Chap' 6 Camera Models

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Chap. 6 Camera Models

• Wonwoo Lee
• 2006. 01. 13
• GIST U-VR Lab.

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Introduction

• In this chapter, we will review
• Geometric model of camera
• Pinhole camera
• Projective camera
• Affine camera

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Pinhole camera model

• A mapping from Euclidean 3-space to Euclidean
  2-space
• In Homogeneous coordinates

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Principal point offset
Principal point

• Generally, the origin of coordinates in the image
  plane is not at the principle point

Camera calibration matrix
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Rotation and Translation

• In general
• A point is expressed in world coordinate system

World coordinate system
Camera coordinate system
Rotation Translation
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Rotation and Translation
World coordinate system
Camera coordinate system
R 3x3 rotration matrix

• 9 DoF
• Parameters in K internal parameters
• Parameters of R and C external parameters

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CCD Camera

• Pinhole camera model assumes that the image
  coordinates are Euclidean coordinates having
  equal scales in both axial directions.
• CCD cameras
• Image coordinates are measured in pixels
• Unequal scale factors in each direction

CCD Charge-Coupled Device
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Finite projective camera

• Adding generality

s skew parameter Finite projective camera 11 DoF
M can be decomposed as a product MKR
Homogeneous 3x4 matrix for which the left hand
3x3 sub-matrix is non-singular
Matrix of finite projective camera
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Summary of camera models

• Simple camera model

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Summary of camera models

• CCD camera model

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Summary of camera models

• Finite projective camera model

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Projective Camera

• A general projective camera P
• Map world points X to image points x
• Geometric entities
• Camera center
• Column points
• Principal plane
• Principal point

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Camera Center
Rank(P) 3, 1D null-space
A line L through C and any other point A in
3-space
(C 4-vector that is PC0)

• All points on L are mapped to the same image
  point PA
• L should be a ray through the camera center

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Column vectors
Vanishing points of the world coordinate x, y
and z
the image of the world origin
Ex)
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Row vectors

• Principal plane
• the plane through the camera centre parallel to
  the image plane
• Consists of the points on line at infinity

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Principal point

• Principal axis
• The line through the camera center C
• Intersect the image plane at the principal point

• Projecting the point on plane at infinity

Principle point
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Cameras at infinity

• Cameras with center lying on the plane at infinity

M is singular

• An affine camera is one that has a camera matrix
  P which the last row is of the form (0,0,0,1)
• Points at infinity are mapped to points at
  infinity

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Affine cameras
Moving backward Camera center moves backward
Zooming Increasing focal length by a factor k
dt/d0
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A hierarchy of affine cameras

• Projection along z-axis

• Orthographic projection
• 5DoF

M Last row zero, The first two rows
orthogonal Unit norm,

• Scaled Orthographic projection
• 6DoF

M Last row zero, The first two rows
orthogonal Equal norm
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A hierarchy of affine cameras

• Weak Perspective Projection
• 7 DoF

M Last row zero, The first two rows
orthogonal Not equal norm
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Affine Camera

• General camera matrix of the affine form
• 8 DoF

• An orthographic projection from 3-space to an
  image
• An affine transformation of the image

• Camera with principal plane being the plane at
  infinity ? Affine camera
• Affine camera maps parallel world lines to
  parallel image lines
• The vector d satisfying M2x3d0 is the direction
  of parallel projection