We present a Gauss-Newton-Krylov solver for large deformation diffeomorphic image registration. We extend the publicly available CLAIRE library to multi-node multi-graphics processing unit (GPUs) systems and introduce novel algorithmic modifications that significantly improve performance. Our contributions comprise (i) a new preconditioner for the reduced-space Gauss-Newton Hessian system, (ii) a highly-optimized multi-node multi-GPU implementation exploiting device direct communication for the main computational kernels (interpolation, high-order finite difference operators and Fast-Fourier-Transform), and (iii) a comparison with state-of-the-art CPU and GPU implementations. We solve a 2563-resolution image registration problem in five seconds on a single NVIDIA Tesla V100, with a performance speedup of 70% compared to the state-of-the-art. In our largest run, we register 20483 resolution images (25B unknowns; approximately 152$\times$ larger than the largest problem solved in state-of-the-art GPU implementations) on 64 nodes with 256 GPUs on TACC’s Longhorn system.

We present SIBIA (Scalable Integrated Biophysics-based Image Analysis), a framework for coupling biophysical models with medical image analysis. It provides solvers for an image-driven inverse brain tumor growth model and an image registration problem, the combination of which can eventually help in diagnosis and prognosis of brain tumors. The two main computational kernels of SIBIA are a Fast Fourier Transformation (FFT) implemented in the library AccFFT to discretize differential operators, and a cubic interpolation kernel for semi-Lagrangian based advection. We present efficiency and scalability results for the computational kernels, the inverse tumor solver and image registration on two x86 systems, Lonestar 5 at the Texas Advanced Computing Center and Hazel Hen at the Stuttgart High Performance Computing Center. We showcase results that demonstrate that our solver can be used to solve registration problems of unprecedented scale, 40963 resulting in ∼ 200 billion unknowns---a problem size that is 64X larger than the state-of-the-art. For problem sizes of clinical interest, SIBIA is about 8X faster than the state-of-the-art.

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 backward 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 10243 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 2563-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.

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 $H^1$- and $H^2$-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 while controlling the determinant of the deformation gradient.

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 (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.