Volumetric data is typically a set S of samples (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 multivalued, with the value representing some measurable property of the data, including, for example, color, density, heat, or pressure.

In general, 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 spacing between samples along each axis is a constant, but there may be three different spacing

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FIGURE 1 Intermixing of a volume-sampled cone with an MRI head using a fuzzy union operation. See also Plate 126.

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. Let the point P lie at location (xp, yp, Zp) within the regular hexahedron, known as a cell, defined by samples A through H. For simplicity, let the distance between samples in all three directions be 1, with sample A at (0,0,0) with a value of vA, and sample H at (1,1,1) with a value of vH. The value vP, according to trilinear interpolation, is then vp = va(1 - xp)(1 - yp)(1 - zp) + ve(1 - xp)(1 - yp)zp + vBxp(1 - yp)(1 - zp) + vFxp(1 - yp )zp + vc(1 - xp )yp(1 - zp) + vg (1 - xp)ypzp

In general, A is at some location (xA, yA, zA), and H is at (xH, yH, zH). In this case, xp in Eq. (1) would be replaced by (xp xA)/(xH xa), with similar substitutions made for yp and zp.

constants for the three axes, 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, rectilinear, curvilinear (structured), or unstructured grids are employed (e.g., [65]). In a rectilinear grid the cells are axis-aligned, yet grid spacings along the axes might be arbitrary. The other types of grids are uncommon in medical imaging.

The array S only defines the value of some measured property of the data, such as density, at discrete locations in space. A function f (x, y, z) may be defined over R3 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 a nearest-neighbor function. The value at any location in R3 is simply the value of the closest sample to that location. With this interpolation method there is a region of constant value around each sample in S. Since the samples in S are regularly spaced, each region is of uniform size and shape. The region of constant value that surrounds each sample is known as a voxel, with each voxel being a rectangular cuboid having 6 faces, 12 edges, and 8 corners.

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