

7 R harts Samle Results True Postve Rate True Postve Rate cheetah False Postve Rate Indeendent Treatment of olor and Teture False Postve Rate Usng Intersecton of olor and Teture Samle Results ont. Samle Results ont. grass lon 7