8. 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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9. Camera Models
Pinhole camera model
Orthographic projection
Scaled orthographic projection
Paraperspective projection
Perspective projection
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10. Pinhole Camera Model
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image
plane
pinhole
virtual image
plane
3D
object
focal length
2D
image
11. Math representation
CS4243 Camera Models and Imaging 11
world frame /
camera frame
image
frame
camera
centre
image
centre
optical
axis
projection
line
12. By default, use right-handed coordinate system.
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13. 3D point P = (X, Y, Z)T projects to
2D image point p = (x, y)T .
By symmetry,
i.e.,
Simplest form of perspective projection.
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14. Orthographic Projection
3D scene is at infinite distance from camera.
All projection lines are parallel to optical axis.
So,
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15. Scaled Orthographic Projection
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Scene depth << distance to camera.
Z is the same for all scene points, say Z0
17. 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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18. 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 optical axis, X = Y = 0, and x = cx, y = cy.
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19. Image frame may not be exactly rectangular.
Let θ denote skew angle between x- and y-axis.
Then,
Combine all parameters yield
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homogeneous coordinates
Intrinsic
parameter
matrix
20. Denoting ρ = Z, we get
Some books and papers absorb ρ into :
giving
Be careful.
A simpler form of K uses skew parameter s:
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21. Extrinsic Parameters
Camera frame is not aligned with world frame.
Rigid transformation between them:
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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frame
object
23. Rotation matrix is orthonormal:
So, .
In particular, let
Then,
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24. Combine intrinsic and extrinsic parameters:
Simpler notation:
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25. Lens Distortion
Lens can distort images,
especially at short focal length.
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barrel pin-cushion fisheye
26. Radial distortion is modelled as
with .
Actual image coordinates
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distorted
coordinates
undistorted
coordinates
distortion
parameters
27. Camera Calibration
Compute intrinsic / extrinsic parameters.
Capture calibration pattern from various views.
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28. Detect inner corners in images.
Run calibration program (available in OpenCV).
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29. Homography
A transformation between projective planes.
Maps straight lines to straight lines.
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30. Homography equation:
Affine is a special case of homography:
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image
point
image
point
homography
31. 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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32. Assembling equations over all points yields
System of linear equations. Easy to solve.
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33. Example
Circles: input corresponding points.
Black dots: computed points, match input points.
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36. Summary
Simplest camera model: pinhole model.
Most commonly used model: perspective model.
Intrinsic parameters:
Focal length, principal point.
Extrinsic parameters:
Camera rotation and translation.
Lens distortion
Homography: maps lines to lines.
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37. 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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38. 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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