Researchers present their findings on geometric estimation problems in computer vision at a conference. The poster, titled "Revisiting Sampson Approximation for Geometric Estimation Problems," explores the geometric and Sampson errors, providing new bounds on approximation. It is authored by Felix Rydell, Angélica Torres, and Viktor Larsson, and affiliated with institutions such as KTH Royal Institute of Technology, Lund University, and the Max Planck Institute. The detailed diagrams and mathematical formulations indicate a sophisticated analysis aimed at refining the error approximations in geometric estimations. Attendees engage with the poster, gaining insights into the researchers' contributions to enhancing computational accuracy in visual data interpretation.
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Live
LUND UNIVERSITY
MAX PLANCK INSTITUTE
FOR MATHEMATICS IN THE SCIENCES
Geometric Error
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Revisiting Sampson Approximation for
Measures how much we must perturb data (image points)
for it to satisfy the model (epipolar constraint).
Example:
Two-view reprojection error min ||II(X) - x₁||2 + ||II(RX+t) - x2||2|
or equivalently formulated as
s.t.
(2, 0) = min |ε2
€
C(ze, 0) = 0
X
X
where C is the epipolar constraint C(z, E) = (x2; 1)E(x1; 1) = 0
Caveat: Does not always has a closed form solution
Sampson Error
fast approximation of the geometric error is the Sampson error:
Linearize constraint
around observed data.
Geometric Estimation Problems
Felix Rydell, Angélica Torres, Viktor Larsson
Linearization 1
C(2)
C(a)+ Je
C(2)+ Je
New bounds on approximation!
Linearization 2
Our contribution: Explicit bounds on the Sampson approximation
in terms of the true geometric error!
Small when constraint is roughly linear
1½ EG ≤ ES ≤ EG +
2
1 || H
2 ||J|
E²
ε (2, 0) =
= min
E
s.t.
C(z) + Jε = 0
Has closed form solution!
C(z)
JT
εξ
(E1)?
||E12|2 + |22||2
Example: Two-view reprojection error
Minimizing Sampson approximation pushes geometric error down,
The right inequality always holds. The left inequality holds if
J0 and ||J||22|C(z)||JHJ|
i.e., constraint is approximately fulfilled or the function is close to linear.
Rose Refi
Goal: Refine car
Uncertainty-weig
min all
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s.t. C(+6
Reprojection
Reprojection
4 Sampson a
Sampson approx. allow
Goal: Minimize
Problem: The m
Approach: G
the Sa
multip
and cons
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case