# Projective geometry, camera models and calibration

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Projective geometry, camera models and calibration

Subhashis BanerjeeDept. Computer Science and Engineering

IIT Delhiemail: suban@cse.iitd.ac.in

January 16, 2008

1

1 The main problems in computer vision

Graphics/Optics

Vision

Image processing

2D Images 3D Descriptions

Correspondence problem: Match image projections of a 3D configuration.

Reconstruction problem: Recover the structure of the 3D configuration from image pro-jections.

Re-projection problem: Is a novel view of a 3D configuration consistent with other views?(Novel view generation)

All of these require camera calibration in some form.

2 An infinitely strange perspective

Parallel lines in 3D space converge in images.

The line of the horizon is formed by infinitely distant points (vanishing points).

Any pair of parallel lines meet at a point on the horizon corresponding to their commondirection.

All intersections at infinity stay constant as the observer moves.

The effects can be modelled mathematically using the linear perspective or

a pin-hole camera (realized first by Leonardo?)

3 The pin-hole camera model

3.1 Standard perspective projection

If the world coordinates of a point are (X,Y, Z) and the image coordinates are (x, y), then

x = fX/Z and y = fY/Z

The model is non-linear.

3.2 In terms of projective coordinates

xy1

=

f 0 0 00 f 0 00 0 1 0

XYZ1

xy1

P2 and

XYZ1

P3

are homogeneous coordinates.The model is linear in projective geometry.

4 Basics of Projective Geometry

4.1 Affine and Euclidean geometries

Given a coordinate system, n-dimensional real affine space is the set of all pointsparameterized by x = (x1, . . . , xn)

t Rn.

An affine transformation is expressed as

x = Ax + b

where A is a n n (usually) non-singular matrix and b is a n 1 vector representinga translation.

In the special case of when A is a rotation (i.e., AAt = AtA = I, the the transformationis Euclidean.

Transformation of one point (or one axis) completely determines an Euclidean trans-formation, an affine transformation in n dimensions is completely determined by amapping of n + 1 points (3 points for a plane).

It is easy to verify that an affine transformation preserves parallelism and ratios oflengths along parallel directions. In fact, coordinates in an affine geometry are definedin terms of these fundamental invariants. An Euclidean transformation, in addition tothe above, also preserves lengths and angles.

Since an affine (or Euclidean) transformation preserves parallelism it cannot be usedto describe a pinhole projection. We need to projective geometry to represent suchtransformations.

4.2 Spherical geometry

Before we introduce projective geometry let us briefly consider spherical geometry (S2), whichis the geometry on the surface of a sphere.

The space S2:S2 =

{x R3 : ||x|| = 1

}

lines in S2: If one begins at a point in S2 and travels straight ahead on the surface,one will trace out a great circle. Viewed as a set in R3 this is the intersection of S2with a plane through the origin. We will call this great circle a line in S2:Let be a unit vector. Then,

l ={x S2 : tx = 0

}

is the line with pole .

l

Two points p and q are antipodal if p = q.

Lines in S2 cannot be parallel. Any two lines intersect at a pair of antipodal points.

A point on a line:lx = 0 or lTx = 0 or xT l = 0

Two points define a line:l = p q

Two lines define a point:x = l m

4.3 Projective geometry

The projective plane P2 is the set of all pairs {x,x} of antipodal points in S2.

Two alternative definitions of P2, equivalent to the preceding one are

1. The set of all lines through the origin in R3.

2. The set of all equivalence classes of ordered triples (x1, x2, x3) of numbers (i.e.,vectors in R3) not all zero, where two vectors are equivalent if they are propor-tional.

The space P2 can be thought of as the infinite plane tangent to the space S2 andpassing through the point (0, 0, 1)t.

P2

x

x

Let : S2 P2 be the mapping that sends x to {x,x}. The is a two-to-one mapof S2 onto P2.

A line of P2 is a set of the form l, where l is a line of S2. Clearly, x lies on l if andonly if tx = 0.

Homogeneous coordinates: In general, points of real n-dimensional projectivespace, Pn,are represented by n+1 component column vectors (x1, . . . , xn, xn+1) Rn+1 such thatat least one xi is non-zero and (x1, . . . , xn, xn+1) and (x1, . . . , xn, xn+1) representthe same point of Pn for all 6= 0.

(x1, . . . , xn, xn+1) is the homogeneous representation of a projective point.

4.4 Canonical injection of Rn into Pn and Points at infinity Affine space Rn can be embedded in Pn by

(x1, . . . , xn) (x1, . . . , xn, 1)

Affine points can be recovered from projective points with xn+1 6= 0 by

(x1, . . . , xn) (x1

xn+1, . . . ,

xnxn+1

, 1) ( x1xn+1

, . . . ,xn

xn+1)

A projective point with xn+1 = 0 corresponds to a point at infinity.

The ray (x1, . . . , xn, 0) can be viewed as an additional ideal point as (x1, . . . , xn)recedes to infinity in a certain direction. For example, in P2,

limT0

(X/T, Y/T, 1) = limT0

(X,Y, T ) = (X,Y, 0)

It is important to note that these are affine concepts.

4.5 Lines and conics in P2

A line equation in R2 isa1x1 + a2x2 + a3 = 0

Substituting by homogeneous coordinates xi = Xi/X3 we get a homogeneous linearequation

(a1, a2, a3) (X1, X2, X3) =3

i=1

aiXi = 0, X P2

A line in P2 is represented by a homogeneous 3-vector (a1, a2, a3).

A point on a line:aX = 0 or aTX = 0 or XTa = 0

Two points define a line:l = p q

Two lines define a point:x = l m

Matrix notation for cross products:The cross product v x can be represented as a matrix multiplication

v x = [v]

x

where [v]

is a 3 3 antisymmetric matrix of rank 2:

[v]

=

0 vz vyvz 0 vxvy vx 0

The line at infinity (l): is the line of equation X3 = 0. Thus, the homogeneousrepresentation of l is (0, 0, 1).

The line (u1, u2, u3) intersects l at the point (u2, u1, 0).

Points on l are directions of affine lines in the embedded affine space (can be extendedto higher dimensions).

Consider the standard hyperbola in the affine space given by equation xy = 1. Totransform to homogeneous coordinates, we substitute x = X/T and y = Y/T to obtainXY = T 2. This is homogeneous in degree 2. Note that both (0, , 0) and (, 0, 0)are solutions. The homogeneous hyperbola crosses the coordinate axes smoothly andemerges from the other side. See the figure.

A conic in affine space (inhomogeneous coordinates) is

ax2 + by2 + cxy + dx + ey + f = 0

Homogenizing this by replacements x = X1/X3 and y = Y1/Y3, we obtain

aX21 + bX22 + cX1X2 + dX1X3 + eX2X3 + fX

23 = 0

which can be written in matrix notation as

XTCX = 0

where C is symmetric and is the homogeneous representation of a conic.

4.6 Planes and lines in P3

The duality that exist between points and lines in P2 exist between points and planes in P3.Thus a plane is defined as a 4-tuple (u1, u2, u3, u4) and the equation of this plane is given as

4

i=1

uixi = 0

Analogous to the line at infinity (l) in P2 we have the plane at infinity () in P3 whoserepresentation is (0, 0, 0, 1)T .

4.6.1 Lines is P3: Plucker coordinates

4.7 Projective basis

Projective basis: A projective basis for Pn is any set of n + 2 points no n + 1 of whichare linearly dependent.

Canonical basis:

10...0

,

01...0

, . . .

points at infinity along each axis

00...1

,

origin

11...1

unit point

Change of basis: Let e1, e2, . . . , en+1, en+2 be the standard basis and a1, a2, . . . , an+1, an+2be any other basis. There exists a non-singular transformation [T](n+1)(n+1) such that:

Tei = iai,i = 1, 2. . . . , n + 2

T is unique up to a scale.

Proof:

From the first n + 1 equations we have that T must be of the form

T =[

1a1 2a2 . . . n+1an+1]

T is non-singular by the linear independence of as.

The final equation gives us:

[1a1 2a2 . . . n+1an+1

]

11...1

= n+2an+2

which is equivalent to:

[a1 a2 . . . an+1

]

12...

n+1

= n+2an+2

Since the matrix on the left hand side of the above equation is of full rank (by linearindependence of xs), the ratios of the i are uniquely determined and no i is 0.

4.8 Collineations

The invertible transformation T : Pn Pn is called a projective transformation orcollineation or homography or perspectivity and is completely determined by n + 2point correspondences.Properties:

4.8.1 Preserves straight lines and cross ratios

Given four collinear points A1, A2,A3 and A4, their cross ratio is defined as

A1A3 A2A4

A1A4 A2A3

if A4 is a point at infinity the the cross ratio is given as

A1A3

A2A3

The cross ratio is independent of the choice of the projective coordinate system.

4.8.2 Illustration of perspectivity

4.8.3 Projective mappings of lines and conics in P2

lines: Let xi be a set of points on a line l and consider the action of a 3 3 projectivetransformation H on the the points. Since the points lie on the line we have

lTxi = 0

One can easily verify tha

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