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Multivariate Tensor-Based Morphometry with a Right-Invariant Riemannian Distance on GL+(n)

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Abstract

Tensor-based morphometry (TBM) studies encode the anatomical information in spatial deformations which are locally characterized by Jacobian matrices. Current methods perform voxel-wise statistical analysis on some features, such as the Jacobian determinant or the Cauchy–Green deformation tensor, which are not complete descriptors of the local deformation. This article introduces a right-invariant Riemannian distance on the GL+(n) group of Jacobian matrices making use of the complete geometrical information of the local deformation. A numerical recipe for the computation of the proposed distance is given. Additionally, experiments are performed on both a synthetic deformation study and a cross-sectional brain MRI study.

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Notes

  1. logm() denotes the matrix logarithm (the inverse of the matrix exponential) and ∥ F the Frobenius norm.

  2. There is no global consensus about what a ‘closed-form’ expression means [16]. However, the matrix exponential function, expm(), can be accurately and efficiently computed by most scientific software packages like Mathematica and Matlab [45].

  3. Let A and B be two \(\text{$n \times n$}\) matrices, then \(\overline{ \mathbf {{A}}\mathbf{{B}} } = ( \mathbf{{B}}^{T} \otimes\mathbf{{I}}_{n} ) \overline{\mathbf{{A}}}\) and therefore \(\mathcal{D}_{\mathbf{{A}}} ( \mathbf{ {A}}\mathbf{{B}} ) = \mathbf{{B}}^{T} \otimes \mathbf{{I}}_{n}\). Similarly, \(\overline{ \mathbf{{A}}\mathbf{ {B}} } = ( \mathbf{{I}}_{n} \otimes \mathbf{{A}} ) \overline{\mathbf{{B}}}\) and therefore \(\mathcal{D}_{\mathbf{{B}}} ( \mathbf{{A}}\mathbf{{B}} ) = \mathbf {{I}}_{n} \otimes\mathbf{{A}}\).

  4. http://www.adni-info.org

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Acknowledgements

This work was funded by research grants TEC2009-14587-C03-01 from CICYT, AMIT project CEN-20101014 from CENIT program, CIM project IPT-2011-1638-900000 from INNPACTO program, Spain.

Data collection and sharing for this project was funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI) (National Institutes of Health Grant U01 AG024904). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; BioClinica, Inc.; Biogen Idec Inc.; Bristol-Myers Squibb Company; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Company; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; GE Healthcare; Innogenetics, N.V.; IXICO Ltd.; Janssen Alzheimer Immunotherapy Research & Development, LLC.; Johnson & Johnson Pharmaceutical Research & Development LLC.; Medpace, Inc.; Merck & Co., Inc.; Meso Scale Diagnostics, LLC.; NeuroRx Research; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Synarc Inc.; and Takeda Pharmaceutical Company. The Canadian Institutes of Health Research is providing funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Disease Cooperative Study at the University of California, San Diego. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of California, Los Angeles. This research was also supported by NIH grants P30 AG010129 and K01 AG030514.

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Correspondence to Salvador Olmos.

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E. Zacur, M. Bossa, S. Olmos, for the Alzheimer’s Disease Neuroimaging Initiative.

This work was funded by research grants TEC2009-14587-C03-01 from CICYT, AMIT project CEN-20101014 from CENIT program, CIM project IPT-2011-1683-900000 from INNPACTO program, Spain.

Data used in preparation of this article were obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.ucla.edu). As such, the investigators within the ADNI contributed to the design and implementation of ADNI and/or provided data but did not participate in analysis or writing of this report. A complete listing of ADNI investigators can be found at: http://adni.loni.ucla.edu/wp-content/uploads/how_to_apply/ADNI_Acknowledgement_List.pdf

Appendices

Appendix A: Fréchet Derivative of the Matrix Exponential

To compute the Fréchet derivative \(\mathcal{D}_{\mathbf{{U}}}\mathrm{Exp}_{\mathbf {{I}}}(\mathbf{{U}})\) given in Eq. (3.6) an expression for \(\mathrm{dexpm}( \mathbf{{M}} ) \equiv \mathcal{D}_{\mathbf{{M}}} \mathrm{expm}( \mathbf{{M}} )\) is required. This is the linear operator containing the derivatives of each element of expm(M) with respect to perturbation a on each element of M and results in an \(\text{$n^{2} \times n^{2}$}\) matrix.

There are different attempts to compute \(\mathcal{D}_{\mathbf {{M}}} \mathrm{expm}( \mathbf{{M}} )\) [3, 4, 38, 48]. In our computations we have used the approach given in [41], where for an analytic matrix function F(M):

$$ \mathbf{{F}}\left( \left( \begin{array}{c@{\quad}|@{\quad}c} \mathbf{{M}} & \mathbf{P} \\ \hline \mathbf{{0}} & \mathbf{{M}} \\ \end{array} \right) \right) = \left( \begin{array}{c@{\quad}|@{\quad}c} \mathbf{{F}}(\mathbf{{M}}) & \textrm{d}_r \mathbf{{F}}( \mathbf{{M}} + r \mathbf {P} ) \\ \hline \mathbf{{0}} & \mathbf{{F}}(\mathbf{{M}}) \end{array} \right) $$

and if P j is the j-th canonical perturbation, then the vectorization of the upper-right submatrix \(\overline{ \textrm{d}_{r} \mathbf{{F}}( \mathbf{{M}} + r \mathbf {P}^{j} ) }\) is the j-th column of the matrix \(\mathcal{D}_{\mathbf{{M}}}\mathbf{ {F}}( \mathbf{{M}} )\).

Appendix B: Convergence and Performance of Descent Procedures

Below, arguments are given for using a descent method to compute the initial velocity of a geodesic starting at the identity I to a target matrix T∈GL+(n). The existence of such geodesic is guaranteed because the set GL+(n) is connected. Moreover, Exp I () in Eq. (3.5) is surjective on GL+(n) and therefore there always exists at least one zero-minimizer of the objective function E(U;T). In fact, it may happen that there exists many (even an infinite number of) different geodesics from the identity to T and all their corresponding initial velocities are zero-minimizers of E(U;Q). Therefore, the objective function E(U;Q) is in general not convex in \(\mathfrak{{gl}}(n)\) and its minimization procedure has many basins of attraction. Nevertheless, it can be shown that all local minima have zero energy and therefore they are also global minima. This can be proved by noticing that the function Exp I () maps open subsets of \(\mathbb{R}^{\text{$n \times n$}}\equiv \mathfrak{{gl}}(n)\) around U to open subsets of \(\mathbb{R}^{\text{$n \times n$}}\supset{\textrm{GL}}^{+}(n)\) around Exp I (U). The same applies for the function \(\mathrm{Exp}_{\mathbf{ {I}}}^{-1}(\boldsymbol{\cdot})\). Moreover, the Frobenius norm is a convex function in \(\mathbb{R}^{\text{$n \times n$}}\). Then, for any \(\mathbf {{U}}\in\mathfrak{{gl}}(n)\) either, E(U;Q) is zero or there is an U′ in a neighborhood of U with a lower value of the objective function.

It is remarkable that for some target matrices there may exists descending valleys in the objective function which extends up to infinity. For example, in the extreme case of , there exists a descent valley in \(\mathfrak{{gl}}(n)\) along the direction whose Riemannian exponential ends up at the null matrix. Although we could not find a simple proof, we conjecture that those attraction basins towards infinity have a zero measure in \(\mathfrak{{gl}}(n)\). For the previous matrix T, descent procedures starting from any symmetric matrix tend to the null matrix, but by slightly perturbing with a skew-symmetric matrix the procedure converges to a global minimum. For a continuous descending path U(t) such that ∥U(t)∥ F →+∞ for t→+∞, it can be proved that lim t→+∞Exp I (U(t)), either it is a singular matrix or it does not exists. Then, to circumvent those valleys towards infinite it is convenient to provide to the descent algorithm with a control in the determinant of Exp I (U k ) (with U k the current iteration of the algorithm) and perform a perturbation if the algorithm is converging to a singular matrix.

To show the performance of descent strategies the following experiment was performed: one thousand \(\text{$3 \times 3$}\) random matrices were generated with positive determinant and with predefined condition numbers; for each target matrix a solution U of the problem (L2) was found by using two descent strategies, along the negated gradient direction and along the Gauss–Newton direction, both performed with an exact line-search; the stopping criterion was set to a relative error in the Frobenius norm smaller than 10−10. Figure 5 shows in the left panel the number of iterations needed to reach the stopping criterion for different condition numbers of the target matrix. Right panel in Fig. 5 shows the relative error for 100 random target matrices with a condition number of 10 for both descent strategies. It is very clear that the Gauss–Newton strategy requires a much smaller number of iterations than gradient descent.

Fig. 5
figure 5

Performance of gradient descent and Gauss–Newton strategies using an exact line-search. Left panel: box-plot showing 5, 25, 50, 75 and 95 percentiles of the number of iterations needed to reach the stopping criterion for target matrices with different condition numbers. Right panel: evolution of the relative error for 100 random target matrices with a condition number of 10

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Zacur, E., Bossa, M. & Olmos, S. Multivariate Tensor-Based Morphometry with a Right-Invariant Riemannian Distance on GL+(n). J Math Imaging Vis 50, 18–31 (2014). https://doi.org/10.1007/s10851-013-0479-7

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