Diffeomorphic Image Registration Research Articles

Coupling brain-tumor biophysical models and diffeomorphic image registration

We present SIBIA (Scalable Integrated Biophysics-based Image Analysis), a framework for joint image registration and biophysical inversion and we apply it to analyze MR images of glioblastomas (primary brain tumors). We have two applications in mind. The first one is normal-to-abnormal image registration in the presence of tumor-induced topology differences. The second one is biophysical inversion based on single-time patient data. The underlying optimization problem is highly non-linear and non-convex and has not been solved before with a gradient-based approach.Given the segmentation of a normal brain MRI and the segmentation of a cancer patient MRI, we determine tumor growth parameters and a registration map so that if we “grow a tumor” (using our tumor model) in the normal brain and then register it to the patient image, then the registration mismatch is as small as possible. This “coupled problem” two-way couples the biophysical inversion and the registration problem. In the image registration step we solve a large-deformation diffeomorphic registration problem parameterized by an Eulerian velocity field. In the biophysical inversion step we estimate parameters in a reaction–diffusion tumor growth model that is formulated as a partial differential equation (PDE). In SIBIA, we couple these two sub-components in an iterative manner. We first presented the components of SIBIA in “Gholami et al., Framework for Scalable Biophysics-based Image Analysis, IEEE/ACM Proceedings of the SC2017”, in which we derived parallel distributed memory algorithms and software modules for the decoupled registration and biophysical inverse problems. In this paper, our contributions are the introduction of a PDE-constrained optimization formulation of the coupled problem, and the derivation of a Picard iterative solution scheme. We perform extensive tests to experimentally assess the performance of our method on synthetic and clinical datasets. We demonstrate the convergence of the SIBIA optimization solver in different usage scenarios. We demonstrate that using SIBIA, we can accurately solve the coupled problem in three dimensions (2563 resolution) in a few minutes using 11 dual-x86 nodes.

CLAIRE: A DISTRIBUTED-MEMORY SOLVER FOR CONSTRAINED LARGE DEFORMATION DIFFEOMORPHIC IMAGE REGISTRATION.

With this work we release CLAIRE, a distributed-memory implementation of an effective solver for constrained large deformation diifeomorphic image registration problems in three dimensions. We consider an optimal control formulation. We invert for a stationary velocity field that parameterizes the deformation map. Our solver is based on a globalized, preconditioned, inexact reduced space Gauss‒Newton‒Krylov scheme. We exploit state-of-the-art techniques in scientific computing to develop an eifective solver that scales to thousands of distributed memory nodes on high-end clusters. We present the formulation, discuss algorithmic features, describe the software package, and introduce an improved preconditioner for the reduced space Hessian to speed up the convergence of our solver. We test registration performance on synthetic and real data. We Demonstrate registration accuracy on several neuroimaging datasets. We compare the performance of our scheme against diiferent flavors of the Demons algorithm for diifeomorphic image registration. We study convergence of our preconditioner and our overall algorithm. We report scalability results on state-of-the-art supercomputing platforms. We Demonstrate that we can solve registration problems for clinically relevant data sizes in two to four minutes on a standard compute node with 20 cores, attaining excellent data fidelity. With the present work we achieve a speedup of (on average) 5× with a peak performance of up to 17× compared to our former work.

Efficient diffeomorphic metric image registration via stationary velocity

Diffeomorphic image registration is a fundamental tool for MRI analysis, and fast growing image data demand highly efficient registration methods. Stationary velocity based method has much faster speed than non-stationary one in producing diffeomorphism, but how to achieve validity for large deformation while keeping its efficiency remains to be solved. To this end, we have proposed a new simplified model for optimal stationary velocity by representing temporal integration form of transformation variation via a single point model, e.g. the middle time point, according to mean value theorem and smoothness of transformation. This model can maintain the same registration accuracy as integral model but reduce the time cost a lot. It also shows as a better strategy than first order approximation model for large deformation mapping. Comparative study has also been conducted between this method and non-stationary approach on both synthesized and real brain images, and this approach demonstrated comparable whole brain mapping accuracy with much faster speed. Results showed efficacy of this approach in reducing complexity of stationary velocity based diffeomorphic registration while achieving validity for large deformation.

Fast Diffeomorphic Image Registration via Fourier-Approximated Lie Algebras

This paper introduces Fourier-approximated Lie algebras for shooting (FLASH), a fast geodesic shooting algorithm for diffeomorphic image registration. We approximate the infinite-dimensional Lie algebra of smooth vector fields, i.e., the tangent space at the identity of the diffeomorphism group, with a low-dimensional, bandlimited space. We show that most of the computations for geodesic shooting can be carried out entirely in this low-dimensional space. Our algorithm results in dramatic savings in time and memory over traditional large-deformation diffeomorphic metric mapping algorithms, which require dense spatial discretizations of vector fields. To validate the effectiveness of FLASH, we run pairwise image registration on both 2D synthetic data and real 3D brain images and compare with the state-of-the-art geodesic shooting methods. Experimental results show that our algorithm dramatically reduces the computational cost and memory footprint of diffemorphic image registration with little or no loss of accuracy.

Efficient Laplace Approximation for Bayesian Registration Uncertainty Quantification.

This paper presents a novel approach to modeling the pos terior distribution in image registration that is computationally efficient for large deformation diffeomorphic metric mapping (LDDMM). We develop a Laplace approximation of Bayesian registration models entirely in a bandlimited space that fully describes the properties of diffeomorphic transformations. In contrast to current methods, we compute the inverse Hessian at the mode of the posterior distribution of diffeomorphisms directly in the low dimensional frequency domain. This dramatically reduces the computational complexity of approximating posterior marginals in the high dimensional imaging space. Experimental results show that our method is significantly faster than the state-of-the-art diffeomorphic image registration uncertainty quantification algorithms, while producing comparable results. The efficiency of our method strengthens the feasibility in prospective clinical applications, e.g., real- time image-guided navigation for brain surgery.

Wavefront sensing with a thin diffuser.

We propose and implement a broadband, compact, and low-cost wavefront sensing scheme by simply placing a thin diffuser in the close vicinity of a camera. The local wavefront gradient is determined from the local translation of the speckle pattern. The translation vector map is computed thanks to a fast diffeomorphic image registration algorithm and integrated to reconstruct the wavefront profile. The simple translation of speckle grains under local wavefront tip/tilt is ensured by the so-called "memory effect" of the diffuser. Quantitative wavefront measurements are experimentally demonstrated, both for the few first Zernike polynomials and for phase-imaging applications requiring high resolution. We finally provided a theoretical description of the resolution limit that is supported experimentally.

Collocation for Diffeomorphic Deformations in Medical Image Registration.

Diffeomorphic deformation is a popular choice in medical image registration. A fundamental property of diffeomorphisms is invertibility, implying that once the relation between two points A to B is found, then the relation B to A is given per definition. Consistency is a measure of a numerical algorithm's ability to mimic this invertibility, and achieving consistency has proven to be a challenge for many state-of-the-art algorithms. We present CDD (Collocation for Diffeomorphic Deformations), a numerical solution to diffeomorphic image registration, which solves for the Stationary Velocity Field (SVF) using an implicit A-stable collocation method. CDD guarantees the preservation of the diffeomorphic properties at all discrete points and is thereby consistent to machine precision. We compared CDD's collocation method with the following standard methods: Scaling and Squaring, Forward Euler, and Runge-Kutta 4, and found that CDD is up to 9 orders of magnitude more consistent. Finally, we evaluated CDD on a number of standard bench-mark data sets and compared the results with current state-of-the-art methods: SPM-DARTEL, Diffeomorphic Demons and SyN. We found that CDD outperforms state-of-the-art methods in consistency and delivers comparable or superior registration precision.

Phase Vector Incompressible Registration Algorithm for Motion Estimation From Tagged Magnetic Resonance Images.

Tagged magnetic resonance imaging has been used for decades to observe and quantify motion and strain of deforming tissue. It is challenging to obtain 3-D motion estimates due to a tradeoff between image slice density and acquisition time. Typically, interpolation methods are used either to combine 2-D motion extracted from sparse slice acquisitions into 3-D motion or to construct a dense volume from sparse acquisitions before image registration methods are applied. This paper proposes a new phase-based 3-D motion estimation technique that first computes harmonic phase volumes from interpolated tagged slices and then matches them using an image registration framework. The approach uses several concepts from diffeomorphic image registration with a key novelty that defines a symmetric similarity metric on harmonic phase volumes from multiple orientations. The material property of harmonic phase solves the aperture problem of optical flow and intensity-based methods and is robust to tag fading. A harmonic magnitude volume is used in enforcing incompressibility in the tissue regions. The estimated motion fields are dense, incompressible, diffeomorphic, and inverse-consistent at a 3-D voxel level. The method was evaluated using simulated phantoms, human brain data in mild head accelerations, human tongue data during speech, and an open cardiac data set. The method shows comparable accuracy to three existing methods while demonstrating low computation time and robustness to tag fading and noise.

Evaluation of five diffeomorphic image registration algorithms for mouse brain magnetic resonance microscopy.

The development of genetically engineered mouse models for neuronal diseases and behavioural disorders have generated a growing need for small animal imaging. High-resolution magnetic resonance microscopy (MRM) provides powerful capabilities for noninvasive studies of mouse brains, while avoiding some limits associated with the histological procedures. Quantitative comparison of structural images is a critical step in brain imaging analysis, which highly relies on the performance of image registration techniques. Nowadays, there is a mushrooming growth of human brain registration algorithms, while fine-tuning of those algorithms for mouse brain MRMs is rarely addressed. Because of their topology preservation property and outstanding performance in human studies, diffeomorphic transformations have become popular in computational anatomy. In this study, we specially tuned five diffeomorphic image registration algorithms [DARTEL, geodesic shooting, diffeo-demons, SyN (Greedy-SyN and geodesic-SyN)] for mouse brain MRMs and evaluated their performance using three measures [volume overlap percentage (VOP), residual intensity error (RIE) and surface concordance ratio (SCR)]. Geodesic-SyN performed significantly better than the other methods according to all three different measures. These findings are important for the studies on structural brain changes that may occur in wild-type and transgenic mouse brains.

ADAPTIVE GRADIENT DESCENT OPTIMIZATION OF INITIAL MOMENTA FOR GEODESIC SHOOTING IN DIFFEOMORPHISMS.

Diffeomorphic image registration algorithms are widely used in medical imaging, and require optimization of a high-dimensional nonlinear objective function. The function being optimized has many characteristics that are relevant for optimization but are typically not well understood. Due to that complexity, most authors have used a simple gradient descent, but it is not often discussed how step sizes are chosen or if line searches are used. Further, if a system is to be robust to a range of input images, that may differ to varying degrees, the optimization must be adaptable. Here, we present two methods of adaptable gradient descent with line searches, and test how they affect image registration. The optimization schemes are deployed for geodesic shooting in diffeomorphisms - an approach that is used to quantify anatomical changes, such as atrophy, in longitudinal image pairs. We evaluate the optimization schemes on their convergence characteristics and based on how well the resulting atrophy scores correlate with diagnostic group and mini mental state exam (MMSE) scores. We find that the Barzilai-Borwein method with a backtracking line search outperforms other optimization schemes in convergence time and adaptability by a wide margin. We also find that the variable optimization schemes do not significantly affect the ability to measure atrophy with clinical significance.

A SEMI-LAGRANGIAN TWO-LEVEL PRECONDITIONED NEWTON-KRYLOV SOLVER FOR CONSTRAINED DIFFEOMORPHIC IMAGE REGISTRATION.

We propose an efficient numerical algorithm for the solution of diffeomorphic image registration problems. We use a variational formulation constrained by a partial differential equation (PDE), where the constraints are a scalar transport equation. We use a pseudospectral discretization in space and second-order accurate semi-Lagrangian time stepping scheme for the transport equations. We solve for a stationary velocity field using a preconditioned, globalized, matrix-free Newton-Krylov scheme. We propose and test a two-level Hessian preconditioner. We consider two strategies for inverting the preconditioner on the coarse grid: a nested preconditioned conjugate gradient method (exact solve) and a nested Chebyshev iterative method (inexact solve) with a fixed number of iterations. We test the performance of our solver in different synthetic and real-world two-dimensional application scenarios. We study grid convergence and computational efficiency of our new scheme. We compare the performance of our solver against our initial implementation that uses the same spatial discretization but a standard, explicit, second-order Runge-Kutta scheme for the numerical time integration of the transport equations and a single-level preconditioner. Our improved scheme delivers significant speedups over our original implementation. As a highlight, we observe a 20× speedup for a two dimensional, real world multi-subject medical image registration problem.

A LAGRANGIAN GAUSS-NEWTON-KRYLOV SOLVER FOR MASS- AND INTENSITY-PRESERVING DIFFEOMORPHIC IMAGE REGISTRATION.

We present an efficient solver for diffeomorphic image registration problems in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM). We use an optimal control formulation, in which the velocity field of a hyperbolic PDE needs to be found such that the distance between the final state of the system (the transformed/transported template image) and the observation (the reference image) is minimized. Our solver supports both stationary and non-stationary (i.e., transient or time-dependent) velocity fields. As transformation models, we consider both the transport equation (assuming intensities are preserved during the deformation) and the continuity equation (assuming mass-preservation). We consider the reduced form of the optimal control problem and solve the resulting unconstrained optimization problem using a discretize-then-optimize approach. A key contribution is the elimination of the PDE constraint using a Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of characteristic curves. We approximate these curves using a fourth-order Runge-Kutta method. We also present an efficient algorithm for computing the derivatives of the final state of the system with respect to the velocity field. This allows us to use fast Gauss-Newton based methods. We present quickly converging iterative linear solvers using spectral preconditioners that render the overall optimization efficient and scalable. Our method is embedded into the image registration framework FAIR and, thus, supports the most commonly used similarity measures and regularization functionals. We demonstrate the potential of our new approach using several synthetic and real world test problems with up to 14.7 million degrees of freedom.

Fast Geodesic Regression for Population-Based Image Analysis.

Geodesic regression on images enables studies of brain development and degeneration, disease progression, and tumor growth. The high-dimensional nature of image data presents significant computational challenges for the current regression approaches and prohibits large scale studies. In this paper, we present a fast geodesic regression method that dramatically decreases the computational cost of the inference procedure while maintaining prediction accuracy. We employ an efficient low dimensional representation of diffeomorphic transformations derived from the image data and characterize the regressed trajectory in the space of diffeomorphisms by its initial conditions, i.e., an initial image template and an initial velocity field computed as a weighted average of pairwise diffeomorphic image registration results. This construction is achieved by using a first-order approximation of pairwise distances between images. We demonstrate the efficiency of our model on a set of 3D brain MRI scans from the OASIS dataset and show that it is dramatically faster than the state-of-the-art regression methods while producing equally good regression results on the large subject cohort.

Frequency Diffeomorphisms for Efficient Image Registration.

This paper presents an efficient algorithm for large deformation diffeomorphic metric mapping (LDDMM) with geodesic shooting for image registration. We introduce a novel finite dimensional Fourier representation of diffeomorphic deformations based on the key fact that the high frequency components of a diffeomorphism remain stationary throughout the integration process when computing the deformation associated with smooth velocity fields. We show that manipulating high dimensional diffeomorphisms can be carried out entirely in the bandlimited space by integrating the nonstationary low frequency components of the displacement field. This insight substantially reduces the computational cost of the registration problem. Experimental results show that our method is significantly faster than the state-of-the-art diffeomorphic image registration methods while producing equally accurate alignment. We demonstrate our algorithm in two different applications of image registration: neuroimaging and in-utero imaging.

A quantitative symmetry‐based analysis of hyperacute ischemic stroke lesions in noncontrast computed tomography

Early identification of ischemic stroke plays a significant role in treatment and potential recovery of damaged brain tissue. In noncontrast CT (ncCT), the differences between ischemic changes and healthy tissue are usually very subtle during the hyperacute phase (< 8 h from the stroke onset). Therefore, visual comparison of both hemispheres is an important step in clinical assessment. A quantitative symmetry-based analysis of texture features of ischemic lesions in noncontrast CT images may provide an important information for differentiation of ischemic and healthy brain tissue in this phase. One hundred thirty-nine (139) ncCT scans of hyperacute ischemic stroke with follow-up magnetic resonance diffusion-weighted (MR-DW) images were collected. The regions of stroke were identified in the MR-DW images, which were spatially aligned to corresponding ncCT images. A state-of-the-art symmetric diffeomorphic image registration was utilized for the alignment of CT and MR-DW, for identification of individual brain hemispheres, and for localization of the region representing healthy tissue contralateral to the stroke cores. Texture analysis included extraction and classification of co-occurrence and run-length texture-based image features in the regions of ischemic stroke and their contralateral regions. The classification schemes achieved area under the receiver operating characteristic [Az] ≈ 0.82 for the whole dataset. There was no statistically significant difference in the performance of classifiers for the data sets with time between 2 and 8 hours from symptom onset. The performance of the classifiers did not depend on the size of the stroke regions. The results provide a set of optimal texture features which are suitable for distinguishing between hyperacute ischemic lesions and their corresponding contralateral brain tissue in noncontrast CT. This work is an initial step toward development of an automated decision support system for detection of hyperacute ischemic stroke lesions on noncontrast CT of the brain.

DR-TAMAS: Diffeomorphic Registration for Tensor Accurate Alignment of Anatomical Structures

In this work, we propose DR-TAMAS (Diffeomorphic Registration for Tensor Accurate alignMent of Anatomical Structures), a novel framework for intersubject registration of Diffusion Tensor Imaging (DTI) data sets. This framework is optimized for brain data and its main goal is to achieve an accurate alignment of all brain structures, including white matter (WM), gray matter (GM), and spaces containing cerebrospinal fluid (CSF). Currently most DTI-based spatial normalization algorithms emphasize alignment of anisotropic structures. While some diffusion-derived metrics, such as diffusion anisotropy and tensor eigenvector orientation, are highly informative for proper alignment of WM, other tensor metrics such as the trace or mean diffusivity (MD) are fundamental for a proper alignment of GM and CSF boundaries. Moreover, it is desirable to include information from structural MRI data, e.g., T1-weighted or T2-weighted images, which are usually available together with the diffusion data. The fundamental property of DR-TAMAS is to achieve global anatomical accuracy by incorporating in its cost function the most informative metrics locally. Another important feature of DR-TAMAS is a symmetric time-varying velocity-based transformation model, which enables it to account for potentially large anatomical variability in healthy subjects and patients. The performance of DR-TAMAS is evaluated with several data sets and compared with other widely-used diffeomorphic image registration techniques employing both full tensor information and/or DTI-derived scalar maps. Our results show that the proposed method has excellent overall performance in the entire brain, while being equivalent to the best existing methods in WM.

Constrained H1-regularization schemes for diffeomorphic image registration.

We propose regularization schemes for deformable registration and efficient algorithms for their numerical approximation. We treat image registration as a variational optimal control problem. The deformation map is parametrized by its velocity. Tikhonov regularization ensures well-posedness. Our scheme augments standard smoothness regularization operators based on H1- and H2-seminorms with a constraint on the divergence of the velocity field, which resembles variational formulations for Stokes incompressible flows. In our formulation, we invert for a stationary velocity field and a mass source map. This allows us to explicitly control the compressibility of the deformation map and by that the determinant of the deformation gradient. We also introduce a new regularization scheme that allows us to control shear. We use a globalized, preconditioned, matrix-free, reduced space (Gauss-)Newton-Krylov scheme for numerical optimization. We exploit variable elimination techniques to reduce the number of unknowns of our system; we only iterate on the reduced space of the velocity field. Our current implementation is limited to the two-dimensional case. The numerical experiments demonstrate that we can control the determinant of the deformation gradient without compromising registration quality. This additional control allows us to avoid oversmoothing of the deformation map. We also demonstrate that we can promote or penalize shear whilst controlling the determinant of the deformation gradient.

Kernel Bundle Diffeomorphic Image Registration Using Stationary Velocity Fields and Wendland Basis Functions.

In this paper, we propose a multi-scale, multi-kernel shape, compactly supported kernel bundle framework for stationary velocity field-based image registration (Wendland kernel bundle stationary velocity field, wKB-SVF). We exploit the possibility of directly choosing kernels to construct a reproducing kernel Hilbert space (RKHS) instead of imposing it from a differential operator. The proposed framework allows us to minimize computational cost without sacrificing the theoretical foundations of SVF-based diffeomorphic registration. In order to recover deformations occurring at different scales, we use compactly supported Wendland kernels at multiple scales and orders to parameterize the velocity fields, and the framework allows simultaneous optimization over all scales. The performance of wKB-SVF is extensively compared to the 14 non-rigid registration algorithms presented in a recent comparison paper. On both MGH10 and CUMC12 datasets, the accuracy of wKB-SVF is improved when compared to other registration algorithms. In a disease-specific application for intra-subject registration, atrophy scores estimated using the proposed registration scheme separates the diagnostic groups of Alzheimer's and normal controls better than the state-of-the-art segmentation technique. Experimental results show that wKB-SVF is a robust, flexible registration framework that allows theoretically well-founded and computationally efficient multi-scale representation of deformations and is equally well-suited for both inter- and intra-subject image registration.

Reduction by Lie Group Symmetries in Diffeomorphic Image Registration and Deformation Modelling

We survey the role of reduction by symmetry in the large deformation diffeomorphic metric mapping framework for registration of a variety of data types (landmarks, curves, surfaces, images and higher-order derivative data). Particle relabelling symmetry allows the equations of motion to be reduced to the Lie algebra allowing the equations to be written purely in terms of the Eulerian velocity field. As a second use of symmetry, the infinite dimensional problem of finding correspondences between objects can be reduced for a range of concrete data types, resulting in compact representations of shape and spatial structure. Using reduction by symmetry, we describe these models in a common theoretical framework that draws on links between the registration problem and geometric mechanics. We outline these constructions and further cases where reduction by symmetry promises new approaches to the registration of complex data types.

An Inexact Newton-Krylov Algorithm for Constrained Diffeomorphic Image Registration.

We propose numerical algorithms for solving large deformation diffeomorphic image registration problems. We formulate the nonrigid image registration problem as a problem of optimal control. This leads to an infinite-dimensional partial differential equation (PDE) constrained optimization problem. The PDE constraint consists, in its simplest form, of a hyperbolic transport equation for the evolution of the image intensity. The control variable is the velocity field. Tikhonov regularization on the control ensures well-posedness. We consider standard smoothness regularization based on H1- or H2-seminorms. We augment this regularization scheme with a constraint on the divergence of the velocity field (control variable) rendering the deformation incompressible (Stokes regularization scheme) and thus ensuring that the determinant of the deformation gradient is equal to one, up to the numerical error. We use a Fourier pseudospectral discretization in space and a Chebyshev pseudospectral discretization in time. The latter allows us to reduce the number of unknowns and enables the time-adaptive inversion for nonstationary velocity fields. We use a preconditioned, globalized, matrix-free, inexact Newton-Krylov method for numerical optimization. A parameter continuation is designed to estimate an optimal regularization parameter. Regularity is ensured by controlling the geometric properties of the deformation field. Overall, we arrive at a black-box solver that exploits computational tools that are precisely tailored for solving the optimality system. We study spectral properties of the Hessian, grid convergence, numerical accuracy, computational efficiency, and deformation regularity of our scheme. We compare the designed Newton-Krylov methods with a globalized Picard method (preconditioned gradient descent). We study the influence of a varying number of unknowns in time. The reported results demonstrate excellent numerical accuracy, guaranteed local deformation regularity, and computational efficiency with an optional control on local mass conservation. The Newton-Krylov methods clearly outperform the Picard method if high accuracy of the inversion is required. Our method provides equally good results for stationary and nonstationary velocity fields for two-image registration problems.