Various camera configurations Single point of fixation where optical axes intersect Optical axes...
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Transcript of Various camera configurations Single point of fixation where optical axes intersect Optical axes...
![Page 1: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/1.jpg)
Various camera configurations
• Single point of fixation where optical axes intersect
• Optical axes parallel (fixation at infinity)
• General case
![Page 2: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/2.jpg)
Epipolar geometry for cameras in general position
![Page 3: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/3.jpg)
Epipolar geometry for cameras in general position
![Page 4: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/4.jpg)
Epipolar geometry for cameras in general position
![Page 5: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/5.jpg)
Epipolar geometry for cameras in general position
![Page 6: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/6.jpg)
Epipolar geometry for cameras in general position
![Page 7: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/7.jpg)
Epipolar geometry for cameras in general position
![Page 8: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/8.jpg)
Epipolar geometry for cameras in general position
![Page 9: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/9.jpg)
Epipolar geometry for cameras in general position
![Page 10: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/10.jpg)
Applications
• Reconstructing a three dimensional object given two or more photos. The relative orientation of the cameras is unknown, such as when using images downloaded from the internet, taken by different people at different times.
• Some examples from Univ. Washington/Microsoft Research:– phototour.cs.washington.edu– photosynth.net– grail.cs.washington.edu/rome/
![Page 11: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/11.jpg)
The approach
•
![Page 12: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/12.jpg)
But first, let us review projective transformations
![Page 13: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/13.jpg)
Projective Transformations
• Under perspective projection, parallel lines can map to lines that intersect. Therefore, this cannot be modeled by an affine transform!
• Projective transformations are a more general family which includes affine transforms and perspective projections.
• Projective transformations are linear transformations using homogeneous coordinates
![Page 14: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/14.jpg)
Homogeneous coordinates
• Instead of using n coordinates for n-dimensional space, we use n+1 coordinates.
![Page 15: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/15.jpg)
Key rule
•
![Page 16: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/16.jpg)
Picking a canonical representative
![Page 17: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/17.jpg)
The projective line
• Any finite point x can be represented as
• Any infinite point can be expressed as
![Page 18: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/18.jpg)
The projective plane
• Any finite point can be represented as
• Any infinite point can be represented as
![Page 19: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/19.jpg)
Lines in homogeneous coordinates
![Page 20: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/20.jpg)
Incidence of points on lines
![Page 21: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/21.jpg)
Incidence of points on lines
![Page 22: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/22.jpg)
Incidence of points on lines
![Page 23: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/23.jpg)
Line incident on two points
![Page 24: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/24.jpg)
Representing affine transformations
![Page 25: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/25.jpg)
Perspective Projection
![Page 26: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/26.jpg)
Projective transformations
![Page 27: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/27.jpg)
The Essential Matrix Constraint
![Page 28: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/28.jpg)
Proof is based on the co-planarity of three rays
![Page 29: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/29.jpg)
Coplanarity of 3-vectors implies triple product is zero
![Page 30: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/30.jpg)
Longuet-Higgins 8 point algorithm
•
![Page 31: Various camera configurations Single point of fixation where optical axes intersect Optical axes parallel (fixation at infinity) General case.](https://reader035.fdocuments.in/reader035/viewer/2022070413/5697bf931a28abf838c8f9b2/html5/thumbnails/31.jpg)
Summary
• The basic module of recovering 3D structure from 2 views with relative orientation (R, t) of cameras unknown can be implemented using the Longuet-Higgins 8 point algorithm.
• The outer loop combines information from all the cameras in a global coordinate system using bundle adjustment. The error that is minimized is the re-projection error. The big idea is that given the guessed 3d positions of a point, one can predict image plane 2d positions in any camera where it is visible. We wish to minimize the squared error between this predicted position and the actual position, summed over all cameras and over all points.
• Lots of engineering has gone into making these approaches work. Read Szeliski’s book, Chapter 7, for more.