A detailed poster presentation on the topic of "Revisiting Sampson Approximation for Geometric Estimation Problems" is displayed at a research conference. The poster, authored by Felix Rydell, Angélica Torres, and Viktor Larsson, delves into measuring and approximating geometric errors in data, emphasizing on the Sampson Error and new bounds on approximation. The poster is visually structured with mathematical equations, diagrams, and graphs for illustration, and is affiliated with notable institutions including KTH, Lund University, and Max Planck Institute. Attendees can be seen in discussion around the poster, with a laptop on a nearby table possibly displaying related supplementary material. The setting appears to be a busy conference hall with overhead lighting.
Text transcribed from the image:
Live
Geometric Error
LUND UNIVERSITY
KTH
VETENSKAP
OCH KONST
MAX PLANCK INSTITUTE
FOR MATHEMATICS IN THE SCIENCES
X
Revisiting Sampson Approximation for
Geometric Estimation Problems
Felix Rydell, Angélica Torres, Viktor Larsson
Linearization ap
C(2)
C(1) + Je
C(2)+ Je
Linearization a2
ose Re
Goal: Refine ca
Uncertainty-wei
min lea
€2.€
S.L. C(+
Reprojection
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||
or equivalently formulated as
s.t.
(2, 0) = min ||=||2
€
C(z+ε, 0) = 0
where C is the epipolar constraint C(z, E) = (x2; 1)TE(x1; 1) = 0
Caveat: Does not always has a closed form solution
Sampson Error
fast approximation of the geometric error is the Sampson error:
(2, 0) = min |ε||2
Linearize constraint
around observed data
€
s.t.
C(z) + Jε = 0
Has closed form solution!
C(z)
||J||2
Example: Two-view reprojection error
εξ
(E)?
E12|2 + ||22||2
New bounds on approximation!
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 +
1 ||H||
2 ||J|
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.
Reprojection
↳ Sampson a
Sampson approx. allow
Goal: Minimize
Problem: The m
App
the