Going Back a littleApplications of RANSAC: Solution for affine parametersAssignmentAnother app. : Automatic Homography H EstimationComputing a HomographyEstimating H: DLT AlgorithmTexture MappingAutomatic H Estimation: Feature ExtractionAutomatic H Estimation: Finding Feature MatchesSlide 10Automatic H Estimation: Initial Match HypothesesAutomatic H Estimation: Applying RANSACAutomatic H Estimation: Outliers & Inliers after RANSACA Short Review of Camera CalibrationPinhole Camera TerminologyCalibrationCalibration and Pose estimation exampleMatlab codeMulti-View GeometrySlide 20StereoSlide 22Stereo ConstraintsSlide 24Slide 25Stereo/Two-View Geometry3D from two-viewsMapping Points between ImagesExample: Two-View GeometryEpipolar GeometryEpipolar LinesEpipolar PencilEpipolar ConstraintExample: Epipolar Lines for Converging CamerasSpecial Case: Translation Parallel to Image PlaneFrom Geometry to AlgebraSlide 37Slide 38Computer Vision : CISC 4/689Going Back a little•Cameras.pptComputer Vision : CISC 4/689Applications of RANSAC: Solution for affine parameters•Affine transform of [x,y] to [u,v]:•Rewrite to solve for transform parameters:Computer Vision : CISC 4/689Assignment•Program-1•info-Link•DataNote: You can generate, bring-in, your own images from www, as long as:For n+1 levels, image must be Mr£2n+1 rows by Mc£2n+1 colsMr and Mc are any +ve integersSunday 10pmComputer Vision : CISC 4/689Another app. : Automatic Homography H Estimation–Homographies describe image transformation of...•General scene when camera motion is rotation about camera center•Planar surfaces under general camera motion•How to get correct correspondences without human intervention?from Hartley & ZissermanComputer Vision : CISC 4/689Computing a Homography•8 degrees of freedom in 3 x 3 matrix H, so at least n = 4 pairs of 2-D points are sufficient to determine it•Use same basic algorithm for P (aka Direct Linear Transformation, or DLT) to compute H–Now stacked matrix A is 2n x 9 vs. 2n x 12 for camera matrix P estimation because all points are 2-D•3 collinear points in either image is a degenerate configuration preventing a unique solution Lets Side-trackComputer Vision : CISC 4/689Estimating H: DLT Algorithm•x0i = Hxi is an equation involving homogeneous vectors, so Hxi and x0i need only be in the same direction, not strictly equal•We can specify “same directionality” by using a cross product formulation:•See Hartley & Zisserman, Chapter 3.1-3.1.1 (linked on course page) for detailsComputer Vision : CISC 4/689Texture Mapping•Needed for nice display when applying transformations (like a homography H) to a whole image•Simple approach: Iterate over source image coordinates and apply x0 = H x to get destination pixel location–Problem: Some destination pixels may not be “hit”, leaving holes•Easy solution: Iterate over destination image and apply inverse transform x = H-1 x0 –Round off H-1 x0 to address “nearest” source pixel value–This ensures every destination pixel is filled inComputer Vision : CISC 4/689Automatic H Estimation: Feature Extraction•Find features in pair of images using corner detection—e.g., eigenvalue threshold of: from Hartley & Zisserman~500 features foundComputer Vision : CISC 4/689Automatic H Estimation: Finding Feature Matches•Best match over threshold within square search window (here §300 pixels) using SSD or normalized cross-correlationfrom Hartley & ZissermanComputer Vision : CISC 4/689Automatic H Estimation: Finding Feature Matches•Best match over threshold within square search window (here §300 pixels) using SSD or normalized cross-correlationfrom Hartley & ZissermanComputer Vision : CISC 4/689Automatic H Estimation: Initial Match Hypotheses268 matched features (over SSD threshold) in left image pointing to locations of corresponding right image featuresfrom Hartley & ZissermanComputer Vision : CISC 4/689Automatic H Estimation: Applying RANSAC•Sampling–Size: Recall that 4 correspondences suffice to define homography, so sample size s = 4–Choice•Pick SSD threshold conservatively to minimize bad matches•Disregard degenerate configurations•Ensure points have good spatial distribution over image•Distance measure–Obvious choice is symmetric transfer error:Computer Vision : CISC 4/689Automatic H Estimation: Outliers & Inliers after RANSAC•43 samples used with t = 1.25 pixels117 outliers (² = 0.44)151 inliersfrom Hartley & ZissermanComputer Vision : CISC 4/689A Short Review of Camera CalibrationComputer Vision : CISC 4/689Pinhole Camera TerminologyCamera center/ pinholePrincipal point/image centerImage pointCamera pointFocal lengthOptical axisImage planeComputer Vision : CISC 4/689Calibration•Slides (calibration.ppt)Computer Vision : CISC 4/689Calibration and Pose estimation example•Recover intrinsic and extrinsic parameters of camera by using calibration board.•3D points are given, can find 2D image coordinates for the corresponding 3D points.•Assume world is located at the folded lower corner, principal point is center of the image, fold is 90 degrees,•Total length and width of board is 9in by 9in. Next 8 slides, courtesy UCF.Computer Vision : CISC 4/689Matlab code•Matlab•fx = 1.5031•fy =1.2773•Rc = -0.0201 -0.2000 -0.9796 0.2198 0.9588 -0.1797 0.9752 -0.2189 0.0247•Tc = (29.0725, -2.8850, 53.4196)•camera position:( -51.0289, 19.7118 26.6985)Computer Vision : CISC 4/689Multi-View GeometryRelates• 3D World Points• Camera Centers• Camera OrientationsComputer Vision : CISC 4/689Multi-View GeometryRelates• 3D World Points• Camera Centers• Camera Orientations• Camera Intrinsic Parameters• Image PointsComputer Vision : CISC 4/689Stereoscene pointscene pointoptical centeroptical centerimage planeimage planeComputer Vision : CISC 4/689Stereo•Basic Principle: Triangulation–Gives reconstruction as intersection of two rays–Requires •calibration•point correspondenceComputer Vision : CISC 4/689Stereo Constraintspp’?Given p in left image, where can the corresponding point p’in right image be?Computer Vision : CISC 4/689Stereo ConstraintsX1Y1Z1O1Image planeFocal planeMpp’Y2X2Z2O2Epipolar LineEpipoleComputer Vision : CISC 4/689Stereo•The geometric information that relates two different viewpoints of the same scene is entirely contained in a mathematical construct known as fundamental matrix.•The geometry of two different