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Measuring Distances Between Images

The most straightforward way to measure the distance between images is to evaluate the mean absolute difference between them, or alternatively treat them as vectors by concatenating pixel/voxel values and take the Euclidean distance. Although this has the merit of simplicity, it does not provide a very robust distance measurement because it is very sensitive to small image misalignments. Robustness can be enhanced by considering a `shuffle distance', inspired by the `shuffle transform' [15]. The idea is to seek correspondence with a wider area around each pixel. Instead of taking the mean absolute difference between exactly corresponding pixels, we take each pixel in one image in turn, and compute the minimum absolute difference between it and pixels in a shuffle neighbourhood of the exactly corresponding pixel in the other image to produce a shuffle difference image $\bigtriangleup S$ (see Figure 3). The shuffle distance is given by $\sum\limits_{j}^{}\bigtriangleup S_{j}$ where $\bigtriangleup S_{j}$ are the elements of $\bigtriangleup S$. This approach is less sensitive to small misalignments, and provides a more robust measure of image distance. The sensitivity to misalignment is determined directly by the size and shape of the shuffle neighbourhood. One obvious choice is a square box around the corresponding pixel, but this is inherently anisotropic. Instead, we consider a shuffle disc, of radius $r$, which contains all pixels within a distance $r$ of the central pixel.

Figure 3: Calculating the shuffle difference image

Figure: Shuffle distance calculation: Left: original image, Right: warped image, Centre, from left to right: shuffle difference images for $r = 1\:\mbox{(abs. diff.)},\:1.5,\: 2.1 \mbox{ \& } 3.7$ respectively.

Figure 4 shows examples of shuffle distance between an original image and a misaligned version evaluation, for varying values of the radius $r$. The effect of the shuffle neighbourhood radius on the sensitivity to misalignment is obvious as the contribution to distance perceivably decreases in areas of limited misalignment, as we go from $r=1$ to $r=3.7$ (roughly equivalent to a $7\times 7$ square window).