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Recovery of affine and metric properties from images in 2D Projective space

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Recovery of affine and metric properties from images in 2D Projective space

2013-03-20

KoDae-Won

Recovery of affine and metric properties from images in 2D Projective space

Affine properties(line at infinity)

- Parallelism
- Parallel length ratios
Metric properties(circular points)

- Angles
- Length ratios
Recover the original shape

Homogeneous coordinates

but only 2DOF

Inhomogeneous coordinates

Recovery of affine and metric properties from images in 2D Projective space

Homogeneous coordinates

equivalence class of vectors, any vector is representative

Set of all equivalence classes in R3(0,0,0)T forms P2

Homogeneous representation of points

on

if and only if

The point x lies on the line l if and only if xTl=lTx=0

Line joining two points

The line through two points and is

Ideal points

Recovery of affine and metric properties from images in 2D Projective space

Points from lines and vice-versa

Intersections of lines

The intersection of two lines and is

Intersections of parallel lines

Line at infinity

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 theorem

Recovery of affine and metric properties from images in 2D Projective space

Duality

or homogenized

or in matrix form

with

Recovery of affine and metric properties from images in 2D Projective space

Conics

Curve described by 2nd-degree equation in the plane

Recovery of affine and metric properties from images in 2D Projective space

Tangent lines to conics

The line l tangent to C at point x on C is given by l=Cx

l

x

C

In general :

Recovery of affine and metric properties from images in 2D Projective space

Dual conics

A line tangent to the conic C satisfies

Dual conics = line conics = conic envelopes

Theorem:

A mapping h:P2P2is a projectivity if and only if there exist a non-singular 3x3 matrix H such that for any point in P2represented by a vector x it is true that h(x)=Hx

Definition: Projective transformation

or

Recovery of affine and metric properties from images in 2D Projective space

Projective transformations

Definition:

A projectivity is an invertible mapping h from P2 to itself such that three points x1,x2,x3lie on the same line if and only if h(x1),h(x2),h(x3) do.

projectivity=collineation=projective transformation=homography

Recovery of affine and metric properties from images in 2D Projective space

The line at infinity

Recovery of affine and metric properties from images in 2D Projective space

Affine properties from images

Recovery of affine and metric properties from images in 2D Projective space

Recovery of affine and metric properties from images in 2D Projective space

Recovery of affine and metric properties from images in 2D Projective space

Distance ratio

Recovery of affine and metric properties from images in 2D Projective space

Distance ratio

Recovery of affine and metric properties from images in 2D Projective space

Recovery of affine and metric properties from images in 2D Projective space

The circular points

Recovery of affine and metric properties from images in 2D Projective space

The circular points

Recovery of affine and metric properties from images in 2D Projective space

Conic dual to the circular points

Recovery of affine and metric properties from images in 2D Projective space

Angles

Recovery of affine and metric properties from images in 2D Projective space

Length ratios

Recovery of affine and metric properties from images in 2D Projective space

Recovery of affine and metric properties from images in 2D Projective space

Length ratios

Recovery of affine and metric properties from images in 2D Projective space

Metric from affine

Recovery of affine and metric properties from images in 2D Projective space

Metric from projective