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.

中文翻译：

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约束微分形图像配准的不精确Newton-Krylov算法。

我们提出了数值算法来解决大变形微晶图像配准问题。我们将非刚性图像配准问题表述为最优控制问题。这导致了无穷维偏微分方程（PDE）约束的优化问题。PDE约束以其最简单的形式包括一个用于图像强度演变的双曲线传递方程。控制变量是速度场。控件上的Tikhonov正则化可确保定型性。我们考虑基于H1-或H2-seminorms的标准平滑度正则化。我们通过限制速度场（控制变量）的发散性来扩展此正则化方案，从而使变形不可压缩（斯托克斯正则化方案），从而确保变形梯度的决定因素等于1，直至数值误差。我们在空间中使用傅立叶伪谱离散化，在时间上使用Chebyshev伪谱离散化。后者使我们能够减少未知数，并使非平稳速度场能够进行时间自适应反演。我们使用预处理的，全局的，无矩阵的，不精确的Newton-Krylov方法进行数值优化。参数连续被设计为估计最佳正则化参数。通过控制变形场的几何特性可确保规则性。全面的，我们得到了一个黑盒求解器，该求解器利用了专门为求解最优系统而量身定制的计算工具。我们研究Hessian的光谱特性，网格收敛，数值精度，计算效率和方案的变形规律。我们将设计的Newton-Krylov方法与全球化的Picard方法（预处理的梯度下降）进行比较。我们及时研究了各种未知数的影响。报告的结果证明了极好的数值精度，保证的局部变形规律性和计算效率，并且可以选择控制局部质量守恒。如果需要反演的高精度，则Newton-Krylov方法明显优于Picard方法。