Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective...
-
date post
20-Dec-2015 -
Category
Documents
-
view
248 -
download
1
Transcript of Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective...
![Page 1: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/1.jpg)
Epipolar Geometry and the Fundamental Matrix F
The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates this
geometry
x F x’ = 0
![Page 2: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/2.jpg)
Epipolar geometry
• The Epipolar geometry depends only on the internal parameters of the cameras and the relative pose.
• A point X in 3 space is imaged in 2 views: x and x’
• X, x, x’ and the camera centre C are coplanar in the plane
• The rays back-projected from x and x’ meet at X
![Page 3: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/3.jpg)
Point correspondence geometry
Fig. 8.1
![Page 4: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/4.jpg)
Point correspondence geometry
![Page 5: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/5.jpg)
Epipolar GeometryFig. 8.2
![Page 6: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/6.jpg)
Epipolar geometry
![Page 7: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/7.jpg)
The geometric entities involved in epipolar geometry
![Page 8: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/8.jpg)
Fig 8.3
![Page 9: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/9.jpg)
Converging cameras
![Page 10: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/10.jpg)
Fig 8.4
![Page 11: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/11.jpg)
Motion parallel to the image plane
![Page 12: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/12.jpg)
Geometric derivationFig. 8.5
![Page 13: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/13.jpg)
Point transfer via a plane
![Page 14: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/14.jpg)
The fundamental matrix F
• x l’
• Geometric Derivation
• Step 1: Point transfer via a plane
There is a 2D homography H mapping
each xi to xi’
Step 2: Constructing the epipolar line
![Page 15: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/15.jpg)
Constructing the epipolar line
![Page 16: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/16.jpg)
Cross products
• If a = ( a1, a2 , a3)T is a 3-vector, then one define a corresponding skew-sysmmetric matrix as follows:
0aa-
a-0a
aa-0
a
12
13
23
X
![Page 17: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/17.jpg)
Cross products 2
• Matrix [a]x is singular and a is its null vector
• a x b = ( a2b3 - a3b2, a3b1 - a1b3 , a1b2 – a2b1)T
• a x b = [a]x b =( aT [b]x )T
![Page 18: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/18.jpg)
Algebraic derivation
![Page 19: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/19.jpg)
Algebraic derivation 2
![Page 20: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/20.jpg)
Example 8.2
![Page 21: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/21.jpg)
Example 8.2 b
![Page 22: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/22.jpg)
Properties of the fundamental matrix (a)
![Page 23: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/23.jpg)
Properties of the fundamental matrix (b)
![Page 24: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/24.jpg)
Summary of the Properties of the fundamental matrix 1
![Page 25: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/25.jpg)
Summary of the properties of the fundamental matrix 2
![Page 26: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/26.jpg)
Epipolar line homography 1
Fig. 8.6a
![Page 27: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/27.jpg)
Epipolar line homography 2
Fig. 8.6 b
![Page 28: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/28.jpg)
Epipolar line homography
![Page 29: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/29.jpg)
The epipolar line homography
![Page 30: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/30.jpg)
A pure camera motion
![Page 31: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/31.jpg)
Pure translation
![Page 32: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/32.jpg)
Fig. 8.8
![Page 33: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/33.jpg)
Pure translation motion
![Page 34: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/34.jpg)
Example of pure translation
![Page 35: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/35.jpg)
General camera motion
Fig. 8.9
![Page 36: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/36.jpg)
General camera motion
![Page 37: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/37.jpg)
Example of general motion
![Page 38: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/38.jpg)
Pure planar motion
![Page 39: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/39.jpg)
Retrieving the camera matricesUsing F to determine the camera matrices of 2 views
• Projective invariance and canonical cameras• Since the relationships l’ = Fx and• x’ F x = 0 are projective relationships• which
![Page 40: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/40.jpg)
Projective invariance and canonical cameras
• The camera matrix relates 3-space measurements to image measurements and so depends on both the image coordinate frame and the choice of world coordinate frame.
• F is unchanged by a projective transformation of 3-space.
![Page 41: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/41.jpg)
Projective invariance and canonical cameras 2
![Page 42: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/42.jpg)
Canonical form camera matrices
![Page 43: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/43.jpg)
Projective ambiguity of cameras given F
![Page 44: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/44.jpg)
Projective ambiguity of cameras given F2
![Page 45: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/45.jpg)
Projective ambiguity of cameras given F3
![Page 46: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/46.jpg)
Canonical cameras given F
![Page 47: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/47.jpg)
Canonical cameras given F 2
![Page 48: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/48.jpg)
Canonical cameras given F 3
![Page 49: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/49.jpg)
Canonical cameras given F 4
![Page 50: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/50.jpg)
The Essential Matrix
![Page 51: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/51.jpg)
Normalized Coordinates
![Page 52: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/52.jpg)
Normalized coordinates 2
![Page 53: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/53.jpg)
Normalized coordinates 3
![Page 54: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/54.jpg)
Properties of the Essential Matrix
![Page 55: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/55.jpg)
Result 8.17 on Essential matrix
![Page 56: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/56.jpg)
Result 8.17 on Essential matrix 2
![Page 57: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/57.jpg)
Extraction of cameras from the Essential Matrix
![Page 58: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/58.jpg)
![Page 59: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/59.jpg)
Determine the t part of the camera matrix P’
![Page 60: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/60.jpg)
Result 8.19
![Page 61: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/61.jpg)
Geometrical interpretation of the four solutions
![Page 62: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/62.jpg)
Geometrical interpretation of the four solutions 2
![Page 63: Epipolar Geometry and the Fundamental Matrix F The Epipolar Geometry is the intrinsic projective geometry between 2 views and the Fundamental Matrix encapsulates.](https://reader033.fdocuments.net/reader033/viewer/2022061618/56649d4b5503460f94a286e4/html5/thumbnails/63.jpg)
The 4 possible solutions for calibrated reconstruction from E