image rectification (stereo) (new book: 7.2.1, old book: 11.1)gerig/cs6320-s2015/materials... ·...
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Image Rectification (Stereo)(New book: 7.2.1,old book: 11.1)
Guido GerigCS 6320 Spring 2015
Credits: Prof. Mubarak Shah, Course notes modified from: http://www.cs.ucf.edu/courses/cap6411/cap5415/, Lecture 25
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Example: converging cameras
courtesy of Andrew Zisserman
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Epipolar Lines in Converging Cameras
Epipolar lines all intersectat epipoles.
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Stereo image rectification
In practice, it is convenient if image scanlines are the epipolar lines.
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Image Rectification
http://en.wikipedia.org/wiki/Image_rectification
If the two cameras are aligned to be coplanar, the search is simplified to one dimension - a horizontal line parallel to the baseline between the cameras
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After translation, P and P’ project onto separate locations.
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After rotation, P and P’ still project onto the same location.
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Stereo image rectification
• Image Reprojection– reproject image planes
onto common plane parallel to line between optical centers
– a homography (3x3 transform)applied to both input images
– pixel motion is horizontal after this transformation
– C. Loop and Z. Zhang. Computing Rectifying Homographies for Stereo Vision. IEEE Conf. Computer Vision and Pattern Recognition, 1999.
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Stereo image rectification: example
Source: Alyosha Efros
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Stereo image rectification: example
Source: Alyosha Efros
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Stereo image rectification: example
Source: Alyosha Efros
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Stereo image rectification: example
Source: Alyosha Efros
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Stereo image rectification: example
Source: Alyosha Efros
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Image Rectification
O O
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Image Rectification
O Oe e
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Image Rectification
P
O Oe e
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Image Rectification
P
O O
p pe e
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Image Rectification
P
O O
l lp p
e e
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Image Rectification
P
O O
l lp p
e e
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Image Rectification
pp
P
O O
l lp p
e e
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ll
Image Rectification
pp
P
O O
l lp p
e e
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ll
Image Rectification
pp
P
O O
l lp p
e e
• Common Image Plane• Parallel Epipolar Lines• Search Correspondences
on scan line• Epipoles → ∞
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Image Rectification
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All epipolar lines are parallel in the rectified image plane.
Image Rectification
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Image Rectification
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Algorithm Rectification
Following Trucco & Verri book pp. 159• known T and R between cameras• Rotate left camera so that epipole el
goes to infinity along horizontal axis• Apply same rotation to right camera to
recover geometry• Rotate right camera by R-1
• Adjust scale
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From: Trucco & Verri, Introductory Techniques for 3-D Computer Vision, pp. 157-161
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Image pair rectification
Goal: Simplify stereo matching by “warping” the images
Apply projective transformation so that epipolar linescorrespond to horizontal scanlines
e
e
map epipole e to (1,0,0)
try to minimize image distortion
problem when epipole in (or close to) the image
He001
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
0Epp
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
0Epp
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
0Epp
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
0Epp
Essential matrix example: parallel cameras
Grauman
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]R[TET
IR
x
]0,0,[ d0 0 00 0 d0 –d 0
0Epp
Essential matrix example: parallel cameras
For the parallel cameras, image of any point must lie on same horizontal line in each image plane.
Grauman
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Example I: compute the fundamental matrix for a parallel camera stereo rig
• reduces to y = y/ , i.e. raster correspondence (horizontal scan-lines)
f
f
X YZ
0 00 00 0 1
R=I t= 00
0 00 0
1
0 0 00 00 0
000 0 1
=0 0 00 00 0
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f
f
X YZ
Geometric interpretation ?
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Image pair rectification
Goal: Simplify stereo matching by “warping” the images
Apply projective transformation so that epipolar linescorrespond to horizontal scanlines
e
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Image pair rectification
Goal: Simplify stereo matching by “warping” the images
Apply projective transformation so that epipolar linescorrespond to horizontal scanlines
e
e
map epipole e to (1,0,0)He001
![Page 45: Image Rectification (Stereo) (New book: 7.2.1, old book: 11.1)gerig/CS6320-S2015/Materials... · Image Rectification (Stereo) (New book: 7.2.1, old book: 11.1) Guido Gerig CS 6320](https://reader035.vdocuments.site/reader035/viewer/2022063022/5fea92c5dce2d50d53583b37/html5/thumbnails/45.jpg)
Image pair rectification
Goal: Simplify stereo matching by “warping” the images
Apply projective transformation so that epipolar linescorrespond to horizontal scanlines
e
e
map epipole e to (1,0,0)
try to minimize image distortionHe
001
![Page 46: Image Rectification (Stereo) (New book: 7.2.1, old book: 11.1)gerig/CS6320-S2015/Materials... · Image Rectification (Stereo) (New book: 7.2.1, old book: 11.1) Guido Gerig CS 6320](https://reader035.vdocuments.site/reader035/viewer/2022063022/5fea92c5dce2d50d53583b37/html5/thumbnails/46.jpg)
Transformations
46
affine
x’, y’: homographies
Homography: Projective Spaces
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Planar rectificationImage Transformations: Findhomographies H’ and H so that after transformations, Fnew
becomes F of parallel cameras:
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Planar rectificationImage Transformations: Findhomographies H’ and H so that after transformations, Fn
becomes F of parallel cameras:
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Planar rectificationImage Transformations: Findhomographies H’ and H so that after transformations, Fn
becomes F of parallel cameras:
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Planar rectification
Bring two views to standard stereo setup(moves epipole to )(not possible when in/close to image)
Image Transformations: Findhomographies H’ and H so that after transformations, Fn
becomes F of parallel cameras:
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More elegant Solution
• Idea: Mapping epipole to infinity → [1,0,0]T
• Factorization of matrix F=SM, where S is skew symmetric and M representing the required homography (projective transformation).
• Use SVD:
http://www.robots.ox.ac.uk/~vgg/hzbook/hzbook2/clarification_rectification.pdf
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Stereo matching with general camera configuration
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Image pair rectification
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Other Material /Code
• Epipolar Geometry, Rectification: http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/FUSIELLO2/rectif_cvol.html
• Fusiello, Trucco & Verri:Tutorial, Matlabcode etc:http://profs.sci.univr.it/~fusiello/demo/rect/
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Run ExampleDemo for stereo reconstruction (out of date): http://mitpress.mit.edu/e-journals/Videre/001/articles/Zhang/CalibEnv/CalibEnv.html
Updated Webpages:http://research.microsoft.com/en-us/um/people/zhang/INRIA/softwares.html
SFM Example:http://research.microsoft.com/en-us/um/people/zhang/INRIA/SFM-Ex/SFM-Ex.html
Software: http://research.microsoft.com/en-us/um/people/zhang/INRIA/softwares.html
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Example: Zhengyou Zhang
Fundamental matrix between the two cameras:
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Points have been extracted using Harris corner detector, point matches via fundamental matrix F and search along epipolar lines.
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Point matches found by a correlation technique
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3D reconstruction represented by a pseudo stereogram
http://research.microsoft.com/en-us/um/people/zhang/INRIA/SFM-Ex/SFM-Ex.html