Class Note for EECS 841 with Professor Potetz at KU 7
Class Note for EECS 841 with Professor Potetz at KU 7
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Date Created: 02/06/15
EECS 841 Computer Vision Brian Potetz Fall 2008 Lecture 27 Optical Flow Suggested Reading Motion Shapiro amp Stockman Chapter 9 Dynamic Pro gramng Ma i mlhinmm 4 ghi ahcdefgk Mama N points have N possible correspondences BUT We might assume the ordeIing of points is the same inthe le amp right eyes Now solve for the best matching of p4 given p3 etc Modern Stereo Algorithms Often use colorbased adaptive windows Adaptive metric oferror between image patch L and R Similarity in mlur of Similarity in color of pixel central pixel x and pixel x o Ax in left a right images ZMWL5L5 Amman Ri A5 dLi A5 Ri A7 zmwltLltixLltiAigtgtwltRltigtgtRm M Sum ur all weights 5 MI Modern Stereo Algorithms Often use colorbased adaptive windows Clean up results using a Global Energy Function ie Minimize DTSFEHWIESE function ufxy Eltdgt Ema Emma 1722 a R Edatad 20Ileftx7y7lrightx t dWMMD 14 24mm ma 7 mg Z ltdltzy 1 7 dltz7ygtgt Motion Field optical Flow Image velocity of a point moving in the scene Motion of brightness pattern in the Image Ideally Optical flow Motion field dr Scene point velocity Va 0 dr dt Image velocity vi 139 dt i I TO Perspective projection Ti f To 39 2 Motion field 8m ro zvo 110 zr0 vi 2 I f 2 875 7390 z Optical Flow Constraint Equation xu6tyv6t D Optical Flow Velocities u v x y x y time t time t 5t Optical Flow 7 Motion Field Displacement 6x6y u 6t v 61 Assume brightness of patch remains same in both images Exu6tyv6tt6t Exyt Motion field exists but no optical flow No motion field but shading changes a lb Aperture Problem Aperture Problem Optical Flow Constraint Barber pole illusion z axis li lil ll l lll ltllliltlli ll llllltlli l ll l lll lli l l l39lllil rlll Barber39s pole Motionme Optical aw Optical Flow Constraint Equation xu6tyv8t L4 Optical Flow Velocities u v Jay Jay Displacement 6x5y u 5tv 5t Assume brightness of patch remains same in both images time t time t or Exu6tyv6tt6t Exyt Optical Flow Constraint Equation xu6tyv5t Optical Flow Velocities uv Jay Jay Displacement 6x5y u 6tv 5t Assume brightness of patch remains same in both images time t time t or Exu6tyv6tt6t Exyt Assume small motion First order Taylor expansion of E Exyt 6x E 5y E 6t E zExyt 6x 6y at Optical Flow Constraint Equation E6y 5t 0 6x 6y 6t Divide by at and take the limit 675 gt 0 E dt 6x dt y at Constraint Equation ExuEyvEr0 6x Optical Flow Constraint Equation 5x 5y 6t 0 6x 6y Divide by at and take the limit 675 gt 0 u deE dy 6E 6E 0 dt 6x dt y at Constraint Equation Ex u Ey v Er 0 NOTE uv must lie on a straight line We can compute Ex Ey Et using gradient operators But uv cannot be found uniquely with this constraint Optical Flow Constraint lntuitively what does this constraint mean The component ofthe ow in the gradient direction is determined The component ofthe ow parallel to an edge is unknown Optical Flow Constraint Equation Constraint Equation ExuEyv l uv must lie on a straight line We can compute Ex E E using gradient operatorsy But uv cannot be found uniquely with this constraint V1417 uiv gum Optical Flow Constraint Equation Constraint Equation ExuEyvE 0 u5v must lie on a straight line We can compute E1 E Et using gradient operatorsly But uv cannot be found uniquely with this constraint VIzWiy uiv gum Optical Flow Constraint Equation Constraint Equation Ex 14 Ey v uv must lie on a straight line We can compute Ex E E using gradient operatorsy But uv cannot be found uniquely with this constraint Wm m gamma 0 Computing Optical Flow Formulate Error in Optical Flow Constraint 2 ea ffExuEyvE dxdy bugs We need additional constraints Smoothness Constraint as in shape from shading and stereo Usually motion eld varies smoothly in he image 30 penalize departure 39om smoothness 2 2 2 2 e f ux uyvx vy dxajz has Find uv at each image pointthat MINIMIZES e eT Lag weignting fa Ctor Discrete Optical Flow Algorithm Considerimage pixel Lj Departure from Smoothness Constraint 1 2 2 5g Z ui1j uij uijl uij 2 vi1j 1 vljl Errorin Optical Flow constraint equation Vij2 if 39v39 9 2 cij ExuijEyvijEr We seek the set amp vythat minimize Discrete Optical Flow Algorithm Differentiating e wrt v amp u and setting to zero 6e i a 2 14 uu2 Efu va E1 Ef o 62 i a 2 v v2x Efu va Ef E o 1 amp a are averages of uv around pixel Ll e22sijicij NOTE u amp v show upinvr39rorethyan one term Update Rule ui Hi 11 7 Ex u Ey v E H n1 n 1ME 2Ef11 n1TEfquotIEviEfl u 1ME Ef Example O O 00 9 90 O 09 0 99 0 9 90 O 90 00 OOQOOOOI JOQOQO 39 I f I IT M it it 00000000060006 000099000000 Optical Flow Result Low Texture Region Bad gradients have small magnitude Edges soso aperture problem a L39 50 100 150 200 250 300 350 large gradients all the same High Textured Region Good Revisiting the Small Motion Assumption gradients are different large magnitudes 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 i i i quot2395p xe s i i i i i i i i x quot125P39Xe39s i it i i i i i i i i i i i39 i i x u5 pixels u10 pixelsquot Gaussian pyramid of image H Gaussian pyramid of image I Coarsetofine Optical Flow Estimation run iterative OF i i upsample run iterative OF i i i i i i i image H Gaussian pyramid of image H Gaussian pyramid of image I
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