Large-Scale Models and Algorithms in Diffeomorphic Shape and Image Registration

包括医学成像、计算机视觉或物理学在内的各个领域的数据通常具有对其理解至关重要的几何属性，并且这些类型的数据在样本量、每个样本量和模式方面都在增加。这种数据集的一个例子，也是这个项目的一个特别重点，是由新的采集技术产生的图像，这些图像在微米尺度上呈现生物组织，同时提供组织中每个细胞的高维信息。几何数据的分析在最近取得了相当大的进展，特别是形状空间概念的数学公式和相关计算方法的设计。然而，这些方法还不能扩展到与新模式相关的海量数据。该项目专注于形状和图像数据，受过去十年最新进展的启发，开发了新的数据科学方法，并开发了处理多尺度表示的新模型。形状分析为理解生物和医学数据提供了主要元素，并且与数据采集技术的改进越来越相关。它已被广泛应用于脑疾病或功能障碍的背景下，如精神分裂症，抑郁症，多动症，自闭症，亨廷顿或阿尔茨海默氏症。该项目将扩展此类研究可用工具的范围，允许以更精细的分辨率分析捕获的数据集，适应最近引入的模式，并使统计调查能够在多个尺度上工作。该项目通过微分同构映射为形状分析开发了新的概念和模型。它描述了三个主要的研究主题：引入多尺度分析的新方法，开发大变形微分同构度量映射算法的随机优化策略，以及研究可能具有不连续或奇点的图像的配准方法。建议的工作包括理论分析、数值发展和实验探索的结合。在第一个主题中研究的多尺度模型提供了一个有吸引力的和以前未开发的范式，具有显着的挑战，特别是在数值方面。通过提供形状变化的丰富分解，并在不同尺度上分离影响数据的各种效应，预计结果将显著影响统计形状分析。第二个主题重温了差分配准的一些基础，以允许使用随机梯度下降进行随机实现。在第三个主题中研究的图像变量具有对携带空间信息的形状进行建模的能力，并且能够通过配准方法处理新的数据模态。该项目将创建适合这些模式的数字表示算法，并探索跨模式注册。该项目将支持一名研究生的教育和本科生的暑期实习。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Data in various fields including medical imaging, computer vision or physics often have geometric properties that are essential for their understanding, and these types of data are increasing in sample size, volume per sample and modalities. An example of such data sets, and a special focus of this project, are images resulting from new acquisition techniques that render biological tissues at the micron scale while providing high-dimensional information on each cell contained in the tissue. The analysis of geometric data has seen considerable progress in the recent past with, in particular, the mathematical formulation of the notion of shape space and the design of associated computational methods. However, these methods do not scale yet to the massive data size that are associated with the new modalities. This project, which focuses on shape and image data, develops new data science approaches inspired by recent advances over the past decade and develops new models to handle multi-scale representations. Shape analysis provides major elements for the understanding of biological and medical data, and becomes increasingly relevant with the refinement of data acquisition technology. It has been extensively applied in the context of brain diseases or dysfunction, such as schizophrenia, depression, ADHD, autism, Huntington or Alzheimer. This project will extend the range of tools available for such studies, allowing for the analysis of datasets captured at finer resolution, accommodating recently introduced modalities, and enabling statistical investigations working at multiple scales.The project develops new concepts and models for shape analysis through diffeomorphic mapping. It describes three main research themes: introducing new approaches for multiscale analysis, developing randomized optimization strategies for the large deformation diffeomorphic metric mapping algorithm, and investigating registration methods for images that may have discontinuities or singularities. The proposed work involves a combination of theoretical analyses, numerical developments, and experimental exploration. The multiscale models studied in the first theme provide an appealing and previously unexplored paradigm, with notable challenges, especially on the numerical side. Results are expected to significantly impact statistical shape analysis by providing an enriched decomposition of shape changes, and enabling the separation of various effects affecting the data when they occur at different scales. The second theme revisits some of the bases underlying diffeomorphic registration to allow for randomized implementations using stochastic gradient descent. Image varifolds, studied in the third theme, have the ability to model shapes carrying spatial information and enable new data modalities to be handled by registration methods. The project will create algorithms, with numerical representation adapted to these modalities, and explore cross-modality registration. The project will support the education of a graduate student and summer internships for undergraduate students.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.