Slide 1ContentsSemanticsSemanticsSemanticsSemanticsSlide 7SemanticsSemanticsRepresentation Theorem for Markov NetworksSemanticsSemanticsSemanticsSemanticsSlide 15Metric MRFsApplications in VisionMRF nodes as pixelsMRF nodes as patchesNetwork joint probabilityApplicationStereoGeometry EstimationGeometry EstimationHW4HW4HW4HW4Slide 29Slide 30Markov NetsDhruv Batra,10-708 Recitation10/30/2008Contents•MRFs–Semantics / Comparisons with BNs–Applications to vision–HW4 implementationSemantics•Bayes NetsSemantics•Markov NetsSemantics•Decomposition–Bayes Nets–Markov NetsSemantics•Factorization•What happens in BNs?Semantics•Active Trails in MNs•What happens in BNs?Semantics•Independence AssumptionsMarkov Nets Bayes NetsGlobal Ind Assumption, d-sepLocal Ind AssumptionsMarkovBlanket10-708 – Carlos Guestrin 2006-200810Representation Theorem for Markov NetworksIf H is an I-map for Pand P is a positive distributionThenThenH is an I-map for PIf joint probability distribution P:joint probability distribution P:Semantics•Factorization•Energy functions•Equivalent representationSemantics•Log Linear ModelsSemantics•Energies in pairwise MRFs•What is encoded on nodes and edges?Semantics•Priors on edges–Ising Prior / Potts Model–Metric MRFsMetric MRFs•Energies in pairwise MRFsApplications in Vision•Image Labelling tasks–Denoising–Stereo–Segmentation, Object Recognition–Geometry EstimationMRF nodes as pixelsWinkler, 1995, p. 32MRF nodes as patchesimage patchesF(xi, yi)Y(xi, xj)imagescenescene patchesNetwork joint probabilitysceneimageScene-scenecompatibilityfunctionneighboringscene nodeslocal observationsImage-scenecompatibilityfunctionÕÕFY=iiijijiyxxxZyxP ),(),(1),(,Application•Motion EstimationStereoGeometry EstimationGeometry EstimationHW4•Interactive Image SegmentationHW4•Pairwise MRFHW4HW4•Step 1: GMMs•Step 2: Adjacency matrix / MRF Structure•Step 3: MRF parameters•Step 4: Loopy BP•Step 5: Segmentation MasksSlide Credits•Bill Freeman, Fredo Durand, Lecture at MIT–http://groups.csail.mit.edu/graphics/classes/CompPhoto06/html/lecturenotes/2006March21MRF.ppt•Charles A. Bouman–https://engineering.purdue.edu/~bouman/ece641/mrf_tutorial/view.pdf•Fast Approximate Energy Minimization via Graph Cuts, Yuri Boykov, Olga Veksler and Ramin Zabih. IEEE PAMI 23(11), November 2001.•Make3D: Learning 3-D Scene Structure from a Single Still Image, Ashutosh Saxena, Min Sun, Andrew Y. Ng, To appear in IEEE PAMI
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