## Assumptions

The following assumptions are made about the measured objects and the measurement process.

(1) Discrete materials. The first assumption is that materials within the objects to be measured are discrete at the sampling resolution. Boundaries need not be aligned with the sampling grid. Figure 6a shows an object with two materials. This assumption is made because the technique is geared toward finding boundaries between materials, and because its input is sampled data, where information about detail finer than the sampling rate is blurred.

This assumption does not preclude homogeneous combinations of submaterials that can be treated as a single material at the sampling resolution. For example, muscle may contain some water, and yet be treated as a separate material from water. This assumption is not satisfied where materials gradually transition from one to another over many samples or are not relatively uniformly mixed; however, the algorithm appears to degrade gracefully even in these cases.

(2) Normally distributed noise. The second assumption is that noise from the measurement process is added to each discrete sample and that the noise is normally distributed. A different variance in the noise for each material is assumed. This assumption is not strictly satisfied for MRI data, but seems to be satisfied sufficiently to classify data well. Note that the sample values with noise added are interpolated to reconstruct the continuous function, p(x). The effect of this band-limited noise is discussed further in Section 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.

(5) Uniform tissue measurements. Measurements of the same material have the same expected value and variance throughout a dataset.

(6) Box filtering for voxel histograms. The spatial measurement kernel, or point-spread function, can be approximated by a box filter for the purpose of deriving histogram basis functions.

(7) Materials identifiable in histogram of entire dataset. The signatures for each material and mixture must be identifiable in a histogram of the entire dataset.

For many types of medical imaging data, including MRI and CT, these assumptions hold reasonably well, or can be satisfied sufficiently with preprocessing [21]. Other types of sampled data, e.g., ultrasound, and video or film images with lighting and shading, violate these assumptions; thus, the technique described here does not apply directly to them.

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