ROBOT VISION
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ROBOT VISION Lesson 4: Camera Models and Calibration Matthias Rüther Slides partial courtesy of Marc Pollefeys Department of Computer Science University of North Carolina, Chapel Hill. Content. Camera Models Pinhole Camera CCD Camera Finite Projective Camera Affine Camera Pushbroom Camera

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ROBOT VISIONLesson 4: Camera Models and CalibrationMatthias RütherSlides partial courtesy of Marc Pollefeys Department of Computer ScienceUniversity of North Carolina, Chapel Hill


Content

Content

  • Camera Models

    • Pinhole Camera

    • CCD Camera

    • Finite Projective Camera

    • Affine Camera

    • Pushbroom Camera

  • Calibration

    • Inner Orientation

    • Nonlinear Distortion

    • Calibration using Planar Targets

    • Calibration using a 3D Target

The Cyclops, 1914 by Odilon Redon


Basic pinhole camera model

Basic Pinhole Camera Model


Basic pinhole camera model1

Basic Pinhole Camera Model


Basic pinhole camera model2

Basic Pinhole Camera Model


Principal point offset

Principal Point Offset

principal point


Principal point offset1

Principal Point Offset

calibration matrix


Camera rotation and translation

Camera Rotation and Translation


Ccd camera

CCD Camera


Finite projective camera

non-singular

Finite Projective Camera

11 dof (5+3+3)

decompose P in K,R,C?

{finite cameras}={P3x4 | det M≠0}

If rank P=3, but rank M<3, then cam at infinity


Action of projective cameras on points

(pseudo-inverse)

Action of Projective Cameras on Points

Forward projection (3D -> 2D)

D…direction

Back-projection (2D -> 3D)


Projective depth of points

Projective Depth of Points

(PC=0)

(dot product)

If ,

then m3 unit vector in positive direction


When is skew non zero

arctan(1/s)

g

1

When is skew non-zero?

for CCD/CMOS, always s=0

Image from image, s≠0 possible

(non coinciding principal axis)

resulting camera:


Moving the camera center to infinity

Moving the Camera Center to Infinity

Camera center at infinity

Affine and non-affine cameras

Definition: affine camera has P3T=(0,0,0,1)


Affine cameras

Affine Cameras


Parallel projection summary

Parallel Projection: Summary

canonical representation

affine calibration matrix

principal point is not defined


A hierarchy of affine cameras

A Hierarchy of Affine Cameras

Orthographic projection

(5dof)

Scaled orthographic projection

(6dof)


A hierarchy of affine cameras1

A Hierarchy of Affine Cameras

Weak perspective projection

(7dof)


A hierarchy of affine cameras2

A Hierarchy of Affine Cameras

Affine camera

(8dof)

  • Affine camera=camera with principal plane coinciding with P∞

  • Affine camera maps parallel lines to parallel lines

  • No center of projection, but direction of projection PAD=0

  • (point on P∞)


Pushbroom cameras

Pushbroom Cameras

(11dof)

Straight lines are not mapped to straight lines!

(otherwise it would be a projective camera)


Line cameras

Line Cameras

(5dof)

Null-space PC=0 yields camera center

Also decomposition


Camera calibration

Camera calibration


Problem statement

Problem Statement


Basic equations

Basic Equations


Basic equations1

minimize subject to constraint

Basic Equations

minimal solution

P has 11 dof, 2 independent eq./points

  • 5½ correspondences needed (say 6)

Over-determined solution

n  6 points


Geometric error

Geometric Error


Gold standard algorithm

Gold Standard algorithm

  • Objective

  • Given n≥6 2D to 2D point correspondences {Xi↔xi’}, determine the Maximum Likelyhood Estimation of P

  • Algorithm

  • Linear solution:

    • Normalization:

    • DLT:

  • Minimization of geometric error: using the linear estimate as a starting point minimize the geometric error:

  • Denormalization:

~

~

~


Example

Example


Exterior orientation

Exterior Orientation

Calibrated camera, position and orientation unkown

 Pose estimation

6 dof  3 points minimal (4 solutions in general)


Nonlinear distortion

Nonlinear Distortion

  • Radial Component

short and long focal length


Nonlinear distortion1

Nonlinear Distortion

  • Radial Component


Correction of radial distortion

Correction of radial Distortion

Correction of radial distortion

Choice of the distortion function and center

  • Computing the parameters of the distortion function

  • Minimize with additional unknowns

  • Straighten lines


Nonlinear distortion2

Nonlinear Distortion

  • Tangential Component

Distortion function:


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