Modeling Statistik untuk Computer Vision sumber: - Forsyth+Ponce Chap. 7. - Standford Vision & Modeling
Modeling Statistik untuk Computer Vision Agenda Statistical Models (baca Forsyth+Ponce Chap. 7.) - Bayesian Decision Theory - Density Estimation PCA (Principal Component Analysis EM (Expectation Maximazation) - title - report on work done together with JM at UCB and together with MC MS at Interval 2 2
Contoh aplikasi model statistik: segmentasi dengan EM Segmentasi Warna
Contoh recognition dengan PCA: Face Recognition dengan PCA (Turk+Pentland, ):
Contoh contour tracking dengan theorema Bayes Snake Tracking E + bW ln p(x|c) + ln p(c)
Statistical Models / Probability Theory Model Statistical : model yg merepresentasikan Uncertainty and Variability Probability Theory: menjelaskan tentang mekanisme dari Uncertainty Lihat contoh2 pada file pdf buku elektronik, pada CD. (Forsyth+Ponce Chap 6) - title - report on work done together with JM at UCB and together with MC MS at Interval 2 2
Statistical Models / Probability Theory Fakta mengakan bahwa Segala sesuatu adalah merupakan Variabel Random - title - report on work done together with JM at UCB and together with MC MS at Interval 2 2
Pengantar Desisi Optimal Bayes Dengan berbagai aplikasi untuk proses klasifikasi
Teori Desisi Bayes (Bayes Decision Theory) Contoh: Character Recognition: Tujuan: Mengklasifikasikan karakter sedemikian rupa sehingga dapat meminimalisasi probabiliti kesalahan klasifikasi (minimize probability of misclassification)
? Teori Desisi Bayes konsep #1: Priors (prob. anggapan awal) P(a)=0.75 P(b)=0.25 a a b a b a a b a b a a a a b a a b a a b a a a a b b a b a b a a b a a
Teori Desisi Bayes # black pixel # black pixel Konsep #2: Conditional Probability / Likelihood # black pixel # black pixel
Teori Desisi Bayes Contoh: X=7
Teori Desisi Bayes Contoh: X=8
Teori Desisi Bayes Contoh: Karena… P(a)=0.75 P(b)=0.25 X=8
Teori Desisi Bayes Contoh: P(a)=0.75 P(b)=0.25 X=9
Teori Desisi Bayes Teorema Bayes :
Teori Desisi Bayes Teorema Bayes :
Teori Desisi Bayes Teorema Bayes : Likelihood x prior Posterior = Normalization factor
Teori Desisi Bayes Contoh:
Teori Desisi Bayes Contoh:
Teori Desisi Bayes Contoh: X>8 sehingga termasuk kelas b
Teori Desisi Bayes Tujuan: Mengklasifikasikan karakter sedemikian rupa sehingga dapat meminimalisasi probabiliti kesalahan klasifikasi (minimize probability of misclassification) Batas2 desisi (Decision boundaries):
Teori Desisi Bayes Batas-batas desisi:
Teori Desisi Bayes Daerah desisi : R3 R1 R2
Teori Desisi Bayes Tujuan: minimize probability of misclassification
Teori Desisi Bayes Tujuan: minimize probability of misclassification
Teori Desisi Bayes Tujuan: minimize probability of misclassification
Teori Desisi Bayes Tujuan: minimize probability of misclassification
Teori Desisi Bayes Mengapa (Posteriori Probability) menjadi sangat-sangat penting ?
Teori Desisi Bayes Mengapa jadi penting sekali ? Contoh #1: Speech Recognition 7 1 8 9 FFT melscale bank = x y e [/ah/, /eh/, .. /uh/] apple, ...,zebra
Teori Desisi Bayes Contoh #1: Speech Recognition /t/ /t/ /t/ /t/ FFT melscale bank /aal/ /aol/ /owl/
Teori Desisi Bayes Contoh #1: Speech Recognition Bagaimana manusia dapat mengenali dengan mudah? Apakah mesin bisa ???
Teori Desisi Bayes Contoh #1: Speech Recognition = x y FFT 7 1 8 9 melscale bank = x y
Teori Desisi Bayes Contoh #1: Speech Recognition = x y Language Model 7 1 8 9 FFT melscale bank = x y P(“wreck a nice beach”) = 0.001 P(“recognize speech”) = 0.02
Teori Desisi Bayes Mengapa penting ? Contoh #2: Computer Vision Low-Level Image Measurements High-Level Model Knowledge
Bayes Mengapa penting? Contoh #3: Curve Fitting E + bW ln p(x|c) + ln p(c)
Bayes Mengapa penting? Contoh #4: Snake Tracking E + bW ln p(x|c) + ln p(c)
Estimasi Densitas (Density Estimation) Statistical Models (Forsyth+Ponce Chap. 6) - Bayesian Decision Theory - Density Estimation - title - report on work done together with JM at UCB and together with MC MS at Interval 2 2
Probability Density Estimation ? Data koleksi: x1,x2,x3,x4,x5,... x Estimasi: x
Probability Density Estimation Beberapa metode estimasi dengan: Parametric Representations Non-Parametric Representations Mixture Models
Probability Density Estimation Parametric Representations - Normal Distribution (Gaussian) - Maximum Likelihood - Bayesian Learning
Normal Distribution
Multivariate Normal Distribution
Multivariate Normal Distribution Mengapa Gaussian, apa istimewanya ? Punya properti sederhana: - linear transformasi Gaussians adalah Gaussian juga - marginal dan conditional densities dari Gaussians adalah Gaussian - Moment dari densitas Gaussian secara explisit merupakan fungsi dari m,s “Good” Model of Nature: - Central Limit Theorem: Mean of M random variables is distributed normally in the limit.
Multivariate Normal Distribution Discriminant functions:
Multivariate Normal Distribution Discriminant functions: equal priors + cov: Jarak Mahalanobis
Multivariate Normal Distribution Bagaimana "Belajar" dari contoh ? Bisa dilakukan dengan : Maximum Likelihood Bayesian Learning
Maximum Likelihood Bagaimana "Belajar" dari contoh ?: ? x ? x
Maximum Likelihood Likelihood dari model densitas q untuk menghasilkan data X:
Maximum Likelihood Likelihood dari model densitas q untuk menghasilkan data X:
Maximum Likelihood “Belajar” = Proses optimasi (maximizing likelihood / minimizing E):
Maximum Likelihood Maximum Likelihood untuk Gaussian density: Solusi singkatnya:
Probability Density Estimation Parametric Representations Non-Parametric Representations Mixture Models