Computer Vision. sumber: - Forsyth+Ponce Chap Standford Vision & Modeling. PCA (Principal Component Analysis EM (Expectation Maximazation)

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1 Modeling Statisti untu omputer Vision sumber: - Forsyth+once hap Standford Vision & Modeling Modeling Statisti untu omputer Vision Agenda Statistical Models baca Forsyth+once hap Bayesian Decision Theory - Density Estimation A rincipal omponent Analysis EM Epectation Maimazation age

2 ontoh apliasi model statisti: segmentasi dengan EM Segmentasi Warna age 2 2

3 ontoh recognition dengan A: Face Recognition dengan A Tur+entland, : ontoh contour tracing dengan theorema Bayes Snae Tracing E + βω ln pc + ln pc age 3 3

4 Statistical Models / robability Theory Model Statistical : model yg merepresentasian Uncertainty and Variability robability Theory: menjelasan tentang meanisme dari Uncertainty Lihat contoh2 pada file pdf buu eletroni, pada D. Forsyth+once hap 6 Statistical Models / robability Theory Fata mengaan bahwa Segala sesuatu adalah merupaan Variabel Random age 4 4

5 engantar Desisi Optimal Bayes Dengan berbagai apliasi untu proses lasifiasi Teori Desisi Bayes Bayes Decision Theory ontoh: : haracter Recognition: Tujuan: Menglasifiasian arater sedemiian rupa sehingga dapat meminimalisasi probabiliti esalahan lasifiasi minimize probability of misclassification age 5 5

6 Teori Desisi Bayes onsep #: riors prob. anggapan awal 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 a=0.75 b=0.25? Teori Desisi Bayes Konsep #2: onditional robability / Lielihood X a X X b # blac piel # blac piel age 6 6

7 Teori Desisi Bayes ontoh: X a X b =? X=7 Teori Desisi Bayes ontoh: X a X b =? X=8 age 7 7

8 Teori Desisi Bayes ontoh: X a X b = a X=8 Karena a=0.75 b=0.25 Teori Desisi Bayes ontoh: X a X b =? X=9 a=0.75 b=0.25 age 8 8

9 9 age 9 Teori Teori Desisi Desisi Bayes Bayes Teorema Bayes : X X X = Teori Teori Desisi Desisi Bayes Bayes Teorema Bayes : X X X = = j j j X X

10 Teori Desisi Bayes Teorema Bayes : osterior = Lielihood prior Normalization factor Teori Desisi Bayes ontoh: X a X b age 0 0

11 Teori Desisi Bayes ontoh: X a a X b b Teori Desisi Bayes ontoh: a X b X X>8 sehingga termasu elas b age

12 Teori Desisi Bayes Tujuan: Menglasifiasian arater sedemiian rupa sehingga dapat meminimalisasi probabiliti esalahan lasifiasi minimize probability of misclassification Batas2 desisi Decision boundaries: > j for all j Teori Desisi Bayes Batas-batas desisi: > j for all j > j j for all j age 2 2

13 Teori Desisi Bayes Daerah desisi : R,..., Rc R R2 R3 Teori Desisi Bayes Tujuan: minimize probability of misclassification error = R2, + R, 2 age 3 3

14 4 age 4 Teori Teori Desisi Desisi Bayes Bayes Tujuan Tujuan: minimize probability of misclassification minimize probability of misclassification,, error 2 2 R R + = R R + = Teori Teori Desisi Desisi Bayes Bayes Tujuan Tujuan: minimize probability of misclassification minimize probability of misclassification,, error 2 2 R R + = R R + = + = R R d p d p

15 Teori Desisi Bayes Tujuan: minimize probability of misclassification = R 2 p d + p 2 2 d R Teori Desisi Bayes Mengapa p menjadi sangat-sangat sangat penting? osteriori robability age 5 5

16 Teori Desisi Bayes Mengapa p jadi penting seali? ontoh #: Speech Recognition FFT melscale ban = y ε [/ah/, /eh/,.. /uh/] apple,...,zebra Teori Desisi Bayes ontoh #: Speech Recognition /t/ /t/ /t/ /t/ FFT melscale ban /aal/ /aol/ /owl/ age 6 6

17 Teori Desisi Bayes ontoh #: Speech Recognition Bagaimana manusia dapat mengenali dengan mudah? Apaah mesin bisa??? Teori Desisi Bayes ontoh #: Speech Recognition FFT melscale ban = y p age 7 7

18 Teori Desisi Bayes ontoh #: Speech Recognition Language Model FFT melscale ban = y wrec a nice beach = 0.00 recognize speech = 0.02 p Teori Desisi Bayes Mengapa p penting? ontoh #2: omputer Vision Low-Level Image Measurements High-Level Model Knowledge p age 8 8

19 Bayes Mengapa p penting? ontoh #3: urve Fitting E + βω ln pc + ln pc Bayes Mengapa p penting? ontoh #4: Snae Tracing E + βω ln pc + ln pc age 9 9

20 Estimasi Densitas Density Estimation Statistical Models Forsyth+once hap. 6 - Bayesian Decision Theory - Density Estimation robability Density Estimation? Data olesi:,2,3,4,5,... Estimasi: p age 20 20

21 robability Density Estimation Beberapa metode estimasi dengan: arametric Representations Non-arametric Representations Miture Models robability Density Estimation arametric Representations - Normal Distribution Gaussian - Maimum Lielihood - Bayesian Learning age 2 2

22 Normal Distribution σ µ σ = mean = variance Multivariate Normal Distribution age 22 22

23 Multivariate Normal Distribution Mengapa Gaussian, apa istimewanya? unya properti sederhana: - linear transformasi Gaussians adalah Gaussian juga - marginal dan conditional densities dari Gaussians adalah Gaussian - Moment dari densitas Gaussian secara eplisit merupaan fungsi dari µ Good Model of Nature: - entral Limit Theorem: Mean of M random variables is distributed normally in the limit. Multivariate Normal Distribution Discriminant functions: y = ln p + ln age 23 23

24 Multivariate Normal Distribution Discriminant functions: y = ln p + ln equal priors + cov: Jara Mahalanobis Multivariate Normal Distribution Bagaimana "Belajar" dari contoh? Bisa dilauan dengan : Maimum Lielihood Bayesian Learning age 24 24

25 Maimum Lielihood Bagaimana "Belajar" dari contoh?:?? Maimum Lielihood Lielihood dari model densitas θ untu menghasilan data X: Lielihood L θ p X N θ = p n= n θ age 25 25

26 Maimum Lielihood Lielihood dari model densitas θ untu menghasilan data X: Lielihood L θ p X N θ = p n= n θ more convenient : E = ln L θ = N n= ln p n θ Maimum Lielihood Belajar = roses optimasi maimizing lielihood / minimizing E: more convenient : E = ln L θ = N n= ln p n θ age 26 26

27 Maimum Lielihood Maimum Lielihood untu Gaussian density: more convenient : E = ln L θ = N n= ln p n θ Solusi singatnya: ˆµ = N n N n= N ˆ = n µ ˆ n µ ˆ N n= T robability Density Estimation arametric Representations Non-arametric Representations Miture Models age 27 27