Camera Models and Imaging

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Camera Models and Imaging
Leow Wee Kheng
CS4243 Computer Vision and Pattern Recognition
Camera Models and Imaging
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Camera Models and Imaging
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Through our eyes…

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Through our eyes we see the world.
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cornea
focus light
lens
retina captures
image
transmit
image
optic
nerves
iris
pupil
control amount
of light entering
the eye
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Camera Models and Imaging
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Through their eyes…
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Camera is computer’s eye.
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
Camera is structurally the same the eye.
sensor | retina
lens
lens | cornea
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lens
aperture | pupil
lens | cornea
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aperture
plates | iris
aperture | pupil
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Imaging

Images are 2D projections of real world scene.

Images capture two kinds of information:
 Geometric:
positions, points, lines, curves, etc.
 Photometric: intensity, colour.

Complex 3D-2D relationships.

Camera models approximate relationships.
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Camera Models

Pinhole camera model
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Orthographic projection

Scaled orthographic projection
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Paraperspective projection
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Perspective projection
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Pinhole Camera Model
image
plane
2D
image
pinhole
virtual image
plane
3D
object
focal length
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
Math representation
world frame /
camera frame
camera
centre
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image
frame
image
centre
Camera Models and Imaging
projection
line
optical
axis
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
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By default, use right-handed coordinate system.
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
3D point P = (X, Y, Z)T projects to
2D image point p = (x, y)T .

By symmetry,
i.e.,
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Simplest form of perspective projection.
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Orthographic Projection
 3D
scene is at infinite distance from camera.
 All projection lines are parallel to optical axis.
 So,
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Scaled Orthographic Projection
depth << distance to camera.
 Z is the same for all scene points, say Z0
 Scene
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Perspective Projection
camera frame
image frame is parallel to
camera’s X-Y plane
world frame
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Intrinsic Parameters

Pinhole camera model

Camera sensor’s pixels not exactly square
 x,
y : coordinates (pixels)
 k, l : scale parameters (pixels/m)
 f : focal length (m or mm)
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 f,
k, l are not independent.
 Can rewrite as follows (in pixels):
 So,
 Image
centre or principal point c may not be at origin.
 Denote location of c in image plane as cx, cy.
 Then,
 Along
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optical axis, X = Y = 0, and x = cx, y = cy.
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 Image
frame may not be exactly rectangular.
 Let θ denote skew angle between x- and y-axis.
 Then,
 Combine
all parameters yield
Intrinsic
parameter
matrix
homogeneous coordinates
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 Denoting
 Some
ρ = Z, we get
books and papers absorb ρ into
:
giving
Be careful.
 A simpler form of K uses skew parameter s:
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Extrinsic Parameters

Camera frame is not aligned with world frame.

Rigid transformation between them:
frame
object
 Coordinates
of 3D scene point in camera frame.
 Coordinates of 3D scene point in world frame.
 Rotation matrix of world frame in camera frame.
 Position of world frame’s origin in camera frame.
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
Translation matrix

Rotation matrix
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
Rotation matrix is orthonormal:

So,

In particular, let
.
Then,
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
Combine intrinsic and extrinsic parameters:
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Simpler notation:
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Lens Distortion

Lens can distort images,
especially at short focal length.
barrel
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pin-cushion
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fisheye
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
Radial distortion is modelled as
distorted
coordinates
with
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undistorted
coordinates
distortion
parameters
.
Actual image coordinates
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Camera Calibration

Compute intrinsic / extrinsic parameters.

Capture calibration pattern from various views.
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
Detect inner corners in images.

Run calibration program (available in OpenCV).
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Homography

A transformation between projective planes.

Maps straight lines to straight lines.
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
Homography equation:
image
point

homography
image
point
Affine is a special case of homography:

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
Scaling H by s does not change equation:

Homography is defined up to unspecified scale.

So, can set h33 = 1.

Expanding homography equation gives
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
Assembling equations over all points yields
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System of linear equations. Easy to solve.
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
Example
 Circles:
input corresponding points.
 Black dots: computed points, match input points.
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Before homographic transform
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After homographic transform
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Summary

Simplest camera model: pinhole model.

Most commonly used model: perspective model.
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Intrinsic parameters:
 Focal

length, principal point.
Extrinsic parameters:
 Camera
rotation and translation.

Lens distortion

Homography: maps lines to lines.
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Further Reading

Camera models: [Sze10] Section 2.1.5, [FP03]
Section 1.1, 1.2, 2.2, 2.3.

Lens distortion: [Sze10] Section 2.1.6, [BK08]
Chapter 11.

Camera calibration: [BK08] Chapter 11, [Zha00].

Image undistortion: [BK08] Chapter 11.
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References

Bradski and Kaehler. Learning OpenCV. O’Reilly, 2008.

D. A. Forsyth and J. Ponce. Computer Vision: A Modern Approach.
Pearson Education, 2003.

R. Szeliski. Computer Vision: Algorithms and Applications. Springer,
2010.

Z. Zhang. A flexible new technique for camera calibration. IEEE
Trans. PAMI, 22(11):1330–1334, 2000.
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