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Triangulation and Multi-View Geometry Class 9. Read notes Section 3.3, 4.3-4.4, 5.1 (if interested, read Triggs’s paper on MVG using tensor notation, see http://www.unc.edu/courses/2004fall/comp/290/089/papers/Triggs-ijcv95.pdf ) . (generate hypothesis). (verify hypothesis).

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triangulation and multi view geometry class 9

Triangulation and Multi-View GeometryClass 9

Read notes Section 3.3, 4.3-4.4, 5.1

(if interested, read Triggs’s paper on MVG using tensor notation, see http://www.unc.edu/courses/2004fall/comp/290/089/papers/Triggs-ijcv95.pdf)

slide2
(generate

hypothesis)

(verify hypothesis)

Automatic computation of F

Step 1. Extract features

Step 2. Compute a set of potential matches

Step 3. do

Step 3.1 select minimal sample (i.e. 7 matches)

Step 3.2 compute solution(s) for F

Step 3.3 determine inliers

until (#inliers,#samples)<95%

Step 4. Compute F based on all inliers

Step 5. Look for additional matches

Step 6. Refine F based on all correct matches

abort verification early
Abort verification early

OIOIIIIOIIIOIOIIIIOOIOIIIIOIOIOIIIIIIII

OOOOOIOOIOOOOOIOOOOOOOIOOOOOIOIOOOOOOOO

Given n samples and an expected proportion of inliers p,how likely is it that I have observed less than T inliers?

abort if P<0.02 (initial sample most probably contained outliers)

(inspired from Chum and Matas BMVC2002)

(use normal approximation to binomial)

To avoid problems this requires to also verify at random!

(but we already have a random sampler anyway)

slide4
Finding more matches

restrict search range to neighborhood of epipolar line

(e.g. 1.5 pixels)

relax disparity restriction (along epipolar line)

slide5
Degenerate cases:
  • Degenerate cases
    • Planar scene
    • Pure rotation
  • No unique solution
    • Remaining DOF filled by noise
    • Use simpler model (e.g. homography)
  • Solution 1: Model selection

(Torr et al., ICCV´98, Kanatani, Akaike)

    • Compare H and F according to expected residual error (compensate for model complexity)
  • Solution 2: RANSAC
    • Compare H and F according to inlier count

(see next slide)

ransac for quasi degenerate cases
RANSAC for (quasi-)degenerate cases

80% in plane

2% out plane

18% outlier

  • Full model (8pts, 1D solution)
  • (accept inliers to solution F)
  • Planar model (6pts, 3D solution)
    • Accept if large number of remaining inliers
  • (accept inliers to solution F1,F2&F3)
  • Plane+parallax model (plane+2pts)
  • closest rank-6 of Anx9 for all plane inliers
  • Sample for out of plane points among outliers
slide7
More problems:
  • Absence of sufficient features (no texture)
  • Repeated structure ambiguity
  • Robust matcher also finds
  • support for wrong hypothesis
  • solution: detect repetition

(Schaffalitzky and Zisserman,

BMVC‘98)

ransac for ambiguous matching
RANSAC for ambiguous matching
  • Include multiple candidate matches in set of potential matches
  • Select according to matching probability (~ matching score)
  • Helps for repeated structures or scenes with similar features as it avoids an early commitment, but also useful in general

(Tordoff and Murray ECCV02)

slide9
geometric relations between two views is fully

described by recovered 3x3 matrix F

two-view geometry

triangulation finally
L2

X

x2

C2

Triangulation (finally!)

x1

C1

L1

Triangulation

  • calibration
  • correspondences
triangulation
L2

x1

C1

X

L1

x2

C2

Triangulation
  • Backprojection
  • Triangulation

Iterative least-squares

  • Maximum Likelihood Triangulation
optimal 3d point in epipolar plane
m1

l1

l1

l2

m1

m2

m2´

m1´

m2

l2

Optimal 3D point in epipolar plane
  • Given an epipolar plane, find best 3D point for (m1,m2)

Select closest points (m1´,m2´) on epipolar lines

Obtain 3D point through exact triangulation

Guarantees minimal reprojection error(given this epipolar plane)

non iterative optimal solution
m1

l2(a)

l1(a)

m2

Non-iterative optimal solution
  • Reconstruct matches in projective frame by minimizing the reprojection error
  • Non-iterative method

Determine the epipolar plane for reconstruction

Reconstruct optimal point from selected epipolar plane

Note: only works for two views

3DOF

(Hartley and Sturm, CVIU´97)

(polynomial of degree 6)

1DOF

backprojection
Backprojection
  • Represent point as intersection of row and column
  • Condition for solution?

Useful presentation for deriving and understanding multiple view geometry

(notice 3D planes are linear in 2D point coordinates)

multi view geometry
Multi-view geometry

(intersection constraint)

(multi-linearity of determinants)

(= epipolar constraint!)

(counting argument: 11x2-15=7)

multi view geometry1
Multi-view geometry

(multi-linearity of determinants)

(3x3x3=27 coefficients)

(= trifocal constraint!)

(counting argument: 11x3-15=18)

multi view geometry2
Multi-view geometry

(multi-linearity of determinants)

(3x3x3x3=81 coefficients)

(= quadrifocal constraint!)

(counting argument: 11x4-15=29)

next class rectification and stereo
Next class: rectification and stereo

image I´(x´,y´)

image I(x,y)

Disparity map D(x,y)

(x´,y´)=(x+D(x,y),y)

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