Markov Nets Dhruv Batra 10 708 Recitation 10302008

Markov Nets Dhruv Batra, 10 -708 Recitation 10/30/2008

Contents • MRFs – Semantics / Comparisons with BNs – Applications to vision – HW 4 implementation

Semantics • Bayes Nets

Semantics • Markov Nets

Semantics • Decomposition – Bayes Nets – Markov Nets

Semantics • Factorization • What happens in BNs?

Semantics • Active Trails in MNs • What happens in BNs?

Semantics • Independence Assumptions Markov Nets Global Ind Assumption, d-sep Local Ind Assumptions Markov Blanket Bayes Nets

Representation Theorem for Markov Networks If joint probability distribution P: Then If H is an I-map for P and Then H is an I-map for P joint probability distribution P: P is a positive distribution 10 -708 – Carlos Guestrin 2006 -2008 10

Semantics • Factorization • Energy functions • Equivalent representation

Semantics • Log Linear Models

Semantics • Energies in pairwise MRFs • What is encoded on nodes and edges?

Semantics • Priors on edges – Ising Prior / Potts Model – Metric MRFs

Metric MRFs • Energies in pairwise MRFs

Applications in Vision • Image Labelling tasks – – Denoising Stereo Segmentation, Object Recognition Geometry Estimation

MRF nodes as pixels Winkler, 1995, p. 32

MRF nodes as patches image patches scene patches F(xi, yi) image Y(xi, xj) scene

Network joint probability 1 P( x, y) = Z scene image ÕY( x , x ) ÕF( x , y ) i j i, j i i Scene-scene Image-scene compatibility function neighboring scene nodes i function local observations

Application • Motion Estimation

Stereo

Geometry Estimation

HW 4 • Interactive Image Segmentation

HW 4 • Pairwise MRF

HW 4

HW 4 • Step 1: GMMs • Step 2: Adjacency matrix / MRF Structure • Step 3: MRF parameters • Step 4: Loopy BP • Step 5: Segmentation Masks

Slide Credits • Bill Freeman, Fredo Durand, Lecture at MIT – http: //groups. csail. mit. edu/graphics/classes/Comp. Photo 06/html/ lecturenotes/2006 March 21 MRF. ppt • Charles A. Bouman – https: //engineering. purdue. edu/~bouman/ece 641/mrf_tutorial/vi ew. pdf • Fast Approximate Energy Minimization via Graph Cuts, Yuri Boykov, Olga Veksler and Ramin Zabih. IEEE PAMI 23(11), November 2001. • Make 3 D: Learning 3 -D Scene Structure from a Single Still Image, Ashutosh Saxena, Min Sun, Andrew Y. Ng, To appear in IEEE PAMI 2008

Questions?