A General and Efficient Framework for Computational Shape Analysis Through Geometric Distributions

通过几何分布进行计算形状分析的通用且有效的框架

• 批准号: 1819131
• 负责人: Nicolas Charon
• 金额: $ 21.5万
• 依托单位: Johns Hopkins University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2020-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1819131&HistoricalAwards=false
• 关键词: General; Efficient; Framework; Computational; Shape

The analysis of shapes and their variability has become an increasingly central problem in multiple areas of data science. In the field of computer vision, shape recognition and classification is often a crucial component of machine learning systems such as self-driving cars. In natural sciences, the recent development of computational anatomy, that is the automatic analysis of anatomical structures by numerical algorithms, provides a fruitful approach in understanding and diagnosing a wide range of pathologies and disorders. Along these different scientific questions, the amount and variety of available data has never ceased to grow. As a result, the concept of shape itself has considerably expanded and may refer to various types of geometric objects, which poses the important challenge of constructing and computing relevant similarity metrics between shapes across all these different modalities. The purpose of this research project is to develop an integrated mathematical model and associated numerical pipeline that allows for morphological analysis of geometric structures in a flexible and efficient way, and explore its possible applications to computational anatomy and computer vision. It will also include a substantial educational component with the training of a graduate student, support for presentations in conferences and workshops, and dissemination of an open-source code to the scientific community.The primal challenge of statistical shape analysis is the rather non-standard and disparate mathematical spaces in which objects belong, whether the shapes in question are raw images, manually or automatically extracted landmarks, curves, surfaces, vector fields or multi-modal objects. While the seminal model proposed by Grenander introduced the idea of comparing any two shapes through the estimation of an optimal deformation (measured by a metric on a certain diffeomorphism group), this model's generality falls short in many real applications where a certain amount of residual dissimilarity is necessary to account for other sources of variability (like noise). This project intends to fill this current gap by introducing a flexible approach to quantify shape similarity which relies on a unified embedding of shape spaces as generalized distributions, following the principles of geometric measure theory. Beyond the past success of these representations for curve and surface registration problems, the objective will be to demonstrate on a mathematical and computational level how it extends to a much wider class of geometric data structures and allows for cross-modality analysis, while pushing the scope of applications to other problems like clustering, classification and sparse representations on shapes. Fast numerical methods for this new framework is also an important aspect of the project, with the objective of making implementations scalable to the current dimensionality of datasets e.g. in medical imaging.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.

形状及其变异性的分析已经成为数据科学多个领域的一个日益核心的问题。在计算机视觉领域，形状识别和分类往往是自动驾驶汽车等机器学习系统的重要组成部分。在自然科学中，计算解剖学的最新发展，即通过数值算法自动分析解剖结构，为理解和诊断广泛的病理和疾病提供了一种卓有成效的方法。随着这些不同的科学问题，可用数据的数量和种类从未停止增长。因此，形状本身的概念得到了相当大的扩展，可以指各种类型的几何对象，这就给构造和计算所有这些不同形态的形状之间的相关相似性度量带来了重要的挑战。该研究项目的目的是开发一个集成的数学模型和相关的数值流水线，允许以灵活和高效的方式对几何结构进行形态分析，并探索其在计算解剖学和计算机视觉中的可能应用。它还将包括一个重要的教育部分，包括对研究生的培训，支持在会议和研讨会上的演讲，以及向科学界传播开放源代码。统计形状分析的主要挑战是对象所属的相当非标准和不同的数学空间，无论所讨论的形状是原始图像、手动或自动提取的地标、曲线、表面、矢量场或多模式对象。虽然Griander提出的开创性模型引入了通过估计最优变形(由某个微分同胚群上的度量来衡量)来比较任意两个形状的想法，但该模型的一般性在许多实际应用中是不够的，在这些应用中，需要一定数量的剩余相异度来解释其他可变性来源(如噪声)。这个项目打算通过引入一种灵活的方法来量化形状相似性来填补目前的空白，该方法依赖于形状空间作为广义分布的统一嵌入，遵循几何度量理论的原则。除了过去在曲线和曲面配准问题上的成功之外，目标是在数学和计算水平上展示它如何扩展到更广泛的几何数据结构类别并允许跨模态分析，同时将应用范围推进到其他问题，如聚集、分类和形状上的稀疏表示。这一新框架的快速数值方法也是该项目的一个重要方面，目标是使实施可扩展到当前数据集的维度，例如在医学成像中。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。

项目成果

An inexact matching approach for the comparison of plane curves with general elastic metrics

平面曲线与一般弹性度量比较的不精确匹配方法

• DOI: 10.1109/ieeeconf44664.2019.9049031
• 发表时间: 2020
• 期刊: and Computers
• 作者: Sukurdeep, Yashil;Bauer, Martin;Charon, Nicolas
• 通讯作者: Charon, Nicolas

Inexact Elastic Shape Matching in the Square Root Normal Field Framework

平方根法向场框架中的不精确弹性形状匹配

• DOI: 10.1007/978-3-030-26980-7_2
• 发表时间: 2019
• 期刊: International Conference on Geometric Science of Information
• 作者: Bauer, Martin;Charon, Nicolas;Harms, Philipp
• 通讯作者: Harms, Philipp

Diffeomorphic Registration of Discrete Geometric Distributions

• DOI: 10.1142/9789811200137_0003
• 发表时间: 2018-01
• 期刊: Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore
• 作者: Hsi-Wei Hsieh;N. Charon

A relaxed approach for curve matching with elastic metrics

• DOI: 10.1051/cocv/2018053
• 发表时间: 2019-11-27
• 期刊: ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
• 影响因子: 1.4
• 作者: Bauer, Martin;Bruveris, Martins;Moller-Andersen, Jakob
• 通讯作者: Moller-Andersen, Jakob

Metrics, Quantization and Registration in Varifold Spaces

多种空间中的度量、量化和配准

• DOI: 10.1007/s10208-020-09484-7
• 发表时间: 2021
• 期刊: Foundations of Computational Mathematics
• 影响因子: 3
• 作者: Hsieh, Hsi-Wei;Charon, Nicolas
• 通讯作者: Charon, Nicolas