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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 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 $\mathbf{I}_1(\mathbf{x})$ and $\mathbf{I}_2(\mathbf{\mathbf{x}})$, then the shuffle distance between them is defined as

$$D_s(\mathbf{I}_1,\mathbf{I}_2)=\frac{1}{n} \sum_{\mathbf{x}} \mat... ...n}_{i}\Vert\mathbf{I}_1(\mathbf{x})-\mathbf{I}_2(\mathbf{N}_i(\mathbf{x}))\Vert$$ (13)

where $\vert\cdot\vert$ is the absolute difference, there are $n$ pixels (or voxels) indexed by $\mathbf{x}$, and $\{\mathbf{N}_i(\mathbf{x})\}$ is the set of pixels in a neighbourhood of radius $r$ around $\mathbf{x}$.

The idea is illustrated in Figure 5. 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 ($r$), as is illustrated in Figure 4.

Figure 4: A comparison between shuffle distance using varying size neighbourhoods (radius $r$). Left: original image, right: warped image, centre, from the left: shuffle distance with $r=1$(Euclidean), $1.5,2.9$ and $3.7$ pixels.

Figure 5: The calculation of a shuffle difference image

It should be noted that the shuffle distance as defined above depends on the direction in which it is measured (see Figure 6), hence is not a true distance. It is trivial to construct a symmetric shuffle distance, by averaging the distance calculated both ways between a pair of images. However, it was found that the improvement obtained using this was not significant, and did not justify the increased computation time. In what follows, we use the asymmetric shuffle distance.