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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


  • 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 Model

Principal Point Offset

principal point

Principal Point Offset

calibration matrix

Camera Rotation and Translation

CCD Camera


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

Forward projection (3D -> 2D)


Back-projection (2D -> 3D)

Projective Depth of Points


(dot product)

If ,

then m3 unit vector in positive direction




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

Camera center at infinity

Affine and non-affine cameras

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

Affine Cameras

Parallel Projection: Summary

canonical representation

affine calibration matrix

principal point is not defined

A Hierarchy of Affine Cameras

Orthographic projection


Scaled orthographic projection


A Hierarchy of Affine Cameras

Weak perspective projection


A Hierarchy of Affine Cameras

Affine camera


  • 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


Straight lines are not mapped to straight lines!

(otherwise it would be a projective camera)

Line Cameras


Null-space PC=0 yields camera center

Also decomposition

Camera calibration

Problem Statement

Basic Equations

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

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:





Exterior Orientation

Calibrated camera, position and orientation unkown

 Pose estimation

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

Nonlinear Distortion

  • Radial Component

short and long focal length

Nonlinear Distortion

  • Radial Component

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 Distortion

  • Tangential Component

Distortion function:

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