Estimating diffeomorphic mappings between templates and noisy data: Variance bounds on the estimated canonical volume form
Authors:
Daniel J. Tward, Partha P. Mitra and Michael I. Miller
Journal:
Quart. Appl. Math. 77 (2019), 467-488
MSC (2010):
Primary 92C55, 49K40
DOI:
https://doi.org/10.1090/qam/1527
Published electronically:
November 20, 2018
MathSciNet review:
3932966
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Abstract: Anatomy is undergoing a renaissance driven by the availability of large digital data sets generated by light microscopy. A central computational task is to map individual data volumes to standardized templates. This is accomplished by regularized estimation of a diffeomorphic transformation between the coordinate systems of the individual data and the template, building the transformation incrementally by integrating a smooth flow field. The canonical volume form of this transformation is used to quantify local growth, atrophy, or cell density. While multiple implementations exist for this estimation, less attention has been paid to the variance of the estimated diffeomorphism for noisy data. Notably, there is an infinite dimensional unobservable space defined by those diffeomorphisms which leave the template invariant. These form the stabilizer subgroup of the diffeomorphic group acting on the template. The corresponding flat directions in the energy landscape are expected to lead to increased estimation variance. Here we show that a least-action principle used to generate geodesics in the space of diffeomorphisms connecting the subject brain to the template removes the stabilizer. This provides reduced-variance estimates of the volume form. Using simulations we demonstrate that the asymmetric large deformation diffeomorphic mapping methods (LDDMM), which explicitly incorporate the asymmetry between idealized template images and noisy empirical images, provide lower variance estimators than their symmetrized counterparts (cf. ANTs). We derive Cramer-Rao bounds for the variances in the limit of small deformations. Analytical results are shown for the Jacobian in terms of perturbations of the vector fields and divergence of the vector field.
References
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References
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- Sarang Joshi, Brad Davis, Matthieu Jomier, and Guido Gerig, Unbiased diffeomorphic atlas construction for computational anatomy, NeuroImage 23 (2004), S151–S160.
- Sarang C. Joshi and Michael I. Miller, Landmark matching via large deformation diffeomorphisms, IEEE Trans. Image Process. 9 (2000), no. 8, 1357–1370. MR 1808275, DOI https://doi.org/10.1109/83.855431
- Bilge Karaçali and Christos Davatzikos, Estimating topology preserving and smooth displacement fields, IEEE Transactions on Medical Imaging 23 (2004), no. 7, 868–880.
- David G. Kendall, A survey of the statistical theory of shape, Statist. Sci. 4 (1989), no. 2, 87–120. With discussion and a reply by the author. MR 1007558
- Michael D. Ketcha, Tharindu De Silva, Runze Han, Ali Uneri, Joseph Goerres, Matthew W. Jacobson, Sebastian Vogt, Gerhard Kleinszig, and Jeffrey H. Siewerdsen, Effects of image quality on the fundamental limits of image registration accuracy, IEEE Transactions on Medical Imaging 36 (2017), no. 10, 1997–2009.
- Ali R Khan, Lei Wang, and Mirza Faisal Beg, Multistructure large deformation diffeomorphic brain registration, IEEE Transactions on Biomedical Engineering 60 (2013), no. 2, 544–553.
- Jan Kybic, Fast no ground truth image registration accuracy evaluation: Comparison of bootstrap and hessian approaches, Biomedical Imaging: From Nano to Macro, 2008. ISBI 2008. 5th IEEE International Symposium on, IEEE, 2008, pp. 792–795.
- Jan Kybic, Bootstrap resampling for image registration uncertainty estimation without ground truth, IEEE Trans. Image Process. 19 (2010), no. 1, 64–73. MR 2729958, DOI https://doi.org/10.1109/TIP.2009.2030955
- M. I. Miller, Computational anatomy: shape, growth, and atrophy comparison via diffeomorphisms, Neuroimage 29 (2004), no. Suppl 1:S, 19–33.
- M. I. Miller and L. Younes, Group Actions, Homeomorphisms, and Matching: A General Framework, International Journal of Computer Vision 41 (2001), no. 4, 61–84.
- Michael I. Miller, Alain Trouvé, and Laurent Younes, On the metrics and euler-lagrange equations of computational anatomy, Annual Review of Biomedical Engineering 4 (2002), no. 1, 375–405.
- Michael I. Miller, Alain Trouvé, and Laurent Younes, Geodesic shooting for computational anatomy, J. Math. Imaging Vision 24 (2006), no. 2, 209–228. MR 2227097, DOI https://doi.org/10.1007/s10851-005-3624-0
- Michael I. Miller, Alain Trouvé, and Laurent Younes, Diffeomorphometry and geodesic positioning systems for human anatomy, Technology 1:36 (2014).
- Michael I. Miller, Alain Trouvé, and Laurent Younes, Hamiltonian systems and optimal control in computational anatomy: 100 years since d’arcy thompson., Annual Review of Biomed Engineering (2015), no. 17, 447–509.
- Susumu Mori, Kenichi Oishi, Hangyi Jiang, Li Jiang, Xin Li, Kazi Akhter, Kegang Hua, Andreia V. Faria, Asif Mahmood, Roger Woods, et al., Stereotaxic white matter atlas based on diffusion tensor imaging in an icbm template, Neuroimage 40 (2008), no. 2, 570–582.
- X. Pennec, From Riemannian Geometry to Computational Anatomy, Elements (2011).
- Petter Risholm, Steve Pieper, Eigil Samset, and William M. Wells, Summarizing and visualizing uncertainty in non-rigid registration, International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2010, pp. 554–561.
- Laurent Risser, François-Xavier Vialard, Robin Wolz, Maria Murgasova, Darryl D. Holm, and Daniel Rueckert, Simultaneous Multi-scale Registration Using Large Deformation Diffeomorphic Metric Mapping., IEEE Transactions on Medical Imaging 30 (2011), no. 10, 1746–59.
- Laurent Risser, François-Xavier Vialard, Habib Y. Baluwala, and Julia A. Schnabel, Piecewise-diffeomorphic image registration: Application to the motion estimation between 3d ct lung images with sliding conditions, Medical image analysis 17 (2013), no. 2, 182–193.
- J. H. Siewerdsen, L. E. Antonuk, Y. El-Mohri, J. Yorkston, W. Huang, J. M. Boudry, and I. A. Cunningham, Empirical and theoretical investigation of the noise performance of indirect detection, active matrix flat-panel imagers (amfpis) for diagnostic radiology, Medical Physics 24 (1997), no. 1, 71–89.
- Amber L. Simpson, Burton Ma, Elvis C. S. Chen, Randy E. Ellis, and A. James Stewart, Using registration uncertainty visualization in a user study of a simple surgical task, International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer, 2006, pp. 397–404.
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Additional Information
Daniel J. Tward
Affiliation:
Center for Imaging Science, Johns Hopkins University, Baltimore, Maryland 21218
Email:
dtward@cis.jhu.edu
Partha P. Mitra
Affiliation:
Cold Spring Harbor Laboratory, Cold Spring Harbor, New York 11724
MR Author ID:
693008
Email:
mitra@cshl.edu
Michael I. Miller
Affiliation:
Department of Biomedical Engineering, Johns Hopkins University, Baltimore, Maryland 21218
MR Author ID:
262997
Email:
mim@cis.jhu.edu
Keywords:
Computational anatomy,
morphometry,
cell density,
Hamiltonian dynamics
Received by editor(s):
September 26, 2018
Received by editor(s) in revised form:
October 8, 2018
Published electronically:
November 20, 2018
Additional Notes:
This work was supported by the National Institutes of Health [P41-EB015909, R01-EB020062, R01-NS102670, R01-MH105660, U19-AG033655, U19-MH114821, U01-MH114824]; National Science Foundation 16-569 NeuroNex contract 1707298; the Kavli Neuroscience Discovery Institute; the Crick-Clay Professorship, CSHL; and the H N Mahabala Chair, IIT Madras.
Dedicated:
Twenty years after Computational Anatomy was hatched by Ulf Grenander and Michael Miller at the Division of Applied Mathematics at Brown University, we revisit the core geodesic equations and nuisance non-identifiable stabilizing subgroup of the deformable template model examining the variance in estimating the fundamental form and associated Cramer-Rao Bound in the small deformation limit. This picks up on a tradition started by Ulf’s thesis advisor, which we would like to think Ulf would have appreciated.
Article copyright:
© Copyright 2018
Brown University
