Volume Rendering using the
Fourier Projection-Slice Theorem
Proc. Graphics Interface '92,
Vancouver, British Columbia, May, 1992,
Canadian Information Processing Society,
The Fourier projection-slice theorem states that the inverse transform of a
slice extracted from the frequency domain representation of a volume yields a
projection of the volume in a direction perpendicular to the slice. This
theorem allows the generation of attenuation-only renderings of volume data in
O(n^2 log N) time for a volume of size n^3. In this paper, we show how more
realistic renderings can be generated using a class of shading models whose
terms are Fourier projections. Models are derived for rendering depth cueing
by linear attenuation of variable energy emitters and for rendering directional
shading by Lambertian reflection with hemispherical illumination. While the
resulting images do not exhibit the occlusion that is characteristic of
conventional volume rendering, they provide sufficient depth and shape cues to
give a strong illusion that occlusion exists.
Additional information available:
Figures 1 through 6 were handdrawn. No online version currently exists.
- PDF without figures
Optical scan with figures
- Lossless TIFF of figures 7 through 13
These images are similar but not identical to those
in the published paper.
conventional volume rendering (similar to figure 7).
Upper-right: attenuation-only (i.e. X-ray) projection (similar to
Lower-left: depth-cued X-ray (similar to figure 10).
Lower-middle: directionally shaded X-ray (similar to figure 11).
Lower-right: depth-cued and directionally shaded X-ray (similar to
All except upper-left were generated using the Fourier
- MPEG movie (99K bytes)
of an occluding conventional volume rendering (left) and a
non-occluding frequency domain volume rendering (right). The frequency
domain rendering includes linear depth cueing and directional diffuse
reflection. The dataset is a 96^3 voxel CT scan of a portion of a
human skull mounted in a lucite head cast.