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# Recovery of affine and metric properties from images in 2D Projective space - PowerPoint PPT Presentation

Recovery of affine and metric properties from images in 2D Projective space. 2013-03-20 Ko Dae -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)

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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

### 1. Recovery of affine properties

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

### 1. Recovery of affine properties

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

### 1. Recovery of affine properties

Duality

or homogenized

or in matrix form

with

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

### 1. Recovery of affine properties

Conics

Curve described by 2nd-degree equation in the plane

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

### 1. Recovery of affine properties

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

### 1. Recovery of affine properties

Dual conics

A line tangent to the conic C satisfies

Dual conics = line conics = conic envelopes

Theorem:

A mapping h:P2P2is 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

### 1. Recovery of affine properties

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

### 1. Recovery of affine properties

The line at infinity

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

### 1. Recovery of affine properties

Affine properties from images

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

### 1. Recovery of affine properties

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

### 1. Recovery of affine properties

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

### 1. Recovery of affine properties

Distance ratio

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

### 1. Recovery of affine properties

Distance ratio

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

### 2. Recovery of metric properties

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

### 2. Recovery of metric properties

The circular points

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

### 2. Recovery of metric properties

The circular points

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

### 2. Recovery of metric properties

Conic dual to the circular points

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

### 2. Recovery of metric properties

Angles

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

### 2. Recovery of metric properties

Length ratios

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

### 2. Recovery of metric properties

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

### 2. Recovery of metric properties

Length ratios

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

### 2. Recovery of metric properties

Metric from affine

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

### 2. Recovery of metric properties

Metric from projective