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```A Simple Method
Centre of Distortion Estimation
Outline
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Introduction
Model and Approach
Further Discussion
Experiments and Results
Conclusions
Introduction
• Lens distortion usually can be classified into three
types :
– decentering distortion
– thin prism distortion
Wang, J., Shi, F., Zhang, J., Liu, Y.: A new calibration model and method of
camera lens distortion.
Introduction
• Method of obtaining the parameters of the
radial distortion function and correcting the
images. These previous works can be divided
roughly into two strategic approaches
– multiple views method
– Single view method
Introduction
• Former approach
– based on the collinearity of undistorted points.
– Need the camera intrinsic parameters and 3D-point
correspondences.
• This paper
– based on single images and the conclusion that distorted
points are concyclic and uses directly the distorted points.
– uses the constraint, that straight lines in the 3D world
project to circular arcs in the image plane, under the single
parameter Division Model
Model and Approach
– PM、DM
• Distorted straight line is a circle
• calibration procedure to estimate the center
and the parameter of the radial distortion
– Circle fitting : LS、LM
,  ∶ distorted image point
,  ∶ undistorted image point
• The Polynomial Model (PM) that describe radial
distortion :
=  1 + λ1  2 + λ2  4 + ⋯
(1)
∶ distances of  ,  to the distorted centre
∶ distances of  ,  to the distorted centre
• The Division Model (DM) that describe radial
distortion :
=

1+λ1  2 +λ2  4 +⋯
(2)
• we use single parameter Division Model as our
distortion model :
=

1+λ  2
(3)
• To simplify equation, we suppose distorted
center  is the origin image coordinates
system, thus :
P (0,0)
=

1+λ  2
,  =

1+λ  2
where  2 =  2 +  2
(4)
The Figure of Distorted Straight Line
• We consider collinear points and their distorted
images.
• Let  =  +  straight line equation
from (4) We have

1+λ 2
=

1+λ
2
+
(5)
=  +  + λ  2 +  2
(6)

λ
(7)
2
2
+  +
−
1

λ
1
λ
+ =0
The Figure of Distorted Straight Line
2
+
2

+
λ
−
1

λ
1
+
λ
=0
(7)
• The graphics of distorted “straight line” is a
circle under the condition of model (3)
we use single parameter Division Model as our
distortion model :

=
(3)
2
1+λ
Estimate the  and λ
• Let 0 , 0 be the coordinates of the distorted
center . From (7) , we have
2
− 0
1
−
λ
2
+
+  − 0
2
+
1
λ

λ
− 0
− 0 + = 0
2
2
(8)

1
+
− 20  + − − 20
λ
λ
2
+0 + 0 −

λ 0
+
1

λ 0
1
λ
+ =0
(9)
Estimate the  and λ
2
+
2

1
+
− 20  + − − 20
λ
λ
2
2
+0 + 0 −
Let
A=

λ

λ 0
+
1

λ 0
− 20 ,  =
2
2
= 0 + 0 −
1
λ
+ =0
1
−
λ

0
λ
+
(9)
− 20
1
0
λ
1
+
λ
2 +  2 +  +  +  = 0
, we have
(10)
Estimate the  and λ
Let

1
A = − 20 ,  = − − 20
λ
λ

1
1
2
2
= 0 + 0 − 0 + 0 +
λ
λ
λ
, we have
2 +  2 +  +  +  = 0
(10)
Base on the relation of , , and  , we have
(圓方程式參數A、B、C 與 radial distortion 參數 P、λ 的關係式)
2
2
1
λ
0 + 0 + 0 + 0 +  − = 0
(11)
Estimate the  and λ
2 +  2 +  +  +  = 0
2
2
0 + 0 + 0 + 0 +
1
−
λ
=0
(10)
(11)
• Obtain 0 , 0 of distorted center
– Extract three “straight line” from image , we can get
,  ,  =1,2,3 by circle fitting from (10)
according to (11) , we have
1 − 2 0 + 1 − 2 0 + 1 − 2 = 0
1 − 3 0 + 1 − 3 0 + 1 − 3 = 0
2 − 3 0 + 2 − 3 0 + 2 − 3 = 0
(12)
• Obtain the parameter of radial distorted λ
– substitute 0 , 0 obtained from 12 to (11)
1
λ
= 0 2 + 0 2 + 0 + 0 +
(13)
Sum up whole algorithm
• Extract ( ≥ 3) “straight line”   = 1,2, ⋯ ,  from the image
• Determine parameter  ,  ,  =1,2,⋯, by fitting every
“straight line”  with a circle according to (10)
2 +  2 +  +  +  = 0
• Calculate the center 0 , 0 of the radial distortion according
to (12)
1 − 2 0 + 1 − 2 0 + 1 − 2 = 0
1 − 3 0 + 1 − 3 0 + 1 − 3 = 0
2 − 3 0 + 2 − 3 0 + 2 − 3 = 0
• Compute the parameter λ of radial distortion according to (13).
1
λ
= 0 2 + 0 2 + 0 + 0 +
Circle fitting
• It is a very important step to fit circle above
algorithm.
– data extracted from image are only short arcs, it is
hard to reconstruct a circle from the incomplete
data.
Distorted
“straight line”
• Method
Circle to fit
Distorted center
– Direct Least Squares Method of Circle Fitting (LS)
– Levenberg-Marquardt Method of Circle Fitting (LM)
Circle fitting - LS
2 +  2 +  +  +  = 0
(10)
• For each point  ,  on the “straight line”, (10) gives
,  , 1

= −  2 +  2
(14)
• Stacking equations from N points together gives
M

=b
(15)
Where M is N×3, b is N×1 matrix
Circle fitting - LS
• Directly using linear least squares fit method,
we can get
, ,
T
T
= M M
−1
MT b
(16)
Circle fitting - LM
Main ideas
• Let the equation of a circle be
1  2 +  2 + 2  + 3  + 4 = 0
Subject to the constraint :
2 2 + 3 2 − 41 4 = 1
•
(17)
(18)
The distance from a point  ,  to the circle
=
2
1+ 1+41
Where  = 1  2 +  2 + 2  + 3  + 4
(19)
(20)
Circle fitting - LM
• From (18) 2 2 + 3 2 − 41 4 = 1 , we can define
an angular coordinate  by
2 =
1 + 41 4 cos ,
3 =
1 + 41 4 sin
(21)
• Apply the standard Levenberg-Marquardt scheme to
minimize the sum of squared distance
ℱ=

2
in the three dimensional parameter space 1 , 4 ,
Further Discussion
• In Algorithm1, we must have ( ≥ 3) “straight lines”,
we relax this constrain and discuss the conditions of
– Only one straight line (L1)
– Only two straight lines (L2)
– Non-square pixels
Only One Straight Line (L1)
• Suppose the distortion center  is the image
center and calculate the distortion parameter
λ by (13)
1
λ
= 0 2 + 0 2 + 0 + 0 +
(13)
Only Two Straight Lines (L2)
• Extract “straight line” 1 、2 from the image;
• Determine parameter  ,  ,  =1,2 by fitting the
“straight line” 1 、2 with a circle according to (10);
2 +  2 +  +  +  = 0
(10)
(12) become
1 − 2 0 + 1 − 2 0 + 1 − 2 = 0
(22)
Only Two Straight Lines (L2)
• Select a suitable interval
=  − 30,  + 30 is suggested, for any 0  ∈ ,
calculating 0  according to (22);
1 − 2 0 + 1 − 2 0 + 1 − 2 = 0
∈
(22)
0  → 0
( ,  )
( − 30,  )
0  , 0  =
( + 30,  )
Only Two Straight Lines (L2)
• Calculate the distortion parameter λ =

(13), for any  = 0 , 0 ;
1
λ
= 0 2 + 0 2 + 1 0 + 1 0 +
1
λ
= 0 2 + 0 2 + 2 0 + 2 0 +  (13)
1
2
λ 1 + λ 2 according to
0  → 0
0  , 0  =
∈
( ,  )
( − 30,  )
( + 30,  )
λ 1
λ 2
1 1
λ = λ + λ 2
2
Only Two Straight Lines (L2)
• Calculate the corresponding corrected points
1 ,  1 (  2 ,  2 ), for any λ ,  and all distorted points
1 ,  1 1 (  2 ,  2 2 ) according to (4);
=

1+λ  2

1+λ  2
2 +  2
,  =
where  2 =
(4)
• Let [d, k] = min = min  1 +  2 , then obtain the optimal estimation
and λ .
1
2
1
2
Non-square Pixels
• Let 0 , 0 : the coordinates of the distorted centre
The distorted radius is given by  2 =  − 0 2 +  2  − 0
• From (8) we have
2
− 0

−
λ

2
+   − 0
2
2
1
λ

+
λ
− 0
− 0 + = 0

+   +
− 20
λ

+0 2 +  2 0 2 − 0
λ
2
2
2
(23)

+ − − 2 2 0
λ

1
+ 0 + = 0
λ
λ

(24)
Non-square Pixels
2 +  2  2 +

−
λ
+0 2 +  2 0 2 −

− 2 2 0
λ

1
0 + = 0
λ
λ
20  + −

λ 0
+

(24)
• Equation (24) shows the graphics of distorted “straight line” is
an ellipse under the condition of model (3).

• Similarly let A = λ
− 20 ,  = − − 2 2 0
λ
= 0 2 +  2 0 2 −

λ 0
+

λ 0
2 +  2  2 +  +  +  = 0
+
1
λ
, we have
(25)
and
1
λ
0 2 +  2 0 2 + 0 + 0 +  − = 0
(26)
Experiments and Results
Experiments and Results
Experiments and Results
Experiments and Results
Experiments and Results
Experiments and Results
Conclusions