Uncalibrated Camera Lecture 5 Uncalibrated Geometry Stratification MASKS

Uncalibrated Camera Lecture 5 Uncalibrated Geometry & Stratification MASKS © 2004 Invitation to 3 D vision

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

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

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

Taxonomy on Uncalibrated Reconstruction • is known, back to calibrated case • is unknown n Calibration with complete scene knowledge (a rig) – estimate n Uncalibrated reconstruction despite the lack of knowledge of n Autocalibration (recover from uncalibrated images) • Use partial knowledge n Parallel lines, vanishing points, planar motion, constant intrinsic • Ambiguities, stratification (multiple views) MASKS © 2004 Invitation to 3 D 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 3 D vision

Calibration with a Rig • Given 3 -D coordinates on known object • Eliminate unknown scales • Recover projection matrix • Factor the into • Solve for translation MASKS © 2004 Invitation to 3 D vision and using QR decomposition

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

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

Properties of the Fundamental Matrix A nonzero matrix is a fundamental matrix if a singular value decomposition (SVD) with for some . There is little structure in the matrix MASKS © 2004 Invitation to 3 D vision except that has

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 3 D 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 3 D vision

Calibrated vs. Uncalibrated Space MASKS © 2004 Invitation to 3 D vision

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

Motion in the distorted space Calibrated space Uncalibrated space • Uncalibrated coordinates are related by • Conjugate of the Euclidean group MASKS © 2004 Invitation to 3 D 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 3 D 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 3 D 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 4 x 4 matrix • Both and are consistent with the epipolar geometry – give the same fundamental matrix MASKS © 2004 Invitation to 3 D 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 ©Euclidean 2004 Structure Invitation to 3 D vision 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 3 D counterparts --> 3 D projective reconstruction => Projective Geometry MASKS © 2004 Invitation to 3 D vision

Homogeneous Coordinates (RBM) 3 -D coordinates are related by: Homogeneous coordinates: Homogeneous coordinates are related by: MASKS © 2004 Invitation to 3 D vision

Homogenous and Projective Coordinates • • Homogenous coordinates in 3 D before – attach 1 as the last coordinate – render the transformation as linear transformation Projective coordinates – all points are equivalent up to a scale 2 D- projective plane 3 D- 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 3 D 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 3 D 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 3 D 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, • In order to preserve the choice of the first reference frame we can restrict some DOF of MASKS © 2004 is of the following form Invitation to 3 D vision

Structure of the Projective Ambiguity • 1 st frame as reference • Choose the projective reference frame then ambiguity is • MASKS © 2004 can be further decomposed as Invitation to 3 D 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 3 D vision

Affine upgrade • Upgrade projective structure to an affine structure • Exploit partial scene knowledge n Vanishing points, no skew, known principal point Special motions n Pure rotation, pure translation, planar motion, rectilinear motion Constant camera parameters (multi-view) • • MASKS © 2004 Invitation to 3 D 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 3 D vision

Affine Upgrade Need at least three vanishing points 3 equations, 4 unknowns (-1 scale) Solve for Set up structure MASKS © 2004 and update the projective Invitation to 3 D vision

Euclidean upgrade • We need to estimate remaining affine ambiguity Alternatives: • 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) • • • MASKS © 2004 Invitation to 3 D 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 3 D vision

Geometric Stratification MASKS © 2004 Invitation to 3 D 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 3 D vision

Pure Rotation - Uncalibrated Case • Calibrated Two views related by rotation only • Conjugate rotation • 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 3 D vision can be estimated

Overview of the methods MASKS © 2004 Invitation to 3 D vision

Summary MASKS © 2004 Invitation to 3 D 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 3 D 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 3 D vision

Summary of (Auto)calibration Methods MASKS © 2004 Invitation to 3 D vision