In studying the network design strate;ies of experts, it was
identified that they employ heuristic knowledge and appear to
use generic networks to overcome the complexity of the sensor
station placement task (Mason,1994). #1 generic network is a
known camera configuration providing te best possible survey
of all points on a particular surface.
The expert system presented by Mason (1994) is based upon
the decomposition of target fields into a number of point
groups that relate to the underlying surfaces of the target fields.
The target fields are decomposed into combinations of surfaces
for which generic camera configurations are known. These
generic networks are then combined into a single, strong
network for the whole object, giving consideration to the
nature of the object, and ensuring the restrictions of the site are
accommodated.
The decomposition of target fields into point groups that are
representative of the surfaces of the object is the main
cognitive operation on which the conceptual model for imaging
geometry configuration is based (Mason 1994). Mason was
able to suggest a partial model for the grouping of points into
the simplest of surfaces for which a generic camera
configuration is known: the plane. Points to be grouped as a
plane must satisfy two criteria (i) proximity - they must be
spatial neighbours; and (ii) uniformity - they must share a
similar surface normal (Mason 1994). These criteria are
appropriate for grouping points into planar regions, however
objects to be surveyed are rarely that simple. Thus, a more
general conceptual model for the grouping of points needs to
be developed (Mason 1994). The work presented in this paper
is part of an investigation to determine the suitability of
proximity and uniformity as criteria for grouping points in
target fields which lie on surfaces other than planes.
2. UNIFORMITY AND PROXIMITY.
Flynn and Jain (1988) claimed that spheres, cylinders and
planes reasonably approximate 85% of manufactured objects.
These three surfaces, along with cones, were chosen as the
primitive surfaces into which target fields are to be
decomposed. The cone was included to increase the range of
objects that can be effectively generalised, or alternatively the
quality of the generalisations. All four of these surfaces belong
to the larger family of quadric surfaces and all can be
represented by the expression:
F(X, Y, Z) = a1X2+ aoŸ” + a3Z” + a4XY + asXZ + asYZ
+ a7X + agŸ + a9Z + ap = 0
X, Y, Z ~ object co-ordinates.
2.1 Uniformity Indicators.
2.1.1 Indicators Reviewed: The uniformity indicators used
for the grouping of points must be applicable to the task of
generalising close range photogrammetric target fields. The
direction of the surface normal at a point has a bearing on the
location of the cameras used to image that point. These surface
normal directions indicate the uniformity of a target field.
Similar surface normal directions suugest points lie on a near
planar surface. Uniformly changing surface normal directions
suggest points lie on the same curved surface. The distance
between neighbouring points on a surface patch could also
6
indicate the uniformity of points. This indicator is, however, of
little value in relation to the intended application. Changes in
the spacing between neighbouring points may bear no relation
to the orientation of the patch on which they are located. The
direction of the surface normal is therefore a potentially useful
quantity for the evaluation of target point uniformity.
A review of uniformity indicators used by researchers in the
field of computer vision and machine intelligence for the
decomposition of complex objects into homogeneous regions
identified a number of potentially useful uniformity indicators.
Krishnapuram and Munshi (1991) trialed a number of
uniformity indicators in their evaluation of image
segmentation techniques. They segmented images using single
uniformity indicators and different combinations of two
indicators (one related to the surface normal and one related to
either the curvature at, or the location of, each point). The five
uniformity indicators were:
Orientation angle of surface normal.
Tilt angle of the surface normal.
Gaussian curvature at a point.
Mean curvature at a point.
Euclidean distance of points from an origin.
Krishnapuram and Munshi (1991) concluded that the
combination of mean curvature at a point and the orientation
angle of the surface normal enabled them to effectively
segment images of both planar and curved objects.
Hoffman and Jain (1987) used three uniformity indicators to
decompose range images in their three dimensional object
recognition system. The uniformity indicators in the minimum
justifiable set that could be effectively utilised in their
application (Hoffman and Jain 1987) are as follows:
Image co-ordinates (r, c) of points,
Range / depth from sensor (F(r,c) = z) of points,
Coefficients of estimated unit surface normal,
vector (Ai+ Bj+ Ck) at points.
The use of these three uniformity indicators requires the
analysis of six parameters, three for the co-ordinates of each
point, and one parameter for each of the three coefficients of
the estimated unit surface normal.
The coefficients of a biquadratic facet model :
Zuv = Bo + B1u+ Bav + B3 u‘ + B4 uv * B5 v^,
evaluated for a surface patch about each point were used by
Jolion et. al. (1991) as a uniformity indicator in the evaluation
of an image segmentation algorithm. As with the uniformity
indicators used by Hoffman and Jain (1987), the use of the
coefficients of the biquadratic facet model requires the
analysis of six parameters.
Flynn and Jain (1988) developed a classification algorithm for
the description of segmented range images, using the
uniformity indicators of minimum curvature and maximum
curvature to discriminate between a sub-set of the quadric
surfaces. The minimum and maximum curvatures are
evaluated at points known to lie on non-planar surfaces, in
order to classify them as lying on spherical, cylindrical, or
conical surface patches. Flynn and Jain made use of the
known distinctive combinations of these curvature measures in
a hierarchical classification process to discriminate between
each surface in the sub-set of quadric surfaces.
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
Surface curv:
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Figure 1. ]
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