Uncalibrated Euclidean Reconstruction

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Uncalibrated Euclidean Reconstruction

• A Review
• (A. Fusiello IVC2000)

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Perspective projection
Point Projection
Projection Matrix
Intrinsic parameters
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2 camera geometry
2 projections of w relate by
Which can be written as the F-matrix
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F- Matrix

• 3x3, Rank 2
• 7 parameters
• Requires 8 or more point projections
• Conveys all the information about 3 uncalibrated
  cameras

Skew-symmetric matrix
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Homographies

• Maps points to points
• Induced by a 3D plane
• Even at infinity...

?
n
d
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Reconstruction

• Given a set of point projections from

cameras, obtain
and such that
and such that
and such that
are also valid for any non singular 4x4 T matrix!

• Ambiguity

• Reconstruction is possible up to a projective
  transformation i.e. Projective Reconstruction

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Projective Reconstruction
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Euclidean from known points

• We want that upgrades to euclidean

• 5 points (or more) with known euclidean coord.
  determine

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Self-calibration

• Idea Obtain Euclidean structure
• Euclidean structure preserves parallelism, angles
  and distances
• Maybank92 proved that
• Constant intrinsics
• 3 or more images

Self calibration is possible
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Kruppa equations

• Kruppa (1913) derived polinomial equations from
  which the intrinsics can be obtained
• Unknown KAAT (symmetric pos. def. thus 5
  d.o.f.)
• Let FUDVT be the SVD of F

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Kruppa equations

• Each F matrix gives 2 independent quadratic
  equations on K (5dof)
• Can be solved with Nonlinear LS for 3 or more
  images, but very unstable! (Faugeras96)
• Assuming known P.P. and zero skew, can be solve
  analytically from just one F-matrix

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Some manipulation...

• Lets assume we performed projective
  reconstruction and have found

• Constant intrinsics

General 3-vector
Let
Has 8 parameters
Affine transformation
Projective transformation
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more manipulation
(Eq 1)
known
5 param
3 param
3 param
Homography at infinity
(Eq 2)
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Affine Structure

• Remembering

• Conclusion If is known then affine
  reconstruction is possible

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Affine Structure

• Can be computed from image measurements
  parallel lines or point correspondences far away

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From Affine to Euclidean

• If is known, then A can be computed easily
• Remember and
• and noting then

• 3 views are required. Solvable with linear LS.

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Different approaches to Self Calib.

• Recent work 92 onwards
• Overview of the approches by
• Hartley (App of Invariance in CV, 93)
• Pollefeys and Van Gool (CVPR 97)
• Heyden and Astrom (ICPR96)
• Triggs (CVPR97)
• No performance comparison...

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Hartleys Approach

• Remember
• then

(Eq 1)
QR decomposition
Known scale factors

• Uses nonlinear LS (Lev-Marq) to minimize

• Estimates a and A from which computes the
  euclidean structure

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Pollefeys and Van Gools Approach

• Remember

(Eq 2)
Conjugated to a scaled Rot matrix
Equal moduli for all eigen vectors

• Characteristic polynomial is

• Which leads to
• This imposes polynomial contraints on a that are
  solved using nonlinear LS

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Heyden and Astroms Approach

• Remember

(Eq 1)

• It can be written as

• Since then

• Minimize a cost function based on

• 3 cameras yield 10 eq. in 8 unknowns

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Triggs Approach

• Based on the Absolut quadric represented by a
  4x4 matrix

• Triggs shows that

• Each camera provides 5 indepedent equations on 14
  unknowns

• Solves using Sequencial quadratic prog. and a
  quasi-linear methods

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Final Comments

• Two important issues in Self-Calibration
• Sensitivity to noise
• Sensitivity to initialization
• At this state further theoretical and
  experimental comparisons are required to choose
  the most appropriate
• Analysis of Uncertainty and degeneracy -
  Uncertainty Analysis of 3D Reconstruction from
  Uncalibrated Views - E. Grossmann 2000