Geometric Camera Calibrationbased on “A Flexible New Technique for Camera

Calibration” by Zhengyou Zhang

Alexander Reuter

2.12.2009

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Motivations

What we know so far

Pinhole cameras are ideal cameras

There doesn’t exist a pinhole camera actually

So there doesn’t exist images obtained by an ideal camera

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Motivations

What is camera calibration?

Find the parameters of a camera that produced several images

Application field

Transform the image to one obtained by an ideal camera

Find the global position and orientation of a camera

Available techniques

Self-calibration

Photogrammetric calibration

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Overview

Overview

Pinhole Camera

Intrinsic parametersExtrinsic parameters

Zhangs Algorithm

Degenerate configurations

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Homography

Properties

Linear projection between two planes

Described by a R3×3 matrix

Advantages

Transformation from a rectangle plane in world space to aquadriliteral in image plane can be described as a homographyproblem

Important tool for camera calibration

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Homography

Homography between image plane and object plane

xcam

ycam

C

zcam

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Homography

example

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Developing intrinsic camera matrix

Motivation

definition of camera properties required to apply cameracalibration

different parameter definitions make introduction necessary

assume naive pinhole camera model

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Pinhole camera

From raw model to an initial transformation matrix

xcam

ycam

C

zcam

Principle Axis

camera centre

f zcamprincipal point

image plane

ximgyimg

xX

f ∗X ′

Z ′

f ∗Y ′

Z ′

X ′

Y ′Z ′

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Pinhole camera

From raw model to an initial transformation matrix

xcam

ycam

C

zcam

Principle Axis

camera centre

f

zcamprincipal point

image plane

ximgyimg

xX

f ∗X ′

Z ′

f ∗Y ′

Z ′

X ′

Y ′Z ′

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Pinhole camera

From raw model to an initial transformation matrix

xcam

ycam

C

zcam

Principle Axis

camera centre

f

zcam

principal point

image plane

ximgyimg

xX

f ∗X ′

Z ′

f ∗Y ′

Z ′

X ′

Y ′Z ′

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Pinhole camera

From raw model to an initial transformation matrix

xcam

ycam

C

zcam

Principle Axis

camera centre

f zcam

principal point

image plane

ximgyimg

xX

f ∗X ′

Z ′

f ∗Y ′

Z ′

X ′

Y ′Z ′

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Development of intrinsic camera matrix

Applying focal length [1]X ′

Y ′

Z ′

1

7−→ fX ′

fY ′

Z ′

=

f 0f 0

1 0

X ′

Y ′

Z ′

1

Focal length examples

Figure: 40mm, 180mm and 600mm focal length

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Development of intrinsic camera matrix

Principal point offset

y

x

ycam

xcamy0

x0

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Development of intrinsic camera matrix

Principal point offsetX ′

Y ′

Z ′

1

7−→ fX ′ + Z ′x0

fY ′ + Z ′y0Z ′

=

f x0 0f y0 0

1 0

X ′

Y ′

Z ′

1

Camera calibration matrix

K :=

f x0f y0

1

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Independent scaling

Unequal scale factors

K :=

αx x0αy y0

1

Skew parameter

K :=

αx s x0αy y0

1

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Development of extrinsic camera matrix

From world to camera coordinates

x

z

xcam

zcam

R, t

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Development of extrinsic camera matrix

Camera rotation and translation

Xcam := object coordinates depending on camera orientation R ∈R3x3 and origin C ∈ R3x1

Getting Xcam

Xcam =

[R −RC0 1

]X ′

Y ′

Z ′

1

=

[R −RC0 1

]X

= R[I | − C ]X

Putting K and Xcam together

x = KXcam = KR[I | − C ]Xt:=−RC

= K [R|t]

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Finite projective camera

Finite projective camera

P ∈ R3×4

P := K [R|t] =

αx s x0αy y0

1

r11 r12 r13 txr21 r22 r23 tyr31 r32 r33 tz

x ′ = PX , where X =

XYZ1

and x ′ =

xyw

⇒ x = x′

w

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Zhangs Algorithm

Prework

Take several images of a planar checkboard

Find checkboard in each image

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Zhangs Algorithm

Prework

Find subpixel corners of eachcheckboard

Get the checkboard coordinate in 2D worldspace of eachcheckboard corner

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Zhangs Algorithm

Algorithm

Estimate a homography for each image

Estimate intrinsic matrix K from the set of homographies

Estimate extrinsic parameters for each checkboard

Estimate coefficients of radial distortion

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Distortion

Two important distortion types

Radialdistortion

Tangentialdistortion

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Estimate coefficients of radial distortion

Distortion in Zhangs Algorithm

Only radial distortion is handled

Estimation provided by minimization

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Zhangs Algorithm

Method Summary

Print a pattern

Take images

Detect feature points

Estimate intrinsic & extrinsic parameters

Estimate coefficients of radial distortion

Refine by minimizing

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Multiple cameras

Scene with several camera positions and orientations

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Multiple cameras

World centered scene

x

z

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Multiple cameras

Scene centered by a selected camera

x

z

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Degenerate configurations

Degenerate configurations

Rotation between two images needed to get homography

Long focal length / small working volume lead to non reliableresults

Outliers in feature point sets not acceptable (RANSAC)

Subpixel accuracy for feature points required

Figure: RANSAC algorithm26 / 28

Geometric Camera Calibration

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Degenerate configurations

Subpixel & Working volume

x

y

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Conclusion

Summary

Camera calibration obtains intrinsic and extrinsic parameters ofa camera

Plane-to-plane transformations, called Homographies are used tomap from object plane to image plane

Many approaches (like Zhangs) estimate distortions

Relative position and orientation of each camera based on theproperties of a selected camera can be obtained

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Homogeneous coordinates

Cartesian coordinates

Standard coordinate system

Standard matrix operation allows rotation, scaling and shearing

Homogeneous coordinates

Enhanced coordinate system

One matrix operation allows translation and perspectivetransformations additionally

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Homogeneous coordinates

From Cartesian coordinates to Homogeneous coordinates

(xy

)→

xy1

and

xyz

→

xyz1

From Homogeneous coordinates to Cartesian coordinates x

yw

→ (xwyw

)and

xyzw

→ x

wywzw

10

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