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Method

The first of the proposed methods for assessing registration quality uses a generalisation of Tanimoto's spatial overlap measure [1]. We start with a manual mark-up of each image, providing an anatomical/tissue label for each voxel, and measure the overlap of corresponding labels following registration. Each label is represented using a binary image, but after warping and interpolation into a common reference frame, based on the results of NRR, we obtain a set of fuzzy label images. These are combined in a generalised overlap score [5]:

$$\mathcal{O} = \frac{ \sum\limits_{\mbox{\small pairs},k}\: \... ...l} \sum\limits_{\mbox{\small voxels},i} MAX(A_{kli},B_{kli}) }$$ (1)

where $i$ indexes voxels in the registered images, $l$ indexes the label and $k$ indexes image pairs. $A_{kli}$ and $B_{kli}$ represent voxel label values in a pair of registered images and are in the range [0, 1]. The $MIN()$ and $MAX()$ operators are standard results for the intersection and union of fuzzy sets. The generalised overlap measures the consistency with which each set of labels partitions the image volume. The parameter $\beta_{l}$ affects the relative weighting of different labels. With $\beta_{l}=1$, label contributions are implicitly volume weighted with respect to one another. We have also considered the cases where $\beta_{l}$ weights for the inverse label volume (which makes the relative weighting of different labels equal), where $\beta_{l}$ weights for the inverse label volume squared (which gives labels of smaller volume higher weighting) and where $\beta_{l}$ weights for a measure of label complexity (which we define arbitrarily as the mean absolute voxel intensity gradient in the label).

Figure 1: Training set and model in hyperspace

The second method assesses registration in terms of the quality of a generative statistical appearance model, constructed from the registered images - for all the experiments reported here, this was an active appearance model (AAM) [3]. The idea is that a correct registration produces an anatomically meaningful dense correspondence between the images, resulting in a better appearance model of the anatomy. We define model quality using two measures - generalisation and specificity. Both are measures of overlap between the distribution of original images and a distribution of images sampled from the model, as illustrated in Figure 1. If we use the generative property of the model to synthesise a large set of images, $\{I_{\alpha}:\alpha=1,\ldots m\}$, we can define Generalisation $G$ as:

$$G=\frac{1}{n} \sum\limits_{i=1}^{n}\min_{\alpha}\vert I_{i}-I_{\alpha}\vert,$$ (2)

where $\vert\cdot\vert$ is a measure of distance between images, $I_{i}$ is the $i^{th}$ training image, and $\min_{\alpha}$ is the minimum over $\alpha$ (the set of synthetic images). That is, Generalisation is the average distance from each training image to its nearest neighbour in the synthetic image set. A good model exhibits a low value of $G$, indicating that the model can generate images that cover the full range of appearances present in the original image set. Given a sufficiently large synthetic set, even registered image with differing brightness levels will be paired with a nearby match. Similarly, we can define Specificity $S$ as:

$$S=\frac{1}{m} \sum\limits_{\alpha=1}^{m}\min_{i}\vert I_{i}-I_{\alpha}\vert.$$ (3)

That is, Specificity is the average distance of each synthetic image from its nearest neighbour in the original image set. A good model exhibits a low value of $S$, indicating that the model only generates synthetic images that are similar to those in the original image set. The uncertainty in estimating $G$ and $S$ can also be computed.

In our experiments we have defined $\vert\cdot\vert$ as the shuffle distance between two images, as illustrated for a single pixel in Figure 2. Shuffle distance images are formed by taking the mean of the minimum absolute difference between each pixel/voxel in one image, and the pixels/voxels in a shuffle neighbourhood of radius $r$ around the corresponding pixel/voxel in a second image. When $r \leq 1$, this is equivalent to the mean absolute difference between corresponding pixels/voxels, but for larger values of $r$ the distance increases more smoothly as the misalignment of structures in the two images increases. The effect on the pixel-by-pixel contribution to the shuffle distance image as $r$ is increased is illustrated in Figure 3.