Spectral Shape Modeling for Medical Image Analysis

本研究旨在为医学图像中的形状分析探索新的方向。目前的挑战在于复杂生物形状的巨大变异性，例如大脑表面。它们的复杂性直接影响到医学图像分析中学习算法的性能。这表明迫切需要更好地利用形状的性质，特别是在分析形状上的数据时。医学成像中的形状分析是今天的，通常基于外部几何信息，例如，从欧几里得坐标推导出来的。因此，传统方法不可避免地需要昂贵的非线性形状归一化，需要长达数小时的计算来对齐形状。另一方面，在医学图像分析中，形状的内在本质经常被过度简化，如果不是被忽视的话。例如，在基于表面的方法中，大脑表面通常被视为简单的球体，增加了计算负担，甚至在体积方法中被忽略，导致表面数据错位。这严重限制了医学成像的研究，因为数据驻留在复杂的表面上。研究方向和方法论一个有希望的途径是通过光谱图理论来研究形状。这为真正的内在形状分析提供了基础，特别是由于其在等距下的不变性。研究表面数据的最新进展和日益增长的需求促使开发了一种新的范式来对复杂的生物形状进行统计。为此，我打算发展频谱形状分析的三个研究方向：(I)形状表示，专注于调和形状建模；(Ii)形状统计，专注于表面数据的学习；(Iii)形状动力学，专注于形状的运动。这个项目将首先关注神经影像数据的结构和功能的变异性，以发现神经退行性疾病的潜在机制。长期的愿景是通过利用形状表示来自动检测生物异常，从而在学习算法中更好地使用医学数据。*影响结果预计将对医学成像产生很高的直接影响，特别是在研究大脑和心脏成像的功能数据方面。光谱框架提供了一个新的范式来对复杂的生物形状进行统计。计算优势预计将为研究功能数据带来更快、更精确的工具，特别是在神经成像领域，这一领域至关重要，需要一个几何感知的统计框架。因此，这为引领未来的功能神经成像研究带来了强大的优势，具有显著扩大未来研究的潜力。光谱框架也适用于数据基本上存在于表面的各种其他领域，包括计算机视觉和机器学习。

Context This research program aims at exploring new directions for the analysis of shapes in medical images. The current challenge resides in the huge variability of complex biological shapes, such as the surface of the brain. Their complexity directly impacts the performance of learning algorithms in medical image analysis. This indicates a crucial need to better exploit the nature of shapes, particularly when data is analyzed on them. Shape analysis in medical imaging is today, often based on extrinsic geometric information, for instance, derived from Euclidean coordinates. As a result, traditional approaches inexorably require costly non-linear shape normalization, up to hours of computation for aligning shapes. On the other side, the intrinsic nature of shapes is often over simplified in medical image analysis, if not ignored. For instance, brain surfaces are typically treated as simple spheres in surface-based methods, increasing computational burden, or even ignored in volumetric methods, leading to misaligned surface data. This severely limits studies in medical imaging where data resides on complex surfaces.******Research Directions and Methodology One promising avenue is to investigate shapes via spectral graph theory. This provides a foundation towards a truly intrinsic shape analysis, notably due to its invariance under isometry. The recent advances and the growing need to study surface data motivate the development of a new paradigm to perform statistics on complex biological shapes. To do so, I intend to develop three axes of research on spectral shape analysis: (i) shape representation, focused on harmonic shape modeling, (ii) shape statistics, focused on the learning of surface data, and (iii) shape dynamics, focused on motion of shapes. This program will first focus on the structural and functional variability of neuroimaging data, in order to discover the underlying mechanisms of neurodegenerative diseases. The long-term vision is to contribute towards a better use of medical data in learning algorithms by exploiting shape representations to detect biological abnormalities automatically.******Impact Outcomes are expected to have a high direct impact in medical imaging, notably for studying functional data in the brain and cardiac imaging. The spectral framework provides a new paradigm to perform statistics on complex biological shapes. The computational advantage is expected to bring faster and more precise tools for studying functional data, notably in neuroimaging, which crucially needs a geometry-aware statistical framework. This brings, therefore, a strong advantage to lead future studies in functional neuroimaging with high potentials to significantly scale up future studies. The spectral framework is also relevant in various other fields where data fundamentally lives on surfaces, including in computer vision and machine learning.