The coordinate system of the space containing the object to be measured is referred to as "spatial coordinates," and points are generally labeled x e X. This space is «-dimensional, where n is 3 for volume data, can be 2 for slices, and is 1 for the example in Fig. 4. Each measurement, which may be a scalar or vector, lies in "feature space" (see Fig. 4), with points frequently denoted as v e V. Feature space is nv-dimensional, where nv is 1 for scalar-valued data, 2 for two-valued vector data, etc. Tables 5 and 6 in Section 9 contain these and other definitions.

(3) Sampling theorem is satisfied. The third assumption is that the sampled datasets satisfy the sampling theorem [4]. The sampling theorem states that if a sufficiently band-limited function is point sampled, the function can be exactly reconstructed from the samples, as demonstrated in Fig. 4b. The band-limiting creates smooth transitions in p(x) as it traverses boundaries where otherwise p(x) would be discontinuous. The intermediate region of Fig. 6b shows a sampling grid and the effect of sampling that satisfies the sampling theorem. Partial-volume mixing of measurements occurs in the region labeled "A&B." Multislice MRI acquisitions do not satisfy this assumption in the through-plane direction. For these datasets the data can be interpolated only within each plane.

(4) Linear mixtures. Each voxel measurement is a linear combination of pure material measurements and measurements of their pairwise mixtures created by band limiting (see Fig. 6).

FIGURE 6 Partial-volume effects. The derivation of the classification technique starts from the assumption that in a real-world object each point is exactly one material, as in (a). The measurement process creates samples that mix materials together; from the samples a continuous, band-limited measurement function, p(x), is reconstructed. Points P1 and P2 lie inside regions of a single material. Point P3 lies near a boundary between materials, and so in (b) lies in the A&B region where materials A and B are mixed. The grid lines show sample spacing and illustrate how the regions may span voxels.

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