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We need to acquire, store, and manipulate images, which is typically achieved through the use of an array of voxels (and the use of computer arithmetic rather than real numbers).  We consider the voxel values to record the value of the image at the exact centre of each voxel (at least in the ITK/ANTs convention).  To compute the value of an image at some point not at a voxel centre, some numerical interpolation amongst the surrounding voxel values must be used.  For instance, in trilinear interpolation one finds the surrounding eight voxel centres and takes a distance-weighted average of their intensities.  Interpolating using splines or other families is also possible.  In (k-) nearest neighbour interpolation, the values are assumed to be discrete, and the nearest (k) voxel values are used for a majority vote for the value of the pointvalue at the point is simply the value at the nearest voxel centre.  (Nearest neighbour interpolation is most often used on image segmentations, discrete-valued images whose values are labels.)  The type of interpolation used (at least continuous vs nearest-neighbour) is often considered a property of a resampling procedure but you might consider it a property of the image itself.  Interpolating an anatomical image using a nearest-neighbour algorithm or a segmentation using trilinear interpolation just doesn't make sense.

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