SIAM Journal on Imaging Sciences, Volume 13, Issue 1, Page 351-380, January 2020.
In medical image analysis, constructing an atlas, i.e., a mean representative of an ensemble of images, is a critical task for practitioners to estimate variability of shapes inside a population, and to characterize and understand how structural shape changes have an impact on health. This involves identifying significant shape constituents of a set of images, a process called segmentation, and mapping this group of images to an unknown mean image, a task called registration, making a statistical analysis of the image population possible. To achieve this goal, we propose treating these operations jointly to leverage their positive mutual influence, in a hyperelasticity setting, by viewing the shapes to be matched as Ogden materials. The approach is complemented by novel hard constraints on the $L^\infty$ norm of both the Jacobian and its inverse, ensuring that the deformation is a bi-Lipschitz homeomorphism. Segmentation is based on the Potts model, which allows for a partition into more than two regions, i.e., more than one shape. The connection to the registration problem is ensured by the dissimilarity measure that aims to align the segmented shapes. A representation of the deformation field in a linear space equipped with a scalar product is then computed in order to perform a geometry-driven Principal Component Analysis (PCA) and to extract the main modes of variations inside the image population. Theoretical results emphasizing the mathematical soundness of the model are provided, among which are existence of minimizers, analysis of a numerical method, asymptotic results, and a PCA analysis, as well as numerical simulations demonstrating the ability of the model to produce an atlas exhibiting sharp edges, high contrast, and a consistent shape.

中文翻译：

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专门用于形状分析的联合分割，配准和图集生成的变分模型

SIAM影像科学杂志，第13卷，第1期，第351-380页，2020年1月。
在医学图像分析中，构建图集（即图像集合的均值表示）对于从业者评估人群内部形状的变异性，表征和了解结构形状变化如何影响健康至关重要。这涉及到识别一组图像的重要形状成分，一个称为分割的过程，以及将这组图像映射到一个未知的平均图像（称为配准）的任务，从而可以对图像总体进行统计分析。为了实现此目标，我们建议通过将要匹配的形状视为Ogden材料，在超弹性设置中共同对待这些操作，以充分利用它们的积极相互影响。该方法还对Jacobian及其逆的$ L ^ \ infty $范数提出了新的硬性约束，确保变形是双Lipschitz同胚。分割基于Potts模型，该模型允许划分为两个以上的区域，即一个以上的形状。旨在对齐分割形状的相异性度量确保了与配准问题的联系。然后计算配备标量积的线性空间中形变场的表示形式，以执行几何驱动的主成分分析（PCA）并提取图像总体内部的主要变化模式。提供了强调模型数学稳定性的理论结果，其中包括最小化器的存在，数值方法的分析，渐近结果和PCA分析，