Abstract
A method for the automatic co-registration of 3D surfaces is
presented. The method utilizes the mathematical model of
Least Squares
2D image matching and extends it for solving
the
3D surface matching problem. The transformation param-
eters of the search surfaces are estimated with respect to a
template surface. The solution is achieved when the sum of
the squares of the
3D spatial (Euclidean) distances between
the surfaces are minimized. The parameter estimation is
achieved using the Generalized Gauss-Markov model. Execu-
tion level implementation details are given. Apart from the
co-registration of the point clouds generated from spaceborne,
airborne and terrestrial sensors and techniques, the proposed
method is also useful for change detection,
3D comparison,
and quality assessment tasks. Experiments using terrain data
examples show the capabilities of the method.
Introduction
With the availability of the various sensors and automated
methods, the production of large numbers of point clouds is
no longer particularly notable. In many cases, the object of
interest is covered by a number of point clouds, which are
referenced in different spatial or temporal datums. Therefore,
the issue of co-registration of point clouds (or surfaces) is an
essential topic in
3D modeling.
In terrestrial laser scanning practice, special targets
provided by the vendors, e.g., Zoller⫹Fröhlich, Leica, and
Riegl, are mostly used for the co-registration of point clouds.
However, such a strategy has several deficiencies with
respect to fieldwork time, personnel and equipment costs,
and accuracy. In a recent study, Sternberg et al. (2004)
reported that registration and geodetic measurements com-
prise 10 to 20 percent of the total project time. In another
study, a collapsed 1,000-car parking garage was documented
in order to assess the damage and structural soundness of the
building. The laser scanning took three days, while the
conventional survey of the control points required two days
(Greaves, 2005). In a project conducted by our research group
at Pinchango Alto (Lambers et al., 2007), two persons set the
targets in the field and measured them using the real-time
kinematic
GPS technique in one and one-half days.
As well as fieldwork time, accuracy is another impor-
tant concern. The target-based registration methods may not
exploit the full accuracy potential of the data. The geodetic
measurements naturally introduce some error, which might
exceed the internal error of the scanner instrument. In
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
March 2010 307
Department of Information Technologies, Isik University,
34980 Sile, Istanbul, Turkey, and formerly with the Institute
of Geodesy and Photogrammetry, ETH Zurich,
Switzerland (akca@isikun.edu.tr).
Photogrammetric Engineering & Remote Sensing
Vol. 76, No. 3, March 2010, pp. 307–318.
0099-1112/10/7603–0307/$3.00/0
© 2010 American Society for Photogrammetry
and Remote Sensing
Co-registration of Surfaces by
3D Least Squares Matching
Devrim Akca
addition, the targets must be kept stable during the whole of
the data acquisition campaign. This might be inconvenient
when the scanning work stretches over more than one day.
On the other hand, one important advantage of the target
based methods should not be ignored. Targets are essentially
required in projects where the absolute orientation to an
object coordinate system is needed.
The surface based registration techniques stand as
efficient and versatile alternatives to the target-based tech-
niques. They simply bring the strenuous additional fieldwork
of the registration task to the computer in the office, at the
same time optimizing the project cost and duration, and
achieving a better accuracy. In the last decade, surface-based
registration techniques have been studied extensively. The
large number of research activities on the topic demonstrates
the relevance of the problem. For an exhaustive literature
review, we refer to Gruen and Akca (2005).
The co-registration is crucially needed wherever spatially
related data sets can be described as surfaces and has to be
transformed to each other. Examples can be found in medicine,
computer graphics, animation, cartography, virtual reality,
industrial inspection and quality control, change detection,
spatial data fusion, cultural heritage, photogrammetry, etc.
We treat the co-registration problem as a Least Squares
matching of overlapping surfaces. Least Squares matching is
a mathematical concept, which was originally developed
for automatic point transfer on stereo or multiple images
(Ackermann, 1984; Pertl, 1984; Gruen, 1985). More recently,
it has been extended to many problem-specific cases, e.g.,
3D voxel matching (Maas and Gruen, 1995) and the line
feature extraction techniques (Gruen, 1996).
This work, called
3D Least Squares surface matching
(
LS3D), is another straightforward extension of the 2D Least
Squares image matching and has the same underlying ideas
and concepts. The next section introduces the basic estima-
tion model. The execution aspects and the implementation
details are extensively elaborated. Particular attention is
given to the surface-to-surface correspondence search, outlier
detection, and the computational acceleration. In previous
work, examples covering the co-registration of the terrestrial
laser scanning data sets were given (Gruen and Akca, 2005;
Akca, 2007a). In the work presented here, the experimenta-
tion concentrates on the topographic data sets, mostly
generated by use of photogrammetric and airborne lidar
techniques. Apart from the co-registration of point clouds, the
proposed method can be utilized for many types of geo-data
308 March 2010
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
analyses. The experiments given in the third section show
examples of how the
3D surface matcher performs more
advantageously than the conventional methods for change
detection and quality assessment tasks.
Least Squares 3D Surface Matching
The Basic Estimation Model
Two 3D surfaces are subject to a co-registration procedure.
The search surface g (x, y, z) is going to be transformed to
the spatial domain of the template surface f (x, y, z). Both f
(x, y, z) and g (x, y, z) are piecewise discrete representa-
tions of the continuous function of the object surface. In
the current implementation, a triangular mesh or a grid
mesh type of representation is used. In the case of the
triangular mesh representation, the piecewise surface is
composed of planar triangle elements; in the same manner,
the grid mesh representation is composed of bi-linear grid
surface elements.
Surface topology is established simply by loading the
data files, e.g., range scanner point clouds or photogrammetric
digital elevation models (
DEM), in row or column order. The
point spacing is by definition irregular, wherefore the regular
grid
DEM is an under-capability case.
Every template surface element is matched to a conjugate
search surface element to establish the surface-to-surface
correspondence. This is achieved by a correspondence
operator. Occlusions and outliers are the perturbation cases,
which are excluded from the system automatically. While all
template surface elements are sought by the operator, some
of the search surface elements might not coincide at all.
If a matching is established between the two surface
elements f (x, y, z) and g (x, y, z), the following equation
holds:
(1)
where e (x, y, z) is a true error vector covering the random
errors of the template and search surfaces, which are
assumed to be uncorrelated. Equation 1 is the observation
equation, which is set up for each template element that
has a valid surface match. The transformation parameters
of the search surface g (x, y, z) are variables to be
estimated.
Here, we have a peculiar case where the search surface
g (x, y, z) is not analytically continuous; rather it is composed
of discrete finite elements in the form of planar triangles or
bilinear grids. As a consequence, the mathematical derivation
operation cannot be performed analytically.
Equation 1 is non-linear. It is linearized by the Taylor
Series expansion:
(2)
with notations:
(3)
where the terms g
x
, g
y
, and g
z
are the numerical first
derivatives of the function g (x, y, z), which are defined
as the components of the local surface normal vector n.
Their calculation depends on the analytical representation
of the search surface elements, i.e., planar triangles or
bilinear grids. The derivative terms are given as the x-y-z
g
x
⫽
0g
0
(x, y, z)
0x
,
g
y
⫽
0g
0
(x, y, z)
0y
,
g
z
⫽
0g
0
(x, y, z)
0z
⫹
0g
0
(x, y, z)
0z
dz ⫺ (f(x, y, z) ⫺ g
0
(x, y, z))
⫺e(x, y, z) ⫽
0g
0
(x, y, z)
0x
dx ⫹
0g
0
(x, y, z)
0y
dy
f(x, y, z) ⫺ e(x, y, z) ⫽ g(x, y, z)
components of the local normal vectors, which are com-
puted at the exact matching location on the respective
search surface elements:
[g
x
g
y
g
z
]
T
⫽ n ⫽ [n
x
n
y
n
z
]
T
. (4)
The terms dx, dy, and dz are the differentiation terms of the
selected
3D transformation model. The geometric relation-
ship is established with a seven-parameter
3D similarity
transformation whose differentiation gives:
(5)
where a
ij
are the coefficient terms. Their expansions are
given in Akca (2007b). The vector [t
x
t
y
t
z
]
T
is the translation
vector, the scalar m is the uniform scale factor, and the
angles
,
, and
are the elements of the orthogonal rotation
matrix R. Depending on the characteristics of the template
and search surfaces, any other higher order transformation
model, e.g., a
3D affine or polynomial model, can be chosen.
By substituting Equations 3 and 5, Equation 2 gives the
following form:
(6)
where g
0
(x, y, z) is the coarsely aligned search surface. The
coarse alignment is performed using the initial approxima-
tions of the transformation parameters (t
0
x
, t
0
y
, t
0
z
, m
0
,
0
,
0
,
and
0
). The term f (x, y, z) ⫺ g
0
(x, y, z) denotes the Euclid-
ean distance between the template and the corresponding
search surface elements.
Equation 6 gives in matrix notation:
(7)
Where A is the design matrix, x ⫽ [dt
x
dt
y
dt
z
dm d
d
d
]
T
is the parameter vector, P ⫽ P
ll
is the priori weight
coefficient matrix, and l ⫽ f (x, y, z) ⫺ g
0
(x, y, z) is the
discrepancy vector.
With the statistical expectation operator E and the
assumptions
, (8)
the system (Equation 7) is a Gauss-Markov estimation model.
The unknown parameters are introduced into the system as
fictitious observations:
(9)
where I is the identity matrix, l
b
is the (fictitious) observa-
tion vector, and P
b
is the associated weight coefficient
matrix. By selecting a very large weight value ((P
b
)
ii
: ⬁), an
individual parameter can be assigned as constant. In com-
monly used topographic data sets, scale differences, even in
some cases the rotational differences do not occur. This
extension is especially useful to avoid such over-parameteri-
zation problems and for the flexible selection of the appro-
priate degree of freedom (
DOF).
The joint system of Equations 7 and 9 is a Generalized
Gauss-Markov model. The Least Squares solution gives the
⫺e
b
⫽ Ix ⫺ l
b
, P
b
E(e) ⫽ 0, E1ee
T
2⫽ s
0
2
P
⫺1
⫺e ⫽ Ax ⫺ l, P
⫺(f (x, y,z) ⫺ g
0
(x,y, z))
⫹ (g
x
a
13
⫹ g
y
a
23
⫹ g
z
a
33
)dk
⫹ (g
x
a
12
⫹ g
y
a
22
⫹ g
z
a
32
)dw
⫹ (g
x
a
11
⫹ g
y
a
21
⫹ g
z
a
31
)dv
⫹ (g
x
a
10
⫹ g
y
a
20
⫹ g
z
a
30
)dm
⫺e(x,y,z) ⫽ g
x
dt
x
⫹ g
y
dt
y
⫹ g
z
dt
z
J
d x
d y
dz
K
⫽
J
d t
x
d t
y
dt
z
K
⫹ d m
J
a
10
a
20
a
30
K
⫹
J
a
11
a
12
a
13
a
21
a
22
a
23
a
31
a
32
a
33
KJ
d v
d w
d k
K
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
March 2010 309
estimated parameter vector and the variance factor as in the
following equations:
(10)
(11)
where stands for the Least Squares estimator, r is the
system redundancy, and the residual vectors v and v
b
are
the Least Squares estimation of the true error vectors ⫺e and
⫺e
b
, respectively.
The solution is iterative. At the end of each iteration,
the search surface is transformed to a new state using the
updated set of transformation parameters:
(12)
(13)
. (14)
At each iteration, the design matrix A and the discrep-
ancies vector l are re-evaluated. The iteration stops if each
element of the vector falls below a certain limit:
(15)
where the criteria c
i
is selected as 1 ppm (⫽10
⫺6
) for the
scale factor, between 1/10 and 1/100 of the least significant
digit for the translation elements, and 10
⫺3
grad for the
rotation elements.
The functional model is non-linear; thus, the initial
approximations of the parameters are required. The initial
approximations should be given or should be computed
prior to the matching. In this paper, the experiments use
geo-datasets, which are crudely aligned. Thus, the initial
approximations of the rotation and translation parameters
are assumed to be zero. This might not be the case for the
terrestrial laser scanning point clouds. In such cases, the
initial approximations can be provided by interactively
selecting three common points on both surfaces before
the matching.
Correspondence Search
For every template surface element, the correspondence
operator seeks a minimum Euclidean distance location on
the search surface. The template surface elements are
represented by the data points. Accordingly, the procedure
becomes a point-to-plane distance or point-to-bilinear
surface distance computation. When a minimum Euclidean
distance is found, in a subsequent step the matching point is
tested to determine whether it is located inside the search
surface element (point-in-polygon test). If not, this element
is disregarded and the operator moves to the next search
surface element with the minimum distance. Hypothetically,
the correspondence criterion searches a minimum magnitude
vector that is perpendicular to the search surface element
and passes through the template point.
In the most straightforward case, the computational
complexity is of order O(N
t
N
s
), where N
t
is the number of
template elements and N
s
is the number of search elements.
This computational expense is reduced by constricting the
search space within
3D boxes. The details are given in the
Computational Acceleration sub-section.
Outlier Detection and Reliability Aspects
Detection of the false correspondences with respect to the
outliers and occlusions is a crucial part of every surface
ƒ
dp
i
ƒ
6 c
i
, dp
i
僆 {dt
x
, dt
y
, dt
z
, dm, dv, dw, dk}
x
N
[vwk]
T
⫽ [v
0
w
0
k
0
]
T
⫹ [d
N
v d
N
w
d
N
k
]
T
m ⫽ m
0
⫹ d
N
m
[t
x
t
y
t
z
]
T
⫽ [t
0
x
t
0
y
t
0
z
]
T
⫹ [d
N
t
x
d
N
t
y
d
N
t
z
]
T
N
s
N
0
2
⫽
v
T
Pv ⫹ v
b
T
P
b
v
b
r
N
x ⫽ (A
T
PA ⫹ P
b
)
⫺1
(A
T
Pl ⫹ P
b
l
b
)
matching method. We use the following strategies in order
to localize and eliminate the outliers and the occluded parts.
A median type of filtering is applied prior to the
matching. For each point, the distances between the
central point and its k-neighborhood points are calculated.
In our implementation, k is selected as 8. If most of those
k-distance values are much greater than the average point
density, the central point is likely to be an erroneous
point on a poorly reflecting surface (e.g., window or glass)
or a range artifact due to surface discontinuity (e.g., points
on the object silhouette). The central point is discarded
according to the number of distances that are greater than
a given distance threshold.
In the course of iterations, a simple weighting scheme
adapted from the robust estimation methods is used:
(16)
The constant value K can be altered according to the
task. If it is an ordinary surface co-registration task, it
should be set to a high value (e.g., K ⬎8 or ⬎10) to reduce
type I errors confidently. Because of the high redundancy of
a typical data set, a certain number of occlusions and/or
smaller outliers, i.e., type II errors, do not have significant
effects on the estimated parameters. If it is a change detec-
tion or deformation study, the constant value K should be
selected based on the a priori knowledge in order that the
changed or deformed parts are excluded from the estimation.
Finally, the correspondences coinciding with mesh
boundaries are excluded from the estimation. The mesh
boundaries represent the model borders, and in addition the
data holes inside the model. The data holes are possibly due
to occlusions. Rejecting the surface correspondences on the
mesh boundaries effectively eliminates the occlusions.
Precision
The quality of the estimated parameters can be checked
through the a posteriori co-variance matrix.
The theoretical precisions of the transformation parame-
ters are optimistic, mainly due to the stochastic properties
of the search surfaces that have not been considered as
such in the estimation model, as is typically done in Least
Squares matching (Gruen, 1985). The omissions are expected
to be minor and do not disturb the solution vector signifi-
cantly. However, the a posteriori covariance matrix will be
affected by the neglected uncertainty of the function values
g (x, y, z). This causes deterioration in the realistic precision
estimates. More details on this issue can be found in Gruen
(1985), Maas (2002), Gruen and Akca (2005), and Kraus
et al. (2006).
Computational Acceleration
Searching for correspondence is guided by an efficient boxing
algorithm (Chetverikov, 1991), which partitions the search
space into voxels. For a given surface element, the correspon-
dence is searched only in the box containing this element
and in the adjacent boxes (Figure 1a). The original publica-
tion concerned
2D point sets. It is straightforwardly extended
here to the
3D case.
Let points a
i
= {x
i
, y
i
, z
i
} S, i ⫽ 0, 1,..., N
s
⫺ 1,
represent the object S
ᑬ
3
, and be kept in list L
1
in
spatially non-ordered form. The boxing data structure
consists of a rearranged point list L
2
and an index matrix
I ⫽ I
u, v, w
whose elements are associated with individual
boxes: u,v,w ⫽ 0,1, ..., M ⫺ 1. The items of L
2
are the
coordinates of N
s
points placed in the order of boxes. The
index matrix I contains integers indicating the beginnings of
the boxes in L
2
(Figure 1b).
僆
僆
1P2
ii
⫽ e
1if|1v2
i
|6 K
N
s
0
0 else
310 March 2010
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
(a) (b)
Figure 1. (a) The 3D boxing bounds all the data points,
and (b) The boxing data structure.
Initialization: Defining the Box Size
Step 1. Recall min, max{x
i
, y
i
, z
i
} of data volume.
Step 2. Define the number of boxes along the x⫺y⫺z
axes. For the sake of simplicity, they are given
the same (M) here.
Pass 1: Computing I
Step 1. Allocate an M ⫻ M ⫻ M size accumulator array
B ⫽ B
u,v,w
, which is to contain the number of
points in each box.
Step 2. Scan L
1
and fill B. For any point a
i
, the box
indices are as follows:
(17)
where stands for the truncation operator, and D
X
, D
Y
,
and D
Z
are dimensions of any box along the x⫺y⫺z axes,
respectively.
Step 3. Fill I using the following recursive formula: I
0,0,0
= 0.
For all
(18)
Pass 2: Filling L
2
Step 1. For all u, v, and w, set B
u,v,w
⫽ 0.
Step 2. Scan L
1
again. Use Equation 17, I and B to fill
L
2
. In L
2
, the first point of the (u,v,w)
th
box is
indexed by I while the address of the subse-
quent points is controlled using B whose value
is incremented each time a new point enters the
box. Finally, release the memory area of B.
The memory requirement is of order O(N
s
) for L
2
and
O(M
3
) for I. For the sake of clarity of the explanation, L
2
is
given as a point list containing the x-y-z coordinate values. If
one wants to keep the L
1
in the memory, then L
2
should only
contain the access indices to L
1
or pointers, which directly
point to the memory locations of the point coordinates.
Access Procedure
Step 1. Using Equation 17, compute the indices u
i
, v
i
,
and w
i
of the box that contains point a
i
.
I
u, v, w
⫽
L
I
u, v, w⫺1
⫹ B
u, v, w⫺1
if w 7 0
I
u, v⫺1, M⫺1
⫹ B
u, v⫺1, M⫺1
else if v 7 0
I
u⫺1, M⫺1, M⫺1
⫹ B
u⫺1, M⫺1, M⫺1
else
(u,v,w) Z (0,0,0)
;:
u
i
⫽
j
x
i
⫺ x
min
D
X
k
, v
i
⫽
j
y
i
⫺ y
min
D
Y
k
, w
i
⫽
j
z
i
⫺ z
min
D
Z
k
Step 2. Use the boxing structure to retrieve the points
bounded by the (u,v,w)
th
box. In L
2
, I indexes
the first point, while the number of points in
the box is given by the following formula:
(19)
The access procedure requires O(q) operations, where
q is the average number of points in the box. One of the
main advantages of the boxing structure is a faster and
easier access mechanism than the tree search-based
methods provide. Construction time of the boxing method
O(N
s
) is less than what the tree search methods need, i.e.,
order of O(N
s
logN
s
) for a k-D tree (Greenspan and Yurick,
2003; Arya and Mount, 2005). On the other hand, the tree
search methods obviously need less storage space, which
is only order of O(N
s
).
The boxing structure, and in general all search struc-
tures, are designed for searching the nearest neighborhood in
the static point clouds. In the
LS3D surface matching case,
the search surface, for which the boxing structure is estab-
lished, is transformed to a new state by the current set of
transformation parameters. Nevertheless, there is no need
either to re-establish the boxing structure or to update the
I and L
2
in each iteration. Only the positions of those four
points (Figure 1a) are updated in the course of iterations:
o ⫽ {x
min
, y
min
, z
min
}, x ⫽ {x
max
, y
min
, z
min
}, y ⫽ {x
min
, y
max
,
z
min
}, z ⫽ {x
min
, y
min
, z
max
}. They uniquely define the boxing
structure under the similarity transformation. The access
procedure is the same, except the following formula is used
for the calculation of indices:
(20)
Where ‘⭈’ stands for a vector dot product. If the transforma-
tion is a similarity rather than a rigid body, the D
X
, D
Y
, and
D
Z
values must also be updated in the iterations:
. (21)
In the current implementation, the correspondence
is searched in the boxing structure during the first few
iterations, and at the same time, its evolution is tracked
across the iterations. Afterwards, the searching process is
carried out only in an adaptive local neighborhood accord-
ing to the previous position and change of correspondence.
In any step of the iteration, if the change of correspondence
for a surface element exceeds a limit value, or oscillates,
the search procedure for this element is returned to the
boxing structure again.
Algorithmic Extensions
Multiple Surface Matching
When more than two point clouds with multiple overlaps
exist, a two step solution is adopted. First, the pairwise
LS3D matchings are run on every overlapping pair and a
subset of point correspondences is saved to separate files.
In the global registration step, all these files are passed to a
block adjustment by the independent models procedure
(Ackermann et al., 1973), which is a well known orienta-
tion procedure in photogrammetry. More details can be
found in Akca (2007b).
D
X
⫽
‘
ox
‘
M
, D
Y
⫽
‘
oy
‘
M
, D
Z
⫽
‘
oz
‘
M
u
i
⫽
j
oa
i
#
ox
‘
ox
‘
D
X
k
, v
i
⫽
j
oa
i
#
oy
‘
oy
‘
D
Y
k
, w
i
⫽
j
oa
i
#
oz
‘
oz
‘
D
Z
k
L
I
u, v, w⫹1
⫺ I
u, v, w
if w 6 M ⫺ 1
I
u, v⫹1, 0
⫺ I
u, v, M⫺1
else if v 6 M ⫺ 1
I
u⫹1, 0, 0
⫺ I
u, M⫺1, M⫺1
else if u 6 M ⫺ 1
N
s
⫺ I
M⫺1, M⫺1, M⫺1
else
PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING
March 2010 311
(a)
(b)
(c)
(d)
Figure 2. Experiment “Tucume”: (a) The shaded view of the final composite surface after
the
LS
3
D
surface matching. Note that the overlapping area is delineated by gray borderlines.
The residuals between the fixed and transformed surfaces are shown; (b) after the
LS
3
D
matching; and (c) after the
ICP
matching. The residuals bar unit (d) is in meters.
Simultaneous Matching of Surface Geometry and Intensity
When the object surface lacks sufficient geometric informa-
tion, i.e., homogeneity or isotropicity of curvatures, the basic
algorithm will either fail or will find a side minimum. In this
extension, available attribute information, e.g., intensity,
color, temperature, etc., is used to form quasi-surfaces in
addition to the actual ones. The matching is performed by
simultaneous use of surface geometry and attribute informa-
tion under a combined estimation model (Akca, 2007a).
Further Conceptual Extensions
The further conceptual extensions are given as: the Least
Squares matching of
3D curves, matching of 3D curves or 3D
sparse points (e.g., ground control points) with a 3D surface,
and a general framework, which can perform the multiple
surface matching, the combined surface geometry and
intensity matching, and georeferencing tasks simultaneously
(Akca, 2007b).
Experimental Results
The algorithm was implemented as a stand-alone MS
Windows
™
application with a graphical user interface.
The software package was developed with the C/C⫹⫹
programming language. The presented examples use solely
the basic model, not the algorithmic extensions.
Tucume
The first experiment is the matching of two photogrammetri-
cally derived digital terrain models (
DTM) of an area in
Tucume (Peru). The horizontal resolution of the
DTMs was
5 m. The
DTMs were manually measured as two independent
models from 1:10 000 scale B/W aerial images in one strip
with an overlap of 60 percent in the flight direction. More
details are given in Sauerbier et al. (2004).
Although it is only a
2.5D model, it is a good example of
the weak data configuration case since the overlapping area is
relatively narrow (along the Y-axis) with little information
regarding the surface geometry (Figure 2a). The
LS3D algorithm
was run in
6-DOF mode with three translation and three
rotation parameters. This version showed a large correlation
coefficient 0.99 between the t
y
and angle, which is an over-
parameterization case. Thus, angle was excluded from the
system, and the second version of the computation was run in
5-DOF mode. The results are successful (Table 1). The compu-
tation takes 1.9 and 2.5 seconds for the plane surface (P) and
bi-linear surface (B) representation versions, respectively.
The ratio between the standard deviations of
and
angles is by factor 14. This difference in angular uncertainty
is due to difference in overlapping areas along the X and Y
axes. The residuals between the fixed and transformed
surfaces show a random distribution pattern, except for some
occasional measurement and modeling errors (Figure 2b).
