EE 6885 Statistical Pattern Recognition
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1 EE 6885 Statistical Patter Recogitio Fall 5 Prof. Shih-Fu Chag Lecture 6 (9/8/5 EE6887-Chag 6- Readig EM for Missig Features Textboo, DHS 3.9 Bayesia Parameter Estimatio ad Sufficiet Statistics Textboo, DHS Applicatio: Bayesia Face Detectio Referece paper (see course web site EE6887-Chag 6-
2 Parameter Estimatio: ML ad EM Give Data D Fid ˆ = arg max pd ( Gaussia ˆ (/ ˆ μ = x Σ= (/ ( x μ( x μ t l = N = log Mixture of Gaussia ( π N( x μ, Σ + π N( μ, Σ x Derive Auxiliary Fuctio N Q( = p( z x, log p( x, z t z t = t+ = arg maxq( t p(x π π x EE6887-Chag 6-3 EM for Missig Feature D = { Dg, Db} Dg : good feature, Db : missig feature i i Q( ; = E [l p( D, D ; D ; ] Margialize over the missig feature Db g b g Example, -D Gaussia * D= { x, x, x3, x4} = {,,, } 4 μ μ = σ σ Q( ; = E [l p( x, x ; x ; ] x4 g 4 g = 3 = [l p( x ] l p( x4 p( x4 ; x4 4 dx + = 4 =.75. x4. p( =. 3 x = 4 4 = [l p( x.938 ] l p( ' 4 dx = x 4 '. p( dx4. 4 E step ad M step? See figure i Sec. 3.9 EE6887-Chag 6-4
3 Bayesia Parameter Estimatio Istead of fixed uow, we assume as radom variable with distributio p( Give samples D, fid the maximal posterior pd ( p( p( D = pd ( = px ( pd ( p( d = Posterior gives the probability distributio of. Whe the sample size icreases, the distributio becomes peay. Differece from ML estimatio? px ( D = px ( p( Dd EE6887-Chag 6-5 Example: Gaussia Uivariate case: μ is the oly uow px ( μ N( μσ, p( μ N( μ, σ posterior p( μ D pd ( μ p( μ = px ( μ p( μ = x μ μ μ = exp ( exp ( πσ σ = πσ σ μ μ = exp ( N( μ, σ πσ σ Reproducig desity σ σ μ ˆ = μ +. μ where ˆ μ = x σ + σ σ + σ ad σ σσ = σ σ + = what if, σ, σ? EE6887-Chag 6-6 3
4 Example: Gaussia Case (Cot. Class coditioal desity px ( D = px ( μ p( μ Dd px ( D Nμ (, σ + σ vs. px ( μ N( μσ, Mea is replaced, variace is icreased. Multi-variate case posterior p( μ D N( μ, Σ Bayesia Classifier: compute posterior px ( D, ω p( ω - - μ ˆ where ˆ μ = =Σ( Σ+ Σ μ + Σ( Σ+ Σ μ x = - Σ =Σ( Σ+ Σ Σ Class coditioal desity px ( D Nμ (, Σ +Σ i i i EE6887-Chag 6-7 Sufficiet Statistics sufficiet statistics s: fuctio of samples that cotais all of the iformatio relevat to estimatio of parameters N Σ Example: Gaussia (, is uow PD x Σ x t ( = exp( / ( ( d = ( π Σ t t t = exp( ( d / / Σ Σ x + x Σ x ( π Σ = exp t t t = + ( exp( ( d / / Σ Σ x = ( π Σ x Σ x = t t g( ˆ μ, exp ( ˆ = Σ Σ μ where ˆ μ = x Sufficiet statistics = hd ( idepedet of EE6887-Chag 6-8 4
5 Sufficiet Statistics for Expoetial Family Gaussia, expoetial, Rayleigh, Poisso, etc. t p( x = α( xexp[ a( + b( c( x] t p( D = α( x exp[ a( + b( c( x ] = g( s, h( D = where s = c( x = = t g(, s = exp[(( a + b( s] hd ( = α( x See Table 3. for the S.S. for differet distributios. = EE6887-Chag 6-9 MAP detectors for face images H. Scheiderma & T. Kaade, Probabilistic modelig of local appearace ad spatial relatioships for object recogitio, CVPR 98. Capture face iformatio from 64x64 regios. Eough details for characterizig faces. Decompose ito 6x6 subregios to simplify models ad capture spatial locatios. Does ot cosider subregio depedecy. Less pealty for face detectio due to cosistet appearace ad locatio of saliet features, e.g., oises ad eyes. Also capture the distictive local appearaces i the lielihood ratio EE6887-Chag 6-5
6 Bayesia face detector (Cot. Simplify the subregio patters by PCA projectio, sparse codig, ad quatizatio 3.8M distictive patters of q Estimate the locatio distributios from traiig data. Reduce the complexity of the above distributios by patter clusterig q =VQ(q Clusterig helps discover saliet locatios Simple model for bacgroud ad ifrequet patters Multi-resolutio scalig to capture the face features at differet scales EE6887-Chag 6- Bayesia face detector (Cot. Fial Bayesia Estimatio Estimate class specific distributio usig traiig frotal face ad o-face images simple frequecy cout For test images, search faces at multiple scales, e.g., 7 scales over 8x8 to 338x338. Key features: joit local appearace-locatio statistical modelig bacgroud modelig discrete o-parameter distributio modelig EE6887-Chag 6-6
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