Triangulation-Based Approaches to Three-Dimensional Scene Reconstruction - 3D Computer Vision: Efficient Methods and Applications - page 26

The metric reconstruction then consists of a determination of the three components
of p and the five independent matrix elements of A (Hartley and Zisserman, 2003 ).

The Basic Equations for Self-calibration and Methods for Their Solution To
determine the basic equations for self-calibration, Hartley and Zisserman ( 2003 )
suggest to denote the camera projection matrices of the projective reconstruction as
P i =[ B i | b i ]

. Combining ( 1.57 ) and ( 1.58 ) then yields
A i R i = B i −

b i p T A 1

(1.62)

A − 1
i

b i p T )A 1 . It follows from

for i = 2 ,...,m , which corresponds to R i = (B i −

the orthonormality of the rotation matrices that RR T = I and thus

b i p T T . (1.63)
This important expression yields the basic equations of self-calibration. Knowledge
about the elements of the camera matrices A i , e.g. the position of the principal point
(u 0 ,v 0 ) or the skew angle θ , yields equations for the eight unknown parameters of
p and A 1 based on ( 1.63 ).
A special case of high practical relevance is the situation where all cameras have
the same intrinsic parameters. Equation ( 1.63 ) then becomes
AA T

= B i − b i p T A 1 A 1 B i −

A i A i

= B i − bp T AA T B i − bp T T .

(1.64)

Since each side of ( 1.64 ) is a symmetric 3 3 matrix and the equation is homoge-

neous, each view apart from the first provides five additional constraints, such that a
solution for the eight unknown parameters and thus for A can be obtained for m ≥

×

3

images (Hartley and Zisserman, 2003 ).
In the context of the basic equations of self-calibration, several geometric entities
are introduced by Hartley and Zisserman ( 2003 ). In projective geometry, the general
representation of curves resulting from the intersection between a plane and a cone,
i.e. circles, ellipses, parabolas, and hyperbolas, is given by a conic. The projective
representation of a conic is a matrix C with x T C x on the conic

C . The dual of a conic is also a conic, because a conic can either be defined by
the points belonging to it, or by the lines (in

˜ x for all points ˜ ˜

3 ) which are dual
to these points and thus form the 'envelope' of the conic (Hartley and Zisserman,
2003 ).
The absolute conic Ω ∞

2 ) or planes (in

P P

π ∞ . An important en-
tity is the dual of the absolute conic Ω ∞ , which is termed absolute dual quadric and
is denoted by Q ∗ ∞

is situated on the plane at infinity ˜

. In a Euclidean coordinate system, Q ∗ ∞ obtains its canonical form

Q ∗ ∞ = I = diag ( 1 , 1 , 1 , 0 ) , while its general form corresponds to Q ∗ ∞ = H IH T

(Hartley and Zisserman, 2003 ).
Two entities which are relevant in the context of metric self-calibration are the
'image of the absolute conic' (IAC) and the 'dual image of the absolute conic'
(DIAC), which are denoted by ω and ω ∗ , respectively. Hartley and Zisserman ( 2003 )
show that they are given in terms of the matrix A comprising the intrinsic camera
parameters (cf. ( 1.11 )) according to

= AA T − 1

A − T A − 1 ω ∗ = ω − 1 AA T . ω = and = (1.65)