Uncertainty Quantification for LDDMM Using a Low-Rank Hessian Approximation
This paper presents an approach to estimate the uncertainty of registration parameters for the large displacement diffeomorphic metric mapping (LDDMM) registration framework. Assuming a local multivariate Gaussian distribution as an approximation for the
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Department of Computer Science, University of North Carolina at Chapel Hill, USA 2 Biomedical Research Imaging Center University of North Carolina at Chapel Hill, USA Abstract. This paper presents an approach to estimate the uncertainty of registration parameters for the large displacement diffeomorphic metric mapping (LDDMM) registration framework. Assuming a local multivariate Gaussian distribution as an approximation for the registration energy at the optimal registration parameters, we propose a method to approximate the covariance matrix as the inverse of the Hessian of the registration energy to quantify registration uncertainty. In particular, we make use of a low-rank approximation to the Hessian to accurately and efficiently estimate the covariance matrix using few eigenvalues and eigenvectors. We evaluate the uncertainty of the LDDMM registration results for both synthetic and real imaging data.
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Introduction
Image registration is critical for many medical image analysis systems to provide spatial correspondences. Consequentially, a large number of image registration methods have been developed which are able to produce high-quality spatial alignments of images. However, most image registration approaches do not provide any measures of registration uncertainty and hence do not allow a user to assess if a registration result is locally “trustworthy” or not. This is particularly problematic for highly flexible registration approaches, such as elastic or fluid registration with very large numbers of parameters to model deformations. Different approaches to address uncertainty quantification in image registration have been proposed. For example, for rigid deformations, physical landmarks have been used to estimate the average registration error for the whole volume. [2]. Nonrigid deformations are challenging as landmarks only capture local aspects of the deformations. Instead, methods have been proposed to assess uncertainty based on probabilistic models of registration and the image itself. For B-spline models, sampling based methods have been proposed for images [5] or the optimal spline parameters [11] to create multiple registrations from which to estimate deformation uncertainty. Simpson [9] uses a variational Bayesian approach to infer the posterior distribution of B-spline parameters. Monte-Carlo sampling methods have also been explored in the context of elastic registration [8] and for LDDMM [12]. Existing methods mainly focus on parametric non-rigid registration methods such as the B-spline model or require large computational effort to sample over c Springer International Publishing Switzerland 2015 N. Navab et al. (Eds.): MICCAI 2015, Part II, LNCS 9350, pp. 289–296, 2015. DOI: 10.1007/978-3-319-24571-3_35
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X. Yang and M. Niethammer
high-dimensional parameter spaces [8,12] as for LDDMM [1]. Here, we develop a method to estimate uncertainty in the deformation parameters of the shooting formulation [10] of LDDMM. We assume a local multivariate Gaussian distribution at the optimal solution, and a
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