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5,3 DSM Postprocessing
Since DSMs are determined by the use of the natural
structure of the teeth due to many degrading effects like
strong reflections, occlusions or overlay of control points
many mismatched DSM points appear, depending on the
quality of the images corresponding to the quality of the
illumination up to more than 30 percent. To overrun this
problem every DSM is postprocessed. In a first step
gross errors are eliminated. In a second step the gravity
point of the DSM is calculated and every DSM point with
a distance from the center that is larger than a certain
threshold is eliminated. The threshold is set manually
and is more less identical with the average diameter of a
tooth corresponding to several millimeters. All DSM
points that were eliminated are interpolated using bilinear
interpolation. In a third step every DSM point is
interpolated using its neighbor points. If the three-
dimensional distance of the original point and the
interpolated one is larger than a threshold that depends
on the sampling rate of the DSM the original point is
skipped and replaced by the interpolated one. In a last
step the DSM is smoothed by using a moving average
algorithm with a window size of 3 x 3 samples.
6. DERIVATION OF DEFORMATION PARAMETERS
The result of DSM generation is a digital surface model
of all teeth of a jaw. The position and orientation in 3D
space is defined by the coordinate system of the control
points on the mirror. For deformation analysis a
reference coordinate system is needed. The definition of
this coordinate system is not very problematic and must
be done by the orthodontist. Only the specialist can
decide what reference coordinate system may be used,
the photogrammetrist provides the tools for definition.
Much more difficult is the definition of object and
reference space for deformation analysis. All teeth that
shall not move during the orthodontic treatment belong to
reference space. Object space means all the teeth the
orthodontist wants to move to a correct position. The 3D
position and rotation changes of the teeth belonging to
object space are calculated with respect to the teeth
belonging to reference space. These teeth have to be
defined by the orthodontist too. He knows what teeth
Shall move and what teeth shell serve as a reference. On
the other hand the photogrammetrist may investigate
whether the teeth belonging to reference space do really
not move and turn. This can be tested applying methods
of deformation analysis coming from engineering
geodesy.
But the main problem is the derivation of position and
orientation changes. Since no artificial target points can
be defined on the teeth's surface deformation parameters
have to be derived using the complete DSM. This is done
by a three-dimensional surface matching algorithm. The
algorithm is similar to least squares 2D greyvalue
template matching. The observation is not a greyvalue
difference, it is the coordinate difference in direction of
the third dimension of the DSMs that shall be matched.
The transformation is done using 6 parameters: 3 shifts
and 3 rotations. A parameter for scale is not used. The
algorithm is fully 3D, this means that any 3D object
described by a DSM can be matched.
251
The z-coordinates of the DSM points can be described in
function of their x and y values using the discrete
functions f and g
Zn f (X). zu (X) (1)
X=(xy) Te(&ntoox) (2a,b)
Defining a vector T of unknown transformation
parameters and a vector X corresponding to the
coordinates of DSM points in xy plane the mathematical
model of 3D DSM matching can be written as
f(X)-e(X)=g(T,X) (3)
where e(X) is a true error vector and g is a function of the
transformation parameters and the planar coordinate
vector X. As equation 3 is nonlinear it must be linearized
leading to the observation equation
jGn-430- gr x) RETR) (4)
With
x" = (dE dn d€ de do dx) (5a)
1=f(X)-g°(X) (5b)
prn 3g T.X) 3g (T.X) ogr.X) agr.x) SE (5c)
9g on at de do EIS
equation 4 results in
-e(X) 2 Ax -1 (6)
Equation 6 forms a set of n DSM correlation equations,
where n is the number of DSM points defining function f.
The system is solved by standard least squares
technique.
Two main problems appear for implementation. Firstly
the derivation of approximations for the transformation
parameters and secondly the choice of an iteration
criterion. The derivation of approximations is done by a
pre-transformation of both DSMs that shall be matched.
In a first step both DSMs are reduced to their gravity
points. As shown in chapter 7 the same DSM was
measured several times. The coordinates of the gravity
points do not change more than 0.1 mm. This leads to
very good approximation values for the shift parameters.
To derive approximations for the rotations and to reduce
correlation of transformation parameters the DSMs are
rotated in a way that their z-coordinates are minimized.
Using this method better results for resampling and
interpolation can be achieved. The choice of a proper
iteration criterion is as problematic as for LSTM. The
main problem is the oscillations of transformation
parameters. The problem is solved the same way as it is
done for LSTM (Beyer 1992).
To get more flexible and to reduce data and computation
time derivation of 3D position and orientation changes is
done in two steps. In a first step every tooth is treated
independently. The DSM of one single tooth is
transformed on the DSM of the same tooth that was
measured in a different time period. In this step every
tooth is described by a feature consisting of four points,
the 'corner points' of the DSM. This has the advantage
that the geometric description of a single tooth
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B5. Vienna 1996
