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Measuring Image Separation
The definitions we have provided for specificity
and generalisation require a measure of separation in image space.
The most straightforward way to measure the distance between
images is to treat each image as a vector formed by concatenating
the pixel/voxel intensity values, then take the Euclidean
distance. This means that each pixel/voxel in one image is
compared against its spatially corresponding pixel/voxel in
another image. Although this has the merit of simplicity, it does
not provide a very wellbehaved distance measure since it
increases rapidly for quite small image misalignments [18].
This observation led us to consider an alternative distance
measure, based on the 'shuffle difference', inspired by the
'shuffle transform' [19]. If we have two images
71#71 and
72#72, then the shuffle distance
between them is defined as
where 53#53 is the absolute difference, there are 43#43 pixels (or voxels) indexed by
15#15, and
74#74 is the set
of pixels in a neighbourhood of radius 66#66 around
15#15.
The idea is illustrated in
Figure . Instead of taking the
sumofsquareddifferences between corresponding pixels, the
minimum absolute difference between each pixel in one image and
the values in a neighbourhood around the corresponding
pixel is used. This is less sensitive to small misalignments, and
provides a betterbehaved distance measure. The tolerance for
misalignment is dependent on the size of the neighbourhood (66#66), as is
illustrated in Figure .
Figure:
The calculation of a shuffle difference
image
75#75

It should be noted that the shuffle distance as defined above
depends on the direction in which it is measured (see
Figure ), hence is not a true distance. It
is trivial to construct a symmetric shuffle distance, by averaging
the distance calculated in both directions between a pair of
images. We found, however, that the improvement obtained was not
significant, and did not justify the increased computation time.
In what follows, we use the asymmetric shuffle distance.
Next: Experimental Validation
Up: ModelBased Evaluation of NRR
Previous: Specificity and Generalisation
Roy Schestowitz
20070311