Fischler and Bolles 1981“Random Sample Consensus” Communications of the ACM, Vol. 24, Number 6, June, 1981
• Cartography application• Interpretation of sensed data• Pre-defined models and Landmarks• Noisy measurements• Averaging/smoothing does not always work• Inaccurate feature extraction• Gross errors vs. Random Noise
camera
world
landmarks
image
position and orientation of camera in world frame
Location Determination Problem
Location Determination Problem
• Variations:– Pose estimation– Inverse perspective– Camera Calibration– Location Determination– Triangulation
Assumptions
• Intrinsic camera parameters are known.
• Location of landmarks in world frame are known.
• Correspondences between landmarks and their images are known.
• Single camera view.
• Passive sensing.
How Many Points are Enough?
• 1 Point: infinitely many solutions.• 2 Points: infinitely many solutions, but bounded.• 3 Points:
– (no 3 colinear) finitely many solutions (up to 4).
• 4 Points:– non coplanar (no 3 colinear): finitely many.– coplanar (no 3 colinear): unique solution!
• 5 Points: can be ambiguous.• 6 Points: unique solutions (“general view”).
Inscribed Angles are Equalhttp://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html
CPCP
CP
A B
C
A
B
CP
LC
LB
LA
BCAB
CA
s1
s3s2
ABBABA LLLLs cos222
1
A’B’
C’
Image Plane
BCCBCB LLLLs cos222
2
ACCACA LLLLs cos222
3
Bezout’s Theorem
• Number of solutions limited by the product of the degrees of the equations: 2x2x2 = 8.
• But, since each term in the equations is of degree 2, each solution L1, L2, L3 generates another solution by taking the negative values -L1, -L2, -L3.
• Therefore, there can be at most 4 physically realizable solutions.
Algebraic Approachreduce to 4th order equation
(Fischler and Bolles, 1981)
044
33
2210 xGxGxGxGG
http://planetmath.org/encyclopedia/QuarticFormula.html
ABBABA LLLLs cos222
1
BCCBCB LLLLs cos222
2
ACCACA LLLLs cos222
3
Iterative Projections
http://faculty.csuci.edu/william.wolfe/csuci/articles/TNN_Perspective_View_3_pts.pdf
The Orthocenter of a Triangle
http://www.mathopenref.com/triangleorthocenter.html
CP
The Danger Cylinder
Why is the Danger Cylinder Dangerous in the P3P Problem? C. Zhang, Z. Hu, Acta Automatiica Sinica, Vol. 32, No. 4, July, 2006.
“A General Sufficient Condition of Four Positive Solutions of the P3p Problem”
C. Zhang, Z. Hu, 2005
“Complete Solution Classification for the Perspective-Three-Point Problem”
X. Gao, X. Hou, J. Tang, H. Cheng
IEEE Trans PAMI Vol. 25, NO. 8, August 2003
4 Coplanar Points(no 3 colinear)
“Passive Ranging to Known Planar Point Sets”
Y. Hung, P. Yeh, D. Harwood
IEEE Int’l Conf Robotics and Automation, 1985.
P0P1
P3P2
L
W
Object
Camera
CP
Q0
Q1
Q3
Q2
P0_Obj = <0,0,0>
P1_Obj = <L,0,0>
P2_Obj = <0,W,0>
P3_Obj = <L,W,0>
P0_Cam = k0*Q0_Cam
P1_Cam = k1*Q1_Cam
P2_Cam = k2*Q2_Cam
P3_Cam = k3*Q3_Cam
P0P1
P3P2
L
W
ObjectCamera
CP
Q0
Q1
Q3
Q2 k0*Q0_Cam = P0_Cam
(P1_Cam - P0_Cam) + (P2_Cam - P0_Cam) = (P3_Cam - P0_Cam)
P2_CamObject
Camera
CP
P3_Cam
P0_Cam
P1_Cam
(k1*Q1_Cam - k0*Q0_Cam) + (k2*Q2_Cam - k0*Q_Cam) =
(k3*Q3_Cam - k0*Q0_Cam)
(P1_Cam - P0_Cam) + (P2_Cam - P0_Cam) = (P3_Cam - P0_Cam)
P0_Cam = k0*Q0_Cam
P1_Cam = k1*Q1_Cam
P2_Cam = k2*Q2_Cam
P3_Cam = k3*Q3_Cam
(k1*Q1_Cam) + (k2*Q2_Cam - k0*Q_Cam) = (k3*Q3_Cam)
let ki’ = ki/k3
(k1’*Q1_Cam) + (k2’*Q2_Cam - k0’*Q0_Cam) = Q3_Cam
-k0’*Q0_Cam + k1’*Q1_Cam + k2’*Q2_Cam = Q3_Cam
Three linear equations in the 3 unknowns: k0’, k1’, k2’
| P3_Obj - P0_Obj | = |P3_Cam - P0_Cam| = | k3*Q3_Cam - k0*Q0_Cam |
k3 = | P3_Obj| / | k0’ * Q0_Cam - Q3_Cam |
P0P1
P3P2
L
W
ObjectCamera
CP
Unit_x = (P1_Cam - P0_Cam)/ | P1_Cam - P0_Cam |
Unit_y = (P2_Cam - P0_Cam) / |P2_Cam - P0_Cam|
Unit_z = Unit_x X Unit_y
Orientation
Summary
• Reviewed location determination problems with 1, 2, 3, 4 points.
• Algebraic vs. Geometric vs. Iterative methods.
• 3 points can have up to 4 solutions.• Iterative solution method for 3 points.• 4 coplanar points has unique solution.• Complete solution for 4 rectangular points.• Many unsolved geometric issues.
References• Random Sample Consensus, Martin Fischler and Robert Bolles,
Communications of the ACM, Vol. 24, Number 6, June, 1981• Passive Ranging to Known Planar Point Sets, Y. Hung, P. Yeh, D. Harwood,
IEEE Int’l Conf Robotics and Automation, 1985.• The Perspective View of 3 Points, W. Wolfe, D. Mathis, C. Sklair, M. Magee.
IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 13, No. 1, January 1991.
• Review and Analysis of Solutions of the Three Point Perspective Pose Estimation Problem. R. Haralick, C. Lee, K. Ottenberg, M. Nolle. Int’l Journal of Computer Vision, 13, 3, 331-356, 1994.
• Complete Solution Classification for the Perspective-Three-Point Problem, X. Gao, X. Hou, J. Tang, H. Cheng, IEEE Trans PAMI Vol. 25, NO. 8, August 2003.
• A General Sufficient Condition of Four Positive Solutions of the P3P Problem, C. Zhang, Z. Hu, J. Comput. Sci. & Technol., Vol. 20, N0. 6, pp. 836-842, 2005.
• Why is the Danger Cylinder Dangerous in the P3P Problem? C. Zhang, Z. Hu, Acta Automatiica Sinica, Vol. 32, No. 4, July, 2006.
Top Related