Download presentation
Presentation is loading. Please wait.
1
Omnidirectional epipolar geometry
Kostas Daniilidis and Chris Geyer University of Pennsylvania
2
Inside-out immersive…
Geyer, Daniilidis ICRA 2003
3
And recently… Geyer, Daniilidis ICRA 2003
4
Omnidirectional input
Porta del Sol, Tiwanaku Bolivia Geyer, Daniilidis ICRA 2003
5
Central Catadioptric Projection
is a double projection:First on the mirror, then on the image plane.
6
Corollary: Conventional cameras are a just a singularity.
Unifying Theorem: All central catadioptric projections are equivalent to double projection through the sphere. Corollary: Conventional cameras are a just a singularity.
7
Equivalence with the sphere
Image of object obtained on image plane identical to catadioptric projection
8
Two facts 1. Parabolic projection = central projection to the sphere then stereo-graphic projection to a plane 2. Perspective projection = central projection to the sphere followed by central projection to a plane from the same center ! Our model covers all conventional perspective cameras!! Geyer, Daniilidis ICRA 2003
9
Inside-Out-Inside: Motion estimation
Geyer, Daniilidis ICRA 2003
10
The projection of a line in space is a conic section and in parabolic mirrors it is a circle.
Geyer, Daniilidis ICRA 2003
11
A new representation of image features
While the projective plane captures both points and lines, we do not have a space suitable for points and circles. We need a CIRCLE SPACE! Geyer, Daniilidis ICRA 2003
12
Lift a circle line projection in parabolic omnicameras
Geyer, Daniilidis ICRA 2003
13
Take inverse stereographic image
Geyer, Daniilidis ICRA 2003
14
Construct cone tangent to locus P is the circle representation
Geyer, Daniilidis ICRA 2003
15
By varying the radius we model points, circles, and imaginary circles!
Geyer, Daniilidis ICRA 2003
16
Not every circle is a line projection (it has to be projection of a great circle). All these feasible lines lie on a plane circle space. Geyer, Daniilidis ICRA 2003
17
Image of the absolute conic
calibrating conic Geyer, Daniilidis ICRA 2003
18
Transformations of circle space
Motivation: In the perspective case the group of transformations is the set of collineations, i.e. non-singular matrices in PGL(3) Goal: find the natural transformation group of circle space. Geyer, Daniilidis ICRA 2003
19
A translation in the plane….
If the sphere has projective quadratic form Then for A to preserve the sphere we must have (Note similarity with ) Geyer, Daniilidis ICRA 2003
20
The Lorentz group What is the general set of 4×4 matrices satisfying
Actually since Q is projective we only need Lorentz Group O(3,1) It is a six dimensional Lie group Geyer, Daniilidis ICRA 2003
21
The Lorentz group rotation about x-axis Geyer, Daniilidis ICRA 2003
22
The Lorentz group rotation about y-axis Geyer, Daniilidis ICRA 2003
23
The Lorentz group rotation about z-axis Geyer, Daniilidis ICRA 2003
24
The Lorentz group x translation Geyer, Daniilidis ICRA 2003
25
The Lorentz group y translation Geyer, Daniilidis ICRA 2003
26
The Lorentz group scale Geyer, Daniilidis ICRA 2003
27
Linear transformation from uncalibrated pixels to calibrated rays.
Such a linear transformation exists and its kernel contains the parameters of this mapping. Geyer, Daniilidis ICRA 2003
28
Scene reconstruction and ego-motion using omnidirectional cameras:
Geyer, Daniilidis ICRA 2003
29
Two view perspective: the essential matrix
Recall that two images p1, p2 of the same space point X satisfy the bilinear constraint where E is a 3×3 rank 2 matrix independent of X, (Tsai-Huang and Longuet-Higgins) Geyer, Daniilidis ICRA 2003
30
Assume p1 and p2 are the catadioptric projections of X
Two views Assume p1 and p2 are the catadioptric projections of X Geyer, Daniilidis ICRA 2003
31
There exists an essential matrix E such that ________
Two views There exists an essential matrix E such that ________ Geyer, Daniilidis ICRA 2003
32
However there exist Lorentz group elements K1 & K2 such that
Two views However there exist Lorentz group elements K1 & K2 such that Geyer, Daniilidis ICRA 2003
33
Catadioptric fundamental matrix
i.e. the lifted image points satisfy a bilinear epipolar constraint!!! F is the 4×4 catadioptric fundamental matrix The kernel of F is the kernel of K. Geyer, Daniilidis ICRA 2003
34
Reconstruction algorithm much simpler than in perspective !
Recover camera parameters with kernel computation and intersection 2. Recover rotation and translation 3. Reconstruct environment or produce novel views. Geyer, Daniilidis ICRA 2003
35
Epipolar circles Geyer, Daniilidis ICRA 2003
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.