Abstract
Modeling the variability of brain structures is a fundamental problem in the neurosciences. In this paper, we start from a dataset of precisely delineated anatomical structures in the cerebral cortex: a set of 72 sulcal lines in each of 98 healthy human subjects. We propose an original method to compute the average sulcal curves, which constitute the mean anatomy in this context. The second order moment of the sulcal distribution is modeled as a sparse field of covariance tensors (symmetric, positive definite matrices). To extrapolate this information to the full brain, one has to overcome the limitations of the standard Euclidean matrix calculus. We propose an affine-invariant Riemannian framework to perform computations with tensors. In particular, we generalize radial basis function (RBF) interpolation and harmonic diffusion PDEs to tensor fields. As a result, we obtain a dense 3D variability map which proves to be in accordance with previously published results on smaller samples subjects. Moreover, leave one (sulcus) out tests show that our model is globally able to recover the missing information when there is a consistentneighboring variability. Last but not least, we propose innovative methods to analyze the asymmetry of brain variability. As expected, the greatest asymmetries are found in regions that includes the primary language areas. Interestingly, such an asymmetry in anatomical variance could explain why there may be greater power to detect group activation in one hemisphere than the other in fMRI studies.
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References
Coulon, O., Alexander, D., Arridge, S.: Diffusion tensor magnetic resonance image regularization. Medical Image Analysis 8(1), 47–67 (2004)
Tschumperlé, D., Deriche, R.: Orthonormal vector sets regularization with pde’s and applications. Int. J. of Computer Vision (IJCV) 50(3), 237–252 (2002)
Fletcher, P.T., Joshi, S.: Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In: Sonka, M., Kakadiaris, I.A., Kybic, J. (eds.) CVAMIA/MMBIA 2004. LNCS, vol. 3117, pp. 87–98. Springer, Heidelberg (2004)
Lenglet, C., Rousson, M., Deriche, R., Faugeras, O.: Statistics on multivariate normal distributions: A geometric approach and its application to diffusion tensor MRI. Research Report 5242, INRIA (2004)
Batchelor, P., Moakher, M., Atkinson, D., Calamante, F., Connelly, A.: A rigorous framework for diffusion tensor calculus. Mag. Res. Mag. Res. in Med. 53, 221–225 (2005)
Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. In: IJCV (2005) (to appear)
Skovgaard, L.T.: A Riemannian geometry of the multivariate normal model. Scand. J. Statistics 11, 211–223 (1984)
Sun, X.: Conditional positive definiteness and complete monotonicity. In: Approximation Theory VIII, vol. 2, pp. 211–234. World Scientific, Singapore (1995)
Mangin, J.-F., Riviere, D., Cachia, A., Duchesnay, E., Cointepas, Y., Papadopoulos- Orfanos, D., Scifo, P., Ochiai, T., Brunelle, F., Regis, J.: A framework to study the cortical folding patterns. NeuroImage 23(Supplement 1), S129–S138 (2004)
Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape modelstheir training and application. Comput. Vis. Image Underst. 61(1), 38–59 (1995)
Trouvé, A., Younes, L.: Diffeomorphic matching problems in one dimension: Designing and minimizing matching functionals. In: Vernon, D. (ed.) ECCV 2000. LNCS, vol. 1842, pp. 573–587. Springer, Heidelberg (2000)
Paulsen, R.R., Hilger, K.B.: Shape modelling using markov random field restoration of point correspondences. In: Taylor, C.J., Noble, J.A. (eds.) IPMI 2003. LNCS, vol. 2732, pp. 1–12. Springer, Heidelberg (2003)
Baumberg, A., Hogg, D.: Learning flexible models from image sequences. In: Eklundh, J.-O. (ed.) ECCV 1994. LNCS, vol. 800, pp. 299–308. Springer, Heidelberg (1994)
Bakircioglu, M., Grenander, U., Khaneja, N., Miller, M.I.: Curve matching on brain surfaces using induced Frenet distance metrics. HBM 6(5), 329–331 (1998)
Pitiot, A., Delingette, H., Toga, A.W., Thompson, P.M.: Learning object correspondences with the observed transport shape measure. In: Taylor, C.J., Noble, J.A. (eds.) IPMI 2003. LNCS, vol. 2732, pp. 25–37. Springer, Heidelberg (2003)
Thompson, P.M., Mega, M.S., Narr, K.L., Sowell, E.R., Blanton, R.E., Toga, A.W.: Brain image analysis and atlas construction. In: Fitzpatrick, M., Sonka, M. (eds.) Handbook of Medical Image Proc. and Analysis, ch. 17. SPIE, San Jose (2000)
Toga, A.W., Thompson, P.M.: Mapping brain asymmetry. Nature Reviews Neuroscience 4(1), 37–48 (2003)
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Fillard, P., Arsigny, V., Pennec, X., Thompson, P.M., Ayache, N. (2005). Extrapolation of Sparse Tensor Fields: Application to the Modeling of Brain Variability. In: Christensen, G.E., Sonka, M. (eds) Information Processing in Medical Imaging. IPMI 2005. Lecture Notes in Computer Science, vol 3565. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11505730_3
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DOI: https://doi.org/10.1007/11505730_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26545-0
Online ISBN: 978-3-540-31676-3
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