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.

中文翻译：

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用于微分同胚图像配准的约束 H1 正则化方案。

我们提出了用于变形配准的正则化方案和用于其数值近似的有效算法。我们将图像配准视为变分最优控制问题。变形图由其速度参数化。吉洪诺夫正则化确保适定性。我们的方案增强了基于 H1- 和 H2-半范数的标准平滑正则化算子，并限制了速度场的散度，这类似于斯托克斯不可压缩流的变分公式。在我们的公式中，我们对静止速度场和质量源图进行反演。这使我们能够明确控制变形图的可压缩性以及变形梯度的决定因素。我们还引入了一种新的正则化方案，使我们能够控制剪切力。我们使用全局化、预处理、无矩阵、缩减空间（高斯）Newton-Krylov 方案进行数值优化。我们利用变量消除技术来减少系统的未知数；我们只迭代速度场的缩减空间。我们当前的实现仅限于二维情况。数值实验表明，我们可以在不影响配准质量的情况下控制变形梯度的决定因素。这种额外的控制使我们能够避免变形图的过度平滑。我们还证明，我们可以在控制变形梯度的决定因素的同时促进或惩罚剪切。