ece
chniques and
ssible. In this
to irregularly-
wn analytical
monoplotting
al unwrapping
tographs fully
vinded plot of
rape-fitting to
f the growing
1991); on the
; sampled by
blish theoreti-
m, 1990; Fo-
id orthophoto
Questions of
herical, have
e recognition
missing third
dition, hence
«e conventio-
Yon-iterative).
uniqeness of
of a develop-
t in unwinded
y the genera-
apping’ of the
may also be
software. In
aining all the
erated. Here,
case of right
y extended to
ING
ric surface is
drical case a
| points mea-
distinction is
given or 'ob-
I; and instan-
eforehand.
on of nine in-
set. General-
ly, these coefficients in fact correspond to the three para-
meters of the canonical form defining shape (i. e. surface
type) and size; and the six elements of rigid body trans-
formation fixing surface position and orientation in the co-
ordinate system. Restle & Stephani (1988) outlined steps
for detecting rotatory surfaces, classification and extract-
ing canonical forms from the full second degree equation.
On the other hand, assumption of specific surface types
allows to directly use their corresponding general equa-
tions. For a right circular cylindrical surface, in particular,
different models have been given (Feltham, 1990; Chand-
ler & Cooper, 1991). In fact, the independent parameters
of this solid are just five (as also pointed out by Robson
et al., 1992). The physical meaning is as follows: radius R
suffices for fixing its canonical form; it has only four de-
grees of freedom in space as translation along and rota-
tion about its axis & does not affect it. Or, put otherwise, a
right circular cylinder is fully known through four elements
that fix its axis £ as a straight line in 3D space and the
common distance (R) of all its points from &.
Indeed, among the three XYZ coordinates fixing an arbi-
trary point K of a line and the three direction numbers L,
M, N defining its direction two corresponding constraints
must be imposed. Amongst possible constraints (Petsa,
1996) and provided that axis orientation is even vaguely
known, it is simplest to set K as belonging to one of the
XY, YZ or ZX planes and to set one (non-zero) direction
number equal to unity. For a cylinder whose axis is not
parallell, say, to the YZ plane, the constraints then are:
Xk = 0, L = 1. These can be directly introduced into the
condition equation employed for surface approximation.
This may equivalently be: either the equation yielding the
distance of a point from a line; or an equation of a circle
on, say, the Y'Z' plane made perpendicular to the cylinder
axis € via two rotations which are functions of its direction
numbers. In the example of the above two constraints, for
instance, the condition used for fitting finally becomes:
X2? - NPSUCY — Y? ENT) rZ - 7: 0M? y
z2X(Y —- Yk)M -2X(2.— 240N 7: 20Y = Yo HZ =. ZicIMN 52.5: 1)
-R?(1-M? - N2) 20
with the five unknowns R, Yk, Zk, M, N which are solved
for in an iterative adjustment process. Initial values may
be found from two points approximately defining a gene-
rating line. This model has been tested in other cases but
not the test object mentioned later whose axis is vertical.
3. DIGITAL MONOPLOTTING
3.1 Basic Equations
With known interior and exterior camera orientations, the
analytic equation of the surface provides a third condition
supplementing the two conventional collinearity equations
- or the direct linear transformation equations (DLT) — for
each image point g(x,y). Hence, space intersection of the
single projective ray |, passing through projective centre
O and q, with the surface results in the 3D coordinates of
digitized image points (Figure 1). If the camera constant c
is unknown, for most practical purposes an approximate
value could be used in space resection and thereafter due
to small object depth. The control points used for exterior
orientation should preferably be six for each photograph
(each three will also appear on the neighbouring images).
291
Figure 1 Geometry of monoplotting cylindrical objects.
The process is simpler if space data are transformed to a
cylinder-centered system XYZ by means of two rotations
resulting, for instance, in § // Y (as in Figure 1); shift of the
origin to the intersection point C of § with the XZ plane will
then lead to & = Y. The two collinearity equations can be
written in the form
X=uZ+U Y=vZ+V (2)
whereby u, v are functions of the image coordinates (x,y),
interior orientation elements (c,xo,yc) and the image rota-
tions (w,@,K) while U, V also depend on the perspective
centre location (X5, Yo,Z;). Combination of the squared X-
equation with that of a circle now describing the cylinder
in the new system gives:
- X24 USD |
E R2 — X2 (3)
u
zZ?
from which X is finally solved for as
. Uzuy(1+u?)R? — U2 (4)
Xu 1+u?
Introduced into the circle equation, each of roots X4,2 pro-
vides two Z-values; each of these results in one Y-value
from the second of Eq. 2. The consequence is that a total
of four points Q4, Qo Q», Qo» are generally obtained, as
illustrated in Figure 1. It is X1= X2 once a projective ray p
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
