Projective Geometry. Projection Vanishing lines m and n.

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Projective Geometry

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

Projective Geometry

Projective plane = S2 with antipodal points identified

Ordinary plane is unbound

Projective plane is bound!

Projective Geometry

Pappus’ Theorem

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

Use circular point as a ruler…

Calibration

• 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

cos sin

sin cos

cos sin

sin cos

Invariance

• Mathematician loves invariance !

• Fixed point theorem

• Eigenvector

( )F x x

Cross Ratio

• Projective line

P = (X,1)t

• Consider

1,1 1,21 1

2,1 2,22 2

t tx X

Cross Ratio

x xx x x

X XX 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

The remaining coefficients can be eliminated by using the fourth point

Pinhole Camera

3x4 projection matrix3x3 intrinsic matrix

Extrinsic matrix

Principle point

Skew factor cot

Pinhole Camera

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)

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:

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

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

Projective Geometry. Projection Vanishing lines m and n.

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