The Perspective View of
3 Points
bill wolfe
CSUCI
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
Example (Fischler and Bolles)
Least Squares
RANDSAC
Poison Point
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
Applications
•
Computer Vision
•
Robotics
•
Cartography
•
Computer Graphics
•
Photogrammetry
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.
Strategy
camera
world
landmarks
measured/
calculated
known
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”).
1 Point
2 Points
CP
A
B
Inscribed Angles are Equal
http://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html
q
q
q
CP
CP
CP
A
B
3 Points
s1
s2
s3
LA
A
B
C
LB
LC
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 Approach
reduce to 4
th
order equation
(Fischler and Bolles, 1981)
http://planetmath.org/encyclopedia/QuarticFormula.html
Iterative Approach
s1
s2
slide
s3
Iterative Projections
http://faculty.csuci.edu/william.wolfe/csuci/articles/TNN_Perspective_View_3_pts.pdf
Geometric Approach
CP
The Orthocenter of a Triangle
http://www.mathopenref.com/triangleorthocenter.html
CP
4 solutions
when CP is
directly over
the orthocenter
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.
P0
P1
P3
P2
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
P0
P1
P3
P2
L
W
Object
Camera
CP
Q0
Q1
Q3
Q2
k0*Q0_Cam = P0_Cam
(P1_Cam

P0_Cam) + (P2_Cam

P0_Cam) = (P3_Cam

P0_Cam)
P2_Cam
Object
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 
P0
P1
P3
P2
L
W
Object
Camera
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
Homgeneous Transformation
Unit_x Unit_y Unit_x P0_Cam
0
0
0
1
H =
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.
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