CS 44957495 Computer Vision

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CS 4495/7495Computer Vision

• Cameras, Geometric Cameras Models,
• and Geometric Camera Calibration
• Frank Dellaert
• Some Slides by Forsyth Ponce, Jim Rehg

Aaron Bobick
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Outline (from Frank)

• Pinhole Cameras
• Cameras with Lenses
• Homogeneous Coordinates
• Joining Points, Lines, and Planes
• Geometric Camera Model
• Intrinsic Parameters
• Extrinsic Parameters
• Projective Cameras
• Geometric Camera Calibration

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Outline (today)

• Pinhole Cameras
• Cameras with Lenses
• Homogeneous Coordinates
• Extrinsic Parameters
• Geometric Camera Model
• Intrinsic Parameters
• Joining Points, Lines, and Planes
• Projective Cameras
• Geometric Camera Calibration

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• Homogeneous Coordinates

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2D Coordinate Frames Points

• coordinates x and y

j
p (x,y)T
i
o
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2D Lines

• Line l axbyc

j
p(x,y)T
c
(a,b)T
i
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Homogeneous Coordinates

• Uniform treatment of points and lines
• Line-point incidence lTp0

j
stay the same when scaled
p(x,y,1)T(kx,kx,k)T
c
l(a,b,c)T(ka,kb,kc)T
(a,b)T
i
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But, more intuitive reason for now

• Use homogenous coordinates to combine rotation
  and translation into same framework matrix
  transformation.
• Allows easy transformation between frames
  common between vision, graphics, and robotics.

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• Extrinsic Parameters

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Camera Pose
In order to apply the camera model, objects in
the scene must be expressed in camera coordinates.
Camera Coordinates
World Coordinates
Calibration target looks tilted from
camera viewpoint. This can be explained as
a difference in coordinate systems.
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Rigid Body Transformations

• Need a way to specify the six degrees-of-freedom
  of a rigid body.
• Why are their 6 DOF?

A rigid body is a collection of points whose
positions relative to each other cant change
Fix one point, three DOF
Fix second point, two more DOF (must
maintain distance constraint)
Third point adds one more DOF, for
rotation around line
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Notations

• Superscript references coordinate frame
• AP is coordinates of P in frame A
• BP is coordinates of P in frame B
• Example (also Page 23)

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Translation
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Translation

• Using homogeneous coordinates, translation can
  be expressed as a matrix multiplication.
• Translation is commutative

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Rotation
means describing frame A in The coordinate system
of frame B
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Rotation
Orthogonal matrix!
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Example Rotation about z axis
What is the rotation matrix?
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Combine 3 to get arbitrary rotation

• Euler angles Z, X, Z
• Heading, pitch roll world Z, new X, new Y
• Three basic matrices order matters, but well
  probably not focus on that

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Rotation in homogeneous coordinates

• Using homogeneous coordinates, rotation can be
  expressed as a matrix multiplication.
• Rotation is not commutative

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Rigid transformations
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Rigid transformations (cont)

• Unified treatment using homogeneous coordinates.

Invertible!
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• Geometric Camera Model

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Perspective Camera Model
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We can see infinity !
Railroad parallel lines
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Affine Camera Model (p.33)
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• Intrinsic Parameters

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Normalized Image coordinates
1
O
uX/Z dimensionless !
P
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Pixel units
Pixels are on a grid of a certain dimension
f
O
uk f X/Z in pixels ! f m (in
meters) k pixels/m
P
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Pixel coordinates
We put the pixel coordinate origin on topleft
f
O
uu0 k f X/Z
P
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Pixel coordinates in 2D
640
(0.5,0.5)
i
(u0,v0)
480
(640.5,480.5)
j
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Important MATLAB Convention
(1,1) !
Just as good as any other convention !
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Summary Intrinsic Calibration
5 Degrees of Freedom !
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• Projective Cameras

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Projective Camera Matrix
56 DOF 11 !
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Projective Camera Matrix
56 DOF 11 !
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Columns Rows of M
m2P0
O
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Homogeneous Coordinates (again!)

• Uniform treatment of points and lines
• Line-point incidence lTp0

j
stay the same when scaled
p(x,y,1)T(kx,kx,k)T
c
l(a,b,c)T(ka,kb,kc)T
(a,b)T
i
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Join cross product !

• Join of two lines is a pointpl1xl2
• Join of two points is a linelp1xp2

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Joining two parallel lines ?

• (a,b,c)

(a,b,c)
(a,b,d)
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Points at Infinity !
(-b,a,0)T
Line at infinity linf(0,0,1)T
j
l(a,b,c)T
i
(-b,a,0)T
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In 3D Same Story

• 3D points (x,y,z,w)T
• 3D planes (a,b,c,d)T
• join of three points plane
• join of three planes point
• plane at infinity (0,0,0,1)T

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• Geometric Camera Calibration

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Affine Calibration
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Affine Calibration
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Affine Camera Matrix
Only 8 DOF !
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Inserting Synthetic Objects
pMP