Jiecai He Jake One thing to take away from this presentation two parallel lines meet at a point at infinity In Projective Geometry The grand problem The left image is with perspective distortion the lines of the windows clearly converge to a finite point How to fix it ID: 269503
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Slide1
Project Geometry
Jiecai He (Jake)
One thing to take away from this presentation:
two parallel lines meet at a point at infinity In Projective GeometrySlide2
The grand problem
The left image is with perspective distortion – the lines of the windows clearly converge to a finite point. How to fix it ?
Only Projective Geometry can help us!!
But wait, what is a perspective distortion??Slide3
perspective distortion
The geometric distortion(including perspective distortion) arises when a plane is imaged by a perspective camera??
How to model a camera in projective geometry? central projectionA central projection of a plane (or section of a plane) is related to the original plane via a projective transformation.
What is a projective transformation? An invertible map from to itself that maps lines to lines.
What is ? The projective planeSlide4
A model for the projective plane
exactly one line through two points
exaclty one point at intersection of two linesSlide5
Removing projective distortion
select four points in a plane with know coordinates
(linear in
h
ij
)
(2 constraints/point, 8DOF
4 points needed
)
Remark: no calibration at all necessary,
better ways to compute (see later)Slide6
Homogeneous coordinates
Homogeneous representation of lines
equivalence class of vectors, any vector is representative
Set of all equivalence classes in
R
3
(0,0,0)
T
forms
P
2
Homogeneous representation of points
on
if and only if
The point
x
lies on the line
l
if and only if
x
T
l
=
l
T
x
=
0
Homogeneous
coordinates
Inhomogeneous
coordinates
but only 2DOFSlide7
Points from lines and vice-versa
Intersections of lines
The intersection of two lines
and is
Line joining two points
The line through two points
and is
ExampleSlide8
Ideal points and the line at infinity
Intersections of parallel lines
Example
Ideal points
Line at infinity
tangent vector
normal direction
Note that in
P
2
there is no distinction
between ideal points and othersSlide9
Duality
Duality principle:
To any theorem of 2-dimensional projective geometry there corresponds a dual theorem, which may be derived by interchanging the role of points and lines in the original theoremSlide10
Projective transformations
A
projectivity is an invertible mapping h from P
2
to itself such that three points
x
1
,x2,x3
lie on the same line if and only if
h
(x
1
),
h
(x
2
),
h
(x3)
do.
Definition:
A mapping
h:P2P
2 is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2 reprented by a vector x it is true that
h(x)=HxTheorem:
Definition:
Projective transformation
or
8DOF
projectivity=collineation=projective transformation=homography