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

包括医学成像、计算机视觉或物理在内的各个领域的数据往往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大小、每个样本的体积和模式方面都在增加。这种数据集的一个例子，也是该项目的一个特别重点，是由新的采集技术产生的图像，这种技术在微米尺度上呈现生物组织，同时提供关于组织中包含的每个细胞的高维信息。几何数据的分析在最近取得了长足的进步，特别是形状空间概念的数学表述和相关计算方法的设计。然而，这些方法还没有扩展到与新的模式相关的海量数据大小。该项目专注于形状和图像数据，受过去十年来最新进展的启发，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，随着数据采集技术的完善，形状分析变得越来越重要。它已被广泛应用于大脑疾病或功能障碍的背景下，如精神分裂症、抑郁症、ADHD、自闭症、亨廷顿或阿尔茨海默氏症。该项目将扩大可用于这类研究的工具范围，允许分析以更高分辨率捕获的数据集，适应最近引入的模式，并使统计调查能够在多个尺度上进行。该项目通过微分同胚图为形状分析开发新的概念和模型。它描述了三个主要的研究主题：引入多尺度分析的新方法，为大变形微分同胚度量映射算法开发随机优化策略，以及研究可能存在不连续或奇异性的图像的配准方法。拟议的工作包括理论分析、数值开发和实验探索相结合。第一个主题中研究的多尺度模型提供了一个吸引人的、以前从未探索过的范例，具有显著的挑战，特别是在数字方面。预计结果将对形状变化进行丰富的分解，从而显著影响统计形状分析，并使在不同尺度上发生的影响数据的各种影响得以分离。第二个主题回顾了差同胚配准的一些基础，以允许使用随机梯度下降的随机化实现。在第三个主题中研究的图像多样性具有对携带空间信息的形状进行建模的能力，并使新的数据模式能够通过配准方法来处理。该项目将创建算法，采用适应这些模式的数字表示法，并探索跨模式注册。该项目将支持研究生的教育和本科生的暑期实习。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.