PDF version of this entire document

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 well-behaved 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

73#73 (13)

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 sum-of-squared-differences 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 better-behaved 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.