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Self-calibration Class 13
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  1. Self-calibrationClass 13 Read Chapter 6

  2. Assignment 3 • Collect potential matches from all algorithms for all pairs • Matlab ASCII format, exchange data • Implement RANSAC that uses combined match dataset • Compute consistent set of matches and epipolar geometry • Report thresholds used, match sets used, number of consistent matches obtained, epipolar geometry, show matches and epipolar geometry (plot some epipolar lines). Due next Tuesday, Nov. 2 naming convention: firstname_ij.dat chris_56.dat [F,inliers]=FRANSAC([chris_56; brian_56; …])

  3. Papers • Each should present a paper during 20-25 minutes followed by discussion. Partially outside of class schedule to make up for missed classes. (When?) • List of proposed papers will come on-line by Thursday, feel free to propose your own (suggestion: something related to your project). • Make choice by Thursday, assignments will be made in class. • Everybody should have read papers that are being discussed.

  4. Papers

  5. 3D photography course schedule

  6. Ideas for a project?

  7. Dealing with dominant planar scenes (Pollefeys et al., ECCV‘02) • USaM fails when common features are all in a plane • Solution: part 1 Model selection to detect problem

  8. Dealing with dominant planar scenes (Pollefeys et al., ECCV‘02) • USaM fails when common features are all in a plane • Solution: part 2 Delay ambiguous computations until after self-calibration (couple self-calibration over all 3D parts)

  9. Non-sequential image collections Problem: Features are lost and reinitialized as new features 3792 points Solution: Match with other close views 4.8im/pt 64 images

  10. Relating to more views • For every view i • Extract features • Compute two view geometry i-1/i and matches • Compute pose using robust algorithm • Refine existing structure • Initialize new structure For every view i Extract features Compute two view geometry i-1/i and matches Compute pose using robust algorithm For all close views k Compute two view geometry k/i and matches Infer new 2D-3D matches and add to list Refine pose using all 2D-3D matches Refine existing structure Initialize new structure Problem: find close views in projective frame

  11. Determining close views • If viewpoints are close then most image changes can be modelled through a planar homography • Qualitative distance measure is obtained by looking at the residual error on the best possible planar homography Distance =

  12. Non-sequential image collections (2) 2170 points 3792 points 9.8im/pt 64 images 4.8im/pt 64 images

  13. Hierarchical structure and motion recovery • Compute 2-view • Compute 3-view • Stitch 3-view reconstructions • Merge and refine reconstruction F T H PM

  14. Stitching 3-view reconstructions Different possibilities 1. Align (P2,P3) with (P’1,P’2) 2. Align X,X’ (and C,C’) 3. Minimize reproj. error 4. MLE (merge)

  15. Refining structure and motion • Minimize reprojection error • Maximum Likelyhood Estimation (if error zero-mean Gaussian noise) • Huge problem but can be solved efficiently (Bundle adjustment)

  16. P1 P2 P3 M U1 U2 W U3 WT V 3xn (in general much larger) 12xm Sparse bundle adjustment Non-linear min. requires to solve Jacobian of has sparse block structure im.pts. view 1 Needed for non-linear minimization

  17. U-WV-1WT WT V 3xn 11xm Sparse bundle adjustment • Eliminate dependence of camera/motion parameters on structure parameters Note in general 3n >> 11m Allows much more efficient computations e.g. 100 views,10000 points, solve 1000x1000, not 30000x30000 Often still band diagonal use sparse linear algebra algorithms

  18. Self-calibration • Introduction • Self-calibration • Dual Absolute Quadric • Critical Motion Sequences

  19. Motivation • Avoid explicit calibration procedure • Complex procedure • Need for calibration object • Need to maintain calibration

  20. Motivation • Allow flexible acquisition • No prior calibration necessary • Possibility to vary intrinsics • Use archive footage

  21. Projective ambiguity Reconstruction from uncalibrated images  projective ambiguity on reconstruction

  22. Stratification of geometry Projective Affine Metric 15 DOF 7 DOF absolute conic angles, rel.dist. 12 DOF plane at infinity parallelism More general More structure

  23. Constraints ? Scene constraints • Parallellism, vanishing points, horizon, ... • Distances, positions, angles, ... Unknown scene  no constraints • Camera extrinsics constraints • Pose, orientation, ... Unknown camera motion  no constraints • Camera intrinsics constraints • Focal length, principal point, aspect ratio & skew Perspective camera model too general  some constraints

  24. Euclidean projection matrix Factorization of Euclidean projection matrix Intrinsics: (camera geometry) (camera motion) Extrinsics: Note: every projection matrix can be factorized, but only meaningful for euclidean projection matrices

  25. Constraints on intrinsic parameters Constant e.g. fixed camera: Known e.g. rectangular pixels: square pixels: principal point known:

  26. Self-calibration Upgrade from projective structure to metric structure using constraintsonintrinsic camera parameters • Constant intrinsics • Some known intrinsics, others varying • Constraints on intrincs and restricted motion (e.g. pure translation, pure rotation, planar motion) (Faugeras et al. ECCV´92, Hartley´93, Triggs´97, Pollefeys et al. PAMI´99, ...) (Heyden&Astrom CVPR´97, Pollefeys et al. ICCV´98,...) (Moons et al.´94, Hartley ´94, Armstrong ECCV´96, ...)

  27. A counting argument • To go from projective (15DOF) to metric (7DOF) at least 8 constraints are needed • Minimal sequence length should satisfy • Independent of algorithm • Assumes general motion (i.e. not critical)

  28. Outline • Introduction • Self-calibration • Dual Absolute Quadric • Critical Motion Sequences

  29. The Dual Absolute Quadric The absolute dual quadric Ω*∞ is a fixed conic under the projective transformation H iff H is a similarity • 8 dof • plane at infinity π∞ is the nullvector of Ω∞ • Angles:

  30. Absolute Dual Quadric and Self-calibration Eliminate extrinsics from equation Equivalent to projection of Dual Abs.Quadric Dual Abs.Quadric also exists in projective world Transforming world so that reduces ambiguity to similarity

  31. * * projection constraints Absolute Dual Quadric and Self-calibration Projection equation: Translate constraints on K through projection equationto constraints on * Absolute conic = calibration object which is always present but can only be observed through constraints on the intrinsics

  32. Constraints on * #constraints condition constraint type

  33. Linear algorithm (Pollefeys et al.,ICCV´98/IJCV´99) Assume everything known, except focal length Yields 4 constraint per image Note that rank-3 constraint is not enforced

  34. Linear algorithm revisited (Pollefeys et al., ECCV‘02) Weighted linear equations assumptions

  35. Projective to metric Compute T from using eigenvalue decomposition of and then obtain metric reconstruction as

  36. Alternatives: (Dual) image of absolute conic • Equivalent to Absolute Dual Quadric • Practical when H can be computed first • Pure rotation(Hartley’94, Agapito et al.’98,’99) • Vanishing points, pure translations, modulus constraint, …

  37. Note that in the absence of skew the IAC can be more practical than the DIAC!

  38. Kruppa equations Limit equations to epipolar geometry Only 2 independent equations per pair But independent of plane at infinity

  39. Refinement • Metric bundle adjustment Enforce constraints or priors on intrinsics during minimization (this is „self-calibration“ for photogrammetrist)

  40. Outline • Introduction • Self-calibration • Dual Absolute Quadric • Critical Motion Sequences

  41. Critical motion sequences (Sturm, CVPR´97, Kahl, ICCV´99, Pollefeys,PhD´99) • Self-calibration depends on camera motion • Motion sequence is not always general enough • Critical Motion Sequences have more than one potential absolute conic satisfying all constraints • Possible to derive classification of CMS

  42. Critical motion sequences:constant intrinsic parameters Most important cases for constant intrinsics Note relation between critical motion sequences and restricted motion algorithms

  43. Critical motion sequences:varying focal length Most important cases for varying focal length (other parameters known)

  44. Critical motion sequences:algorithm dependent Additional critical motion sequences can exist for some specific algorithms • when not all constraints are enforced (e.g. not imposing rank 3 constraint) • Kruppa equations/linear algorithm: fixating a point Some spheres also project to circles located in the image and hence satisfy all the linear/kruppa self-calibration constraints

  45. Non-ambiguous new views for CMS (Pollefeys,ICCV´01) • restrict motion of virtual camera to CMS • use (wrong) computed camera parameters

  46. Next class: shape from silhouettes