a simple method of radial distortion correction with centre of distortion estimation

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 :– 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.

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

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

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

Radial Distortion Models

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

(1)

Radial Distortion Models

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

(2)

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

(3)

Radial Distortion Models

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

, (4)

P (0,0)

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)

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)

Estimate the and

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

(9)

Estimate the and

(9)

Let , we have

(10)

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)

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)

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).

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”

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

Circle fitting - LS

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

(16)

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)

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

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

Only One Straight Line (L1)

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

(13)

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)

(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

(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 )

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

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)

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)

Experiments and Results

Experiments and Results

Experiments and Results

Experiments and Results

Experiments and Results

Experiments and Results

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.