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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 extent. 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. The best performing windows are Blackman, Kaiser and Gaussian windows. The latter two are particularly useful since both have a parameter to control their shape, 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. Optimal values for function and first derivative reconstruction for window widths of two, three, four and five are presented explicitly.

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BibTeX

@techreport{Theussl-2000-MWI,
  title =      "Mastering Windows: Improving Reconstruction",
  author =     "Thomas Theu{\ss}l and Helwig Hauser and Meister Eduard
               Gr\"{o}ller",
  year =       "2000",
  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 extent.
               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. The best
               performing windows are Blackman, Kaiser and Gaussian
               windows. The latter two are particularly useful since both
               have a parameter to control their shape, 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. Optimal values for
               function and first derivative reconstruction for window
               widths of                  two, three, four and five are
               presented explicitly.",
  month =      apr,
  number =     "TR-186-2-00-08",
  address =    "Favoritenstrasse 9-11/E193-02, A-1040 Vienna, Austria",
  institution = "Institute of Computer Graphics and Algorithms, Vienna
               University of Technology ",
  note =       "human contact: technical-report@cg.tuwien.ac.at",
  keywords =   "Taylor series expansion, frequency response, windowing,
               ideal reconstruction",
  URL =        "https://www.cg.tuwien.ac.at/research/publications/2000/Theussl-2000-MWI/",
}