When the decentering distortion profiles from the Nikonos
camera were examined, a large discrepancy at a radial distance
r= 18 mm, of 3 um from the plumb-line technique versus
37 um from the self-calibrating bundle adjustment was noted.
The explanation lies in the high correlation between the values
for the parameters P, P? and xp, yp. In situations where a high
degree of projective coupling exists between parameters, the
self-calibration model can be thought of as over-parameterised
in that either set of parameters can adequately describe this
component of the systematic error signal.
To prove this hypothesis of high correlation, another bundle
adjustment was run where P, and P2 were constrained to their
values computed from the plumb-line technique. The result
was large changes of 0.14 mm and 0.33 mm in x, and yp and
no alteration to the object point co-ordinates. The important
conclusion is that camera calibration must be viewed not as an
end in itself, but rather as a step towards achieving the goal of
obtaining the best possible object point co-ordinates. Similarly,
the actual values of the parameters Kj, K2, K5, P1 and P2 may
appear to differ from one determination to another, but the
shapes of the profiles of radial and decentering distortion must
be examined to see if there is any significant difference.
2.4 Radial Distortion and Stereophotogrammetry
The effects of lens distortions, especially radial, has been
acknowledged for decades by mapmakers using aerial cameras
(for example, Ekelund, 1956). A textbook on photogrammetry
published in 1960 (Hallert, 1960, p. 60), describes the
stereoscopic photography of a plane surface and how “... from
measurements of the deformations of the surface, the systematic
errors which caused the deformations of the bundle of rays can
be determined numerically".
In the case of aerial photogrammetry, the radial distortion
includes the combined effects of earth curvature and refraction
as well as lens distortion. In the close-range situation only the
latter is relevant and this discussion has been included in this
paper to alert researchers and practitioners involved with close—
range stereophotogrammetric situations such as archaelogical,
architectural, medical, etc., to a potential error source.
It has been mathematically demonstrated (Fryer and Mitchell,
1987) that a relative orientation may be made on photographs
incorporating radial distortion and all y—parallaxes can be
removed. No residual x—parallaxes will be present in the
corners of the overlap region but an appreciable amount will left
undetected in the central region of the model. In fact this
amount was shown to be of magnitude 1.25 b? K1, where b is
the base distance (in mm) between the left and right hand
principal points and Kj is the first term of radial distortion.
Unresolved x—parallaxes are equivalent to a height difference
and for a typical 70 mm camera, a heighting error of 14 mm for
a camera—object distance of 2 m has been reported.
The physical appearance of this effect for a flat surface, such as
a building facade, is to have a “hump” in the middle of the
stereomodel. Most check points in relative and absolute
orientations are placed near the periphery of stereomodels and
this effect therefore will pass undetected. On objects such as a
building facade the effect will be detected visually but cannot be
eliminated by any amount of repeating the orientation process.
A specific radial distortion correction must be applied to the
analytical stereoplotter's camera calibration files. If the object
under examination is itself a curved surface, for example a
human back or an archaelogical artefact, then the effect may not
be detected.
This Section was specifically included in this paper to highlight
the need for the increasing numbers of non-metric camera
users, who may not have a complete understanding of the
uncompensated systematic errors which may be present in
stereophotogrammetry. to proceed with caution in their use of
analytical and digital plotting equipment.
3. COMMENTS ON THE OFFSETS OF THE
PRINCIPAL POINTS
The importance, or otherwise, of an exact knowledge of the
offsets of the principal point, xp and yp, from the intersection of
the fiducial axes are examined. in this Section. In Section 2.3,
the high correlation between decentering distortion and xp, yp
was demonstrated. In the study described, it was observed on
each iteration of a self—calibrating bundle adjustment that the
values of P1, P2? as opposed to xp, yp would alternatively
increase and decrease in proportion. The values for the co—
ordinates of the object points remained unaltered during this
process. When the values of Pj, P2 were constrained to their
values as determined by the plumb-line techniques, xp and yp
altered by up to 0.33 mm but again the object co-ordinates were
unaltered.
Perhaps an important feature of these tests was that neither
camera had “proper” fiducial marks, but rather the edges of the
format were used to establish pseudo fiducial corners. Film
stretch and unflatness have been shown to cause up to 100 um
of difference in distance between corners on 35 mm frames
(Donnelly, 1988), so an exact knowledge of xp, yp can be
purely “academic” and not really essential to achieving accurate
co-ordinates on the object.
Some recent (1991) adjustments of photography taken with a
125 mm by 125 mm image format camera (a 1943 F-24
reconnaissance camera which has been refurbished with a
90 mm Nikkor lens) has provided some further discussion on
this topic. This camera is fitted with a glass reseau and
therefore has fiducial marks. Probably due to the refurbishment
procedure, it was noted that the decentering distortion
parameters Pj, P2 were larger than usually expected. The
bundle adjustment was re—run with P1, P2 constrained and the
values for xp, yp were computed as —0.42 mm and +0.14 mm
respectively. More interestingly, the rms values for the
residuals on the six station, twelve photograph solution reduced
from 5 um to 4 um. The object photographed was a large
water storage dam which is almost planar in shape.
Although the precision of the object co-ordinates on the dam
wall did not significantly improve, this experience has tempted
the author to offer the following tentative advice. For small
format non-metric photogrammetric exercises of low to medium
accuracy, say « 1:5000, there appears little benefit in
incorporating xp, yp in bundle adjustments. This is especially
so if no fiducial marks are present and the frame edges are used
as pseudo fiducials. The decentering distortion parameters P1,
P^ appear to suffice. On the other hand, for more accurate tasks
with medium-sized camera formats and with cameras
possessing fiducial marks, the determination of xp, yp and their
application in conjunction with P, P^, rather than the sole use
of P1, P5, is recommended. If the values of Xp, yp approach or
exceed 0.5 mm, then their application is also recommended
rather than reliance on P, P» alone.
4. FILM UNFLATNESS AND STRETCH
4.] Film Deformation From Planar
The mathematics of all analytical photogrammetry is based on
the assumption that the image points are co-planar. This
implies that the film in the image plane must be flat during
exposure. Non-metric cameras usually do not possess a film
flattening mechanism and the shape which film takes has been
studied by several researchers, notably Fraser (1982) for
70 mm cameras and Donnelly (1988) for 35 mm cameras.
Fraser (1982) used the set of AP's described by equations (7)
and (8) to study the film unflatness in a 500 ELM Hasselblad
used to photograph a "cube" of targets from four exposure
stations arranged for convergent imaging. As increasing
numbers of AP's were used, the rms values of the residuals for
the plate co-ordinates were reduced, but the rms error of the
object point co-ordinates increased. The dilemma of reducing
