Chapter 3 Volumetric Data

Volumetric data is typically a set of samples S(x, y, z, v), representing the value v of some property of the data, at a 3D location (x, y, z). If the value is simply a 0 or a 1, with a value of 0 indicating background and a value of 1 indicating the object, then the data is referred to as binary data. The data may instead be multi-valued, with the value representing some measurable property of the data, including, for example, color, density, heat or pressure. The value v may even be a vector, representing, for example, velocity at each location. In general, the samples may be taken at purely random locations in space, but in most cases the set S is isotropic containing samples taken at regularly spaced intervals along three orthogonal axes. When the spacing between samples along each axis is a constant, then S is called isotropic, but there may be three different spacing constants for the three axes. In that case the set S is anisotropic. Since the set of samples is defined on a regular grid, a 3D array (called also volume buffer, cubic frame buffer, 3D raster) is typically used to store the values, with the element location indicating position of the sample on the grid. For this reason, the set S will be referred to as the array of values S(x, y, z), which is defined only at grid locations. Alternatively, either rectilinear, curvilinear (structured) [12], or unstructured grids, are employed (Figure 1.1) [13].

Figure 1.1: Grid types in volumetric data. a. Cartesian grid,    b. Regular grid, c. Rectilinear grid, d. Curvilinear grid, e. Block structured grid, and   f. Unstructured grid

In a rectilinear grid the cells are axis-aligned, but grid spacing along the axes are arbitrary. When such a grid has been non-linearly transformed while preserving the grid topology, the grid becomes curvilinear. Usually, the rectilinear grid defining the logical organization is called computational space, and the curvilinear grid is called physical space. Otherwise the grid is called unstructured or irregular. An unstructured or irregular volume data is a collection of cells whose connectivity has to be specified explicitly. These cells can be of an arbitrary shape such as tetrahedra, hexahedra, or prisms [14].

The array S only defines the value of some measured property of the data at discrete locations in space. A function f(x, y, z) may be defined over the volume in order to describe the value at any continuous location. The function f(x, y, z) = S(x, y, z) if (x, y, z) is a grid location, otherwise f(x, y, z) approximates the sample value at a location (x, y, z) by applying some interpolation function to S. There are many possible interpolation functions. The simplest interpolation function is known as zero-order interpolation, which is actually just a nearest-neighbor function [15]. The value at any location in the volume is simply the value of the closest sample to that location. With this interpolation method there is a region of a constant value around each sample in S. Since the samples in S are regularly spaced, each region is of a uniform size and shape. The region of the constant value that surrounds each sample is known as a voxel with each voxel being a rectangular cuboid having six faces, twelve edges, and eight corners.

Higher-order interpolation functions can also be used to define f(x, y, z) between sample points. One common interpolation function is a piecewise function known as first-order interpolation, or trilinear interpolation. With this interpolation function, the value is assumed to vary linearly along directions parallel to one of the major axes [14].