Camera Models class 8 Multiple View Geometry Comp 290-089 Marc Pollefeys

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  • Camera Modelsclass 8Multiple View GeometryComp 290-089Marc Pollefeys

  • ContentBackground: Projective geometry (2D, 3D), Parameter estimation, Algorithm evaluation.Single View: Camera model, Calibration, Single View Geometry.Two Views: Epipolar Geometry, 3D reconstruction, Computing F, Computing structure, Plane and homographies.Three Views: Trifocal Tensor, Computing T.More Views: N-Linearities, Multiple view reconstruction, Bundle adjustment, auto-calibration, Dynamic SfM, Cheirality, Duality

  • Multiple View Geometry course schedule(subject to change)

  • N measurements (independent Gaussian noise s2) model with d essential parameters(use s=d and s=(N-d))

    RMS residual error for ML estimator

    RMS estimation error for ML estimatornXSM

  • Backward propagation of covarianceOver-parameterization

    Forward propagation of covarianceMonte-Carlo estimation of covariance

  • s=1 pixel S=0.5cm(Crimisi97)Example:

  • Single view geometryCamera model

    Camera calibration

    Single view geom.

  • Pinhole camera model

  • Pinhole camera model

  • Principal point offsetprincipal point

  • Principal point offsetcalibration matrix

  • Camera rotation and translation

  • CCD camera

  • Finite projective camera11 dof (5+3+3)decompose P in K,R,C?{finite cameras}={P4x3 | det M0}If rank P=3, but rank M
  • Camera anatomyCamera centerColumn pointsPrincipal planeAxis planePrincipal pointPrincipal ray

  • Camera centernull-space camera projection matrixFor all A all points on AC project on image of A, therefore C is camera centerImage of camera center is (0,0,0)T, i.e. undefinedFinite cameras: Infinite cameras:

  • Column vectorsImage points corresponding to X,Y,Z directions and origin

  • Row vectorsnote: p1,p2 dependent on image reparametrization

  • The principal point

  • The principal axis vectorvector defining front side of camera(direction unaffected)because

  • Action of projective camera on pointForward projectionBack-projection

  • Depth of points(dot product)(PC=0)If , then m3 unit vector in positive direction

  • Camera matrix decompositionFinding the camera center(use SVD to find null-space)Finding the camera orientation and internal parameters(use RQ decomposition ~QR)(if only QR, invert)

  • When is skew non-zero?1garctan(1/s)for CCD/CMOS, always s=0Image from image, s0 possible(non coinciding principal axis)resulting camera:

  • Euclidean vs. projectivegeneral projective interpretationMeaningfull decomposition in K,R,t requires Euclidean image and space Camera center is still valid in projective space Principal plane requires affine image and space Principal ray requires affine image and Euclidean space

  • Cameras at infinityCamera center at infinityAffine and non-affine camerasDefinition: affine camera has P3T=(0,0,0,1)

  • Affine cameras

  • Affine camerasmodifying p34 corresponds to moving along principal ray

  • Affine camerasnow adjust zoom to compensate

  • Error in employing affine cameraspoint on plane parallel with principal plane and through origin, then general points

  • Affine imaging conditionsApproximation should only cause small error

    D much smaller than d0Points close to principal point (i.e. small field of view)

  • Decomposition of Pabsorb d0 in K2x2alternatives, because 8dof (3+3+2), not more

  • Summary parallel projectioncanonical representationcalibration matrixprincipal point is not defined

  • A hierarchy of affine camerasOrthographic projectionScaled orthographic projection(5dof)(6dof)

  • A hierarchy of affine camerasWeak perspective projection(7dof)

  • Affine camera=camera with principal plane coinciding with PAffine camera maps parallel lines to parallel linesNo center of projection, but direction of projection PAD=0(point on P)A hierarchy of affine camerasAffine camera(8dof)

  • Pushbroom camerasStraight lines are not mapped to straight lines!(otherwise it would be a projective camera)(11dof)

  • Line cameras(5dof)Null-space PC=0 yields camera center

    Also decomposition

  • Next class: Camera calibration

    Note that these are lower bound for residual error against which a particular algorithm may be measured