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Consider a toy example in 1D, as shown in figure 1. In the
example there are two shapes, and , each of different size
(length) which results in the generation of four different
vectors: to . Note that at the interface between
the shapes, the values of are between 0.0 and 1.0, representing
the partial volume fraction. The transformation, , effects both
the global position (and stretching) of the shapes in the image as
well as the partial volume fractions.
Figure 1:
Example of image model formation in 1D. The
two shapes, and , cover ten image voxels (indicated by
vertical dashed lines) and generate four different vectors:
which represents the mean intensity (partial volume
component) of ; which represents the a linear
intensity change across (in the xdirection); and similarly for
. The middle voxel is a partial volume voxel and shows how both
the mean and linear components are multiplied by the appropriate
partial volume fraction. Furthermore, note that the linear components are
zero mean. Note that in 3D there would also be linear
intensity changes across the ydirection and zdirection, represented
by and .

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