Symmetric diffeomorphic image registration with cross-correlation: evaluating automated labeling of elderly and neurodegenerative brain

Abstract

One of the most challenging problems in modern neuroimaging is detailed characterization of neurodegeneration. Quantifying spatial and longitudinal atrophy patterns is an important component of this process. These spatiotemporal signals will aid in discriminating between related diseases, such as frontotemporal dementia (FTD) and Alzheimer's disease (AD), which manifest themselves in the same at-risk population. Here, we develop a novel symmetric image normalization method (SyN) for maximizing the cross-correlation within the space of diffeomorphic maps and provide the Euler-Lagrange equations necessary for this optimization. We then turn to a careful evaluation of our method. Our evaluation uses gold standard, human cortical segmentation to contrast SyN's performance with a related elastic method and with the standard ITK implementation of Thirion's Demons algorithm. The new method compares favorably with both approaches, in particular when the distance between the template brain and the target brain is large. We then report the correlation of volumes gained by algorithmic cortical labelings of FTD and control subjects with those gained by the manual rater. This comparison shows that, of the three methods tested, SyN's volume measurements are the most strongly correlated with volume measurements gained by expert labeling. This study indicates that SyN, with cross-correlation, is a reliable method for normalizing and making anatomical measurements in volumetric MRI of patients and at-risk elderly individuals.

Figures

Fig. 1

An illustration of a SyN geodesic path between images. The images in the top row are the original images, I and J, at the initialization of the method. After the SyN solution converges, (second row) these images deform in time along the series of diffeomorphisms that connect them. The deforming grids associated with these diffeomorphisms are shown in the bottom row. The maps, φ1 and φ2 are of equivalent length and map I and J to the mean shape between the images. The full path, φ and φ−1, are found by joining the paths φ1 and φ2. The symmetric nature of this problem (proven in [40]) is due to the interchange-ability of the labels I and J in our problem formulation.

Fig. 2

The symmetric normalization method is represented, at top, by its two components, φ1 and φ2, meeting at the middle of the normalization domain. Note that each sub-path may be traversed either from the middle to the end or from an end to the middle. Alternatively, the ICIR method is shown in a schematic at the bottom panel of the figure. The correspondence defining vector fields associated with ICIR are called h and g. In ICIR, all four deformation fields overlap in time and may, in fact, be different from each other. The inverse of h may not be its true inverse. Further, the inverse of h may not be equivalent to g.

Fig. 3

The local cross correlation measure allows robust matching of images despite the presence of a strong bias field affecting the image quality. The smoothness of the grid is also unaffected by the bias.

Fig. 4

The atlas was initially aligned to these FTD images via a rigid plus uniform scaling transformation. The subsequent Demons registration to each image, used for labeling, is in the right column. The SyN result is in the center column, while the corresponding original images are in the left column. The Demons method does a reasonable normalization, but leaves the ventricles and other smaller structures only partly normalized. The quadratic elastic penalty prevents the remaining shape differences from being captured. A similar loss of resolution in the mapping is seen in the elastic cross-correlation mappings. These are illustrative images from our previous study, [49], and were not used as actual study data in this exposition.

Fig. 5. Normalization Results

All images in each row should look similar to the first image in the row. The atlas was initially aligned to these elderly (top three) and FTD (bottom three) images via a rigid plus uniform scaling transformation. These examples are from our current study. We have highlighted (with a circle) the type of small scale difference one may see in the registration quality. Larger scale differences are also clear. In particular, the Demons and elastic cross-correlation have problems both shrinking and expanding the ventricles (the second elderly image). In addition, the Demons intensity consistency assumption causes significant errors in the first FTD image, a case where this assumption does not hold.

Fig. 6. Normalization Results Difference Images

This figure shows the same results as figure 4, but with absolute value of the image difference, after normalization. One can observe that small sulci may not be captured by any of the methods. Under the intensity difference metric, the Demons method should be expected to perform best, as it explicitly minimizes this error.

Fig. 7. Normalization Results Label Error Images

Here, we show three individual images along with the manual, SyN, elastic and Demons labels. This bottom row for each individual shows the difference of the algorithmic and manual labels. The majority of the errors are due to a shift or a curvature in the boundary definition gained algorithmically, with respect to the manual definition. Manually defined lobar boundaries tend to be planar. Further, when one compares the (in particular, parietal) boundaries determined by the labeler on these three images, it becomes apparent that there may be some inconsistency. Secondary errors are due to lack of sulcal definition in the template labels, with respect to the individual labeled ground truth.

Fig. 8. Performance comparison and average Dice statistic for each method and each structure

Example brain labelings mark the structures over which we evaluated the three algorithms. The final results showed that, overall, SyN > Elastic > Demons for automatically labeling these lobar structures. The P-values for the difference in performance were computed using non-parametric permutation testing based on the paired T-test on vectors of Dice statistics. We used 10,000 permutations per example, thus limiting our minimum P-value to 0.0001. Extremely small P-values indicate that the method uniformly and strongly outperformed the method with which it was paired.

Fig. 9

Temporal lobe volume, in cubic centimeters, as measured by each of the three methods. The algorithmic measures are plotted against the manually measured volume.

Fig. 10

Frontal lobe volume, in cubic centimeters, as measured by each of the three methods. The algorithmic measures are plotted against the manually measured volume.

Fig. 11

Parietal lobe volume, in cubic centimeters, as measured by each of the three methods. The algorithmic measures are plotted against the manually measured volume.