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stable object can be used for calibration, there is no need 
for an accurate calibration normal. Nonetheless, since ab- 
solute measurements are required, one accurate distance is 
given from a scale bar to fix the scale. 
2.3 Sensor Performance 
To be able to assess the accuracy of our sensor system and 
to determine the amount and nature of noise we have per- 
formed a series of tests using a certified test object. The ob- 
ject is a precise sphere with a diameter of 100 mm which is 
certified to 0.001 mm in shape. We acquired several shots 
of the object with the sensor. The range data was then fit 
to a sphere. We have used an implicit polynomial for the 
fitting. Least squares adjustment was performed using an 
eigenvalue approach. The minimization criterion we used 
is the algebraic distance. On a measurement area of about 
300 x 300 mm we have found the standard deviation of the 
error of fit to be 0.02 mm. The sampling distance on the 
object's surface is approximately 2 tenth of a millimeter. 
Of the 25000 points tested 99.8% were below 0.07 mm in 
deviation. The exact distribution of deviations is given in 
figure 2. 
More important than the magnitude of noise is the nature 
of the noise on the object’s surface. Since the system uses 
two components incorporating a grid structure we expe- 
rience Moiré effects on the surface data. They result in 
concentric ripples across the surface (see figure 2). The 
amplitude of the ripples is approximately one hundredth 
of a millimeter. This type of noise is actually worse than 
purely statistical noise since it is more difficult to filter out 
and it locally changes surface characteristics. However the 
sensors accuracy is still more than adequate and allows to 
capture shape in great detail. 
3 THREE-DIMENSIONAL SHAPE 
In order to find an initial grouping of the pixels of a range 
image we have to establish quantities characterizing the lo- 
cal behavior of a surface. The following describes the most 
fundamental quantities and gives the mathematical formu- 
las to compute these quantities (do Carmo, 1976). 
3.1 Fundamentals 
Any parameterized surface X in three-dimensional space 
is given by the projection of an open set U over 3? into the 
space X? : 
Xi VER AN 
There are three fundamental ways to describe a surface: 
e using a vector-function 
e using an explicit function 
X: = Flu,0) 
e using an implicit expression 
X: F(u,v,w)=0 
The partial derivatives a and t are noted as X,, and 
X,. In the case of a vector function these partial deriva- 
tives are easily computed as: 
Oz(u,) Oy(uv) Oz(u,v) 
  
  
Xu e Ou ^" Ou ? Ou 
T = dz (u,v) Oy(u,v) Oz(u,v) 
v ov ^ Qu ^ Ou 
Every explicit function can be converted to vector notation: 
X = (u,v, Flu,v)) 
and its partial derivatives are given by: 
+ lia OF (u,v) f OF (u,v) 
ou ov 
It is important to note, that every surface can be locally 
described by an explicit function. 
3.2 Fundamental Forms 
A differentiable surface X is given with the condition XX 
X, # 0. The unit normal vector N is then given as 
RT Xu vU 
Ne 
[Xu x Xo 
The expression 
I = dX dX 
= (Xadu+ X dv) - (X udu + X dv) 
Edu? + 2Fdudv + Gdv? 
with 
Fey. XPS Ze =X 5 
is called first fundamental form. 
The expression 
IH - dX-dN 
m -Oduu Xue): (N. du3 N,dv) 
— Ldu? + 2Mdudv + Ndv? 
with 
= T > = Rr ull: 
L--—X.NSM = "(Ka Not Xo- Nu), N = Xu: N 
is called second fundamental form. The above condition 
can be rewritten as 
L = Xuu N, M = Xu N, N= Ku: N 
—139- 
———OMZ