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大小，每个样本的体积和模态方面都在增加。这类数据集的一个例子，也是该项目的一个特别重点，是新的采集技术产生的图像，这些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。几何数据的分析已经看到了相当大的进步，在最近的过去，特别是，形状空间的概念和相关的计算方法的设计的数学公式。然而，这些方法尚未扩展到与新模式相关联的海量数据大小。该项目专注于形状和图像数据，开发了受过去十年最新进展启发的新数据科学方法，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，并且随着数据采集技术的改进而变得越来越重要。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。该项目将扩大可用于此类研究的工具范围，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工作。该项目通过形态映射开发新的概念和模型，用于形状分析。它描述了三个主要的研究主题：引入新的方法进行多尺度分析，开发随机优化策略的大变形几何度量映射算法，并调查可能有不连续性或奇异性的图像配准方法。建议的工作涉及理论分析，数值模拟的发展，和实验探索相结合。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探索的范式，具有显着的挑战，特别是在数值方面。结果预计将显着影响统计形状分析，提供了丰富的形状变化的分解，并使分离的各种影响数据时，他们发生在不同的尺度。第二个主题回顾了一些基本的同构注册，以允许使用随机梯度下降的随机实现。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来处理新的数据模态。该项目将创建算法，采用适合这些模式的数字表示，并探索跨模式配准。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.