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Abstract

The Fast Hartley Transform (FHT), a discrete version of the Hartley Transform (HT), has been studied in various papers and shown to be faster and more convenient to implement and handle than the corresponding Fast Fourier Transform (FFT). As the HT is not as nicely separable as the FT, a multidimensional version of the HT needs to perform a final correction step to convert the result of separate HTs for each dimension into the final multi-dimensional transform. Although there exist algorithms for two and three dimensions, no generalization to arbitrary dimensions can be found in the literature. We demonstrate an easily comprehensible and efficient implementation of the fast HT and its multi-dimensional extension. By adapting this algorithm to volume rendering by the projection-slice theorem and by the use for filter analysis in frequency domain we further demonstrate the importance of the HT in this application area.

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@techreport{Theussl-1999-MDH,
  title =      "The Multi-Dimensional Hartley Transform as a Basis for
               Volume
Rendering",
  author =     "Thomas Theu{\ss}l and Robert F. Tobler and Meister Eduard
               Gr{\"o}ller",
  year =       "1999",
  abstract =   "The Fast Hartley Transform (FHT), a discrete version of the
               Hartley Transform (HT), has been studied in various papers
               and shown to be faster and more convenient to implement and
               handle than the corresponding Fast Fourier Transform (FFT). 
               As the HT is not as nicely separable as the FT, a
               multidimensional version of the HT needs to perform a final
               correction step to convert the result of separate HTs for
               each dimension into the final multi-dimensional transform.
               Although there exist algorithms for two and three
               dimensions, no generalization to arbitrary dimensions can be
               found in the literature. We demonstrate an easily
               comprehensible and efficient implementation of the fast HT
               and its multi-dimensional extension.  By adapting this
               algorithm to volume rendering by the projection-slice
               theorem and by the use for filter analysis in frequency
               domain we further demonstrate the importance of the HT in   
                             this application area.",
  address =    "Favoritenstrasse 9-11/186, A-1040 Vienna, Austria",
  institution = "Institute of Computer Graphics and Algorithms, Vienna
               University of Technology",
  note =       "human contact: technical-report@cg.tuwien.ac.at",
  month =      oct,
  number =     "TR-186-2-99-21",
  keywords =   "volume rendering, Fourier transform, Hartley transform",
  URL =        "http://www.cg.tuwien.ac.at/research/publications/1999/Theussl-1999-MDH/",
}