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Uncalibrated reconstruction

Uncalibrated reconstruction. Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial scene knowledge. S. Soatto, J. Kosecka. Uncalibrated Camera. Calibrated camera. Image Plane Coordinates

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Uncalibrated reconstruction

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  1. Uncalibrated reconstruction Calibration with a rig Uncalibrated epipolar geometry Ambiguities in image formation Stratified reconstruction Autocalibration with partial scene knowledge S. Soatto, J. Kosecka

  2. Uncalibrated Camera Calibrated camera • Image Plane Coordinates • Perspective projection Uncalibrated camera • Pixel coordinates • Camera intrinsic parameters • Projection Matrix ICRA 2003

  3. Taxonomy of uncalibrated reconstruction • K is known, back to calibrated case • K is unknown • Calibration (with a rig) – estimate K • Uncalibrated reconstruction despite the lack of knowledge of K • Autocalibration (recover K from images) • Use complete scene knowledge (a rig) • Use partial knowledge • Parallel lines, vanishing points, planar motion, constant intrinsic • Ambiguities, stratification (multiple views) ICRA 2003

  4. 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 ICRA 2003

  5. Uncalibrated camera vs. distorted space • Inner product in Euclidean space : compute distances and angles • Calibration K transforming spatial coordinates • Transformation induced on the inner product • S (the metric of the space) and K are equivalent ICRA 2003

  6. Calibrated vs. Uncalibrated Space ICRA 2003

  7. Calibrated vs. Uncalibrated Space Distances and angles are modified by S ICRA 2003

  8. Motion in the distorted space Calibrated space Uncalibrated space • Uncalibrated coordinates are related by • Conjugate of the Euclidean group ICRA 2003

  9. Uncalibrated Camera or Distorted Space • Uncalibrated camera with Calibration matrix K viewing points in Euclidean space and moving with (R,T) is equivalent to calibrated camera viewing points in distorted space governed by S and moving with motion conjugate to (R,T) ICRA 2003

  10. Uncalibrated Epipolar Geometry • Epipolar Constraint • Fundamental matrix • equivalent forms of F

  11. Properties of the fundamental matrix • Epipolar lines • Epipoles Image correspondences ICRA 2003

  12. Characterization 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 ICRA 2003

  13. Decomposing the fundamental matrix • Decomposition of the Fundamental matrix into skew • symmetric matrix and nonsingular matrix • Decomposition of F is not unique • Unknown parameters - ambiguity • Corresponding projection matrix ICRA 2003

  14. What does F tell us? • F can be inferred from point matches (8-point algorithm) • Cannot extract motion, structure and calibration from one fundamental matrix (2 views) • F allows reconstruction up to a projective transformation (as we will see soon) • F encodes all the geometric information among two views when no additional information is available ICRA 2003

  15. 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 ICRA 2003

  16. Ambiguities in the image formation Structure of the Motion Parameters ICRA 2003

  17. Ambiguities in the image formation Structure of the projection matrix • For any invertible 4 x 4 matrix H • In the uncalibrated case we cannot distinguish between camera imaging word from camera imaging distorted world • In general, H is of the following form • In order to preserve the choice of the first reference frame we can restrict some DOF of H ICRA 2003

  18. Structure of the projective ambiguity • For i-th frame • 1st frame as reference • Choose the projective reference frame then ambiguity is • H can be further decomposed as ICRA 2003

  19. Geometric stratification (contd.) ICRA 2003

  20. Projective reconstruction • From points, extract F; from F extract P, X • Canonical decomposition • Projection matrices ICRA 2003

  21. Projective reconstruction • Given projection matrices recover projective structure • Given 2 images and no prior information, the scene can • be recovered up a a 4-parameter family of solutions. • This is the best one can do! ICRA 2003

  22. 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) ICRA 2003

  23. Affine upgrade using vanishing points Maps the points To points with affine coordinates ICRA 2003

  24. Vanishing point estimation ICRA 2003

  25. Euclidean upgrade • Exploit special motions (e.g. pure rotation) • If Euclidean is the goal, perform Euclidean reconstruction directly (no stratification) • Multi-view reconstruction (tomorrow) • Autocalibration (Kruppa) ICRA 2003

  26. Overview of the methods ICRA 2003

  27. Direct stratification for multiple views Absolute Quadric Constraint ICRA 2003

  28. Direct methods - Summary ICRA 2003

  29. Summary ICRA 2003

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