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Slide 1RoadmapLast time: StereoToday: SFMStructure from motionMultiple-view geometry questionsStructure from motionStructure from motion ambiguityStructure from motion ambiguityProjective ambiguityProjective ambiguityAffine ambiguityAffine ambiguitySimilarity ambiguitySimilarity ambiguityHierarchy of 3D transformationsStructure from motionRecall: Orthographic ProjectionAffine camerasAffine camerasAffine structure from motionAffine structure from motionAffine structure from motionAffine structure from motionFactorizing the measurement matrixFactorizing the measurement matrixFactorizing the measurement matrixFactorizing the measurement matrixFactorizing the measurement matrixAffine ambiguityEliminating the affine ambiguityAlgorithm summaryReconstruction resultsDealing with missing dataDealing with missing dataFurther Factorization workStructure from motion of multiple moving objectsStructure from motion of multiple moving objectsShape interaction matrixArticulated motion subspacesResultsArticulated shape and motion factorizationStructure from motion of deforming objectsRepresenting dynamic shapesResultsDynamic SfM factorizationNon-rigid 3D subspace flowResultsResultsProjective structure from motionProjective structure from motionProjective SFM: Two-camera caseProjective factorizationSequential structure from motionSequential structure from motionSequential structure from motionBundle adjustmentSelf-calibrationSummary: Structure from motionSlide 64Slide 65“Rome in a day”: Coliseum video“Rome in a day”: Trevi video“Rome in a day”: St. Peters videoSlide 69Today: SFMRoadmapC280, Computer VisionProf. Trevor [email protected] 11: Structure from MotionRoadmap•Previous: Image formation, filtering, local features, (Texture)…•Tues: Feature-based Alignment –Stitching images together–Homographies, RANSAC, Warping, Blending–Global alignment of planar models•Today: Dense Motion Models–Local motion / feature displacement–Parametric optic flow•No classes next week: ICCV conference•Oct 6th: Stereo / ‘Multi-view’: Estimating depth with known inter-camera pose•Oct 8th: ‘Structure-from-motion’: Estimation of pose and 3D structure–Factorization approaches–Global alignment with 3D point modelsLast time: Stereo•Human stereopsis & stereograms•Epipolar geometry and the epipolar constraint–Case example with parallel optical axes–General case with calibrated cameras•Correspondence search•The Essential and the Fundamental Matrix•Multi-view stereoToday: SFM•SFM problem statement•Factorization•Projective SFM•Bundle Adjustment•Photo Tourism•“Rome in a day:Structure from motionLazebnikMultiple-view geometry questions•Scene geometry (structure): Given 2D point matches in two or more images, where are the corresponding points in 3D?•Correspondence (stereo matching): Given a point in just one image, how does it constrain the position of the corresponding point in another image?•Camera geometry (motion): Given a set of corresponding points in two or more images, what are the camera matrices for these views?LazebnikStructure from motion•Given: m images of n fixed 3D points xij = Pi Xj , i = 1, … , m, j = 1, … , n •Problem: estimate m projection matrices Pi and n 3D points Xj from the mn correspondences xijx1jx2jx3jXjP1P2P3LazebnikStructure from motion ambiguity•If we scale the entire scene by some factor k and, at the same time, scale the camera matrices by the factor of 1/k, the projections of the scene points in the image remain exactly the same:It is impossible to recover the absolute scale of the scene!)(1XPPXx kkLazebnikStructure from motion ambiguity•If we scale the entire scene by some factor k and, at the same time, scale the camera matrices by the factor of 1/k, the projections of the scene points in the image remain exactly the same •More generally: if we transform the scene using a transformation Q and apply the inverse transformation to the camera matrices, then the images do not change  QXPQPXx-1LazebnikProjective ambiguity  XQPQPXx P-1 PLazebnikProjective ambiguityLazebnikAffine ambiguity  XQPQPXx A-1 AAffineLazebnikAffine ambiguityLazebnikSimilarity ambiguity  XQPQPXxS-1SLazebnikSimilarity ambiguityLazebnikHierarchy of 3D transformationsvTvtAProjective15dofAffine12dofSimilarity7dofEuclidean6dofPreserves intersection and tangencyPreserves parallellism, volume ratiosPreserves angles, ratios of length10tAT10tRTs10tRTPreserves angles, lengths•With no constraints on the camera calibration matrix or on the scene, we get a projective reconstruction•Need additional information to upgrade the reconstruction to affine, similarity, or EuclideanLazebnikStructure from motion•Let’s start with affine cameras (the math is easier)center atinfinityLazebnikRecall: Orthographic ProjectionSpecial case of perspective projection•Distance from center of projection to image plane is infinite•Projection matrix:ImageWorldSlide by Steve SeitzLazebnikOrthographic ProjectionParallel ProjectionAffine camerasLazebnikAffine cameras•A general affine camera combines the effects of an affine transformation of the 3D space, orthographic projection, and an affine transformation of the image:•Affine projection is a linear mapping + translation in inhomogeneous coordinates10bAP1000]affine44[100000100001]affine33[22322211131211baaabaaaxXa1a2bAXx 21232221131211bbZYXaaaaaayxProjection ofworld originLazebnikAffine structure from motion•Given: m images of n fixed 3D points: xij = Ai Xj + bi , i = 1,… , m, j = 1, … , n •Problem: use the mn correspondences xij to estimate m projection matrices Ai and translation vectors bi, and n points Xj •The reconstruction is defined up to an arbitrary affine transformation Q (12 degrees of freedom):•We have 2mn knowns and 8m + 3n unknowns (minus 12 dof for affine ambiguity)•Thus, we must have 2mn >= 8m + 3n – 12•For two views, we need four point correspondences1XQ1X,Q10bA10bA1LazebnikAffine


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Berkeley COMPSCI C280 - Structure from Motion

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