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The Multi-Dimensional Hartley Transform as a Basis for Volume Rendering

by T. Theußl, R. F. Tobler, and E. 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 data sets of the human head and the human kidney are courtesy Tiani Medgraph GesmbH, Vienna.

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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 Fourier Transform (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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Source code

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Figures in the paper (JPEG)

[Fig. 1a][Fig. 1b]Figure 1a, Figure 1b:
Several function reconstruction filters (a) : the ideal function reconstruction filter, the sinc function, the Hamming windowed sinc function (where the Hamming window itself is also depicted) and the tent function, which corresponds to linear interpolation, and the corresponding frequency responses (b).
[Fig. 2a][Fig. 2b]Figure 2a, Figure 2b:
Two images generated with frequency domain volume rendering: a CT data set of a human head (a) and a MRI data set of a human kidney (b).

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Thomas Theußl, last update on January 5, 2000.