homogeneous coordinates en slides

8/10/2019 Homogeneous Coordinates en Slides

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Introduction

Homogeneous coordinates are an augmented representation of points andlines in Rn spaces, embedding them in Rn+1, hence using n + 1 parameters.

This representation is useful in dealing with perspective and projectivetransformation (computer graphics, etc.) and for rigid displacementrepresentation.

We start introducing the concept on 2D spaces.

Given a point p in R2

, represented as P = (p 1, p 2), i.e., the vectorp =

p 1 p 2

Tits homogeneous representation (using homogeneous

coordinates) is

p = p 1 p 2 p 3T; with 0 0 0T

not allowed

The vector representation is obtained dividing the first n homogeneouscomponents by the (n + 1)-th, that is often called scale.

p 1 = p 1/p 3; p 2 = p 2/p 3

Basilio Bona (DAUIN-Politecnico di Torino) Rigid motions July 2009 2 / 32

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A geometric interpretation of the homogeneous coordinates is given in thefollowing Figure.

Figure: Geometric interpretation of homogeneous coordinates.


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We embed the plane π : z = 1 in a 3D space and we consider the planarcoordinates as the intersection of a projection line that goes through theorigin of the RF {i, j, k}.

As z goes to ∞ we can represent the origin, while as z goes to 0, we canrepresents points at infinity, along a well defined direction. The set of allpoints at infinity represents the line at infinity.

Figure: Points at ∞.


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The same approach is valid for 3D homogeneous coordinates: given a

point p in R3

, represented as P = (p 1, p 2, p 3), or else by a vector

p =

p 1 p 2 p 3T

its homogeneous representation is

p =

p 1 p 2 p 3 p 4T

Hencep 1 = p 1/p 4; p 2 = p 2/p 4; p 3 = p 3/p 4

In robotics it is common to set p 4 = 1. We will adhere to this convention.


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Some author treats differently the homogeneous representation of a point(geometric vector vp ) by that of an oriented segment (physical vector vab ).

In the first case

vp = vp

1 while in the second case

vab = vb

1 − va

1 = vab

0 Not all textbooks adhere to this convention, since in this case the sum of two geometric vectors produces

vp + vq = vp

1 + vq

1 = vp + vq

2 This sum, once transformed back in 3D space, becomes not the usualparallelogram sum of segments, but the midpoint of the parallelogram (seealso Affine Spaces).


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Figure: Homogeneous sum of two geometric vectors.


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Affine spaces

Briefly, affine spaces are spaces where the origin is not considered to be aspecial point, and homogeneous coordinates are used to characterize the

affine elements (points and planes).

Figure: Linear and affine subspaces.


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Rigid displacements and homogeneous coordinates

Using homogeneous representation, the generic displacement (R, t) is

given by the following homogeneous transformation matrix

H ≡ H(R, t) =

R t

0T 1

since

Hv =

R t

0T 1

v1

= Rv + t

Hence, the SE (3) group transformation g = (R, t) ∈ SE (3), is represented

by the 4 × 4 homogeneous matrix H through the use of homogeneouscoordinates:

pa = gab pb ↔ pa = Hab (Rab , tab )pb


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Transformation composition gab gbc is obtained by the homogeneousmatrix product

Hab (Rab , tab )Hbc (Rbc , tbc )i.e.,

Rab tab

0T 1

Rbc tbc

0T 1

=

Rab Rbc Rab tbc + tab

0T 1

Often it is convenient to write the time derivative of a homogeneousvector as

˙p =

p 1p 2p 3

0

This notation will prove useful when dealing with twists.


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Introduction to projective/perspective geometry

Homogeneous coordinates p are used for describing point in projectivespaces P n of dimension n.

Two points a and b are “equal” (a ∼ b) if there exist a real λ = 0 suchthat

a = λb → b ∼ λb collinear points are equal

this also means that vectors are equal up to a scale factor.

We say that a and b belong to the same line, i.e., to the same affinesubspace of Rn+1.

Consider the planar (2D) projection through a pinhole camera of a 3D

point P , p =

x y z

T

. The ray of light intercepts the image plane inthe 2D image point P i , whose homogeneous coordinates on the imageplane are

pi =

x

y f

→ pi =

x /f y /f

T

= 1

f

x y

T

where f is the focal distanceBasilio Bona (DAUIN-Politecnico di Torino) Rigid motions July 2009 11 / 32

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Figure: Example of projection through a pinhole camera.


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Considering the similar triangles in Figure, we have:

x

f =

X

Z ;

y

f =

Y

Z

If we set λ = Z /f , where f is known and the distance Z from π of thepoint p is unknown, we have

X = λx , Y = λy , Z = λf

Hence, if we compute the cartesian coordinates of pi on π fromhomogeneous coordinates, we have

pi = x /f y /f ↔ pi =

fX fY

Z

∼ λ

x y

1

∼

x y

f

and every 3D point with homogeneous coordinates λ

x y f T

isprojected onto the same point pi of the image plane π.

Notice that the above transformation is a nonlinear one.


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The nonlinear mapping R3 → R2 can be expressed as a (linear) matrix

transformation between the homogeneous spaces R4 → R3 as

fX

fY Z ∼ λx

y 1 = f 0 0 0

0 f 0 00 0 1 0

P

X

Y Z 1

⇔ λpi = P p1

Nonsingular (i.e., invertible) mappings between projective spaces aredescribed by products of full rank square matrices T by homogeneouscoordinates, pa = Tab pb

An invertible mapping from P n onto itself preserves collinearities, i.e., if two points belong to a line, also the transformed points will belong to a

line.Affine transformations preserve not only collinearities, but also parallelismbetween lines and ratios of distances.

Similarity transformations preserves also angles, but not scales, whileEuclidean (rigid) transformations preserve also scales


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Figure: Examples of different mappings.


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Hyperplanes

A hyperplane in Rn is defined by the equation between a set of points p i

and a set of (real) coefficients ℓi

ℓ1p 1 + ℓ2p 2 + · · · + ℓnp n + ℓn+1 = 0

Using the homogeneous coordinates we obtain an homogeneous linearequation in P n

ℓTp = 0

where

ℓ =

ℓ1...

ℓnℓn+1

; p =

p 1...

p n1

The two homogeneous vectors can exchange their roles; we say that theyare dual.


Remember that given a vector p ∈ V(F) the set of all linear

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Remember that, given a vector p ∈ V (F ), the set of all lineartransformations f (p) : V → R1 form another vector space V ∗, andf ∈ V ∗ are called dual vectors or 1−forms.

For example, the total time derivative of a scalar function f (p) on R3 is

df (p(t ))dt

= ∇pφ(p)p(t )

is a dual vector wrt p. Another dual vector is the divergence

∇ · p = ∂ ∂ p 1

· · · ∂ ∂ p n

T p 1

...p n

or the Laplacian (gradient of the divergence )

∇2f = ∇ · ∇f =

∂

∂ p 1· · ·

∂

∂ p n

T

∂ f ∂ p 1...

∂ f

∂ p n


D lit i j ti

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Duality in projective spaces

The space of all hyperplanes in Rn defines another perspective space,called the dual of the original P n

Duality principle: for all projective results established using points andplanes, a symmetrical result holds interchanging the role of planes andpoints.

Example:

two points define a line ↔ two lines define a point

Notice that the projective plane P 2 whose point are defined byhomogeneous coordinates, has more point that the usual R2 plane, since itincludes also points at ∞.


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Points and lines



Points and lines

We restrict our analysis on the 2D projective space P 2; for the sake of

clarity, we drop the ˜ sign on top of the homogeneous vectors, i.e., a isnow indicated as a.

A line ℓ =

ℓ1 ℓ2 ℓ3

can be characterized by three parameters:

1 the slope −ℓ1/ℓ2

2 the x -intercept −ℓ3/ℓ1

3 the y -intercept −ℓ3/ℓ2

Points p and lines ℓ are related by

ℓT

p = 0

in the sense that a point is on a line or a line include a point iff theprevious relation holds.


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Duality

A line ℓ passing for two points p1 and p2 is defined as

ℓ = p1 × p2 = S(p1)p2

A point p belonging to two lines (intercept) ℓ1 and ℓ2 is defined as

p = ℓ1 × ℓ2 = S(ℓ1)ℓ2

Points at infinity or ideal points have the following representation

p∞

=

p 1 p 2 0T

All ideal points lie on the ideal line, represented by

ℓ∞ =

0 0 1T


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Three points p1, p2, p3 lie on the same line (they are collinear) iff

det

p1 p2 p3

= 0

Since the duality principle holds, three lines ℓ1, ℓ2, ℓ3 meet at the same

point (they are concurrent) iff

detℓ1 ℓ2 ℓ3

= 0


Parallelism

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Parallelism

Two lines

m = m1

m2m3

; n = n1

n2n3

are parallel when the intersection point is at infinity, i.e.,

p = m × n = p 1p 20

This reduces to the conditions

m1m2

= n1n2

or det m1 n1m2 n2

= 0

The direction k

p 1p 2

points toward the point at infinity.


Summary

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Summary

point p =

p 1 p 2 p 3T

line ℓ =

ℓ1 ℓ2 ℓ3T

incidence pTℓ = 0 incidence ℓ

Tp = 0

line ℓ = p1 × p2 point p = ℓ1 × ℓ2

collinearity det

p1 p2 p3

= 0 collinearity det

ℓ1 ℓ2 ℓ3

= 0

∞ point p 1 p 2 0T

∞ line 0 0 ℓ3T


Transformations in P2

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Transformations in P

The generic projective transformation (also called homographies) in P 2 isrepresented by a non singular 3 × 3 matrix (acting on homogeneous

vectors), whose elements are

T =

t 11 t 12 t 13

t 21 t 22 t 23

t 31 t 32 t 33

Since the transformations are equivalent up to a scale factor, the numberof dof in T is 8.

According to structural constraint, we have a hierarchy of transformations,from the most general to the least one; in particular

Transformations in P 2

(most general) projective → affine → similarity → Euclidean (rigid)


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The affine transformation must preserve the ideal line and the ideal points

(as well as parallelism); hence, for any arbitrary scaling factor λ

λ

p 1p 20

= T

p 1p 20

that implies

T =

t 11 t 12 t 13

t 21 t 22 t 23

0 0 t 33

→ dof=6


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The similarity transformation must preserve angles and ratios of lengths;that implies

T =

cos θ − sin θ t 13

sin θ cos θ t 23

0 0 t 33

→ dof=3

with t 33 = 1.

The effects of the last column in T is to produce a shear i.e., a scalevariation that is non uniform in all directions.


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The Euclidean (rigid) transformation must preserve also lengths; thatimplies

T = cos θ − sin θ t 13

sin θ cos θ t 23

0 0 1 → dof=3


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From homogeneous coordinates to quaternions



g q

There exist a link between homogeneous coordinates and quaternions that

goes through the Mobius transformation.Given two complex numbers x = x 1 + jx 2, y = y 1 + jy 2 ∈ C, the orderedpair of complex numbers

z = x y T∈ C

2, z = 0 0T

are the homogeneous coordinates of z = z 1 + jz 2

z = x

y = z 1 + jz 2 =

x 1 + jx 2

y 1 + jy 2

To every pair of complex numbers (x , y ) a unique complex numbers z corresponds, while for every complex number z there is an infinite numberof complex pairs (projective transformation).


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Transformation between homogeneous representations

z = x

y → w = u

v where u

v = a b

c d x

y = ax + by

cx + dy where a, b , c , d are complex numbers.

We can compute the transformed complex number as

w = u v

= ax + by cx + dy

=a x

y + b

c x y

+ d = az + b

cz + d

This is nothing else that the Mobius transformation.

Mobius transformation

w = az + b

cz + d ; a, b , c , d , w , z ∈ C


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If we take a particular form of the Mobius transformation, due to Gauss:

w = az + b

−b ∗z + a∗

we know that every rotation of the sphere can be expressed by a complexfunction as the one above, and we notice that it can be represented by thematrix of its coefficients

a b −b ∗ a∗ = α + jβ γ + jδ −γ + jβ α − jβ

= α

1 00 1

+ β

j 00 j

+ γ

0 1−1 0

+ δ

0 j j 0

= α1 + β i + γ j + δ k

= a quaternion

For further details see: T. Needham, Visual Complex Analysis, OUP, 1997.


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homogeneous coordinates en slides

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