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Mastering Windows: Improving Reconstruction
by
Thomas Theußl,
Helwig Hauser, and
Meister Eduard Gröller.
Project Duration: 1999 
This page assembles some results (figures)
of work that is part of our visualization research.
The figures are provided in JPEG format.
This work has been funded by the V^{is}M^{ed}
project. V^{is}M^{ed} is supported by Tiani Medgraph, Vienna and the Forschungsförderungsfonds
für die gewerbliche Wirtschaft, Austria. The medical
data sets are courtesy Tiani Medgraph GesmbH, Vienna.
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Abstract
Ideal reconstruction filters, for function or arbitrary derivative
reconstruction, have to be bounded in order to be practicable since
they are infinite in their spatial extend. This can be accomplished by
multiplying them with windowing functions. In this paper, we discuss
and assess the quality of commonly used windows and show that most of
them are unsatisfactory in terms of numerical accuracy. Particularly
useful are the Kaiser and Gaussian windows since both have a parameter
to control the shape of the window, which, on the other hand, requires
to find appropriate values for these parameters. We show how to derive
optimal parameter values for Kaiser and Gaussian windows using a
Taylor series expansion of the convolution sum.
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Additional images can be found at:
Figures in the paper (JPEG)

Figure 1:
A CT scan of a head reconstructed with (a) linear interpolation and
central differences with linear interpolation, (b) CatmullRom spline
and derivative and (c) Kaiser windowed sinc and cosc of width three
with numerically optimal parameters.


Figure 2:
Rectangular, Bartlett, Welch, Parzen, Hann, Hamming, Blackman and
Lanczos windows of width two on top, below the frequency responses of
corespondingly windowed sincs and coscs.


Figure 3:
Kaiser and Gaussian windows of width two with varying parameters on
top, below again the frequency responses of corespondingly windowed
sincs and coscs.


Figure 4:
Coefficient plot of Taylor series expansion for windowed cosc filters
with width three.


Figure 5:
Coefficient plot of Taylor series expansion for Kaiser windowed sinc
and cosc with varying parameters and window width two. The right image
is a closeup to the minima of these functions.


Figure 6:
Marschner Lobb data set
(a  f) and data set of a human kidney (g  l) reconstructed with
(a,g) linear interpolation and central differences and linear
interpolation, (b,h) CatmullRom spline and derivative, (c,i)
truncated (rectangular window) sinc and cosc of width three, (d,j)
Blackman windowed sinc and cosc with window width three, (e,k) Kaiser
windowed sinc and cosc with window width three and numerically optimal
parameters and (f,l) Gaussian windowed sinc and cosc with window width
three and numerically optimal parameters.

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Thomas Theußl, last update on March 30, 2000.