Branch and Bound in Rotation Space (ICCV 2007)
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Transcript of Branch and Bound in Rotation Space (ICCV 2007)
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U
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Direction (unit) vectors from cameras (blue) to points (black) are given : Find the positions of the cameras and points.
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Branch and Bound in Rotation Space (ICCV 2007)
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Essential Matrix Estimation
Encodes the relative displacement between two cameras.
•Rotation
•Translation
• Needs at least 5 points
X
x1 x2
(R, t)
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2-view SfM with known rotations
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Best current error
We can eliminate all rotations within the ball of radius 0.3 about trial.
Rotation Space
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theta
v
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Angle between two quaternions is half the angle between the corresponding rotations, defined by
All rotations within a delta-neighbourhood of a reference rotation form a circle on the quaternion sphere.
Isometry of Rotations and Quaternions
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Flatten out the meridians (longitude lines)
Azimuthal Equidistant Projection
Angle-axis representation of Rotations
Rotations are represented by a ball of radius pi in 3-Dimensional space.
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Subdividing and testing rotation space
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Numbers of cubes left at each iteration (Log-10 scale)
Remaining Volume at each iteration (Log-10 scale in cubic radians).
Performance
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V’
t
V
C’
C
X
Point correspondence in two views
Coplanarity constraint with uncertainty
Linear Programming, not SOCP
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Multi-Camera Systems (Non-overlapping) – L inf Method
Translation direction lies in a polyherdron (Green) from point correspondences
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Multi-Camera Systems (Non-overlapping) – L inf Method
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Each point correspondence gives two LP constraints on the direction t (epipolar direction).
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Essential Matrix Calculated from 3 points (above) or 4 points (below)
Possible rotations.
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Timing Examples
29 correspondences : 2.9 seconds
794 correspondences : 75 seconds.
6572 correspondeces : 3m 30 seconds
Timing (in milliseconds) for E-matrix computation – 360 degree camera.
0
50
100
150
200
250
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350
0 500 1000 1500 2000
360 degree camera
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Further Application – 1D camera (e.g. robot moving in a plane)
Joint work with Kalle Astrom, Fredrik Kahl, Carl Olsson and Olof Enquist
Complete structure and motion problem for “planar motion”
Optimal solution in L-infinity norm.
Same idea of searching in rotation space.
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Original and dual problems
Reconstructed points and path
Hockey Rink Data
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Method works also for rigidly placed multi-camera systems.
•Can be considered as a single “generalized” camera
•One rotation, one translation to be estimated.
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Robust 6DOF motion estimation from Non-overlapping images, Multi-camera systems
4 images from the right 4 images from the left
(Images: Courtesy of UNC-Chapel Hill)
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Generalized Cameras (Non-overlapping)
Ladybug2 camera(The locally-central case)
5 cameras (horizontal)1 camera (top)
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Generalized Cameras (Non-overlapping)
Experiment setup
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Generalized Cameras (Non-overlapping)
An Infinity-like path which the Ladybug2 camera follows(total 108 frames)
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Robust 6DOF motion estimation from Non-overlapping images, Multi-camera systems
Critical configuration
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Generalized Cameras (Non-overlapping) – Linear Method
Estimated path (Linear Method) vs. Ground truth
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Generalized Cameras (Non-overlapping) – Linear Method
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Generalized Cameras (Non-overlapping) – Linear Method
Demo video : 16 sec (Click to play)
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Multi-Camera Systems (Non-overlapping) – SOCP Method
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Multi-Camera Systems (Non-overlapping) – L inf Method
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Multi-Camera Systems (Non-overlapping) – L inf Method
E+SOCP : Motion of multi-camera rigs using SOCP methodBB+LP : Motion of multi-camera rigs using L inf method
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Multi-Camera Systems (Non-overlapping) – L inf Method
E+SOCP : Motion of multi-camera rigs using SOCP methodBB+LP : Motion of multi-camera rigs using L inf method
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Multi-Camera Systems (Non-overlapping) – L inf Method
Estimated path (L inf Method) vs. Ground truth
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Multi-Camera Systems (Non-overlapping) – L inf Method
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Multi-Camera Systems (Non-overlapping) – L inf Method
Demo video : 16 sec (Click to play)
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Obtaining an initial region
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277,000 3D points triangulated.
All but 281 proved by simple test to be minima.
All except 153 proved to be global minima by more complex test.
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Hardy: Pure mathematics is on the whole distinctly more useful than applied. For what is useful above all is technique, and mathematical technique is taught mainly through pure mathematics.