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Uncalibrated Geometry & Stratification Sastry and YangUncalibrated Camera Uncalibrated Geometry & Stratification Sastry and Yang MASKS © 2004 Invitation to 3D vision
Uncalibrated Camera calibrated coordinates Linear transformation pixelMASKS © 2004 Invitation to 3D vision
Overview Calibration with a rig Uncalibrated epipolar geometryAmbiguities in image formation Stratified reconstruction Autocalibration with partial scene knowledge MASKS © 2004 Invitation to 3D vision
Uncalibrated Camera Calibrated camera Uncalibrated cameraImage plane coordinates Camera extrinsic parameters Perspective projection Calibrated camera Pixel coordinates Projection matrix Uncalibrated camera MASKS © 2004 Invitation to 3D vision
Taxonomy on Uncalibrated Reconstructionis known, back to calibrated case is unknown Calibration with complete scene knowledge (a rig) – estimate Uncalibrated reconstruction despite the lack of knowledge of Autocalibration (recover from uncalibrated images) Use partial knowledge Parallel lines, vanishing points, planar motion, constant intrinsic Ambiguities, stratification (multiple views) MASKS © 2004 Invitation to 3D vision
Calibration with a Rig Use the fact that both 3-D and 2-D coordinates of feature points on a pre-fabricated object (e.g., a cube) are known. MASKS © 2004 Invitation to 3D vision
Calibration with a Rig Given 3-D coordinates on known objectEliminate unknown scales Recover projection matrix Factor the into and using QR decomposition Solve for translation MASKS © 2004 Invitation to 3D vision
Uncalibrated Epipolar GeometryEpipolar constraint Fundamental matrix Equivalent forms of MASKS © 2004 Invitation to 3D vision
Properties of the Fundamental MatrixEpipolar lines Epipoles Image correspondences MASKS © 2004 Invitation to 3D vision
Properties of the Fundamental MatrixA nonzero matrix is a fundamental matrix if has a singular value decomposition (SVD) with for some There is little structure in the matrix except that MASKS © 2004 Invitation to 3D vision
Estimating Fundamental MatrixFind such F that the epipolar error is minimized Pixel coordinates Fundamental matrix can be estimated up to scale Denote Rewrite Collect constraints from all points MASKS © 2004 Invitation to 3D vision
Two view linear algorithm – 8-point algorithmSolve the LLSE problem: Solution eigenvector associated with smallest eigenvalue of Compute SVD of F recovered from data Project onto the essential manifold: cannot be unambiguously decomposed into pose and calibration MASKS © 2004 Invitation to 3D vision
Calibrated vs. Uncalibrated SpaceMASKS © 2004 Invitation to 3D vision
Calibrated vs. Uncalibrated SpaceDistances and angles are modified by S MASKS © 2004 Invitation to 3D vision
Motion in the distorted spaceCalibrated space Uncalibrated space Uncalibrated coordinates are related by Conjugate of the Euclidean group MASKS © 2004 Invitation to 3D vision
What Does F Tell Us? can be inferred from point matches (eight-point algorithm) Cannot extract motion, structure and calibration from one fundamental matrix (two views) allows reconstruction up to a projective transformation (as we will see soon) encodes all the geometric information among two views when no additional information is available MASKS © 2004 Invitation to 3D vision
Decomposing the Fundamental MatrixDecomposition of the fundamental matrix into a skew symmetric matrix and a nonsingular matrix Decomposition of is not unique Unknown parameters - ambiguity Corresponding projection matrix MASKS © 2004 Invitation to 3D vision
Projective ReconstructionFrom points, extract , followed by computation of projection matrices and structure Canonical decomposition Given projection matrices – recover structure Projective ambiguity – non-singular 4x4 matrix Both and are consistent with the epipolar geometry – give the same fundamental matrix MASKS © 2004 Invitation to 3D vision
Projective ReconstructionGiven projection matrices recover projective structure This is a linear problem and can be solve using linear least-squares Projective reconstruction – projective camera matrices and projective structure MASKS © 2004 Invitation to 3D vision Euclidean Structure Projective Structure
Euclidean vs Projective reconstructionEuclidean reconstruction – true metric properties of objects lenghts (distances), angles, parallelism are preserved Unchanged under rigid body transformations => Euclidean Geometry – properties of rigid bodies under rigid body transformations, similarity transformation Projective reconstruction – lengths, angles, parallelism are NOT preserved – we get distorted images of objects – their distorted 3D counterparts --> 3D projective reconstruction => Projective Geometry MASKS © 2004 Invitation to 3D vision
Homogeneous Coordinates (RBM)3-D coordinates are related by: Homogeneous coordinates: Homogeneous coordinates are related by: MASKS © 2004 Invitation to 3D vision
Homogenous and Projective CoordinatesHomogenous coordinates in 3D before – attach 1 as the last coordinate – render the transformation as linear transformation Projective coordinates – all points are equivalent up to a scale 2D- projective plane 3D- projective space Each point on the plane is represented by a ray in projective space Ideal points – last coordinate is 0 – ray parallel to the image plane points at infinity – never intersects the image plane MASKS © 2004 Invitation to 3D vision
Vanishing points – points at infinityRepresentation of a 3-D line – in homogeneous coordinates When -> 1 - vanishing points – last coordinate -> 0 Similarly in the image plane MASKS © 2004 Invitation to 3D vision
Ambiguities in the image formationPotential Ambiguities Ambiguity in K (K can be recovered uniquely – Cholesky or QR) Structure of the motion parameters Just an arbitrary choice of reference frame MASKS © 2004 Invitation to 3D vision
Ambiguities in Image FormationStructure of the (uncalibrated) projection matrix For any invertible 4 x 4 matrix In the uncalibrated case we cannot distinguish between camera imaging word from camera imaging distorted world In general, is of the following form In order to preserve the choice of the first reference frame we can restrict some DOF of MASKS © 2004 Invitation to 3D vision
Structure of the Projective Ambiguity1st frame as reference Choose the projective reference frame then ambiguity is can be further decomposed as MASKS © 2004 Invitation to 3D vision
Stratified (Euclidean) ReconstructionGeneral ambiguity – while preserving choice of first reference frame Decomposing the ambiguity into affine and projective one Note the different effect of the 4-th homogeneous coordinate MASKS © 2004 Invitation to 3D vision
Affine upgrade Upgrade projective structure to an affine structureExploit partial scene knowledge Vanishing points, no skew, known principal point Special motions Pure rotation, pure translation, planar motion, rectilinear motion Constant camera parameters (multi-view) MASKS © 2004 Invitation to 3D vision
Affine upgrade using vanishing pointsHow to compute Maps the points To points with affine coordinates Vanishing points – last homogeneous affine coordinate is 0 MASKS © 2004 Invitation to 3D vision
Affine Upgrade Need at least three vanishing points3 equations, 4 unknowns (-1 scale) Solve for Set up and update the projective structure MASKS © 2004 Invitation to 3D vision
Euclidean upgrade We need to estimate remaining affine ambiguityIn the case of special motions (e.g. pure rotation) – no projective ambiguity – cannot do projective reconstruction Estimate directly (special case of rotating camera – follows) Multi-view case – estimate projective and affine ambiguity together Use additional constraints of the scene structure (next) Autocalibration (Kruppa equations) Alternatives: MASKS © 2004 Invitation to 3D vision
Euclidean Upgrade using vanishing pointsVanishing points – intersections of the parallel lines Vanishing points of three orthogonal directions Orthogonal directions – inner product is zero Provide directly constraints on matrix S – has 5 degrees of freedom, 3 vanishing points – 3 constraints (need additional assumption about K) Assume zero skew and aspect ratio = 1 MASKS © 2004 Invitation to 3D vision
Geometric StratificationMASKS © 2004 Invitation to 3D vision
Special Motions – Pure RotationCalibrated Two views related by rotation only Mapping to a reference view – rotation can be estimated Mapping to a cylindrical surface MASKS © 2004 Invitation to 3D vision
Pure Rotation - Uncalibrated CaseCalibrated Two views related by rotation only Conjugate rotation can be estimated Given C, how much does it tell us about K (or S) ? Given three rotations around linearly independent axes – S, K can be estimated using linear techniques Applications – image mosaics MASKS © 2004 Invitation to 3D vision
Overview of the methodsMASKS © 2004 Invitation to 3D vision
Summary MASKS © 2004 Invitation to 3D vision
Direct Autocalibration MethodsThe fundamental matrix satisfies the Kruppa’s equations If the fundamental matrix is known up to scale Under special motions, Kruppa’s equations become linear. Solution to Kruppa’s equations is sensitive to noises. MASKS © 2004 Invitation to 3D vision
Direct Stratification from Multiple ViewsFrom the recovered projective projection matrix we obtain the absolute quadric contraints Partial knowledge in (e.g. zero skew, square pixel) renders the above constraints linear and easier to solve. The projection matrices can be recovered from the multiple-view rank method to be introduced later. MASKS © 2004 Invitation to 3D vision
Summary of (Auto)calibration MethodsMASKS © 2004 Invitation to 3D vision
More on single-view geometry
Lecture 11: Two-view geometry
Geometry 2: A taste of projective geometry Introduction to Computer Vision Ronen Basri Weizmann Institute of Science.
MASKS © 2004 Invitation to 3D vision Lecture 7 Step-by-Step Model Buidling.
Projective Geometry- 3D
Two-View Geometry CS Sastry and Yang
Recovering metric and affine properties from images
Camera calibration and epipolar geometry
Structure from motion.
Multiple View Geometry Marc Pollefeys University of North Carolina at Chapel Hill Modified by Philippos Mordohai.
3D reconstruction class 11
Epipolar geometry. (i)Correspondence geometry: Given an image point x in the first view, how does this constrain the position of the corresponding point.
Structure from motion. Multiple-view geometry questions Scene geometry (structure): Given 2D point matches in two or more images, where are the corresponding.
Epipolar Geometry and the Fundamental Matrix F
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