Abstract
This paper proposes an original approach for the statistical analysis of longitudinal shape data. The proposed method allows the characterization of typical growth patterns and subject-specific shape changes in repeated time-series observations of several subjects. This can be seen as the extension of usual longitudinal statistics of scalar measurements to high-dimensional shape or image data. The method is based on the estimation of continuous subject-specific growth trajectories and the comparison of such temporal shape changes across subjects. Differences between growth trajectories are decomposed into morphological deformations, which account for shape changes independent of the time, and time warps, which account for different rates of shape changes over time. Given a longitudinal shape data set, we estimate a mean growth scenario representative of the population, and the variations of this scenario both in terms of shape changes and in terms of change in growth speed. Then, intrinsic statistics are derived in the space of spatiotemporal deformations, which characterize the typical variations in shape and in growth speed within the studied population. They can be used to detect systematic developmental delays across subjects. In the context of neuroscience, we apply this method to analyze the differences in the growth of the hippocampus in children diagnosed with autism, developmental delays and in controls. Result suggest that group differences may be better characterized by a different speed of maturation rather than shape differences at a given age. In the context of anthropology, we assess the differences in the typical growth of the endocranium between chimpanzees and bonobos. We take advantage of this study to show the robustness of the method with respect to change of parameters and perturbation of the age estimates.
Similar content being viewed by others
Notes
Source: www.bordalierinstitute.com.
We notice that in the time-specific paradigm, this would be \(S^{\prime \prime }(t) = \chi _{\psi (t)}(\phi (S_0))\) (see Sect. 2).
References
Aljabar, P., Bhatia, K., Murgasova, M., Hajnal, J., Boardman, J., Srinivasan, L., et al. (2008). Assessment of brain growth in early childhood using deformation-based morphometry. NeuroImage, 39(1), 348–358.
Allassonnière, S.,& Kuhn, E. (2009). Stochastic algorithm for bayesian mixture effect template estimation. ESAIM Probability and Statistics (in press).
Chandrashekara, R., Rao, A., Sanchez-Ortiz, G., Mohiaddin, R. H.,& Rueckert, D. (2003). Construction of a statistical model for cardiac motion analysis using nonrigid image registration. In Information processing in medical imaging (Vol. 2732, pp. 599–610). Springer, Lecture Notes in Computer Science.
Courchesne, E., Campbell, K.,& Solso, S. (2011). Brain growth across the life span in autism: Age-specific changes in anatomical pathology. Brain Research, 1380, 138–145 (the Emerging Neuroscience of Autism Spectrum Disorders).
de Craene, M., Camara, O., Bijnens, B. H.,& Frangi, A. F. (2009). Large diffeomorphic FFD registration for motion and strain quantification from 3D-US sequences. In Proceedings of functional imaging and modeling of the heart (Vol. 5528, pp. 437–446). Spinger, LNCS.
Davis, B., Fletcher, P., Bullitt, E.,& Joshi, S. (2007). Population shape regression from random design data. In Proceedings of international conference on computer vision (ICCV), pp. 1–7.
Declerck, J., Feldman, J.,& Ayache, N. (1998). Definition of a 4D continuous planispheric transformation for the tracking and the analysis of LV motion. Medical Image Analysis, 4(1), 1–17.
de Waal, F. B. M. (1995). Bonobo sex and society. Scientific American, 272, 82–88.
Dupuis, P., Grenander, U.,& Miller, M. (1998). Variational problems on flows of diffeomorphisms for image matching. Quaterly of Applied Mathematics, 56(3), 587–600.
Durrleman, S. (2010). Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. Thèse de sciences (phd thesis), Université de Nice-Sophia Antipolis.
Durrleman, S., Pennec, X., Trouvé, A.,& Ayache, N. (2009a). Statistical models of sets of curves and surfaces based on currents. Medical Image Analysis, 13(5), 793–808.
Durrleman, S., Pennec, X., Trouvé, A., Gerig, G.,& Ayache, N. (2009b) Spatiotemporal atlas estimation for developmental delay detection in longitudinal datasets. In Medical image computing and computer-assisted intervention—MICCAI (Vol. 5761, pp. 297–304). Springer, LNCS.
Durrleman, S., Fillard, P., Pennec, X., Trouvé, A.,& Ayache, N. (2011). Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. NeuroImage, 55(3), 1073–1090.
Ehrhardt, J., Werner, R., Schmidt-Richberg, A., Schulz, B.,& Handels, H. (2008). Generation of a mean motion model of the lung using 4D-CT image data. In Proceedings of eurographics workshop on visual computing for biomedicine (pp. 69–76). Eurographics Association.
Fishbaugh, J., Durrleman, S.,& Gerig, G. (2011). Estimation of smooth growth trajectories with controlled acceleration from time series shape data. In Medical image computing and computer-assisted intervention—MICCAI. Springer, LNCS (to appear).
Gerber, S., Tasdizen, T., Fletcher, T. P.,& Whitaker, R. (2010). Manifold modeling for brain population analysis. Medical Image Analysis 14(5), 643–653 (special Issue on the 12th International Conference on Medical Image Computing and Computer-Assisted Intervention (MICCAI) 2009).
Gerig, G., Davis, B., Lorenzen, P., Xu, S., Jomier, M., Piven, J.,& Joshi, S. (2006). Computational anatomy to assess longitudinal trajectory of brain growth. In Third international symposium on 3D data processing, visualization, and transmission (pp. 1041–1047).
Glaunès, J. (2005). Transport par difféomorphismes de points, de mesures et de courants pour la comparaison de formes et l’anatomie numérique. PhD thesis, Université Paris 13, http://cis.jhu.edu/joan/TheseGlaunes.pdf.
Gogtay, N., Lu, A., Leow, A., Klunder, A., Lee, A., Chavez, A., et al. (2008). 3D growth pattern abnormalities visualized in childhood-onset schizophrenia using tensor-based morphometry. Proceedings of the National Academy of Sciences, 105(41), 15979–15984.
Grenander, U., Srivastava, A.,& Saini, S. (2007). A pattern-theoretic characterization of biological growth. IEEE Transactions on Medical Imaging, 26(5), 648–659.
Hart, G., Shi, Y., Zhu, H., Sanchez, M., Styner, M.,& Niethammer, M. (2010). DTI longitudinal atlas construction as an average of growth models. In Proceedings of international workshop on spatio-temporal image analysis for longitudinal and time-series image data.
Hazlett, H., Poe, M., Gerig, G., Smith, R., Provenzale, J., Ross, A., et al. (2005). Magnetic resonance imaging and head circumference study of brain size in autism. The Archives of General Psychiatry, 62, 1366–1376.
Hazlett, H., Poe, M., Styner, M., Chappell, C., Smith, R., Vachet, C., et al. (2011). Early brain overgrowth in autism associated with an increase in cortical surface area before age 2 years. Journal of Archives of General Psychiatry, 68(5), 467–476.
Jian, B.,& Vemuri, B. C. (2005). A robust algorithm for point set registration using mixture of Gaussians. In: 10th IEEE international conference on computer vision (ICCV 2005), 17–20 October 2005, Beijing, China, pp. 1246–1251. http://gmmreg.googlecode.com.
Joshi, S.,& Miller, M. (2000). Landmark matching via large deformation diffeomorphisms. IEEE Transaction on Image Processing, 9(8), 1357–1370.
Kano, T. (1992). The last ape: Pygmy chimpanzee behavior and ecology. Stanford: Stanford University Press.
Khan, A.,& Beg, M. (2008). Representation of time-varying shapes in the large deformation diffeomorphic framework. In 5th IEEE international symposium on biomedical imaging ISBI (pp. 1521–1524).
Kinzey, W. G. (1984). The dentition of the pygmy chimpanzee, Pan paniscus. New York: Plenum.
Kuroda, S. (1989). Developmental retardation and behavioural characteristics of pygmy chimpanzees. Cambridge: Harvard University Press.
Mansi, T., Durrleman, S., Bernhardt, B., Sermesant, M., Delingette, H., Voigt, I., et al. (2009). A statistical model of right ventricle in tetralogy of fallot for prediction of remodelling and therapy planning. In Proceedings of medical image computing and computer assisted intervention (MICCAI) (Vol. 5761, pp. 214–221). Springer, LNCS.
Miller, I. M., Trouvé, A.,& Younes, L. (2002). On the metrics and euler-lagrange equations of computational anatomy. Annual Review of Biomedical Engineering, 4, 375–405.
Miller, M., Trouvé, A.,& Younes, L. (2006). Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision, 24(2), 209–228.
Pennec, X., Fillard, P.,& Ayache, N. (2006). A Riemannian framework for tensor computing. International Journal of Computer Vision, 66(1), 41–66.
Perperidis, D., Mohiaddin, R. H.,& Rueckert, D. (2005). Spatio-temporal free-form registration of cardiac MRI sequences. Medical Image Analysis, 9(5), 441–456.
Peyrat, J. M., Delingette, H., Sermesant, M., Pennec, X., Xu, C.,& Ayache, N. (2008). Registration of 4D time-series of cardiac images with multichannel diffeomorphic demons. In Proceedings of medical image computing and computer assisted intervention (MICCAI) (Vol. 5242, pp. 972–979). Springer, LNCS.
Qiu, A., Younes, L., Miller, M.,& Csernansky, J. (2008). Parallel transport in diffeomorphisms distinguishes the time-dependent pattern of hippocampal surface deformation due to healthy aging and the dementia of the Alzheimer’s type. NeuroImage, 40, 68–76.
Qiu, A., Albert, M., Younes, L.,& Miller, M. I. (2009). Time sequence diffeomorphic metric mapping and parallel transport track time-dependent shape changes. NeuroImage, 45(1 Supplement 1), S51–S60.
Shea, B. (1989). Heterochrony in human evolution: The case for neoteny reconsidered. Yearbook of Physical Anthropology, 32, 69–101.
Thompson, P. M., Giedd, J. N., Woods, R. P., MacDonald, D., Evans, A. C.,& Toga, A. W. (2000). Growth patterns in the developing human brain detected by using continuum-mechanical tensor maps. Nature, 404, 6774.
Trouvé, A. (1998). Diffeomorphisms groups and pattern matching in image analysis. International Journal of Computer Vision, 28(3), 213–221.
Trouvé, A., Vialard, F. X. (2010). Shape splines and stochastic shape evolutions: A second order point of view. Quaterly of Applied Mathematics (to appear). http://arxiv.org/abs/1003.3895.
Vaillant, M., Miller, M., Younes, L.,& Trouvé, A. (2004). Statistics on diffeomorphisms via tangent space representations. NeuroImage, 23, 161–169.
Xie, Y., Ho, J.,& Vemuri, B. C. (2010). Image atlas construction via intrinsic averaging on the manifold of images. In The twenty-third IEEE conference on computer vision and pattern recognition, CVPR 2010 (pp. 2933–2939). San Francisco, CA, USA: IEEE.
Yushkevich, P. A., Piven, J., Ho, S., Gee, J. C.,& Gerig, G. (2006). User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability. Neuroimage, 31(3), 1116–1128.
Acknowledgments
We would like to thank B. Combès (IRISA, France) for preprocessing the endocast data, J. Piven, director of Carolina Institute for Developmental Disabilities at UNC Chapel Hill, for providing imaging data related to autism research, and M. Styner (Psychiatry UNC Chapel Hill) for processing the subcortical structures. We thank W. Van Neer and E. Gilissen the previous and current curator of the “Musée de l’Afrique Centrale” at Tervuren (Belgium). We are indebted to Chems Touati for his help for creating figures and movies and James Fishbaugh for his kind proofreading of the manuscript, both at the Scientific Computing and Imaging Institute, University of Utah. This work has been funded in part by the INRIA ARC 3D-Morphine (PI: Sylvain Prima), the European IP Project Health-e-child (IST-2004-027749) and Microsoft Research.
Author information
Authors and Affiliations
Corresponding author
Electronic supplementary material
Below is the link to the electronic supplementary material.
Appendices
A Differentiation of the Temporal Shape Regression Criterion
1.1 A.1 Matrix Notations
For the sake of simplicity, we introduce matrix notations: \(\mathbf{x}_{0}=\{x_{p}\}_{p=1,\ldots ,N}\) denotes the \(3N\) vector which is the concatenation of the coordinates of \(N\) vertices of the baseline shape \(M_0\). Using the notations of Sect. 3.2.2, we denote the moving points \(\mathbf{x}(t)\) (resp. the parameterizing vectors \(\varvec{\alpha }(t)\)) the \(3N\) vector: \((x_p(t))_{p=1,\ldots ,N}\) (resp. \((\alpha _p(t))_{p=1,\ldots ,N}\)). We denote also \(\mathbf{K}^\chi (\mathbf{x}(t),\mathbf{x}(t))\) the \(3N\)-by-\(3N\) block matrix whose block \(p,q\) is given by the 3-by-3 matrix \((K^\chi (x_p(t),x_q(t)))\). This matrix is symmetric, positive definite by definition of the kernel \(K^\chi \).
Thanks to these notations, the norm of the speed vector \(v_t\) is written: \(\left\Vert{v_t}\right\Vert_{V^\chi }^2 = \varvec{\alpha }(t)^t \mathbf{K}^\chi (\mathbf{x}(t),\mathbf{x}(t)) \varvec{\alpha }(t)\). For \(A\), a function from \(\mathbb R ^3\) to \(\mathbb R \), we denote by \(d_xA\) its Jacobian matrix at point \(x\), so that for any vector \(V\): \((d_x A)V = (\nabla _x A)^t V\). By extension, \(\nabla _{\mathbf{x}}A\) denotes the \(3N\) vector \((\nabla _{x_1}A,\ldots ,\nabla _{x_N}A)\).
With these notations, the regression criterion (17) becomes:
subject that:
where we denote:
For the sake of simplicity, we will denote in the sequel, \(f(t)\) and \(L^\chi (t)\) instead of \(f(\mathbf{x}(t),\varvec{\alpha }(t))\) and \(L^\chi (\mathbf{x}(t),\varvec{\alpha }(t))\).
1.2 A.2 Gradient in a matrix form
Let \(\delta E\) be a variation of the criterion \(E\) with respect to a variation \(\delta \varvec{\alpha }(t)\) of the momenta \(\varvec{\alpha }(t)\):
where \(\delta \mathbf{x}(t)\) denotes the variations of the positions \(\mathbf{x}(t)\) with respect to the variations of the momenta \(\varvec{\alpha }(t)\). The differentiation of the flow Eq. (40) shows that these variations \(\delta \mathbf{x}(t)\) satisfy a linear ODE with source term:
We introduce the flow \(R_{ut}\) for \(u,t\in [0,T]\) which is solution of the homogeneous equation:
The method of the variations of the parameters leads to the following solution of the ODE:
In particular, this shows that we can write the variations \(\delta \mathbf{x}(t_i)\) as:
where \(\mathbf{1}_{\{t\le t_i\}}=1\) if \(t\le t_i\) and 0 otherwise.
Now, we can plug these last two equations into (42). Using Fubini’s theorem, which implies that \(\int _0^T \int _0^t F(u,t)dudt = \int _0^T \int _u^T F(u,t)dt du = \int _0^T \int _t^T F(t,u)dudt\) for any \(L^2\) function \(F(u,t)\), this leads to:
This gives the gradient of \(E\) with respect to the \(L^2\) metric as:
where we denote the auxiliary variable \(\varvec{\eta }(t)\):
The auxiliary variable \(\varvec{\eta }(u)\) depends on the flows \(R_{ut}\) and therefore satisfies an ODE. To make this ODE explicit, we write the inverse flow \(R_{ut}\) in integral form. Noticing that \(R_{tu}R_{ut} = \text{ Id}\), we have \(\frac{d R_{ut}}{du} = -\partial _{\mathbf{x}}f(u) R_{ut}\), which gives in integral form (noticing that \(R_{ut}\) and \(f\) commute):
Now, we can plug this equation into the definition of \(\varvec{\eta }(t)\) in (49). Writing \(R_{tt_i}= \text{ Id}+ \int _t^T R_{ut_i}\partial _{\mathbf{x}}f(u)\mathbf{1}_{\{u\le t_i\}} du\) and noticing that for any \(L^2\) function \(F(u,s)\), the Fubini’s theorem implies that \(\int _t^T \int _t^u F(u,s)dsdu = \int _t^T \int _u^T F(s,u) dsdu\), this leads to:
Now, we notice that \(t\le u\) within the integral, which implies that \(\mathbf{1}_{\{t\le t_i\}}\mathbf{1}_{\{u\le t_i\}} = \mathbf{1}_{\{u\le t_i\}}\). Hence, \((\star )\) is exactly equal to \(\varvec{\eta }_u\). Therefore, \(\varvec{\eta }_t\) is the solution of the integral equation (integrated upstream in time):
1.3 A.3 Gradient in Coordinates
Due to the definition of the functions \(f\) and \(L^\chi \) in (41), we have:
Therefore, the gradient of the regression criterion with respect to the \(L^2\) metric given in (48) is now equal to:
The matrix \(\mathbf{K}^\chi (\mathbf{x}(t),\mathbf{x}(t))\) is precisely the Sobolev metric induced by the kernel on the set of \(L^2\) functions (see Sect. 3.2.2), so that the gradient with respect to this metric is given by:
The auxiliary variable \(\varvec{\eta }(t)\) satisfies the ODE (52), now written as:
The \(3N\) vector \(\nabla _{\mathbf{x}_{t_i}}A_i\) is equal to \((\nabla _{x_1(t_i)} A_i,\ldots ,\nabla _{x_N(t_i)} A_i)\). For generic \(3N\) vectors \(\mathbf{x},\,\mathbf{y}\) and \(\varvec{\alpha }\), the \(k\) th coordinate of the \(3N\)-vector \(\mathbf{K}^\chi (\mathbf{x},\mathbf{y})\varvec{\alpha }\) is given as: \((\mathbf{K}^\chi (\mathbf{x},\mathbf{y})\varvec{\alpha })_k = \sum _{p=1}^N K^\chi (x_k,y_p)\alpha _p\). The kernel \(K^\chi \) is scalar, namely of the form \(K^\chi (x,y)=k^\chi (x,y)\text{ Id}\) for a scalar function \(k^\chi \). We have therefore for every \(i,j=1,\ldots ,N\):
Therefore, for a generic \(3N\)-vector \(\varvec{\beta }\), we have:
Now, we can apply this equation with \(\mathbf{y}= \mathbf{x}\) and \(\varvec{\beta }= \gamma _\chi \varvec{\alpha }+ \varvec{\eta }\) and combine it with (55). Noticing that for a symmetric kernel, we have \(\nabla _1 k(x,y) = \nabla _2 k(y,x)\), we get eventually the set of ODEs satisfied by the functions \(\eta _p(t)\) as given in (19).
B Differentiation of the Spatiotemporal Matching Criterion
1.1 B.1 Matrix Notations
Let \(\mathbf{t}_0 = \{t_j\}_{j=1,\ldots ,N_\mathrm{target}}\) be the vector of time-points associated to the target shapes. The 1D diffeomorphism \(\psi _u\) changes \(\mathbf{t}_0\) into \(\mathbf{t}(u) = \{t_j(u)\}_{j=1,\ldots ,N_\mathrm{target}}\) for \(u\in [0,1]\). This vector satisfies the ODE: \(\tfrac{d \mathbf{t}}{du}(u) = \mathbf{K}^\psi (\varvec{\beta }(u),\varvec{\beta }(u))\mathbf{t}(u)\) with \(\mathbf{t}(0)=\mathbf{t}_0\), where \(\varvec{\beta }(u)\) is the concatenation of the vectors \(\beta _j(u)\) defined in (23), \(\mathbf{K}^\psi (\varvec{\beta }(u),\varvec{\beta }(u))\) is the block matrix whose block \((i,j)\) is given by: \(K^\psi (\beta _i(u),\beta _j(u))\).
Similarly, we denote \(\mathbf{x}_0(t) = \{x_p(t)\}_{p=1,\ldots ,N}\) be the concatenation of the positions of all the points of the source evolution \(S(t)\) for any time-point \(t\) and \(\mathbf{x}_0(\mathbf{t}(1))\) the concatenation of the \(\mathbf{x}_0(t_j(1))\) for \(j=1,\ldots ,N_\mathrm{target}\). The diffeomorphism \(\phi _u\) maps this vector to \(\mathbf{x}(u)\), which satisfies the ODE: \(\frac{d\mathbf{x}}{du} = \mathbf{K}^\phi (\mathbf{x}(u),\mathbf{x}(u))\varvec{\alpha }(u)\) with initial condition: \(\mathbf{x}(0) = \mathbf{x}_0(\mathbf{t}(1))\) (which depends on the final time-points \(\mathbf{t}(1)\)), where \(\varvec{\alpha }(u)\) is the concatenation of the vectors \(\alpha _{p,j}(u)\) defined in (22) for \(p=1,\ldots ,N\) and \(j=1,\ldots ,N_\mathrm{target}\).
Therefore, we can write the matching criterion (4) as:
subject to:
where we denote:
For the sake of simplicity, we will write in the sequel \(f(u),\,g(u),\,L^\phi (u)\) and \(L^\psi (u)\) instead of \(f(\mathbf{x}(u),\varvec{\alpha }(u))\), \(g(\mathbf{t}(u),\varvec{\beta }(u))\), \(L^\phi (\mathbf{x}(u),\varvec{\alpha }(u))\) and \(L^\psi (\mathbf{t}(u),\varvec{\beta }(u))\) respectively.
1.2 B.2 Gradient in a Matrix Form
Now, let \(\delta E\) be a variation of the criterion \(E\) induced by a variation of the momenta \(\delta \varvec{\alpha }(u)\) and \(\delta \varvec{\beta }(u)\):
where we denote \(\delta \mathbf{x}(u)\) and \(\delta \mathbf{t}(u)\) the variations of the path \(\mathbf{x}(u)\) and \(\mathbf{t}(u)\) induced by the variations of the momenta \(\delta \varvec{\alpha }(u)\) and \(\delta \varvec{\beta }(u)\). These vectors satisfy the linear ODEs with source term derived from (59):
We introduce the flows \(P_{su}\) and \(R_{su}\) for all \(s,u\in [0,1]\), which are solution of the homogeneous equations:
The method of variations of the parameters leads to the following solution of the ODEs:
where the variations of the initial condition \(\delta \mathbf{x}(0) = \delta \mathbf{x}_0(\mathbf{t}(1))\) equals:
according to (64).
Plugging (64) into (61) leads to the variation of the criterion (noticing that for any \(L^2\) function \(F(s,u)\) we have that \(\int _0^1\int _0^u F(s,u)dsdu = \int _0^1\int _s^1 F(s,u)duds = \int _0^1\int _u^1 F(u,s)dsdu\)):
Now, we denote,
which appears twice in (66) as \(\varvec{\eta }(0)\) and \(\varvec{\eta }(u)\). Given the expression of \(\delta \mathbf{x}(0)\) in (65), we have:
Denoting
we end up with the gradient of the criterion with respect to the \(L^2\) metric written as:
The auxiliary variables \(\varvec{\eta }(u)\) and \(\varvec{\xi }(u)\) depend on the flows \(R_{us}\) and \(P_{us}\). Therefore they satisfy a ODE, which we need to make explicit now. The inverse flows are written in integral form as:
so that the auxiliary variable \(\varvec{\eta }(u)\) satisfies:
where we denote \(\nabla _{\mathbf{x}}A = \left(d_{\mathbf{x}}A\right)^t\) for any scalar function \(A\).
Since we have for any \(L^2\) functions \(F(r,s)\), \(\int _u^1\int _u^sF(r,s) drds = \int _u^1\int _s^1 F(s,r) dr ds\) by permuting the two integrals, we have:
The term in the brackets is exactly \(\varvec{\eta }(s)\), so that the integral equation satisfied by \(\varvec{\eta }(u)\) is eventually given by:
Similar computations using the integral form of the flow \(R_{us}\) leads to the integral equation satisfied by \(\varvec{\xi }(u)\):
1.3 B.3 Gradient in Coordinates
Given the definition of the functions \(f,\,g,\,L^\phi \) and \(L^\psi \), we have:
so that the gradient with respect to the Sobolev metric (the matrices \(\mathbf{K}^\phi (\mathbf{x},\mathbf{x})\) and \(\mathbf{K}^\psi (\mathbf{t},\mathbf{t})\) factorize in (70)) is given as:
where
and
The \(3NN_\mathrm{target}\) vector \(\nabla _{\mathbf{x}(1)}A\) is the concatenation of the vectors \(\nabla _{x_p(t_j(1))}A\) for \(p=1,\ldots ,N\) and \(j=1,\ldots ,N_\mathrm{target}\). Similarly, the \(3NN_\mathrm{target}\) vector is the concatenation of the vectors \(\{x_p(t_j(1))\}_{p=1,\ldots ,N,j=1,\ldots ,N_\mathrm{target}}\). \(d_{\mathbf{t}(1)}\mathbf{x}_0\) is the \(3NN_\mathrm{target}\)-by-\(N_\mathrm{target}\) matrix: \((d_{t_1(1)}\mathbf{x}_0,\ldots ,d_{t_{N_\mathrm{target}}}\mathbf{x}_0)\). In the vector \(d_{t_j(1)}\mathbf{x}_0\) almost every coordinate vanishes except the ones corresponding at the jth block of size \(3N\): \((d_{t_j(1)}x_1(t_j(1)),\ldots ,d_{t_j(1)}x_N(t_j(1)))\) (since \(d_{t_j(1)}x_p (t_i(1))\!=\!0\) when \(i\ne j\)). Therefore, we have: \((d_{t_j(1)}\mathbf{x}_0)^t\eta = \sum _{p=1}^N \left(\frac{d x_p}{dt}(t_j(1))\right)^t\eta _{p,j}\), which is the jth coordinate of the \(N_\mathrm{target}\) vector \((d_{\mathbf{t}(1)}\mathbf{x}_0)^t\eta \).
Eventually, using the generic expression (57) for scalar kernels \(K^\phi (x,y)=k^\phi (x,y)\text{ Id}\) and \(K^\psi (x,y) = k^\psi (x,y)\text{ Id}\), the evolution of \(\varvec{\eta }(u)\) and \(\varvec{\xi }(u)\) in (78) are written in coordinates as in (28) and (29).
C Algorithms
Rights and permissions
About this article
Cite this article
Durrleman, S., Pennec, X., Trouvé, A. et al. Toward a Comprehensive Framework for the Spatiotemporal Statistical Analysis of Longitudinal Shape Data. Int J Comput Vis 103, 22–59 (2013). https://doi.org/10.1007/s11263-012-0592-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11263-012-0592-x