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A Simple Method of Radial Distortion Correction with Centre of Distortion Estimation

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A Simple Method of Radial Distortion Correction with Centre of Distortion Estimation. Outline. Introduction Model and Approach Further Discussion Experiments and Results Conclusions. Introduction. Lens distortion usually can be classified into three types : - PowerPoint PPT Presentation
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A Simple Method of Radial Distortion Correction with Centre of Distortion Estimation
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Page 1: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

A Simple Method of Radial Distortion Correction with Centre of

Distortion Estimation

Page 2: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Outline

• Introduction• Model and Approach• Further Discussion• Experiments and Results• Conclusions

Page 3: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Introduction

• Lens distortion usually can be classified into three types :– radial distortion (predominant)– decentering distortion– thin prism distortion

Wang, J., Shi, F., Zhang, J., Liu, Y.: A new calibration model and method of camera lens distortion.

Page 4: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

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

Page 5: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Introduction

Correct the radial distortion• 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

Page 6: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Model and Approach

• Radial Distortion Models– 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

Page 7: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Radial Distortion Models

• The Polynomial Model (PM) that describe radial distortion :

(1)

Page 8: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Radial Distortion Models

• The Division Model (DM) that describe radial distortion :

(2)

• we use single parameter Division Model as our distortion model :

(3)

Page 9: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Radial Distortion Models

• To simplify equation, we suppose distorted center is the origin image coordinates system, thus :

, (4)

P (0,0)

Page 10: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

The Figure of Distorted Straight Line

• We consider collinear points and their distorted images.

• Let straight line equation from (4) We have

(5)

(6) (7)

Page 11: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

The Figure of Distorted Straight Line

(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)

Page 12: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Estimate the and

• Let be the coordinates of the distorted center . From (7) , we have (8)

(9)

Page 13: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Estimate the and

(9)

Let , we have

(10)

Page 14: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Estimate the and

Let , we have

(10)

Base on the relation of , we have ( 圓方程式參數 A 、 B 、 C 與 radial distortion 參數 P 、 的關係式 )

(11)

Page 15: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Estimate the and

(10) (11)

• Obtain of distorted center – Extract three “straight line” from image , we can get

by circle fitting from (10)

according to (11) , we have (12)

(13)

Page 16: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Sum up whole algorithm

• Extract “straight line” from the image• Determine parameter by fitting every “straight line”

with a circle according to (10)

• Calculate the center of the radial distortion according to

(12)

• Compute the parameter λ of radial distortion according

to (13).

Page 17: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

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.

• Method– Direct Least Squares Method of Circle Fitting (LS)– Levenberg-Marquardt Method of Circle Fitting (LM)

Circle to fit

Distorted center

Distorted “straight line”

Page 18: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Circle fitting - LS

(10) • For each point on the “straight line”, (10) gives

(14)

• Stacking equations from N points together gives b (15)

Where M is N3, b is N1 matrix

Page 19: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Circle fitting - LS

• Directly using linear least squares fit method, we can get

(16)

Page 20: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Circle fitting - LM

Main ideas• Let the equation of a circle be

(17)

Subject to the constraint : (18)

• The distance from a point to the circle (19)

Where (20)

Page 21: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Circle fitting - LM

• From (18) , we can define an angular coordinate by , (21)

• Apply the standard Levenberg-Marquardt scheme to minimize the sum of squared distance

in the three dimensional parameter space

Page 22: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Further Discussion

• In Algorithm1, we must have “straight lines”, we relax this constrain and discuss the conditions of– Only one straight line (L1)– Only two straight lines (L2)– Non-square pixels

Page 23: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Only One Straight Line (L1)

• Suppose the distortion center is the image center and calculate the distortion parameter by (13)

(13)

Page 24: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Only Two Straight Lines (L2)

• Extract “straight line” from the image;

• Determine parameter by fitting the “straight line” with a circle according to (10); (10)

(12) become (22)

Page 25: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

(𝐶 𝑥 ,𝐶 𝑦 )

(𝐶 𝑥+30 ,𝐶 𝑦)(𝐶 𝑥−30 ,𝐶 𝑦)

𝑷 𝒋∈𝑫

Only Two Straight Lines (L2)• Select a suitable interval is suggested, for any , calculating according to (22); (22)

(𝑥0 𝑗 , 𝑦0 𝑗 )=𝑃 𝑗

𝑥0 𝑗→ 𝑦0 𝑗

Page 26: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

(𝐶 𝑥 ,𝐶 𝑦 )

(𝐶 𝑥+30 ,𝐶 𝑦)(𝐶 𝑥−30 ,𝐶 𝑦)

𝑷 𝒋∈𝑫

Only Two Straight Lines (L2)• Calculate the distortion parameter according to (13), for any ;

(13)

(𝑥0 𝑗 , 𝑦0 𝑗 )=𝑃 𝑗

𝑥0 𝑗→ 𝑦0 𝑗

λ 𝑖1 λ 𝑖

2

λ 𝑖=12 ( λ 𝑖1+λ𝑖2 )

Page 27: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Only Two Straight Lines (L2)• Calculate the corresponding corrected points , for any ,

and all distorted points according to (4); , (4)

• Let [d, k] = min = min, then obtain the optimal estimation and .

𝑙1

𝑙2

𝑑𝑖2

𝑑𝑖1

Page 28: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Non-square Pixels• Let : the coordinates of the distorted centre : pixel aspect radio The distorted radius is given by • From (8) we have (23)

(24)

Page 29: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Non-square Pixels (24)• Equation (24) shows the graphics of distorted “straight line” is

an ellipse under the condition of model (3).• Similarly let , we have (25)

and (26)

Page 30: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Experiments and Results

Page 31: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Experiments and Results

Page 32: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Experiments and Results

Page 33: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Experiments and Results

Page 34: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Experiments and Results

Page 35: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation
Page 36: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Experiments and Results

Page 37: A Simple Method  of Radial Distortion Correction with Centre of Distortion Estimation

Conclusions

• Advantage– Neither information about the intrinsic camera

parameters nor 3D-point correspondences are required.

– based on single image and uses the distorted positions of collinear points.

– Algorithm is simple, robust and non-iterative.• Disadvantage– It needs straight lines are available in the scene.


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