Projective cameras
• Motivation• Elements of Projective Geometry• Projective structure from motion
Planches :– http://www.di.ens.fr/~ponce/geomvis/lect3.ppt
– http://www.di.ens.fr/~ponce/geomvis/lect3.pdf
Weak-Perspective Projection Model
r(p and P are in homogeneous coordinates)
p = A P + b (neither p nor P is in hom. coordinates)
p = M P (P is in homogeneous coordinates)
Affine Structure from Motion
Reprinted with permission from “Affine Structure from Motion,” by J.J. (Koenderink and A.J.Van Doorn, Journal of the Optical Society of America A,8:377-385 (1990). 1990 Optical Society of America.
Given m pictures of n points, can we recover• the three-dimensional configuration of these points?• the camera configurations?
(structure)(motion)
The Projective Structure-from-Motion Problem
Given m perspective images of n fixed points P we can write
Problem: estimate the m 3x4 matrices M andthe n positions P from the mn correspondences p .
i
j ij
2mn equations in 11m+3n unknowns
Overconstrained problem, that can be solvedusing (non-linear) least squares!
j
The Projective Ambiguity of Projective SFM
If M and P are solutions, i j
So are M’ and P’ wherei j
and Q is an arbitrary non-singular 4x4 matrix.
When the intrinsic and extrinsic parameters are unknown
Q is a projective transformation.
When do m+1 points define a p-dimensional subspace Y of ann-dimensional projective space X equipped with some coordinateframe?
Writing that all minors of size (p+1)x(p+1) of D are equal tozero gives the equations of Y.
Rank ( D ) = p+1, where
mnnn
m
m
xxx
xxx
xxx
D
...
............
...
...
10
11101
01000
Hyperplanes and duality
This can be rewritten as u0x0+u1x1+…+unxn = 0, orT P = 0, where = (u0,u1,…,un)T.
Consider n+1 points P0, … , Pn-1, P in a projective space X ofdimension n. They liein the same hyperplanewhen Det(D)=0.
nnn xxx
xxx
xxx
D
...
............
...
...
10
11101
01000
Hyperplanes form a dual projective space X* of X, and anytheorem that holds for points in X holds for hyperplanes in X*.
What is the dual of a straight line?
Cross-Ratios
Collinear points
Pencil of coplanar lines Pencil of planes
{A,B;C,D}=sin(+)sin(+)
sin(++)sin
Cross-Ratios and Projective Coordinates
Along a line equipped with the basis
In a plane equipped with the basis
In 3-space equipped with the basis
*
Projective Transformations
Bijective linear map:
Projective transformation:( = homography )
Projective transformations map projective subspaces ontoprojective subspaces and preserve projective coordinates.
Projective transformations map lines onto lines andpreserve cross-ratios.
Projective Shape
Two point sets S and S’ in some projective space X are projectively equivalent when there exists a projective transformation : X X such that S’ = ( S ).
Projective structure from motion = projective shape recovery.
= recovery of the corresponding motion equivalence classes.
Geometric SceneReconstruction
Idea: use (A,B,C,D,F) as a projective basis and reconstruct O’ and O’’, assuming that the epipolesare known.
A
B
CD F
GH
IJ
K
E
O’
O’’
Geometric SceneReconstruction II
Idea: use (A,O”,O’,B,C)as a projective basis, assuming again that theepipoles are known.
Epipolar Constraint
• Potential matches for p have to lie on the corresponding epipolar line l’.
• Potential matches for p’ have to lie on the corresponding epipolar line l.
Properties of the Essential Matrix
• E p’ is the epipolar line associated with p’.
• E p is the epipolar line associated with p.
• E e’=0 and E e=0.
• E is singular.
• E has two equal non-zero singular values (Huang and Faugeras, 1989).
T
T
Properties of the Fundamental Matrix
• F p’ is the epipolar line associated with p’.
• F p is the epipolar line associated with p.
• F e’=0 and F e=0.
• F is singular.
T
T
Non-Linear Least-Squares Approach (Luong et al., 1993)
Minimize
with respect to the coefficients of F , using an appropriate rank-2 parameterization.
The Normalized Eight-Point Algorithm (Hartley, 1995)
• Center the image data at the origin, and scale it so themean squared distance between the origin and the data points is 2 pixels: q = T p , q’ = T’ p’.
• Use the eight-point algorithm to compute F from thepoints q and q’ .
• Enforce the rank-2 constraint.
• Output T F T’.T
i i i i
i i
Wit
hout
nor
mal
izat
ion
Wit
h no
rmal
izat
ion
Mean errors:10.0pixel9.1pixel
Mean errors:1.0pixel0.9pixel
Trinocular Epipolar Constraints: Transfer
Given p and p , p can be computed
as the solution of linear equations.
1 2 3
Trifocal Constraints: 3 Points
Pick any two lines l and l through p and p .
Do it again.2 3 2 3
T( p , p , p )=01 2 3
Properties of the Trifocal Tensor
Estimating the Trifocal Tensor
• Ignore the non-linear constraints and use linear least-squaresa posteriori.
• Impose the constraints a posteriori.
• For any matching epipolar lines, l G l = 0.
• The matrices G are singular.
• They satisfy 8 independent constraints in theuncalibrated case (Faugeras and Mourrain, 1995).
2 1 3T i
1i
For any matching epipolar lines, l G l = 0. 2 1 3T i
The backprojections of the two lines do not define a line!