Geometric Fundamentals in Robotics Homogeneous Coordinates Basilio - - PowerPoint PPT Presentation

Geometric Fundamentals in Robotics Homogeneous Coordinates

Basilio Bona

DAUIN-Politecnico di Torino

July 2009

Basilio Bona (DAUIN-Politecnico di Torino) Rigid motions July 2009 1 / 32

Introduction

Homogeneous coordinates are an augmented representation of points and lines in Rn spaces, embedding them in Rn+1, hence using n +1 parameters. This representation is useful in dealing with perspective and projective transformation (computer graphics, etc.) and for rigid displacement representation. We start introducing the concept on 2D spaces. Given a point p in R2, represented as P = (p1, p2), i.e., the vector p = p1 p2 T its homogeneous representation (using homogeneous coordinates) is ˜ p = ˜ p1 ˜ p2 ˜ p3 T ; with 0T not allowed The vector representation is obtained dividing the first n homogeneous components by the (n + 1)-th, that is often called scale. p1 = ˜ p1/˜ p3; p2 = ˜ p2/˜ p3

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A geometric interpretation of the homogeneous coordinates is given in the following 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 planar coordinates as the intersection of a projection line that goes through the

• rigin of the RF {i, j, k}.

As z goes to ∞ we can represent the origin, while as z goes to 0, we can represents points at infinity, along a well defined direction. The set of all points 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 = (p1, p2, p3), or else by a vector p = p1 p2 p3 T its homogeneous representation is ˜ p = p1 p2 p3 p4 T Hence p1 = ˜ p1/˜ p4; p2 = ˜ p2/˜ p4; p3 = ˜ p3/˜ p4 In robotics it is common to set ˜ p4 = 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
• 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 usual

parallelogram sum of segments, but the midpoint of the parallelogram (see also 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 a special 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

H˜ v =

• R

t 0T 1 v 1

• = 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 homogeneous coordinates: pa = gabpb ↔ ˜ pa = Hab(Rab, tab)˜ pb

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Transformation composition gabgbc is obtained by the homogeneous matrix product Hab(Rab, tab)Hbc(Rbc, tbc) i.e.,

• Rab

tab 0T 1 Rbc tbc 0T 1

• =
• RabRbc

Rabtbc + tab 0T 1

• Often it is convenient to write the time derivative of a homogeneous

vector as ˙ ˜ p =    ˙ p1 ˙ p2 ˙ p3    This notation will prove useful when dealing with twists.

Basilio Bona (DAUIN-Politecnico di Torino) Rigid motions July 2009 10 / 32

Introduction to projective/perspective geometry

Homogeneous coordinates ˜ p are used for describing point in projective spaces Pn of dimension n. Two points ˜ a and ˜ b are “equal” (˜ a ∼ ˜ b) if there exist a real λ = 0 such that ˜ a = λ˜ b → ˜ b ∼ λ˜ b 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 affine subspace of Rn+1. Consider the planar (2D) projection through a pinhole camera of a 3D point P, p = x y

• zT. The ray of light intercepts the image plane in

the 2D image point Pi, whose homogeneous coordinates on the image plane are ˜ pi =   x y f   → pi =

• x/f

y/f T = 1 f

• x

y T where f is the focal distance

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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 the point p is unknown, we have X = λx, Y = λy, Z = λf Hence, if we compute the cartesian coordinates of pi on π from homogeneous 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 is projected 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   ∼ λ   x y 1   =   f f 1  

• P

   X Y Z 1    ⇔ λ˜ pi = P

• p

1

• Nonsingular (i.e., invertible) mappings between projective spaces are

described by products of full rank square matrices T by homogeneous coordinates, ˜ pa = Tab˜ pb An invertible mapping from Pn 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 parallelism between lines and ratios of distances. Similarity transformations preserves also angles, but not scales, while Euclidean (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 pi and a set of (real) coefficients ℓi ℓ1p1 + ℓ2p2 + · · · + ℓnpn + ℓn+1 = 0 Using the homogeneous coordinates we obtain an homogeneous linear equation in Pn ˜ ℓ

T˜

p = 0 where ˜ ℓ =     ℓ1 . . . ℓn ℓn+1     ; ˜ p =     p1 . . . pn 1     The two homogeneous vectors can exchange their roles; we say that they are dual.

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Remember that, given a vector p ∈ V(F), the set of all linear transformations f (p) : V → R1 form another vector space V∗, and f ∈ 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 = ∂ ∂p1 · · · ∂ ∂pn T   p1 . . . pn  

• r the Laplacian (gradient of the divergence)

∇2f = ∇ · ∇f = ∂ ∂p1 · · · ∂ ∂pn T       ∂f ∂p1 . . . ∂f ∂pn      

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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 Pn Duality principle: for all projective results established using points and planes, a symmetrical result holds interchanging the role of planes and points. Example: two points define a line ↔ two lines define a point Notice that the projective plane P2 whose point are defined by homogeneous coordinates, has more point that the usual R2 plane, since it includes also points at ∞.

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A collineation given by a full rank matrix T transforms points as ˜ pa = Tab˜ pb A hyperplane ˜ ℓ

T b ˜

pb = 0 is transformed as ˜ ℓ

T b ˜

pb = 0 ⇔ ˜ ℓ

T a ˜

pa = 0 hence ˜ ℓ

T a Tab˜

pb ⇒ ˜ ℓ

T b = ˜

ℓ

T a Tab

yielding ˜ ℓa = T−

ab T˜

ℓ

T b

Matrix T−

ab T is called the dual of Tab

Comparing this last equation with the first one, we conclude that points and lines can be transformed, but using different (dual) collineation matrices Tab and T−

ab T.

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Points and lines

We restrict our analysis on the 2D projective space P2; for the sake of clarity, we drop the ˜ sign on top of the homogeneous vectors, i.e., ˜ a is now 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 ℓTp = 0 in the sense that a point is on a line or a line include a point iff the previous 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∞ = p1 p2 0T All ideal points lie on the ideal line, represented by ℓ∞ = 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 det ℓ1 ℓ2 ℓ3

• = 0

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Parallelism

Two lines m =   m1 m2 m3   ; n =   n1 n2 n3   are parallel when the intersection point is at infinity, i.e., p = m × n =   p1 p2   This reduces to the conditions m1 m2 = n1 n2

• r

det

• m1

n1 m2 n2

• = 0

The direction k

• p1

p2

• points toward the point at infinity.

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Summary

point p = p1 p2 p3 T line ℓ = ℓ1 ℓ2 ℓ3 T incidence pTℓ = 0 incidence ℓTp = 0 line ℓ = p1 × p2 point p = ℓ1 × ℓ2 collinearity det p1 p2 p3

• = 0

collinearity det ℓ1 ℓ2 ℓ3

• = 0

∞ point p1 p2 0T ∞ line ℓ3 T

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Transformations in P2

The generic projective transformation (also called homographies) in P2 is represented by a non singular 3 × 3 matrix (acting on homogeneous vectors), whose elements are T =   t11 t12 t13 t21 t22 t23 t31 t32 t33   Since the transformations are equivalent up to a scale factor, the number

• f 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 P2

(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   p1 p2   = T   p1 p2   that implies T =   t11 t12 t13 t21 t22 t23 t33   → dof=6

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The similarity transformation must preserve angles and ratios of lengths; that implies T =   cos θ − sin θ t13 sin θ cos θ t23 t33   → dof=3 with t33 = 1. The effects of the last column in T is to produce a shear i.e., a scale variation that is non uniform in all directions.

Basilio Bona (DAUIN-Politecnico di Torino) Rigid motions July 2009 27 / 32

The Euclidean (rigid) transformation must preserve also lengths; that implies T =   cos θ − sin θ t13 sin θ cos θ t23 1   → dof=3

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Historical notes

Homogeneous coordinates were introduced in 1827 by M¨

• bius in the

context of barycentric coordinates. Given a triangle, the barycentric coordinates of a point P are the weights required at the triangle vertices such that P become the center of gravity of the triangle. The point is computed as p = wax wby wczT, with wa + wb + wc = 1

Figure: Example of barycentric coordinates.

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From homogeneous coordinates to quaternions

There exist a link between homogeneous coordinates and quaternions that goes through the M¨

• bius transformation.

Given two complex numbers x = x1 + jx2, y = y1 + jy2 ∈ C, the ordered pair of complex numbers ˜ z = x yT ∈ C2, ˜ z = 0T are the homogeneous coordinates of z = z1 + jz2 z = x y = z1 + jz2 = x1 + jx2 y1 + jy2 To every pair of complex numbers (x, y) a unique complex numbers z corresponds, while for every complex number z there is an infinite number

• f complex pairs (projective transformation).

Basilio Bona (DAUIN-Politecnico di Torino) Rigid motions July 2009 30 / 32

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 M¨

• bius transformation.

M¨

• bius 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 M¨

• bius transformation, due to Gauss:

w = az + b −b∗z + a∗ we know that every rotation of the sphere can be expressed by a complex function as the one above, and we notice that it can be represented by the matrix of its coefficients

• a

b −b∗ a∗

• =
• α + jβ

γ + jδ −γ + jβ α − jβ

• =

α

• 1

1

• + β
• j

j

• + γ
• 1

−1

• + δ
• j

j

• =

α1 + βi + γj + δk = a quaternion For further details see: T. Needham, Visual Complex Analysis, OUP, 1997.

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