Distributed-memory large deformation diffeomorphic 3D image registration

We present a parallel distributed-memory algorithm for large deformation diffeomorphic registration of volumetric images that produces large isochoric deformations (locally volume preserving). Image registration is a key technology in medical image analysis. Our algorithm uses a partial differential equation constrained optimal control formulation. Finding the optimal deformation map requires the solution of a highly nonlinear problem that involves pseudo-differential operators, biharmonic operators, and pure advection operators both forward and back- ward in time. A key issue is the time to solution, which poses the demand for efficient optimization methods as well as an effective utilization of high performance computing resources. To address this problem we use a preconditioned, inexact, Gauss-Newton- Krylov solver. Our algorithm integrates several components: a spectral discretization in space, a semi-Lagrangian formulation in time, analytic adjoints, different regularization functionals (including volume-preserving ones), a spectral preconditioner, a highly optimized distributed Fast Fourier Transform, and a cubic interpolation scheme for the semi-Lagrangian time-stepping. We demonstrate the scalability of our algorithm on images with resolution of up to $1024^3$ on the "Maverick" and "Stampede" systems at the Texas Advanced Computing Center (TACC). The critical problem in the medical imaging application domain is strong scaling, that is, solving registration problems of a moderate size of $256^3$---a typical resolution for medical images. We are able to solve the registration problem for images of this size in less than five seconds on 64 x86 nodes of TACC's "Maverick" system.Comment: accepted for publication at SC16 in Salt Lake City, Utah, USA; November 201

Finite element surface registration incorporating curvature, volume preservation, and statistical model information

We present a novel method for nonrigid registration of 3D surfaces and images. The method can be used to register surfaces by means of their distance images, or to register medical images directly. It is formulated as a minimization problem of a sum of several terms representing the desired properties of a registration result: smoothness, volume preservation, matching of the surface, its curvature, and possible other feature images, as well as consistency with previous registration results of similar objects, represented by a statistical deformation model. While most of these concepts are already known, we present a coherent continuous formulation of these constraints, including the statistical deformation model. This continuous formulation renders the registration method independent of its discretization. The finite element discretization we present is, while independent of the registration functional, the second main contribution of this paper. The local discontinuous Galerkin method has not previously been used in image registration, and it provides an efficient and general framework to discretize each of the terms of our functional. Computational efficiency and modest memory consumption are achieved thanks to parallelization and locally adaptive mesh refinement. This allows for the first time the use of otherwise prohibitively large 3D statistical deformation models

An Efficient Numerical Method for Mean Curvature-Based Image Registration Model

Mean curvature-based image registration model firstly proposed by Chumchob-Chen-Brito (2011) offered a better regularizer technique for both smooth and nonsmooth deformation fields. However, it is extremely challenging to solve efficiently this model and the existing methods are slow or become efficient only with strong assumptions on the smoothing parameter β. In this paper, we take a different solution approach. Firstly, we discretize the joint energy functional, following an idea of relaxed fixed point is implemented and combine with Gauss-Newton scheme with Armijo's Linear Search for solving the discretized mean curvature model and further to combine with a multilevel method to achieve fast convergence. Numerical experiments not only confirm that our proposed method is efficient and stable, but also it can give more satisfying registration results according to image quality

Symmetric and Transitive Registration of Image Sequences

This paper presents a method for constructing symmetric and transitive algorithms for registration of image sequences from image registration algorithms that do not have these two properties. The method is applicable to both rigid and nonrigid registration and it can be used with linear or periodic image sequences. The symmetry and transitivity properties are satisfied exactly (up to the machine precision), that is, they always hold regardless of the image type, quality, and the registration algorithm as long as the computed transformations are invertable. These two properties are especially important in motion tracking applications since physically incorrect deformations might be obtained if the registration algorithm is not symmetric and transitive. The method was tested on two sequences of cardiac magnetic resonance images using two different nonrigid image registration algorithms. It was demonstrated that the transitivity and symmetry errors of the symmetric and transitive modification of the algorithms could be made arbitrary small when the computed transformations are invertable, whereas the corresponding errors for the nonmodified algorithms were on the order of the pixel size. Furthermore, the symmetric and transitive modification of the algorithms had higher registration accuracy than the nonmodified algorithms for both image sequences

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 $H^1$- or $H^2$-seminorms. We augment this regularization scheme with a constraint on the divergence of the velocity field rendering the deformation incompressible 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. 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. 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 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.Comment: 32 pages; 10 figures; 9 table

An improved discontinuity-preserving image registration model and its fast algorithm

Recently, Chumchob–Chen(2010) proposed a discontinuity-preserving image registration model which was more flexible than those common techniques such as the diffusion and total variation based regularization techniques. However, each component of displacement field is regularized separately in this model and then the nonlinear diffusion processes resulting from the first variation of the discontinuity-preserving regularization do not enforce coupling between the primary components of the displacement field. Thus the discontinuity-preserving model may prevent to obtain a good registration in some situations, for example non-smooth registration problems with non-axis-aligned discontinuities. To utilize interdependence between the primary components of the deformation field for smooth and non-smooth registration problems, we propose an improved discontinuity-preserving image registration model in this paper, second we propose an idea of relaxed fixed point combining with Gauss–Newton scheme with Armijo's line search for solving the new model and further to combine with a multilevel method to achieve fast convergence. Numerical experiments not only confirm that our proposed method is efficient and stable, but also it can give more satisfying registration results according to image quality

An Augmented Lagrangian Method for Solving a New Variational Model based on Gradients Similarity Measures and High Order Regularization for Multimodality Registration

In this work we propose a variational model for multi-modal image registration. It minimizes a new functional based on using reformulated normalized gradients of the images as the fidelity term and higher-order derivatives as the regularizer. We first present a theoretical analysis of the proposed model. Then, to solve the model numerically, we use an augmented Lagrangian method (ALM) to reformulate it to a few more amenable subproblems (each giving rise to an Euler-Lagrange equation that is discretized by finite difference methods) and solve iteratively the main linear systems by the fast Fourier transform; a multilevel technique is employed to speed up the initialisation and avoid likely local minima of the underlying functional. Finally we show the convergence of the ALM solver and give numerical results of the new approach. Comparisons with some existing methods are presented to illustrate its effectiveness and advantages