4D Atlas: Statistical Analysis of the Spatiotemporal Variability in Longitudinal 3D Shape Data (2101.09403v2)

Abstract: 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, also called 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 where the 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 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; and (4) the synthesis of random 4D surfaces.

• The paper presents a novel framework that employs elastic Riemannian metrics and SRNF to simplify the statistical analysis of evolving 3D shapes.
• It leverages advanced spatiotemporal registration techniques to compute geodesics, Karcher means, and principal modes, capturing dynamic deformations effectively.
• The framework supports efficient synthesis of 4D surfaces, offering practical applications in fields such as medical diagnosis, computer graphics, and deep learning.

4D Atlas: Statistical Analysis of Spatiotemporal Variability in Longitudinal 3D Shape Data

The paper introduces a comprehensive framework for the analysis of spatiotemporal variability in longitudinal 3D shape data. The authors focus on 4D surfaces, which are essentially trajectories traced by 3D shapes as they evolve over time. Such data often arise in applications like facial expression analysis, body motion capture, and growth modeling in biology or medicine. These 4D constructs pose significant challenges due to their inherent spatiotemporal variability, necessitating sophisticated registration methods to facilitate meaningful analysis.

Framework Overview

The core methodology consists of treating surfaces as points in a shape space equipped with an elastic Riemannian metric. This technical approach efficiently addresses the complexity associated with different parameterizations of surfaces and their diverse execution rates through time. A pivotal innovation in the framework is the use of Square-Root Normal Fields (SRNF), which simplifies the complex elastic metric into a computationally manageable one. This represents a significant reduction in complexity, as it allows the problem to be recast into a Euclidean space, making it amenable to straightforward statistical analysis and optimization.

Methodological Contributions

The paper details several contributions to the analysis of longitudinal shape data:

1. Spatiotemporal Registration: The authors employ a Riemannian approach to facilitate spatial registration of surfaces, accommodating large elastic deformations and variations in execution speeds. This aspect is particularly crucial for ensuring accurate alignment of shape data, which serves as a preprocessing step before statistical analysis.
2. Geodesics and Statistical Summaries: The formulation allows for efficient computation of geodesic paths between 4D surfaces, and the calculation of statistical summaries such as Karcher means and principal modes of variation. These summaries enable insights into typical deformation patterns and variabilities within a dataset.
3. Efficient Synthesis: The framework supports the generation of new 4D surfaces through both random and controlled synthesis. This capability is vital for applications that require extensive datasets, such as training deep learning models.

Practical Implications and Future Directions

The implications of this research are broad and impactful, significantly enhancing the capacity to analyze dynamic 3D shapes in various domains. The robust framework could be applied to medical diagnosis, such as distinguishing between normal growth and pathological changes, or in computer graphics to generate and animate realistic human figures.

Looking forward, the framework invites several potential advancements. Exploration of the approach's applicability to higher-genus surfaces and its extension to account for topological changes could further broaden its utility. Integration with data collected from partial or noisy scans could enhance real-world robustness, particularly in contexts where complete data acquisition is challenging.

The authors envision their framework having profound applications in both practical and theoretical contexts, with the possibility of driving future research in AI where understanding and modeling complex dynamic shapes are essential.