There is two principal components, projected into the 2D space of the population, computed from each of the 7 different estimated centres marked in red for the exact centre, in yellow for IC1, in purple for IC2 and in green for PW. This figure shows that the 6 different tangent spaces projected into the 2D space of shapes, are very similar, even if the centres have different positions.
Overall, for this synthetic dataset, the 6 estimated centres give very similar PCA results. We then studied the impact of different centres on the approximated distance matrices. We computed the seven approximated distance matrices corresponding to the seven centres, and the direct pairwise distance matrix computed by matching all subjects to each other. Computation of the direct distance matrix took 1, min 17 h for this synthetic dataset of 50 subjects. In the following, we denote as aM C the approximated distance matrix computed from the centre C.
To quantify the difference between these matrices, we used the following error e :. Results are reported in Table 3. For visualization of the errors against the pairwise computed distance matrices, we also computed the error between the direct distance matrix, by computing pairwise deformations 23h h of computation per population , for 3 populations randomly selected. Figure 4 shows scattered plots between the pairwise distances matrices and the approximated distances matrices of IC1 and T IC1 for the 3 randomly selected populations.
The errors between aM IC 1 matrices and the pairwise distance matrices of each of the populations are 0. We can observe a subtle curvature of the scatter-plot, which is due to the curvature of the shape space. This figure illustrates the good approximation of the distances matrices, regarding to the pairwise estimation distance matrix.
The variational templates are getting slightly closer to the identity line, which is expected as for the better ratio values since they are extra iterations to converge toward a centre of the population, however the estimated centroids from the different algorithms, still provide a good approximation of the pairwise distances of the population.
In conclusion for this set of synthetic population, the different estimated centres have also a little impact on the approximation of the distance matrices. Figure 4. Left Scatter plot between the approximated distance matrices aM IC 1 of 3 different populations, and the pairwise distances matrices of the corresponding populations.
Right Scatter plot between the approximated distance matrices aM T IC 1 of 3 different populations, and the pairwise distances matrices of the corresponding populations. For both plots, the matrix error equation 27 and Pearson correlation coefficients only here to support the visualization of the scatter plots are indicated in color.
The red line corresponds to the identity. We now present experiments on the real dataset RD For this dataset, the exact center of the population is not known, neither is the distribution of the population and meshes have different numbers of vertices and different connectivity structures.
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We estimated our 3 centroids IC1 75 min IC2 min and PW 88 min , and the corresponding variational templates, which took respectively , , and min. For comparison of computation time, we also computed a template using the standard initialization the whole population as initialisation which took min Extra computation time was needed to estimate initial momentum vectors from IC1, IC2, and PW to each subject of the population, for subsequent analysis in the tangent spaces of each centres.
The computation time is thus 94 min for IC1, for IC2, and min for PW which is considerably faster than estimating the variational template 1, min. As for the synthetic dataset, we assessed the centring of these six different centres. The ratios are higher than for the synthetic dataset indicating that centres are less centred. This was predictable since the population is not built from one surface via geodesic shootings as the synthetic dataset. In order to better understand these values, we computed the ratio for each subject of the population after matching each subject toward the population , as if each subject was considered as a potential centre.
For the whole population, the average ratio was 0. These ratios are larger than the one computed for the estimated centres and thus the 6 estimated centres are closer to a critical point of the Frechet functional than any subject of the population. Figure 5 and Table 4 show the proportion of cumulative explained variance for different number of modes. We can note that for any given number of modes, all PCAs result have really similar proportions of explained variance. The highest differences in cumulative explained variance, at a given number of component, between all 6 centres is 0.
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Real dataset RD Proportion of cumulative explained variance for Kernel PCAs computed from the 6 different centres, with respect to the number of dimensions. Vertical red lines indicate values displayed at Table 4. As for the synthetic dataset, we then studied the impact of these different centres on the approximated distance matrices. A direct distance matrix was also computed around 90 h of computation time. We compared the approximated distance matrices of the different centres to: i the approximated matrix computed with IC1; ii the direct distance matrix.
We computed the errors e M 1 , M 2 defined in equation Results are presented in Table 5. Errors are small and with the same order of magnitude. Figure 6 -left shows scatter plots between the direct distance matrix and the six approximated distance matrices. Interestingly, we can note that the results are similar to those obtained by Yang X. Figure 6 -right shows scatter plots between the approximated distance matrix from IC1 and the five others approximated distance matrices.
The approximated matrices thus seem to be largely independent of the chosen centre. Figure 6. Left 6 scatter plots between direct distance matrix and approximated distance matrices from the six centres. Right 5 scatter plots between the approximated distance matrix aM IC 1 computed from IC 1 and the 5 others approximated distance matrices.
Results on the real dataset RD50 and the synthetic SD showed that results were highly similar for the 6 different centres. In light of these results and because of the large size of the real dataset RD, we only computed IC1 for this last dataset. The computation time was about min The ratio R of equation 26 computed from the IC1 centroid was 0. We then performed a Kernel PCA on the initial momentum vectors from this IC1 centroid to the 1, shapes of the population.
The proportions of cumulative explained variance from this centroid are 0. In addition, we explored the evolution of the cumulative explained variance when considering varying numbers of subjects in the analysis. Results are displayed in Figure 7. We can first note that about 50 dimensions are sufficient to describe the variability of our population of hippocampal shapes from healthy young subjects.
Moreover, for large number of subjects, this dimensionality seems to be stable. When considering increasing number of subjects in the analysis, the dimension increases and converges around Figure 7. Proportion of cumulative explained variance of K-PCA as a function of the number of dimensions in abscissa and considering varying number of subjects.
The dark blue curve was made using subjects, the blue , the light blue , the green curve subjects, the yellow one , very close to the dotted orange one which was made using 1, subjects. Finally, we computed the approximated distance matrix. Its histogram is shown in Figure 8. It can be interesting to note that, as for RD50, the average pairwise distance between the subject is around 12, which means nothing by itself, but the points cloud on Figure 6 -left and the histogram on Figure 8 , show no pairwise distances below 6, while the minimal pairwise distance for the SD50 dataset - generated by a 2D space - is zero.
This corresponds to the intuition that, in a space of high dimension, all subjects are relatively far from each other. Figure 8. Left side shows histogram of the approximated distances of the large database RD estimated from the computed centroid IC1 shown on the right side. All those 1, subjects have an IHI score ranking from 0 to 8 Cury et al. The IHI score is based on the roundness of the hippocampal body, the medial positionning of the hippocampus within the brain, and the depth of the colateral sulcus and the occipito-temporal sulcus, which both are close to the hippocampus.
Here we selected only hippocampi with good segmentation, we therefore removed strong IHI which could not be properly segmented. We now apply our approach to predict incomplete hippocampal inversions IHI from hippocampal shape parameters. Specifically, we predict the visual IHI score, which corresponds to the sum of the individual criteria as defined in Cury et al. We studied whether it is possible to predict the IHI score using statistical shape analysis on the RD dataset composed of 1, healthy subjects left hippocampus.
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The deformation parameters characterizing the shapes of the population are the eigenvectors computed from the centroid IC1, and they are the independent variables we will use to predict the IHI scores. We simply used a multiple linear regression model Hastie et al. The R a d j 2 coefficient, unlike R 2 , takes into account the number of variables and therefore does not increase with the number of variables.
One can note that R is the coefficient of correlation of Pearson. We then tested the significance of each model by computing the F statistic. So for each number of variables i.
For each model, we computed 10, k -fold cross validation and displayed the mean and the standard deviation of MSE corresponding to the model. Results are given at Figure 9 , and display the coefficient of determination of each model. The cross validation is only computed on models with a coefficient of correlation higher than 0. Figure 9D presents results of cross validation; for each model computed from 20 to 40 dimensions we computed the mean of the 10, MSE of the fold and its standard deviation.
To have a point of comparison, we also computed the MSE between the IHI scores and random values which follow a normal distribution with the same mean and standard deviation as the IHI scores red cross on the Figure. The firsts principal components between 1 and 20 represent general variance maybe characteristic of the normal population, the shape differences related to IHI appear after.
It is indeed expected that the principal i. Figure 9. Results for prediction of IHI scores. A Values of the adjusted coefficient of determination using from 1 to 40 eigenvectors resulting from the PCA.
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B the coefficient correlation corresponding to the coefficient of determination of A. D Cross validation of the models using 20 to 40 dimensions by fold. The red cross indicates the MSE of the model predicted using random values, and the errorbar corresponds to the standard deviation of MSE computed from 10, cross validations for each model, the triangle corresponds to the average MSE. In this paper, we proposed a method for template-based shape analysis using diffeomorphic centroids. This approach leads to a reasonable computation time making it applicable to large datasets.
It was thoroughly evaluated on different datasets including a large population of 1, subjects. The results demonstrate that the method adequately captures the variability of the population of hippocampal shapes with a reasonable number of dimensions. In particular, Kernel PCA analysis showed that the large population of left hippocampi of young healthy subjects can be explained, for the metric we used, by a relatively small number of variables around Moreover, when a large enough number of subjects was considered, the number of dimensions was independent of the number of subjects.
The comparisons performed on the two small datasets show that the different centroids or variational templates lead to very similar results. This can be explained by the fact that in all cases the analysis is performed on the tangent space to the template, which correctly approximates the population in the shape space. Moreover, we showed that the different estimated centres are all close to the Frechet mean of the population. While all centres centroids or variational templates yield comparable results, they have different computation times.
Thus, for the study of hippocampal shape, IC1 or PW algorithms seem to be more adapted than IC2 or the variational template estimation. However, it is not clear whether the same conclusion would hold for more complex sets of anatomical structures, such as an exhaustive description of cortical sulci Auzias et al.
Besides, as mentioned above in section 5. This requires N more matchings, which almost doubles the computation time. Even with this additional step, centroid-based shape analysis stills leads to a competitive computation time about 26 h for the complete procedure on the large dataset of 1, subjects. Here we computed for all our datasets the initial momentum vectors from the mean shape to each subject of the population to apply a kernel PCA on the mean shape. But this is not a mandatory step, for example, in Cury et al.
It is also possible, for visualization purposes, to estimate different templates within a population controls vs. In future work, this approach could be improved by using a discrete parametrisation of the LDDMM framework Durrleman et al. The control points number and position are independent from the shapes being deformed as they do not require to be aligned with the shapes' vertices. Even if the method accepts any kind of topology, for more complex and heavy meshes like the cortical surface which can have more than vertices per subjects , we could also improve the method presented here by using a multiresolution approach Tan and Qiu, An other interesting point would be to study the impact of the choice of parameters on the number of dimensions needed to describe the variability population in this study the parameters were selected to optimize the matchings.
Finally we can note that this template-based shape analysis can be extended to data types such as images or curves. Hamburg: Ethics board, Hamburg Chamber of Phsyicians. Berlin: ethics committee of the Faculty of Psychology. And Mannheims ethics committee approved the whole study.
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Study concepts and design or data acquisition or data analysis and interpretation, all authors; manuscript drafting or manuscript revision for important intellectual content, all authors; approval of final version of submitted manuscript, all authors. The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
It should be noted that a part of this work has been presented for the first time in CC's Ph. Afsari, B. Riemannian Lp center of mass: Existence, uniqueness, and convexity. On the convergence of gradient descent for finding the Riemannian center of mass. SIAM J. Control Optimizat. Arnaudon, M. Stochastic algorithms for computing means of probability measures. Their Appl. Ashburner, J. Identifying global anatomical differences: deformation-based morphometry. Brain Mapp. PubMed Abstract Google Scholar. Auzias, G. Diffeomorphic brain registration under exhaustive sulcal constraints. Baulac, M.
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