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Optimal Regular Volume Sampling

by Thomas Theußl, Torsten Möller, and Meister Eduard Gröller.

Project Duration: 2001 -

This page assembles some results (figures) of work that is part of our visualization research. The figures are provided in JPEG format.



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Abstract

The classification of volumetric data sets as well as their rendering algorithms are typically based on the representation of the underlying grid. Grid structures based on a Cartesian lattice are the de-facto standard for regular representations of volumetric data. In this paper we introduce a more general concept of regular grids for the representation of volumetric data. We demonstrate that a specific type of regular lattice - the so-called \emph{body-centered cubic} - is able to represent the same data set as a Cartesian grid to the same accuracy but with 29.3% less samples. This speeds up traditional volume rendering algorithms by the same ratio, which we demonstrate by adopting a splatting implementation for these new lattices. We investigate different filtering methods required for computing the normals on this lattice. The lattice representation results also in lossless compression ratios that are better than previously reported. Although other regular grid structures achieve the same sample efficiency, the body-centered cubic is particularly easy to use. The only assumption necessary is that the underlying volume is isotropic and band-limited - an assumption that is valid for most practical data sets.



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Figures in the paper (JPEG)

[Fig. 1] [Fig. 1]
[Fig. 1] [Fig. 1]
Figure 1:
CT scans of a lobster and a tooth, represented on Cartesian and body-centered cubic grids (left and right images, respectively). The representation via body-centered cubic grids requires approximately 30% less samples.
  spatial
domain 		frequency
domain
rectangular
sampling 		[Fig. 2] [Fig. 2]
hexagonal
sampling 		[Fig. 2] [Fig. 2]
Figure 2:
Regular rectangular and hexagonal sampling in spatial and frequency domain.
[Fig. 3] Figure 3:
One cell of an fcc lattice with base vectors ui. The black dots mark additional sample points (in the center of the faces) as compared to a simple cubic cell.
[Fig. 4] Figure 4:
One cell of a bcc structure with base vectors bi. The only difference to a simple cubic cell is one additional sample point right in the center of the cell, marked with a black dot.
[Fig. 5] Figure 5:
Different indexing schemes. The image on the left corresponds to Eq.3. The figure on the right corresponds to Eq.20.
[Fig. 6] Figure 6:
A bcc grid interpreted as two inter-penetrating Cartesian grids.
[Fig. 7]
(a)
[Fig. 7]
(b)
[Fig. 7]
(c)
Figure 7:
Difference images of analytically calculated gradients to our gradient estimation schemes (see Sec. 3.3), first two columns, and central differences with linear interpolation, third column, for (a) sphere, (b) Sinc, and (c) simplified Marschner-Lobb function. The top rows show the error in magnitude whereas the bottom rows show the angular error.(a) error in magnitude by 30% and an angular error of 15 degrees corresponds to white, (b) amplitude error of 60% and an angular error of 30 degrees corresponds to white, (c) amplitude error of 10% and an angular error of 5 degrees corresponds to white.
[Fig. 8] [Fig. 8]
Figure 8:
Marschner-Lobb data set rendered by splatting with a Cartesian grid on the left and a bcc grid on the right.
Cartesian grid body-centered
cubic grid
[Fig. 9] [Fig. 9]
[Fig. 9] [Fig. 9]
[Fig. 9] [Fig. 9]
Figure 9:
Images generated via splatting on a Cartesian grid on the left respectively a body-centered cubic grid on the right. The body-centered cubic grids require approximately 30% less samples. Only small quality differences are visible, that likely are caused by pre-classification.



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Thomas Theußl, last update on April 4, 2001.