Uncalibrated Geometry & Stratification Sastry and Yang Uncalibrated Camera Uncalibrated Geometry & Stratification Sastry and Yang MASKS © 2004 Invitation to 3D vision

Uncalibrated Camera calibrated coordinates Linear transformation pixel MASKS © 2004 Invitation to 3D vision

Overview Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial scene knowledge MASKS © 2004 Invitation to 3D vision

Uncalibrated Camera Calibrated camera Uncalibrated camera Image plane coordinates Camera extrinsic parameters Perspective projection Calibrated camera Pixel coordinates Projection matrix Uncalibrated camera MASKS © 2004 Invitation to 3D vision

Taxonomy on Uncalibrated Reconstruction is 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 object Eliminate unknown scales Recover projection matrix Factor the into and using QR decomposition Solve for translation MASKS © 2004 Invitation to 3D vision

Uncalibrated Epipolar Geometry Epipolar constraint Fundamental matrix Equivalent forms of MASKS © 2004 Invitation to 3D vision

Properties of the Fundamental Matrix Epipolar lines Epipoles Image correspondences MASKS © 2004 Invitation to 3D vision

Properties of the Fundamental Matrix A 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 Matrix Find 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 algorithm Solve 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 Space MASKS © 2004 Invitation to 3D vision

Calibrated vs. Uncalibrated Space Distances and angles are modified by S MASKS © 2004 Invitation to 3D vision

Motion in the distorted space Calibrated 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 Matrix Decomposition 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 Reconstruction From 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 Reconstruction Given 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 reconstruction Euclidean 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 Coordinates Homogenous 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 infinity Representation 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 formation Potential 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 Formation Structure 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 Ambiguity 1st frame as reference Choose the projective reference frame then ambiguity is can be further decomposed as MASKS © 2004 Invitation to 3D vision

Stratified (Euclidean) Reconstruction General 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 structure Exploit 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 points How 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 points 3 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 ambiguity In 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 points Vanishing 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 Stratification MASKS © 2004 Invitation to 3D vision

Special Motions – Pure Rotation Calibrated 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 Case Calibrated 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 methods MASKS © 2004 Invitation to 3D vision

Summary MASKS © 2004 Invitation to 3D vision

Direct Autocalibration Methods The 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 Views From 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 Methods MASKS © 2004 Invitation to 3D vision