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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 VisMed project. VisMed 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)

[Fig. 1] Figure 1:
A CT scan of a head reconstructed with (a) linear interpolation and central differences with linear interpolation, (b) Catmull-Rom spline and derivative and (c) Kaiser windowed sinc and cosc of width three with numerically optimal parameters.
[Fig. 2] [Fig. 2]
[Fig. 2] [Fig. 2]
[Fig. 2] [Fig. 2] 		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.
[Fig. 3] [Fig. 3]
[Fig. 3] [Fig. 2]
[Fig. 3] [Fig. 3] 		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.
[Fig. 4] [Fig. 4]
[Fig. 4] [Fig. 4]
 Figure 4:
Coefficient plot of Taylor series expansion for windowed cosc filters with width three.
[Fig. 5] [Fig. 5] 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.
[Fig. 6] [Fig. 6] [Fig. 6]
[Fig. 6] [Fig. 6] [Fig. 6]
[Fig. 6] [Fig. 6] [Fig. 6]
[Fig. 6] [Fig. 6] [Fig. 6]
 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) Catmull-Rom 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.