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Component Analysis for PR & HS Component Analysis for PR & HS Computer Vision & Image Processing Computer Vision & Image Processing Structure from motion. Structure from motion. Structure from motion


  1. Component Analysis for PR & HS Component Analysis for PR & HS • Computer Vision & Image Processing • Computer Vision & Image Processing – Structure from motion. – Structure from motion. Structure from motion – Spectral graph methods for segmentation. – Spectral graph methods for segmentation. – Appearance and shape models. – Appearance and shape models. – Fundamental matrix estimation and calibration. – Fundamental matrix estimation and calibration. – Compression. – Compression. – Classification. – Classification. – Dimensionality reduction and visualization. – Dimensionality reduction and visualization. • Signal Processing • Signal Processing – Spectral estimation, system identification (e.g. Kalman filter), sensor – Spectral estimation, system identification (e.g. Kalman filter), sensor array processing (e.g. cocktail problem, eco cancellation), blind source array processing (e.g. cocktail problem, eco cancellation), blind source separation, - separation, - • Computer Graphics • Computer Graphics – Compression (BRDF), synthesis,- – Compression (BRDF), synthesis,- • Speech, bioinformatics, combinatorial problems. • Speech, bioinformatics, combinatorial problems. ������������������ �������� �������� ������� ���������������������� PAVIS school on CV and PR 7 ������������������ �������� �������� ������� ���������������������� PAVIS school on CV and PR 8 Component Analysis for PR & HS Component Analysis for PR & HS • Computer Vision & Image Processing • Computer Vision & Image Processing – Structure from motion. – Structure from motion. – Spectral graph methods for segmentation. Spectral graph methods for segmentation. – Spectral graph methods for segmentation. – Appearance and shape models. – Appearance and shape models. Appearance and shape models – Fundamental matrix estimation and calibration. – Fundamental matrix estimation and calibration. – Compression. – Compression. – Classification. – Classification. – Dimensionality reduction and visualization. – Dimensionality reduction and visualization. • Signal Processing • Signal Processing – Spectral estimation, system identification (e.g. Kalman filter), sensor – Spectral estimation, system identification (e.g. Kalman filter), sensor array processing (e.g. cocktail problem, eco cancellation), blind source array processing (e.g. cocktail problem, eco cancellation), blind source separation, - separation, - • Computer Graphics • Computer Graphics – Compression (BRDF), synthesis,- – Compression (BRDF), synthesis,- • Speech, bioinformatics, combinatorial problems. • Speech, bioinformatics, combinatorial problems. PAVIS school on CV and PR PAVIS school on CV and PR 10 ������������������ �������� �������� ������� ���������������������� 9 ������������������ �������� �������� ������� ���������������������� 1

  2. Component Analysis for PR & HS Component Analysis for PR & HS • Computer Vision & Image Processing • Computer Vision & Image Processing – Structure from motion. – Structure from motion. – Spectral graph methods for segmentation. – Spectral graph methods for segmentation. – Appearance and shape models. – Appearance and shape models. – Fundamental matrix estimation and calibration. – Fundamental matrix estimation and calibration. – Compression. – Compression. – Classification. – Classification. – Dimensionality reduction and visualization. Dimensionality reduction and visualization – Dimensionality reduction and visualization. • Signal Processing • Signal Processing – Spectral estimation, system identification (e.g. Kalman filter), sensor – Spectral estimation, system identification (e.g. Kalman filter), sensor array processing (e.g. cocktail problem, eco cancellation), blind source array processing (e.g. cocktail problem, eco cancellation), blind source cocktail problem separation, - separation, - • Computer Graphics • Computer Graphics – Compression (BRDF), synthesis,- – Compression (BRDF), synthesis,- • Speech, bioinformatics, combinatorial problems. • Speech, bioinformatics, combinatorial problems. ������������������ �������� �������� ������� ���������������������� PAVIS school on CV and PR 11 ������������������ �������� �������� ������� ���������������������� PAVIS school on CV and PR 12 Why CA for PR & HS? Independent Component Analysis (ICA) Sound • Learn from high dimensional data and few samples. Source 1 – Useful for dimensionality reduction specially when functions are Mixture 1 smooth. • Natural geometric interpretation Output 1 Sound Source 2 • Easy to formulate, to solve and to extend Mixture 2 I – Non@linearities (Kernel methods) (Scholkopf & Smola,2002; Shawe@Taylor & C Cristianini,2004) Output 2 A – Probabilistic (latent variable models) (Everitt,1984) – Multi@factorial (tensors) (Paatero & Tapper, 1994 ;O’Leary & Peleg,1983; Vasilescu & Terzopoulos,2002; Vasilescu & Terzopoulos,2003) Output 3 – Exponential family PCA (Gordon,2002; Collins et al. 01) Mixture 3 Sound • Efficient methods O( d n< <n 2 ) Source 3 samples features PAVIS school on CV and PR 13 PAVIS school on CV and PR 14 ������������������ �������� �������� ������� ���������������������� ������������������ �������� �������� ������� ���������������������� 2

  3. Outline Are CA methods popular/useful/used? • About 28% of CVPR@07 papers use CA. • Google: • Introduction (15 min) – Results 1 @ 10 of about 1,870,000 for "principal component • Generative models (40 min) Generative models (40 min) analysis" . – (PCA, k@means, spectral clustering, NMF, ICA, MDS) (PCA, k@means, spectral clustering, NMF, ICA, MDS) – Results 1 @ 10 of about 506,000 for "independent component analysis" • Discriminative models (40 min) – Results 1 @ 10 of about 273,000 for "linear discriminant – (LDA, SVM, OCA, CCA) analysis" • Standard extensions of linear models (30 min) – Results 1 @ 10 of about 46,100 for "negative matrix factorization" – (Kernel methods, Latent variable models, Tensor factorization ) – Results 1 @ 10 of about 491,000 for "kernel methods" • Unified view (20 min) • Still work to do – Results 1 � 10 of about 83,000 for “ Spanish crisis " – Results 1 @ 10 of about 287,000,000 for "Britney Spears" ������������������ �������� �������� ������� ���������������������� PAVIS school on CV and PR 15 ������������������ �������� �������� ������� ���������������������� PAVIS school on CV and PR 16 Generative models Principal Component Analysis (PCA) (Pearson, 1901; Hotelling, 1933;Mardia et al., 1979; Jolliffe, 1986; Diamantaras, 1996) D ≈ BC � # • Principal Component Analysis/Singular Value   � � " � � ! Decomposition Σ =  $ � # �  +*+=+   +*+=+ Σ = � � ! � � �     • Non@Negative Matrix Factorization � # � #   +*@=@ • Independent Component Analysis +*@=@ � $ • K@means and spectral clustering @*@ =+ @*@ =+ • Multi@dimensional Scaling • PCA finds the directions of maximum variation of the data • PCA decorrelates the original variables PAVIS school on CV and PR 17 PAVIS school on CV and PR 18 ������������������ �������� �������� ������� ���������������������� ������������������ �������� �������� ������� ���������������������� 3

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