Xue Zhong, Shen Dinggang, Karacali Bilge, Davatzikos Christos
Section of Biomedical Image Analysis, Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104, USA.
Med Image Comput Comput Assist Interv. 2005;8(Pt 2):500-8. doi: 10.1007/11566489_62.
This paper proposes an approach to effectively representing the statistics of high-dimensional deformations, when relatively few training samples are available, and conventional methods, like PCA, fail due to insufficient training. Based on previous work on scale-space decomposition of deformation fields, herein we represent the space of "valid deformations" as the intersection of three subspaces: one that satisfies constraints on deformations themselves, one that satisfies constraints on Jacobian determinants of deformations, and one that represents smooth deformations via a Markov Random Field (MRF). The first two are extensions of PCA-based statistical shape models. They are based on a wavelet packet basis decomposition that allows for more accurate estimation of the covariance structure of deformation or Jacobian fields, and they are used jointly due to their complementary strengths and limitations. The third is a nested MRF regularization aiming at eliminating potential discontinuities introduced by assumptions in the statistical models. A randomly sampled deformation field is projected onto the space of valid deformations via iterative projections on each of these subspaces until convergence, i.e. all three constraints are met. A deformation field simulator uses this process to generate random samples of deformation fields that are not only realistic but also representative of the full range of anatomical variability. These simulated deformations can be used for validation of deformable registration methods. Other potential uses of this approach include representation of shape priors in statistical shape models as well as various estimation and hypothesis testing paradigms in the general fields of computational anatomy and pattern recognition.
本文提出了一种方法，用于在可用训练样本相对较少且传统方法（如主成分分析（PCA））因训练不足而失效的情况下，有效地表示高维变形的统计信息。基于先前关于变形场尺度空间分解的工作，在此我们将“有效变形”空间表示为三个子空间的交集：一个满足对变形本身的约束，一个满足对变形雅可比行列式的约束，还有一个通过马尔可夫随机场（MRF）表示平滑变形。前两个是基于PCA的统计形状模型的扩展。它们基于小波包基分解，能够更准确地估计变形或雅可比场的协方差结构，并且由于它们互补的优势和局限性而联合使用。第三个是嵌套的MRF正则化，旨在消除统计模型假设引入的潜在不连续性。通过在这些子空间中的每一个上进行迭代投影，直到收敛（即满足所有三个约束），将随机采样的变形场投影到有效变形空间上。变形场模拟器使用此过程生成不仅逼真而且能代表整个解剖变异性范围的变形场随机样本。这些模拟变形可用于验证可变形配准方法。该方法的其他潜在用途包括在统计形状模型中表示形状先验，以及在计算解剖学和模式识别的一般领域中的各种估计和假设检验范式。
