A Formulation for Unknown Focal Length
A Formulation for Unknown Focal Length
Changchang wu
Directly Solving for Focal Length?
●
There is a two fold redundancy
●
Leads to 2X solutions, and slower minimal solver
●
The problem exists in both absolute pose and relative pose
G
e
t
t
i
n
g
A
r
o
u
n
d
s
u
c
h
R
e
d
u
n
d
a
n
c
i
e
s
?
●
Sometimes there is a pattern of f^2
○
Relative pose with unknown focal length
●
Modeling the problems differently
○
Absolute pose with unknown focal length (P4P)
■
3D Point distance (ratio) -> over-determined
■
Null space -> not general
●
Sometimes you are unlucky
○
Absolute pose with one also unknown radial distortion
12 meaningful solutions out of 24 in
Josephson et al. CVPR 2009
A
N
e
w
P
a
r
a
m
e
t
e
r
i
z
a
t
i
o
n
●
Separate the rotation around the
Z
axis
●
Combine the rotation around the Z axis with focal length
●
The same number of parameters (Dof) without the redundancy
●
Possible degeneracy with
f = 0
(any
θ
) is reduced to (
fc = fs = 0
)
A
p
p
l
i
c
a
t
i
o
n
:
P
4
P
-
>
P
3
.
5
P
(
W
u
C
V
P
R
2
0
1
5
)
●
Truly minimal (using exactly 7 coordinates)
○
Allows to work with single coordinates
○
Exactly the minimal 10 solution
○
Extra coordinate can be used for solution filtering
●
Simpler polynomial system & faster GB solver
O
t
h
e
r
A
p
p
l
i
c
a
t
i
o
n
s
●
Relative pose between generalized camera and uncalibrated pinhole
○
See our “Structure-less resection” poster at 3A-1
○
Reduces the number of solutions by half (e.g. 100->50 for 5+2)
●
Feels like symmetry?
●
Absolute pose with unknown focal length & radial distortion
○
Reduces the number of solutions from 24 to 12
○
Using a different radial distortion model.
Thank you!
A detailed exploration of solving for unknown focal length in camera pose estimation problems, addressing redundancies, introducing a new parameterization method, and discussing novel applications. The formulation allows for more efficient and accurate solutions in scenarios with varying complexities.
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Presentation Transcript
A Formulation for Unknown Focal Length Changchang wu Google
Directly Solving for Focal Length? There is a two fold redundancy Negated focal length additional rotation around z by 180 Leads to 2X solutions, and slower minimal solver The problem exists in both absolute pose and relative pose
Getting Around such Redundancies? Sometimes there is a pattern of f^2 Relative pose with unknown focal length Modeling the problems differently Absolute pose with unknown focal length (P4P) 3D Point distance (ratio) -> over-determined Null space -> not general Sometimes you are unlucky Absolute pose with one also unknown radial distortion 12 meaningful solutions out of 24 in Josephson et al. CVPR 2009
A New Parameterization Separate the rotation around the Z axis Combine the rotation around the Z axis with focal length The same number of parameters (Dof) without the redundancy Possible degeneracy with f = 0 (any ) is reduced to (fc = fs = 0)
Application : P4P -> P3.5P (Wu CVPR 2015) Truly minimal (using exactly 7 coordinates) Allows to work with single coordinates Exactly the minimal 10 solution Extra coordinate can be used for solution filtering Simpler polynomial system & faster GB solver
Other Applications Relative pose between generalized camera and uncalibrated pinhole See our Structure-less resection poster at 3A-1 Reduces the number of solutions by half (e.g. 100->50 for 5+2) Absolute pose with unknown focal length & radial distortion Reduces the number of solutions from 24 to 12 Using a different radial distortion model. Feels like symmetry?
