n
ırch’s condi-
1 the resection
pair of rays in
same as that
e system. This
-^nj;-—nj:ngg
ripts of angles
| in the image
vely. Analytic
are presented
)4, 240 resp.],
r solutions.
that the prob-
e-forming rays
instead of the
ying the coan-
ritten in terms
nvestigate the
lar product of
> dual angle 0
lar joining the
ut the perpen-
= Rn; + es; x
yy distributing
:S; x Rnj).
(8)
t — that the
ys at the focal
x Rn;
< Rn;
X nj)
(9)
A vector be-
inar with vec-
se projections,
.1s more often
om. The con-
r the resection
esent 1S orien-
n; x n;||, the
ation R to be
(10)
tor a, calculated
ct, calculated as
S
Figure 2: Coplanarity of image rays
Since achieving this the author has discovered a simi-
lar minimisation equation to (10) in [Liu et al., 1990],
but it was found through observation of problem ge-
ometry rather than analytically as shown above.
6 Rotation solution
The elements of an orthonormal rotation matrix R
are non-linear functions of the unknown Euler angles
of rotation w, @, x. This makes equation (10) highly
non-linear in terms of the three unknowns and thus a
closed-form solution for them is improbable. Various
approaches to minimisation are outlined in [Shevlin,
1996]. The one presented here uses the quaternion
q parameterisation of rotation [Horn, 1987]. Equa-
tion (10) can be rewritten,
n
D (8500)? =D (gcssa)”. (11)
i=1
Let N; = G/ C; where G; and C; are orthogonal ma-
trices formed from vectors g; and c; (see [Horn, 1987]).
Let N — 5, Ni. Writing the quaternion q as a vec-
tor q — [go q1 42 42 q3] and denoting quaternion mul-
tiplication as a matrix by vector product gives the fol-
lowing matrix expression equivalent to equation (11),
(a” Na)”. (12)
This could be minimised by finding solving the
quadratic form F(q) for the four quaternion variables
(subject to the unit quaternion constraint q3 + a? +
Q3 4 + q3 = 1),
F(q): q' Nq - 0. (13)
Instead of attempting to solve a quadratic in four vari-
ables subject to a constraint, it was decided to form
an overdetermined system of simultaneous non-linear
equations F;(q): q' N;q — 0 (for each observation
i =1,...,n). Each of the F;(q) can be linearised in
the neighbourhood of a known q using the first two
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
terms of a Taylor’s series expansion,
OF;
F;(q + Aq) = Fi(q) + CD Aq, c
OF; OF; OF;
Un + — CU + (@ Ags (14)
0q1 0q2 03
The Newton-Raphson method can be used to solve it-
eratively for corrections Aq which minimise the least-
squared error.
An important advantage of using the quaternion pa-
rameterisation in the solution of the problem (as op-
posed to Euler angles and hence rotation matrices as
used in [Liu et al., 1990]) is the ease with which suc-
cessful initial values for the iterative solution can be
found. The following set of coefficients provide a reg-
ular tesselation of the unit quaternion hemisphere and
it can be shown that convergence to the four possible
solutions is guarenteed,
051. 03005 05
051 10571 1005| 203
05/105 17105171051
051 os os zo
054 1051 105% [05
enses sies]: 05
05 dd 0 24-05 (15)
ipe mov e a le:
The negation of the elements of this set provide
diametrically-opposed points on the other hemisphere
and so could also be used as starting values. The
solutions found through this linearisation may then
be used as start points in a non-linear optimisation
search to minimise equation (12). A very accurate
and efficient means to perform such a search on the
unit quaternion sphere is the spherical optimisation
search outlined in [Kanatani, 1993, p. 123].
7 Translation solution
Once orientation has been found the least-squared er-
ror solution for position p can be found in closed-form
using the pseudo-inverse method. The position of the
i'^ image-forming ray is specified with the scene con-
trol point s; in the Plücker line 1; = n;+€s; x n;. Since
each image-forming ray passes through the focal point
p,
p x n; = s; x N;. (16)
An overdetermined system of linear equations in terms
of the unknown p can be formed,
P Xn = 81 X11
P XN =82 X19
(17)
PXM = Sn X Mn
Rewriting this using a skew-symmetric matrix product
instead of the cross product on the left-hand side and
