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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 :

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outline
Outline
  • Introduction
  • Model and Approach
  • Further Discussion
  • Experiments and Results
  • Conclusions
introduction
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.

introduction1
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
introduction2
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
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
Radial Distortion Models
  • The Polynomial Model (PM) that describe radial distortion :

(1)

radial distortion models1
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 models2
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
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 line1
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
Estimate the and
  • Let be the coordinates of the distorted

center . From (7) , we have

(8)

(9)

estimate the and1
Estimate the and

(9)

Let

, we have

(10)

estimate the and2
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 and3
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
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
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)

Distorted “straight line”

Circle to fit

Distorted center

circle fitting ls
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 ls1
Circle fitting - LS
  • Directly using linear least squares fit method,

we can get

(16)

circle fitting lm
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 lm1
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
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
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
Only Two Straight Lines (L2)
  • Extract “straight line” from the image;
  • Determine parameter by fittingthe “straight line” with a circle according to(10);

(10)

(12) become

(22)

only two straight lines l21
Only Two Straight Lines (L2)
  • Select a suitable interval

is suggested, for any ,

calculating according to (22);

(22)

only two straight lines l22
Only Two Straight Lines (L2)
  • Calculate the distortion parameter according to (13), for any ;

(13)

only two straight lines l23
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 .
non square pixels
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 pixels1
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)

conclusions
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.