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Image Difference Measures

There are various ways of measuring a perceived similarity between two images and selection of a method depends on the images at hand [] as well as dimensionality. Sum of pixel- or voxel-wise differences (also sum-of-squared-differences or even the mean rather than a summation) emerges as the most intuitive method, which merely accumulates pixel- or voxel-wise differences between the images. This measure, however, performs rather poorly in situations where slight miscorrespondence results in very high localised differences. If areas of the images to be aligned are not smooth (strong edges), sum-of-squared-differences will be sensitive to miscorrespondence. But if edges get blurred, the measure will become insensitive to miscorrespondence. Moreover, sum-of-squared-differences is unable to deal with cases of multi-modality where intensities are inherently different and where mutual information-based methods may be more appropriate.

Sum-of-Squared-Differences.  One of the most intuitive and least resource-intensive approaches is the sum of differences and its variants. Pixels are being compared in two images, one pixel at a time, and their (potentially squared) grey-level differences are calculated. A sum over all pixel-wise differences is accumulated or averaged over, which gives a measure that is based on the sum-of-squared-differences (SSD). Mean-of-squared-differences (MSD) is merely the case where the differences are averaged over, rather than summed up. Other variants include the case where absolute differences are not raised to the power of two (squared). This method is usually powerful if the two images which are being compared are also closely aligned and their intensity values are relatively continuous and low in contrast. In cases where images are resistant enough to noise, MSD/SSD will tolerate a low level of locally-situated difference, while contrariwise, MI and NMI properly handle greater dispersion of pixels in some localised region [].

Mutual Information (MI). Viola [] developed a method2.4 of measuring similarity between two images by repeatedly comparing histograms of pairs of images [,]. When measuring mutual information, one computes informational overlap across images. The basic idea is that if two images are properly aligned, image values in one image can be used to predict to some extent image values in the second image. This means that the joint histogram for the two images will tend to contain sharp peaks. Under the complementary case of mis-registration, one will be unable to predict values in one image from values in the other image, which corresponds to a joint histogram without sharp peaks.

Let there be two images $A$ and $B$. By defining a joint information (or entropy) to be $H(A,B)$ and the information contained in a single histogram $A$ to be $H(A)$, then the mutual information (MI) is given by: $H(A)+H(B)-H(A,B)$. There are variants thereof [], but the prime idea is that joint information is subtracted from the sum of information present in the two individual images.

One important strength of MI-based measures is that they are able to handle images that are obtained by different imaging techniques; hence, unlike SSD, they are not restricted to images of a single modality but can perform multimodal registration.

Normalised Mutual Information (NMI). Studholme [] and Maes [] suggested that normalisation should be applied to mutual information. Several steps are involved in this normalisation process2.5. The main difference is that the expression used for MI is significantly extended and divided by a normalisation term. The method is predominantly used in non-rigid registration as it is generic, adaptable to new data, and yields better results.

Correlation Ratio. Another method for measuring similarity makes use of the correlation ratio, the principle of which goes back over half a century ago []. The correlation ratio is used to solve a broad range of problems, but in this case it estimates a measure of the relationship between the dispersion in an image set and the dispersion across a larger sample. It takes into consideration a small set of images - however they may be defined - and compares that to a larger set of images. The correlation ratio takes values in the range of 0 to 1.

Roy Schestowitz 2010-04-05