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PSU CSE/EE 486 - Planar Homographies

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1CSE486, Penn StateRobert CollinsLecture 16:Planar HomographiesCSE486, Penn StateRobert CollinsMotivation:Points on Planar SurfacexyCSE486, Penn StateRobert CollinsReview : Forward ProjectionUVWXYZxyuvWorldCoordsCameraCoordsFilmCoordsPixelCoordsMextMprojMaffMintUVWXYZuvMUVWuv342414332313312212312111mmmmmmmmmmmmCSE486, Penn StateRobert CollinsWorld to Camera TransformationXXYYZZPCUUVVWWPWTranslate by - C (align origins)CRotate toalign axesRPC = R ( PW - C ) = R PW + TCSE486, Penn StateRobert CollinsPerspective Matrix Equation(Camera Coordinates)PM p ⋅=int 10001000000'''=ZYXffzyxCZYfyZXfx==CSE486, Penn StateRobert CollinsFilm to Pixel Coords 10001000000'''=ZYXffzyx100a13a12a11a23a22a21w’v’u’2D affine transformation from filmcoords (x,y) to pixel coordinates (u,v):u = Mint PC = Maff Mproj PCMprojMaff2CSE486, Penn StateRobert CollinsProjection of Points on Planar SurfacexyPoint on planeRotation +TranslationPerspectiveprojectionFilm coordinatesCSE486, Penn StateRobert CollinsProjection of Planar PointsCSE486, Penn StateRobert CollinsProjection of Planar Points (cont)Homography H(planar projectivetransformation)CSE486, Penn StateRobert CollinsProjection of Planar Points (cont)Homography H(planar projectivetransformation)Punchline: For planar surfaces, 3D to 2D perspectiveprojection reduces to a 2D to 2D transformation.Punchline2: This transformation is INVERTIBLE!CSE486, Penn StateRobert CollinsSpecial Case : Frontal PlaneWhat if the planar surface is perpendicular tothe optic axis (Z axis of camera coord system)?Then world rotation matrix simplies:CSE486, Penn StateRobert CollinsFrontal PlaneSo the homography for a frontal plane simplies:Similarity Transformation!3CSE486, Penn StateRobert CollinsConvert to Pixel Coordsuv100a13a12a11a23a22a21pixelsInternal cameraparametersCSE486, Penn StateRobert CollinsPlanar Projection DiagramHere’s wheretransformationgroups get useful!CSE486, Penn StateRobert CollinsGeneral Planar ProjectionH1A1HCSE486, Penn StateRobert CollinsGeneral Planar ProjectionH1A1H-1CSE486, Penn StateRobert CollinsFrontal Plane ProjectionS1A1ACSE486, Penn StateRobert CollinsFrontal Plane ProjectionS1A1A-14CSE486, Penn StateRobert CollinsGeneral Planar ProjectionH2H1HCSE486, Penn StateRobert CollinsSummary: Planar ProjectionxyPoint on planeRotation + TranslationPerspectiveprojectionPixel coordsuvInternalparamsHomographyCSE486, Penn StateRobert CollinsApplying Homographies to RemovePerspective Distortionfrom Hartley & Zisserman4 point correspondences suffice forthe planar building facadeCSE486, Penn StateRobert CollinsHomographies forBird’s-eye Viewsfrom Hartley & ZissermanCSE486, Penn StateRobert CollinsHomographies for Mosaicingfrom Hartley & ZissermanCSE486, Penn StateRobert CollinsTwo Practical IssuesHow to estimate the homography given four or more point correspondences (will derive L.S. solution now) How to (un)warp image pixel values to produce a new picture (last class)5CSE486, Penn StateRobert CollinsEstimating a HomographyMatrix Form:Equations:CSE486, Penn StateRobert CollinsDegrees of Freedom?There are 9 numbers h11,…,h33 , so are there 9 DOF?No. Note that we can multiply all hij by nonzero kwithout changing the equations:CSE486, Penn StateRobert CollinsEnforcing 8 DOFOne approach: Set h33 = 1.Second approach: Impose unit vector constraintSubject to theconstraint:CSE486, Penn StateRobert CollinsL.S. using Algebraic DistanceSetting h33 = 1Multiplying through by denominatorRearrangeCSE486, Penn StateRobert CollinsAlgebraic Distance, h33=1 (cont)Point 1additionalpoints2N x 8 8 x 1 2N x 1Point 4Point 3Point 2CSE486, Penn StateRobert CollinsAlgebraic Distance, h33=1 (cont)A h = b2Nx8 8x1 2Nx1LinearequationsSolve:AT A h = AT b 2Nx8 8x1 2Nx18x2N8x2N(AT A) h = (AT b)8x18x8 8x1h = (AT A) (AT b)-1Matlab: h = A \ b6CSE486, Penn StateRobert CollinsCaution: Numeric ConditioningR.Hartley: “In Defense of the Eight Point Algorithm”Observation: Linear estimation of projective transformation parameters from point correspondences often suffer from poor“conditioning” of the matrices involves. This means the solutionis sensitive to noise in the points (even if there are no outliers).To get better answers, precondition the matrices by performinga normalization of each point set by:• translating center of mass to the origin • scaling so that average distance of points from origin is sqrt(2).• do this normalization to each point set independentlyCSE486, Penn StateRobert CollinsHartley’s PreConditioningH?PointSet1PointSet2Scale so averagepoint dist is sqrt(2)S1S2Translatecenter of massto originT1T2Estimate HnormH = T2-1 S2-1 Hnorm S1 T1CSE486, Penn StateRobert CollinsA More General ApproachWhat might be wrong with setting h33 = 1?If h33 actually = 0, we can’t get the right answer.CSE486, Penn StateRobert CollinsAlgebraic Distance, ||h||=1||h|| = 1Multiplying through by denominatorRearrange= 0= 0CSE486, Penn StateRobert CollinsAlgebraic Distance, ||h||=1 (cont)additionalpoints2N x 9 9 x 1 2N x 14 POINTSCSE486, Penn StateRobert CollinsAlgebraic Distance, ||h||=1 (cont)A h = 02Nx9 9x1 2Nx1HomogeneousequationsSolve:SVD of ATA = U D UTLet h be the column of U (unit eigenvector) associated with the smallest eigenvalue in D. (if only 4 points, that eigenvalue will be 0)AT A h = AT 02Nx9 9x1 2Nx19x2N 9x2N(AT A) h =


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