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3D Computer Vision II

ReminderCamera Models

Nassir Navabbased on a course given at UNC by Marc Pollefeys & the book Multiple View Geometry by Hartley & Zisserman

October 27, 2009

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Outline Camera Models

# Geometric Parameters of a Finite Camera# Projective Camera Model

!# Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray!# Decomposition of Camera Matrix

# Cameras at Infinity# Other Camera Models (Pushbroom and Line Cameras)

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Pinhole Camera Model

# Mapping between 3D world and 2D image# Central projection# Models are described in matrices with particular properties

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Homogeneous Coordinates

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Central Projection

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are the coordinates of the principal point

Principal Point Offset

where

principal point(perpendicular intersection point of

principal axis and image plane)

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is called camera calibration

matrix

Principal Point Offset

where

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Inhomogeneous

coordinates

where represents the point in

world coordinates

represents the same point in

camera coordinates

represents the coordinates of the camera

origin in the world coordinate frame

Camera Rotation and Translation

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Homogeneous

coordinates

projection to image plane from camera coordinates

projection to image plane from world coordinates

Camera Rotation and Translation

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where

projection matrix of a general

pinhole camera with 9 DOF

intrinsic camera parameters with 3 DOF

extrinsic camera parameters with each 3 DOF

(camera orientation, position in world coordinates)

Extrinsic and Intrinsic Parameters

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No explicit

camera center

where

from

Camera Rotation and Translation

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CCD Cameras Non-Square Pixels

number of pixels per unit distance

4 DOF

10 DOF

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Skew Parameter

skew parameter

5 DOF

finite projective camera with11 DOF

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non-singular

Finite Projective Camera Summary

projection matrix

11 DOF (5+3+3)

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decompose projection matrix P in K,R,C

Finite Projective Camera Decomposition of P

non-singular 3x3 matrix (8 DOF)

QR matrix

decomposition

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Finite Projective Camera Summary

# Camera matrices Pare identical with the set of homogeneous 3x4matrices for which the left 3x3 sub-matrix is non-singular

# If rank(P) = 3, but rank(M) < 3, then camera at infinity# if rank(P) < 3 the matrix mapping will be a line or a point and not a

plane (not a 2D image)

where

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Outline Camera Models

# Geometric Parameters of a Finite Camera# Projective Camera Model

!# Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray!# Decomposition of Camera Matrix

# Cameras at Infinity# Other Camera Models (Pushbroom and Line Cameras)

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Camera Anatomy

# Camera center# Column vectors# Principal plane# Axis plane# Principal point# Principal ray

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P has a 1D null-space

Finite cameras:

Infinite cameras:

Camera Center

we will prove that the 4-vector C is the camera center

points on a line through A and C

since

All 3D points on the line are mapped on the same 2D image point, and

thus the line is a ray through the camera center

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Column vectors are the image points which project the

axis directions (X,Y,Z) and the origin

Column Vectors

Example for the image

of the y-axis

is the image of the world origin

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Row Vectors

Represent geometrically particular world planes.

row vectors

column vectors

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Row Vectors of the Projection Matrix

p1is defined by the camera center and the

line x=0 on the image.

p2is defined by the camera center and the

line y=0 on the image.

Example p2

respectively for p1

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Principal Plane

Plane through camera center and

parallel to the image plane.

points Xare imaged on the line at

infinity

if X is on the principle plane

especially

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The line through camera center and

perpendicular to principal plane is the

principal axis.

The intersection of the principal

axis with the image plane is the

principal point.

normal direction to the principal plane

principal point

where

third row of Mand

Principal Point

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Ambiguity that principal axis points

towards the front of the camera (positive

direction)

direction unaffected by scaling

since

Principal Axis Vector

towards the front of the camera

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Maps a point in space on the image plane

Forward Projection

Vanishing points

Only M affects the projection of vanishing points

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w can be interpreted

as the dot product of

the ray CX with the

principal ray direction

(PC=0)

If ,

then m3is a unit vector pointing in positive axis direction

Suppose . Then

Depth of Points

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Depth of Points: Examples

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Outline Camera Models

# Geometric Parameters of a Finite Camera# Projective Camera Model

!# Camera Center, Principal Plane, Axis Plane, Principal Point, Principal Ray!#

Decomposition of Camera Matrix

# Cameras at Infinity# Other Camera Models (Pushbroom and Line Cameras)

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Finding the camera center C

numerically: find right null-space by SVD of P

Algebraically:

where

Camera Matrix Decomposition

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Finding the camera center C

Any plane !going through C will be a linear combination of the three

planes defined by the rows of P. Therefore:

where

Camera Matrix Decomposition

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Finding the camera orientation and internal parameters

using RQ decomposition

Q R=( )-1= -1 -1QR

Decompose

Ambiguity removed by enforcing positive diagonal entries

Camera Matrix Decomposition

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1

arctan(1/s)

for CCD/CMOS, always s=0

Image from image, s"0 possible

(non coinciding principal axis)

resulting camera:

When is Skew Non-zero?

where H is a 3x3 homography

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General projective interpretation

# Meaningful decomposition in K,R,t requiresEuclidean image