projective geometry. projection vanishing lines m and n
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Projective Geometry
Projective Geometry
Projective Geometry
Projective Geometry
Projective Geometry
Projection
Projection
Vanishing lines m and n
Projective Plane (Extended Plane)
Projective Plane
How???Ordinary plane
Point Representation
A point in the projective plane is represented as a ray in R3
Projective Geometry
Homogeneous coordinates
0 cbyax 01 x,y,a,b,cT
0,~ ka,b,cka,b,cTT
Homogeneous representation of 2D points and lines
equivalence class of vectors, any vector is representative
Set of all equivalence classes in R3(0,0,0)T forms P2
0,1,,~1,, kyxkyx TT
The point x lies on the line l if and only if
Homogeneous coordinates
Inhomogeneous coordinates Tyx, T321 ,, xxx but only 2DOF
Note that scale is unimportant for incidence relation
0xl T
Projective Geometry
Projective Geometry
Projective plane = S2 with antipodal points identified
Ordinary plane is unbound
Projective plane is bound!
Projective Geometry
Projective Geometry
Pappus’ Theorem
Pappus’ Theorem
Pappus’ Theorem
Conic Section
Conic Section
Conic Section
Conic Section
Conic Section
Conic Section
Conic Section
Conic Section
Form of Conics
Transformation
• Projective: incidence, tangency
• Affine : plane at infinity, parallelism
• Similarity : absolute conics
Circular Point
Circular points
Euclidean Transformation
Any transformation of the projective plane which leaves the circular points fixed is a Euclidean transformation, and
Any Euclidean transformation leaves the circular points fixed.
A Euclidean transformation is of the form:
Euclidean Transformation
Calibration
Calibration
Use circular point as a ruler…
Calibration
Today
• Cross ratio
• More on circular points and absolute conics
• Camera model and Zhang’s calibration
• Another calibration method
Transformation
• Let X and X’ be written in homogeneous coordinates, when X’=PX
• P is a projective transformation when…..
• P is an affine transformation when…..
• P is a similarity transformation when…..
Transformation
Projective
Affine
Similarity
Euclidean
Matrix Representation
x x x
x x x
x x x
0 0 1
x x x
x x x
cos sin
sin cos
0 0 1
a a b
a a c
cos sin
sin cos
0 0 1
b
c
Invariance
• Mathematician loves invariance !
• Fixed point theorem
• Eigenvector
( )F x x
A x x
Cross Ratio
• Projective line
P = (X,1)t
• Consider
1,1 1,21 1
2,1 2,22 2
t tx X
t tx X
TXx
Cross Ratio
11 2
2
, /1
x xx x x
x
x
11 2
2
, /1
X XX X X
X
X
Cross RatioConsider determinants:
Rewritting
So we have
Consider 12 1 2| ( )( ) |d T T P P
Cross Ratio
How do we eliminate |T| and the coefficients
The idea is to use the ratio. Consider
and
The remaining coefficients can be eliminated by using the fourth point
Pinhole Camera
Pinhole Camera
3x4 projection matrix3x3 intrinsic matrix
Extrinsic matrix
Principle point
Skew factor cot
Pinhole Camera
Absolute Conic
Absolute Conic
Absolute ConicImportant: absolute conic is invariant to any rigid transformation
We can write and
That is,
and obtain
Absolute ConicNow consider the image of the absolute conic
It is defined by
Typical Calibration
1. Estimate the camera projection matrix from correspondence between scene points and image points (Zhang p.12)
2. Recover intrinsic and extrinsic parameters
Typical Calibration
P[3][4], B[3][3], b[3]
Calibration with IAC
Can we calibrate without correspondence?
(British Machine Vision)
Calibration with IAC
Calibration with IAC
From Zhang’s, the image of the absolute conic is the conic
Let’s assume that the model plane is on the X-Y plane of the world coordinate system, so we have:
Calibration with IAC
Points on the model plane with t=0 form the line at infinity
It is sufficient to consider model plane in homogeneous coordinates
We know that the circular points I = (1,i,0,0)T and J = (1,-i,0,0)T must satisfy
Let the image of I and J be denoted by
Calibration with IACConsider the circle in the model plane with center (Ox,Oy,1) and radius r.
This circle intersects the line at infinity when
or
Any circle (any center, any radius) intersects line at infinity in the two circular points
The image of the circle should intersect the image of the line at infinity (vanishing line) in the image of the two circular points
Calibration with IAC
1C
Calibration with IAC