cee598 - visual sensing for civil infrastructure eng. & mgmt....p c m p w k >r t @ p w...
TRANSCRIPT
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CEE598 - Visual Sensing for
Civil Infrastructure Eng. & Mgmt.
Session 4 – Camera Calibration
Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign
Mani Golparvar-Fard Department of Civil and Environmental Engineering
3129D, Newmark Civil Engineering Lab
e-mail: [email protected]
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Outline
Camera Calibration
• Review Camera Parameters
• Camera Calibration Problem
• Example
2
CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. © Mani Golparvar-Fard, 2013 Some slides in this lecture are courtesy to Profs. S. Savarese, J.
Ponce & F-F Li
Reading: [FP] Chapter 3
[HZ] Chapter 7
[S] Chapter 6
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Pinhole perspective projection f
Oc
f = focal length
Projective camera
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Pinhole perspective projection
x
y
xc
yc
C=[uo, vo]
f
Oc
f = focal length
uo, vo = offset
Projective camera
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f
Oc
Units: k,l [pixel/m]
f [m]
[pixel] , Non-square pixels
f = focal length
uo, vo = offset
non-square pixels ,
Projective camera
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x
y
xc
yc
C=[uo, vo]
f
Oc
f = focal length
uo, vo = offset
non-square pixels , = skewness
Projective camera
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f
Oc
1
z
y
x
0100
0v0
0ucot
P o
o
sin
K has 5 degrees of freedom!
Pc
P’
f = focal length
uo, vo = offset
non-square pixels , = skewness
Projective camera
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f
Oc
Pc
Ow
iw
kw
jw
R,T
wc PTRP
P’
f = focal length
uo, vo = offset
non-square pixels , = skewness
R,T = rotation, translation
Projective camera
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f = focal length
uo, vo = offset
non-square pixels ,
f
Oc
P
Ow
iw
kw
jw
R,T
wPMP
wPTRK
Internal parameters
External parameters
= skewness
R,T = rotation, translation
P’
Projective camera
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wPMP wPTRK
Internal parameters
External parameters
Goal of calibration
Estimate intrinsic and extrinsic parameters
from 1 or multiple images
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wPMP wPTRK
Note: To simplify
notation let P = Pw
100
v0
ucot
K o
o
sin
T
3
T
2
T
1
R
r
r
r
z
y
x
t
t
t
T
Goal of calibration
Estimate intrinsic and extrinsic parameters
from 1 or multiple images
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•P1… Pn with known positions in [Ow,iw,jw,kw]
•p1, … pn known positions in the image
Goal: compute intrinsic and extrinsic parameters
jC
Calibration rig
Calibration Problem
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jC
Calibration rig
How many correspondences do we need?
•P has 11 unknown • We need 11 equations • 6 correspondences would do it
Calibration Problem
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image jC
Calibration rig
In practice: user may need to look at the
image and select the n>=6 correspondences
Calibration Problem
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jC
ii PMP
i
i
iv
up
3
2
1
M
m
m
m
i3
i2
i3
i1
P
P
P
P
m
m
m
m
in pixels
Calibration Problem
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i3
i1i
P
Pu
m
m
i2i3i P)P(v mm
i1i3i P)P(u mm
i3
i2i
P
Pv
m
m
i
i
v
u
i3
i2
i3
i1
P
P
P
P
m
m
m
m
0Pv
ui
3i2
3i1
mm
mm
Calibration Problem
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0Pv
ui
3i2
3i1
mm
mm
0Pv
un
3n2
3n1
mm
mm
…
…
0Pv
u1
312
311
mm
mm
Calibration Problem
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2221
1211
2221
1211
BB
BBB
AA
AAA
What is AB ?
2222122121221121
2212121121121111
BABABABA
BABABABAAB
Review on Block Matrix Multiplication
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2n x 12 12x1
1x4
0Pv
ui
3i2
3i1
mm
mm
T
3
T
2
T
1def
m
m
m
m
4x1
0Pv
un
3n2
3n1
mm
mm
…
…
Homogenous linear system
known unknown
Calibration Problem
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CEE598 Visual Sensing for Civil Infrastructure Eng. & Mgmt. © Mani Golparvar-Fard, 2013
Homogeneous M x N Linear Systems
A
A
x
x 0
0 =
=
Square system (M=N):
Rectangular system (M>N)
• 0 is always a solution
• Rank(A) = N
Minimize |Ax|2
under the constraint |x|2 =1
M x N
M=number of equations
N=number of unknown
• noisy measurements
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How do we solve this homogenous linear system?
Calibration Problem
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ixix
H
ii xHx
0xHx ii 0Ai h
Function of measurements
unknown
DLT algorithm (Direct Linear Transformation)
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1212T
121212n2 VDU
Last column of V gives m
iPM ip
General Calibration Problem
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A
T
3
T
2
T
1
A
a
a
a
TRK
3
1
a
3
2
1
b
b
b
b
Estimated values
)(u 21
2
o aa
)(v 32
2
o aa
3231
3231cosaaaa
aaaa
Intrinsic
b
100
v0
ucot
K o
o
sin
Extracting camera parameters
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Theorem (Faugeras, 1993)
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A
T
3
T
2
T
1
A
a
a
a
TRK
3
2
1
b
b
b
b
Estimated values
Intrinsic
sin31
2aa
sin32
2aa
b
f
Extracting camera parameters
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A
T
3
T
2
T
1
A
a
a
a
b
TRK
3
2
1
b
b
b
b
Estimated values
Extrinsic
32
321
aa
aar
3
3
1
ar
132 rrr b1KT
Extracting camera parameters
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•Pi’s cannot lie on the same plane!
• Points cannot lie on the intersection curve of two
quadric surfaces
Degenerate cases
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Taking lens distortions into account
Chromatic Aberration
Spherical Aberration
Radial Distortion
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No distortion
Pin cushion
Barrel
Radial Distortion
• Caused by imperfect lenses
• Deviations are most noticeable for rays that pass through the
edge of the lens
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Radial Distortion
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i
i
i
i pv
uPM
100
00
00
1
1
d
v
vucvbuad 222
u
3
1p
2p
pdκ1λ
Polynomial function
Distortion coefficient
To model radial behavior
Radial Distortion
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Estimating m1 and m2…
i
i
iv
up
i3
i2
i3
i1
P
P
P
P
1
m
m
m
m
How to do that?
d
v
u Hint: slopev
u
i
i
Radial Distortion
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Estimating m1 and m2…
i
i
iv
up
i3
i2
i3
i1
P
P
P
P
1
m
m
m
m
0)()( 121111 PuPv mm
0)()( 21 iiii PuPv mm
0)()( 21 nnnn PuPv mm…
0Q n
2
1
m
mn
Tsai technique [87]
i
i
i
i
i
i
i
i
P
P
P
P
P
P
v
u
2
1
3
2
3
1
)(
)(
)(
)(
m
m
m
m
m
m
Radial Distortion
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Once that m1 and m2 are estimated…
i
i
iv
up
i3
i2
i3
i1
P
P
P
P
1
m
m
m
m
3m is non linear function of 1m
2m
There are some degenerate configurations for which m1 and m2 cannot be computed
Radial Distortion
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i
i
iv
up
3
2
1
Q
q
q
q
i3
i2
i3
i1
P
P
P
P
q
q
q
q
i
i
i
i pv
uPM
100
00
00
1
1
Q
PPv
PPu
2i3i
i1i3i
Non-linear system of equations
Radial Distortion
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Radial Distortion
Conversion from distorted coordinates to
undistorted coordinates
• p' = f * r(p) * p
(conversion to pixel coordinates)
r(p) is a function that computes a scaling factor to
undo the radial distortion:
• r(p) = 1.0 + k1 * ||p||^2 + k2 * ||p||^4.
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)(PfX
measurement parameter
f( ) is nonlinear
-Newton Method
-Levenberg-Marquardt Algorithm
• Iterative, starts from initial solution
• May be slow if initial solution far from real solution
• Estimated solution may be function of the initial solution
• Newton requires the computation of J, H
• Levenberg-Marquardt doesn’t require the computation of H
General Calibration Problem
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A possible algorithm
1. Solve linear part of the system to find approximated solution
2. Use this solution as initial condition for the full system
3. Solve full system using Newton or L.M.
)(PfX
measurement parameter
f( ) is nonlinear
General Calibration Problem
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Typical assumptions:
- zero-skew, square pixel
- uo, vo = known center of the image
- no distortion
Just estimate f
and R, T
)(PfX
measurement parameter
f( ) is nonlinear
General Calibration Problem
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Camera Calibration Toolbox for Matlab
J. Bouguet – [1998-2000]
http://www.vision.caltech.edu/bouguetj/calib_doc/index.html#examples
Calibration Procedure
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Calibration Procedure
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Calibration Procedure
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Calibration Procedure
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Calibration Procedure
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Calibration Procedure
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Calibration Procedure
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Calibration Procedure
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Next lecture Single view reconstruction
• Assignment 1 will be online Next Tuesday
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