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.

图像配准是一个固有的不适定问题，它缺乏被配准的两个图像的体素之间的唯一映射所需的约束。因此，必须规范注册以实现物理上有意义的转换。正则化惩罚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。然而，数值方法的计算成本取决于图像大小，因此已经开发了计算效率高的分析框架。使用三次 B 样条作为配准变换，我们开发了一个支持五种不同正则化器的广义数学框架：扩散、曲率、线弹性、三阶和总位移。我们通过在准确性方面将每个方法与其数值对应物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相比，解析解的运行速度要快得多——最多两个数量级。