A Simple Method - PowerPoint Presentation

Slide1

A Simple Method of Radial Distortion Correction with Centre of Distortion EstimationSlide2

OutlineIntroduction

Model and Approach

Further Discussion

Experiments and Results

ConclusionsSlide3

Introduction

Lens distortion usually can

be classified

into three

types :

radial distortion (predominant)decentering distortionthin prism distortionWang, J., Shi, F., Zhang, J., Liu, Y.: A new calibration model and method of camera lens distortion.Slide4

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 Slide5

Introduction

Correct the radial distortion

Former approach

based

on the

collinearity of undistorted points.Need the camera intrinsic parameters and 3D-point correspondences.This paperbased 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 ModelSlide6

Model and Approach

Radial Distortion

Models

PM

、

DMDistorted straight line is a circlecalibration procedure to estimate the center and the parameter of the radial distortionCircle fitting : LS、LMSlide7

Radial Distortion Models

The

Polynomial

Model (PM)

that describe

radial distortion

:

(1)

 Slide8

Radial Distortion Models

The

Division Model (DM)

that describe radial distortion :

(2)

we

use single parameter Division Model as our distortion

model :

(3)

 Slide9

Radial Distortion Models

To simplify equation, we suppose distorted center

is the origin image coordinates system, thus :

,

(4)

P

(0,0)Slide10

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)

 Slide11

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)

 Slide12

Estimate the

and

Let

be the coordinates of the distorted

center

. From (7) , we have

(8)

(9)

 Slide13

Estimate the

and

(9)

Let

, we have

(10)

 Slide14

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)

 Slide15

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)

 Slide16

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).

 Slide17

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.MethodDirect Least Squares Method of Circle Fitting (LS)Levenberg-Marquardt Method of Circle Fitting (LM)

Circle to fit

Distorted center

Distorted “straight line”Slide18

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 Slide19

Circle fitting - LS

Directly using linear least squares fit method,

we can get

(16)

 Slide20

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)

 Slide21

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

 Slide22

Further Discussion

In Algorithm1, we must have

“straight

lines”,

we relax this constrain and discuss the conditions ofOnly one straight line (L1)Only two straight lines (L2)Non-square pixels

 Slide23

Only One Straight Line (L1)

Suppose the distortion center

is the image center and calculate the distortion parameter

by (13)

(13)

 Slide24

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) Slide25

Only

Two

Straight

Lines

(

L2)

Select

a suitable interval

is suggested,

for any

,

calculating

according

to

(22

)

;

(22)

 Slide26

Only

Two

Straight

Lines

(

L2)

Calculate the distortion parameter

according to

(13)

, for

any

;

(13)

 Slide27

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

.

 Slide28

Non-square Pixels

Let

: the coordinates of

t

he distorted

centre

: pixel aspect radio The distorted radius is given by

From (8) we have

(23)

(24)

 Slide29

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)

 Slide30

Experiments and ResultsSlide31

Experiments and ResultsSlide32

Experiments and ResultsSlide33

Experiments and ResultsSlide34

Experiments and ResultsSlide35
Slide36

Experiments and ResultsSlide37

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

.DisadvantageIt needs straight lines are available in the scene.