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Tolerant Similarity Measure

Further to an explanation in §2, which makes a mentioning of image similarity, we seek a measure of similarity that is robust to slight localised differences between images. If an image gets treated merely as an ordered collection of pixels, then strict pixel-to-pixel difference can be derived, which is Euclidean. This does not, however, account for any spatial relationship between pixels. A slight mismatch can entail a significant penalty, which is uncalled for. Consequently, the sensitivity of our similarity measure to change is so high that the Euclidean measure becomes almost meaningless.

A better method of computing image similarity should take into account additional spatial properties (e.g. movements, bends and elasticity). This seems rather natural once we realise that corresponding pixels across the images may lie in the vicinity of geometrically corresponding points in the images. As a result we are then inclined to use the shuffle distance, which can cover a large disc-shaped region where the best match for any given point is to be identified (see Fig. 2). The effects of varying the 'scope' of the shuffle distance are shown in Fig. 3.

Fig. 2. The symmetric calculation of a shuffle difference image. Each pixel is compared to its closest match in the other image.

Fig. 3. Surveying image distances for evaluation of brain registration. On the left and the right: distinct yet similar images; Across the centre (from left): shuffle distance with $r=0$ (Euclidean distance), $1.5,2.9$ and $3.7$.