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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B7. Istanbul 2004
There have been some studies on the absolute orientation of
stereo models using DEMS as control information. This work is
known as DEM matching. The absolute orientation of the
models using DTMs as control information was first proposed
by Ebner and Mueller (1986), and Ebner and Strunz (1988).
Afterwards, the functional model of DEM matching has been
formulated by Rosenholm and Torlegard (1988). This method
basically estimates the 3D similarity transformation parameters
between two DEM patches, minimizing the least square
differences along the Z axes. Several applications of DEM
matching have been reported (Karras and Petsa, 1993, Pilgrim,
1996, Mitchell and Chadwick, 1999, Xu and Li, 2000). Maas
(2000) successfully applied a similar method to register
airborne laser scanner strips, among which vertical and
horizontal discrepancies generally show up due to GPS/INS
accuracy problems. Another similar method has been presented
for registering surfaces acquired using different methods, in
particular, laser altimetry and photogrammetry (Postolov,
Krupnik, and McIntosh, 1999). Furthermore, techniques for
2.5D DEM surface matching have been developed, which
correspond mathematically with Least Squares Image Matching.
The DEM matching concept can only be applied to 2.5D
surfaces, whose analytic function is described in the explicit
form, i.e. z = f(x,y). Of course, this formulation has several
problems in the matching of solid (3D) surfaces.
Although the registration of 3D point clouds is a very active
research area in both Computer Vision and Photogrammetry,
there is not such a method that has a complete capability to the
following three properties: matching of multi-scale data sets,
matching of real 3D surfaces without any limitation, fitting the
physical reality of the problem statement as good as possible.
The proposed work completely meets these requirements.
The Least Squares Matching concept had been applied to many
different types of measurement and feature extraction problems
due to its high level of flexibility and its powerful mathematical
model: Adaptive Least Squares Image Matching (Gruen, 1984,
Gruen, 1985a), Geometrically Constrained Multiphoto
Matching (Gruen and Baltsavias, 1988), Image Edge Matching
(Gruen and Stallmann, 1991), Multiple Patch Matching with 2D
images (Gruen, 1985b), Multiple Cuboid (voxel) Matching with
3D images (Maas, 1994, Maas and Gruen, 1995), Globally
Enforced Least Squares Template Matching (Gruen and
Agouris, 1994), Least Squares B-spline Snakes (Gruen and Li,
1996). For a detailed survey the author refers to (Gruen, 1996).
If 3D point clouds derived by any device or method represent
an object surface, the problem should be defined as a surface
matching problem instead of the 3D point cloud matching. In
particular, we treat it as least squares matching of overlapping
3D surfaces, which are digitized/sampled point by point using a
laser scanner device, the photogrammetric method or other
surface measurement techniques. This definition allows us to
find a more general solution for the problem as well as to
establish a mathematical model in the context of LSM.
Our proposed method, Least Squares 3D Surface Matching
(LS3D), estimates the 3D transformation parameters between
two or more fully 3D surface patches, minimizing the Euclidean
distances between the surfaces by least squares. This
formulation gives the opportunity of matching arbitrarily
oriented 3D surface patches. An observation equation is written
for each surface element on the template surface patch, i.e. for
cach sampled point. The geometric relationship between the
conjugate surface patches is defined as a 7-parameter 3D
similarity transformation. This parameter space can be extended
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or reduced, as the situation demands it. The constant term of the
adjustment is given by the observation vector whose elements
are Euclidean distances between the template and search surface
elements. Since the functional model is non-linear, the solution
is iteratively approaching to a global minimum. The unknown
transformation parameters are treated as stochastic quantities
using proper weights. This extension of the mathematical model
gives control over the estimation parameters. The details of the
mathematical modeling of the proposed method, the
convergence behaviour, and the statistical analysis of the
theoretical precision of the estimated parameters are explained
in the following section. The experimental results based on
registration of close-range laser scanner and photogrammetric
point clouds are presented in the third section. The conclusions
are given in the last section.
2. LEAST SQUARES 3D SURFACE MATCHING
2.1 The Estimation Model
Assume that two different surfaces of the same object are
digitized/sampled point by point, at different times (temporally)
or from different viewpoints (spatially). f(x,y,z) and g(X,y.z) are
conjugate regions of the object in the left and right surfaces
respectively. The problem statement is finding the
correspondent part of the template surface patch f(x,y,z) on the
search surface g(x,y, z).
Ê(X,y,z) = g(x, y, z) (1)
According to Equation (1) each surface element on the template
surface patch f(x,y,z) has an exact correspondent surface
element on the search surface patch g(x,y,z), if both of the
surface patches would be continuous surfaces. In order to model
the random errors, which come from the sensor, environmental
conditions or measurement method, a true error vector e(X.y,z)
has to be added.
f(x,y,z)re(x,v.2) 2: gix, y,z) (2)
The matching is achieved by minimizing a goal function, which
measures the Euclidean distances between the template and the
search surface elements. Equation (2) is considered observation
equations, which functionally relate the observations f(x, y.z) to
the parameters of g(x,y,z). The final location is estimated with
respect to an initial position of g(x,y,z), the approximation of
the conjugate search surface patch 2'(x.y,z).
To express the geometric relationship between the conjugate
surface patches, a 7-parameter 3D similarity transformation is
used. Depending on the deformation between the template and
the search surfaces, the geometric relationship could be defined
using any other type of 3D transformation methods, e.g. 12-
parameter affine, 24-parameter tri-linear, or 30-parameter
quadratic family of transformations.
X-t,-*m(njXo t n;yo t n3zo)
ysl,tm(rnX,-tr05yg* 032.) (3)
Z-it,tm(nXo t yo t rz,
where rj = R(o,p,K) are the elements of the orthogonal rotation
matrix, [t, t, t,]" is the translation vector, and m is the central
dilation.
In order to perform least squares estimation, Equation (2) must
. . . . . d .
be linearized by Taylor expansion, ignoring 2" and higher order
terms.
