U

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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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We can eliminate all rotations within the ball of
radius 0.3 about trial.
Rotation Space
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Isometry of Rotations and Quaternions
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.
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Angle-axis representation of Rotations
Flatten out the meridians (longitude lines)
Azimuthal Equidistant Projection
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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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Linear Programming, not SOCP
X
V
V
t
C
C
Point correspondence in two views
Coplanarity constraint with uncertainty
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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
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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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Hockey Rink Data
Reconstructed points and path
Original and dual problems
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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
ESOCP Motion of multi-camera rigs using SOCP
method BBLP Motion of multi-camera rigs using
L inf method
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Multi-Camera Systems (Non-overlapping) L inf
Method
ESOCP Motion of multi-camera rigs using SOCP
method BBLP 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.