Image registration is an inherently ill-posed problem that lacks the constraints needed for a unique mapping between voxels of the two images being registered. As such, one must regularize the registration to achieve physically meaningful transforms. The regularization penalty is usually a function of derivatives of the displacement-vector field, and can be calculated either analytically or numerically. The numerical approach, however, is computationally expensive depending on the image size, and therefore a computationally efficient analytical framework has been developed. Using cubic B-splines as the registration transform, we develop a generalized mathematical framework that supports five distinct regularizers: diffusion, curvature, linear elastic, third-order, and total displacement. We validate our approach by comparing each with its numerical counterpart in terms of accuracy. We also provide benchmarking results showing that the analytic solutions run significantly faster - up to two orders of magnitude - than finite differencing based numerical implementations.

中文翻译：

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均匀三次b样条位移场解析正则化的广义框架。

图像配准是一个固有的不适定问题，缺少在要配准的两个图像的体素之间进行唯一映射所需的约束。因此，必须规范化注册过程，以实现对身体有意义的转换。正则化罚分通常是位移矢量场的导数的函数，可以通过解析或数值计算得出。然而，数值方法取决于图像大小在计算上是昂贵的，因此已经开发了一种计算有效的分析框架。使用三次B样条作为配准变换，我们开发了一个通用的数学框架，该框架支持五个不同的正则化器：扩散，曲率，线性弹性，三阶和总位移。我们通过将每种方法与其数值对应物在准确性方面进行比较来验证我们的方法。我们还提供了基准测试结果，表明与基于有限差分的数值实现方案相比，解析解决方案的运行速度显着提高（最多两个数量级）。