Gross - 6 
This application exemplifies the flexibility of B-splines for 3D shape reconstruction and 
for visualization. Both graphics and vision techniques had been employed to generate pic 
tures like those in fig. 3. 
3 The Power of Hierarchies: Wavelet-based Models 
One of the striking disadvantages of any B-spline is the ability to handle very large data 
sets. The exponentially growing amount of data in scientific visualization therefore re 
quires adaptive methods, where level-of-detail and center of focus can be controlled effi 
ciently. It is clear, that we have to set up some hierarchical model descriptions with the 
constraints of well-defined error bounds, localization properties and a sufficiently low al 
gorithmic complexity. Fortunately, functional analysis provided us during the last years 
with wavelets and with the multiresolution analysis, that are well suited for the above mod 
eling tasks [10]. 
Wavelets successfully combine some interesting properties, such as 
• Local support. 
• (Bi)-Orthogonality. 
• L 2 -error bounds. 
• Hierarchical organization. 
• Low computational complexity of the FWT. 
• Provision of local spectral estimates of the data. 
This collection of interesting features can be harvested for both data approximation and 
coding in visualization and vision, as well as for data analysis [6]. 
The following pictures illustrate the local and global approximation behavior of the WT 
applied on a digital terrain model. Let ip a ,b( x >y) be a 2D wavelet basis function, which is 
derived from a mother wavelet xp by scaling and shifting with 
(3) 
a x , a y , b x , by: scaling and shifting parameters. 
A parametric shape approximation of any shape x(s,t) is figured out by