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# Class Note for EECS 841 with Professor Potetz at KU 10

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Date Created: 02/06/15

EECS 841 Computer Vision Brian Potetz Fall 2008 Lecture 28 Optical Flow Optical Flow Motion of brightness pattern in the image Ideally Optical flow Motion field Aperture Problem Aperture Problem Optical Flow Constraint Equation xu8tyv5t 1mg Optical Flow Velocities u 1 Jay Ly Displacement time 1 time t at Gx y u 6t v St Assume brightness of patch remains same in both images Exu6tyv6tt5t Exyt Assume small motion First order Taylor expansion of E Exyt 6x 6y E6t E mExyt 0x y at Optical Flow Constraint Equation g6y 6t 0 6x y at Divide by at and take the limit 6t gt 0 eggyg dt 6x dt y at x Constraint Equation ExuEyvE 0 Optical Flow Constraint Equation 6E BE 5xi i 6x ayay Divide by 5t and take the limit 6t a 0 u EQEE0 d1 61 d By a Constraint Equatlun ExuEy vE 0 quot39 NOTE uv must lie on a straight line We can compute Ex Ey 5 using gradient operators But uv cannot be found uniquely with this constraint Optical Flow Constraint Intuitively what does this constraint mean The component ofthe flow in the gradient direction is determined The component ofthe flow parallel to an edge is unknown Computing Optical Flow Formulate Error lrl Optical Flow Constraint 2 ExuEyvE2dxdy Wu We need additional constraintsi Smoothness Constraint as in snape trom snading and stereo Usually motion rield varies smoothly in ne image so penalize departure rrom smootnness 2 2 2 2 e f ux uyvx vy dxdy WK Find u v at each image point tnat MlNlMlZES e e M Weighting taotoi Discrete Optical Flow Algorithm Considerimage pixel 13 Departure trom Smoothness Constraint 1 Z Wt1 391 Lin 39 iJ 2 2 1 virlJ 1 vllvIJ Error in Optical Flow constraint equation cg EV ug EV vv E902 We seellttne set 14y amp vy hat minimize e 2 2 v A 0v Sprinter one term Discrete Optical Flow Algorithm Differentiating e Wrt v5 4 u and settingto zero izmu nzxwfuunzfv 15 0 quotH g2v E2AEfu EfvEfiEquot 0 u E amp a are averages or uv around pixel kl Update Rule HT H in H 41 7 Ex nEy 15 u quot39 39quot 391MEf E 1 m T EfgEquEtu u v v quot quot 1ME39 E39 Example Optical Flow Result Low Texture Region Bad gradients have small magnitude Edges soso aperture problem large gradients all the same High Textured Region Good gradients are different large magnitudes Revisiting the Small Motion Assumption 77 7 i 39 l 1 Is this motion small enough Probably not it s much larger than one pixel 2ncl order terms dominate How might we solve his problem Revisiting the Small Motion Assumption Is this motion small enough Probably not it s much larger than one pixel 2ncl order terms dominate How might we solve this problem Reduce the Resolution Coarsetofine Optical Flow Estimation u125 pixels quot2395P Xe s u5 pixels u10 pixelsquot Gaussian pyramid of image H Gaussian pyramid of image I Coarsetofine Optical Flow Estimation run iterative OF Jupsample l i i I i i x x I i i i ll i h gt run Iterative OF 4 i i l I 39 if i i X 1 I g X 39 l l l x l t t i l i 39 t i i image H Gaussian pyramid of image H Gaussian pyramid of image I Future Lectures Concepts useful to many class projects Statistical methods in computer vision statistical models of natural images statistical inference approaches to energy minimization Modern Object Recognition Systems SFT features Segmentation Superpixe approaches ScaleInvariant Feature Transform SIFT Superpixels OverSegmentation Probabilistic Methods for Vision Probabilistic approaches are useful in several ways Setting parameters according to the statistics of real world problems Provides natural ways to decompose problems into subproblems Bayes rule chain rule conditional independence Approach optimization with techniques from statistical inference Typical Snake Energies 1 Elasticity EezasticvO syn5945 1 1 Stiffness Estiffnessv O maidsnug 1 Edge Proximity EedgevO VIS7ys 2ds User interaction Elmo O1 Userzsysds Smoothness Constraint for SFS In nature objects are cohesive and typically have smooth surfaces Smoothness constraint relates surface normals of neighboring surface points Minimize Esmoothness l l penalize rapid changes in surface normals over the image Modern Stereo Algorithms Often use colorbased adaptive windows Clean up results using a Global Energy Function ie Minimize DTSFEHWIESE function ufxy E01 Edutad t Esmoothd 1732 H 73 Edatad 20Ileftxay7lrightx t dWMMD any 7 24mm 7 ma 7 mg Z ltdltzy 1 7 dltziygtgt S m o ot h n e ss P ri 0 rs SSEESISSELTZSSIIIS I ltAdgt quot 7 7 MM Gaussian Approach Lorentzian Approach 7 Z dx Ly 7 mm Z ltdltm 1 7 my m Every Energy Function has a Probabilistic Interpretation Recall the energy function we minimized for stereo Ed Em AEsmcth Every Energy Function has a Probabilistic Interpretation Recall the energy function we minimized for stereo Ed Em AEsmcth EM Ileft lugn Edamltdgt Ileft lugn Es rvwathd Every Energy Function has a Probabilistic Interpretation Recall the energy function we minimized for stereo Ed Em AEsmcth EdIiethmgm EdatadgtIleftgtImgI n Es rvwathd 1 Pdl122f lugn Z 6XPEd122fImghz Every Energy Function has a Probabilistic Interpretation Recall the energy function we minimized for stereo Ed Em AEsmcth EM Ileft lugn Edamltdgt Ileft lugn Es rvwathd 1 Pdl122f lugn Z 6XPEd Iiefz LagLn 1 Z expEdatadgt Ileft Imam expEs7vwathd Probability A Review ofthe Basics Points I d like to cover Large joint discrete distributions Marginals Condit39ionals Bayes Rule Factorizing large probability distributions

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