Chen Mingli, Lu Weiguo, Chen Quan, Ruchala Kenneth J, Olivera Gustavo H
TomoTherapy, Inc., 1240 Deming Way, Madison, Wisconsin 53717, USA.
Med Phys. 2008 Jan;35(1):81-8. doi: 10.1118/1.2816107.
Inversion of deformation fields is applied frequently to map images, dose, and contours between the reference frame and the study frame. A prevailing approach that takes the negative of the forward deformation as the inverse deformation is oversimplified and can cause large errors for large deformations or deformations that are composites of several deformations. Other approaches, including Newton's method and scatter data interpolation, either require the first derivative or are very inefficient. Here we propose an iterative approach that is easy to implement, converges quickly to the inverse when it does, and works for a majority of cases in practice. Our approach is rooted in fixed-point theory. We build a sequence to approximate the inverse deformation through iterative evaluation of the forward deformation. A sufficient but not necessary convergence condition (Lipschitz condition) and its proof are also given. Though this condition guarantees the convergence, it may not be met for an arbitrary deformation field. One should always check whether the inverse exists for the given forward deformation field by calculating its Jacobian. If nonpositive values of the Jacobian occur only for few voxels, this method will usually converge to a pseudoinverse. In case the iteration fails to converge, one should switch to other means of finding the inverse. We tested the proposed method on simulated 2D data and real 3D computed tomography data of a lung patient and compared our method with two implementations in the Insight Segmentation and Registration Toolkit (ITK). Typically less than ten iterations are needed for our method to get an inverse deformation field with clinically relevant accuracy. Based on the test results, our method is about ten times faster and yet ten times more accurate than ITK's iterative method for the same number of iterations. Simulations and real data tests demonstrated the efficacy and the accuracy of the proposed algorithm.
形变场反演经常用于在参考帧和研究帧之间映射图像、剂量和轮廓。一种常见的方法是将正向形变的负值作为反向形变，但这种方法过于简单，对于大形变或由几种形变合成的形变可能会导致较大误差。其他方法，包括牛顿法和散射数据插值法，要么需要一阶导数，要么效率非常低。在此，我们提出一种迭代方法，该方法易于实现，在收敛时能快速收敛到逆形变，并且在实际中适用于大多数情况。我们的方法基于不动点理论。我们构建一个序列，通过对正向形变的迭代评估来近似逆形变。还给出了一个充分但非必要的收敛条件（李普希茨条件）及其证明。尽管这个条件保证了收敛性，但对于任意形变场可能并不满足。对于给定的正向形变场，应该始终通过计算其雅可比行列式来检查逆形变是否存在。如果雅可比行列式的非正值仅出现在少数体素上，这种方法通常会收敛到一个伪逆。万一迭代不收敛，就应该改用其他方法来求逆。我们在模拟的二维数据和一位肺部患者的真实三维计算机断层扫描数据上测试了所提出的方法，并将我们的方法与Insight分割与配准工具包（ITK）中的两种实现方法进行了比较。通常，我们的方法只需不到十次迭代就能得到具有临床相关精度的逆形变场。基于测试结果，在相同迭代次数下，我们的方法比ITK的迭代方法快约十倍，且精度高约十倍。模拟和真实数据测试证明了所提算法的有效性和准确性。
