4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data
4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data
复制标题
DOI：
10.1109/tpami.2022.3163720
复制
发表时间：
2021-01
影响因子：
23.6
通讯作者：
Hamid Laga;Marcel Padilla;Ian H. Jermyn;S. Kurtek;Bennamoun;A. Srivastava
中科院分区：
文献类型：
--
作者：
Hamid Laga;Marcel Padilla;Ian H. Jermyn;S. Kurtek;Bennamoun;A. Srivastava
我们提出了一个新的框架来学习纵向3D形状数据集的时空变化，其中包含随着时间的推移而演变和变形的对象的观察。这个问题是具有挑战性的，因为表面来与任意参数化，因此，他们需要空间注册。此外，不同的变形对象（下文中称为4D表面）以不同的速度演变，因此它们需要在时间上对准。我们解决这个时空配准问题，使用黎曼方法。我们将3D表面视为形状空间中的一个点，该形状空间配备了弹性黎曼度量，该度量测量表面所经历的弯曲和拉伸的量。一个4D表面可以被看作是这个空间中的一个轨迹。有了这个公式，4D表面的统计分析可以被转换为分析嵌入在非线性黎曼流形中的轨迹的问题。然而，在这样的非线性空间上执行时空配准以及随后计算统计并不简单，因为它们依赖于复杂的非线性优化。我们的核心贡献是将曲面映射到平方根正规场（SRNF）空间，其中$\mathbb {L}^{2}$L2度量等价于曲面空间中的部分弹性度量。因此，通过解决SRNF空间中的空间配准，分析4D表面的问题变成分析嵌入在具有欧几里德结构的SRNF空间中的轨迹的问题。在本文中，我们开发的积木，使这种分析。其中包括：（1）任意参数化的4D表面的时空配准，即使在存在大的弹性变形和它们的执行速率的大的变化的情况下;（2）4D表面之间的测地线的计算;（3）4D表面的集合的统计概要的计算，例如变化的均值和模式;以及（4）随机4D表面的合成。我们使用4D面部表面和4D人体形状展示了所提出的框架的性能。
We propose a novel framework to learn the spatiotemporal variability in longitudinal 3D shape data sets, which contain observations of objects that evolve and deform over time. This problem is challenging since surfaces come with arbitrary parameterizations and thus, they need to be spatially registered. Also, different deforming objects, hereinafter referred to as 4D surfaces, evolve at different speeds and thus they need to be temporally aligned. We solve this spatiotemporal registration problem using a Riemannian approach. We treat a 3D surface as a point in a shape space equipped with an elastic Riemannian metric that measures the amount of bending and stretching that the surfaces undergo. A 4D surface can then be seen as a trajectory in this space. With this formulation, the statistical analysis of 4D surfaces can be cast as the problem of analyzing trajectories embedded in a nonlinear Riemannian manifold. However, performing the spatiotemporal registration, and subsequently computing statistics, on such nonlinear spaces is not straightforward as they rely on complex nonlinear optimizations. Our core contribution is the mapping of the surfaces to the space of Square-Root Normal Fields (SRNF) where the $\mathbb {L}^{2}$L2 metric is equivalent to the partial elastic metric in the space of surfaces. Thus, by solving the spatial registration in the SRNF space, the problem of analyzing 4D surfaces becomes the problem of analyzing trajectories embedded in the SRNF space, which has a euclidean structure. In this paper, we develop the building blocks that enable such analysis. These include: (1) the spatiotemporal registration of arbitrarily parameterized 4D surfaces even in the presence of large elastic deformations and large variations in their execution rates; (2) the computation of geodesics between 4D surfaces; (3) the computation of statistical summaries, such as means and modes of variation, of collections of 4D surfaces; and (4) the synthesis of random 4D surfaces. We demonstrate the performance of the proposed framework using 4D facial surfaces and 4D human body shapes.