Manifold Bayesian Sequential Partitioning and Manifold Contraction Correction

Manifold Bayesian Sequential Partitioning and Manifold Contraction Correction Shortest Distance Merging Student: Po-Hsu Shih Advisor: Sheng-Jyh Wang 1

Outline • • • Introduction Background Manifold Bayesian Sequential Partitioning Manifold Contraction Correction Shortest Distance Merging Discussion Conclusion & Future Works 2

Introduction • Complicated Feature, High Dimension • Data (exponential growth) • The curse of dimensionality • Appearance • Dimensionality Reduction • Manifold Learning • Visualization • Low dimensional manifold embedded in the high dimensional space Ref : The Manifold Ways of Perception 中央研究院週報 (第 1058期) 流行學習與人臉辨識 3

Introduction • Manifold Learning Methods • • • Isometric Feature Mapping (Isomap) Locally Linear Embedding (LLE) Maximum Variance Unfolding (MVU) Laplacian Eigenmap …… Ref : Advanced Introduction to Machine Learning , CMU-10715 Modern Multidimensional Scaling: Theory and Applications A Global Geometric Framework for Nonlinear Dimensionality Reduction by Locally Linear Embedding Unsupervised learning of image manifolds by semidefinite programming Laplacian eigenmaps for dimensionality reduction and data representation 4

Introduction • Ref : Modern Multidimensional Scaling: Theory and Applications 古典多維標度法 (MDS) , 線代啟示錄 Advanced Introduction to Machine Learning , CMU-10715 A Global Geometric Framework for Nonlinear Dimensionality Reduction

Outline • • • Introduction Background Manifold Bayesian Sequential Partitioning Manifold Contraction Correction Shortest Distance Merging Discussion Conclusion & Future Works 6

Background • • Segmentation a sequel of optimization pixel-based + grid-based region-based hierarchical Segmentation number • Iterative Contraction and Merging (ICM) 40 30 20 MNCut GBIS Mshift ICM 7 Ref : Hierarchical Image Segmentation based on Iterative Contraction and Merging

Background • Affinity regularization window number color mean covariance identity • Energy Function • Optimal Solution 8 Ref : Hierarchical Image Segmentation based on Iterative Contraction and Merging

Background • region data probability parameter multinomial Dirichlet Beta volume 9 Ref : Multivariate Density Estimation by Bayesian Sequential Partitioning

Background • cut path 10 Ref : Multivariate Density Estimation by Bayesian Sequential Partitioning

Background • Ref : Unsupervised Hierarchical Image Segmentation based on Bayesian Sequential Partitioning 11

Background • precision mean parameters Noticeable Difference Unnoticeable Difference variance region channel number Lab color Ref : Unsupervised Hierarchical Image Segmentation based on Bayesian Sequential Partitioning 12

Outline • • • Introduction Background Manifold Bayesian Sequential Partitioning Manifold Contraction Correction Shortest Distance Merging Discussion Conclusion & Future Works 13

Manifold Bayesian Sequential Partitioning • Ref : Gaussian Conjugate Prior Cheat Sheet (Tom SF Haines) Bayesian Data Analysis (Gelman, Carlin, Stern and Rubin) 14

Manifold Bayesian Sequential Partitioning Score Function Ref : Gaussian Conjugate Prior Cheat Sheet (Tom SF Haines) Bayesian Data Analysis (Gelman, Carlin, Stern and Rubin) 15

Manifold Bayesian Sequential Partitioning Segmentation = 30 , N = 1000 Segmentation = 70 , N = 999 • Experiment 1 • Circles Log Posterior

Manifold Bayesian Sequential Partitioning Segmentation = 40 , N = 1000 Segmentation = 20 , N = 1000 Segmentation = 70 , N = 1000 • Experiment 2 • Swiss rolls Log Posterior 17

Manifold Bayesian Sequential Partitioning Experiment 3 Segmentation = 2 Segmentation = 3 Segmentation = 4 Segmentation = 1 , N = 1000 Log Posterior Log Posterior Group Score = 2. 664 Group Score = 0. 010851 Group Score = 1. 6708 Group Score = 2. 7852 Group Score = 0. 5564 18

Manifold Bayesian Sequential Partitioning • Experiment 4 Segmentation = 40 , N = 1000 Segmentation = 8 , N = 1000 Partial Enlarged Picture • Outliers Log Posterior 19

Outline • • • Introduction Background Manifold Bayesian Sequential Partitioning Manifold Contraction Correction Shortest Distance Merging Discussion Conclusion & Future Works 20

Manifold Contraction Correction Shortest Distance Merging • Shortest Distance Merging Cluster = 4 Cluster = 3 Cluster = 2 Segmentation = 5 , N = 1000 Log Posterior Algorithm: 1. Region Distance Matrix (5 x 5) 2. Shortest Distance Merging 3. Region Distance Matrix (4 x 4) 4. …… 21

Manifold Contraction Correction Shortest Distance Merging • Shortest Distance Merging Segmentation = 40 , N = 1000 Log Posterior Cluster = 2 22

Manifold Contraction Correction Shortest Distance Merging • 23

Manifold Contraction Correction Shortest Distance Merging Segmentation = 8 , N = 1000 Log Posterior • Experiment 1 Cluster = 2 24

Manifold Contraction Correction Shortest Distance Merging Segmentation = 40 , N = 1000 Log Posterior Partial Enlarged Picture • Experiment 2 Cluster = 2 25

Manifold Contraction Correction Shortest Distance Merging • Experiment 3 26

Manifold Contraction Correction Shortest Distance Merging • Experiment 3 27

Manifold Contraction Correction Shortest Distance Merging • Experiment 4 28

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Outline • • • Introduction Background Manifold Bayesian Sequential Partitioning Manifold Contraction Correction Shortest Distance Merging Discussion Conclusion & Future Works 30

Discussion Manifold Bayesian Sequential Partitioning Score Function Ref : Gaussian Conjugate Prior Cheat Sheet (Tom SF Haines) Bayesian Data Analysis (Gelman, Carlin, Stern and Rubin) 31

Segmentation = 70 , N = 1000 Log Posterior Group Score 32

Discussion Segmentation = 70 , N = 1000 Noise Log Posterior Noise = 0. 2 Noise = 1. 8 Noise = 0. 6 Noise = 1. 4 Noise = 2. 2 33

Outline • • • Introduction Background Manifold Bayesian Sequential Partitioning Manifold Contraction Correction Shortest Distance Merging Discussion Conclusion & Future Works 34

Conclusion • Manifold Bayesian Sequential Partitioning • can separate different manifolds in data space ∵ enough cuts • outliers appear ∵ not enough cuts & noise • Manifold Contraction Correction Shortest Distance Merging • can remerge outliers ∵ MBSP manifolds roughly provides different • hard to decide distance limit ∵ without visual observation 35

Future Works • • More examples 3 -dimensional data Real Sequential images ( high-dimensional data ) Speed up 36

Reference • The Manifold Ways of Perception • 中央研究院週報 (第 1058期) 流行學習與人臉辨識 • Advanced Introduction to Machine Learning , CMU-10715 • Modern Multidimensional Scaling: Theory and Applications • A Global Geometric Framework for Nonlinear Dimensionality Reduction • Nonlinear Dimensionality Reduction by Locally Linear Embedding • Unsupervised learning of image manifolds by semidefinite programming • Laplacian eigenmaps for dimensionality reduction and data representation • Modern Multidimensional Scaling: Theory and Applications • 古典多維標度法 (MDS) , 線代啟示錄 • Hierarchical Image Segmentation based on Iterative Contraction and Merging • Multivariate Density Estimation by Bayesian Sequential Partitioning • Unsupervised Hierarchical Image Segmentation based on Bayesian Sequential Partitioning 37

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