Topologically invariant manifold learning for medical imaging

科学数据分析在工程问题中的一个重要目标是了解一个复杂系统的行为，生物过程或物理状态的变化，给出一些观察样本。这就需要综合大量的多元数据，并提出了数据简化的基本问题：如何发现高维数据的紧凑和有意义的表示？虽然在流形学习领域已经取得了重大进展，但这些方法仍然受到基本挑战的阻碍，例如缺乏等距，现实世界样本所展示的高度可变的结构拓扑以及处理非常大量的数据以有效地学习相关的表示。我们建议采取必要的步骤来弥合理论与应用之间的现有鸿沟。该研究计划的总体目标是开发一种新的流形学习技术范式，其数据表示对底层拓扑结构是不变的。这将是面向医学成像和计算机视觉中遇到的应用问题。该方法将能够减少大量的复杂的，高维的医学图像分解成联合嵌入和保持平滑特性的结构。拟议的计算框架将纳入高度创新的理论发展，即：（1）使用联合格拉斯曼嵌入从独立于固有数据结构的高维训练集重建流形和非流形表面的算法，（2）用于执行环境空间和流形空间之间的映射的参数化模型，以及（3）使用更高级的顺序马尔可夫随机场从底层空间推断新模型。这一创新平台可提供处理大量多参数数据所需的工具，对许多应用（如医学成像和计算机视觉）具有巨大价值。它有可能为神经系统疾病的早期预测提供新的见解和迹象，并在计算解剖学，肿瘤生长和器官建模方面开辟新的线索。

An important goal of scientific data analysis in engineering problems is to understand the behaviour of a complex system, biological process or physical-state alterations given some observation samples. This introduces the need to synthesize large amounts of multivariate data and raises the fundamental question of data reduction: how to discover compact and meaningful representations of high-dimensional data? While significant progress has been made in the field of manifold learning, these approaches are still hamstrung by fundamental challenges such as lack of isometry, highly variable structural topologies demonstrated by real-world samples and handling very large amounts of data to efficiently learn the associated representation. We propose to take the required steps to bridge the existing chasm between theory and application. The overall objective of the research program is to develop a new paradigm of manifold learning techniques, which data representation is invariant to the underlying topology. This will be geared towards applied problems encountered in medical imaging and in computer vision. The approach will be able to reduce large amounts of complex, high-dimensional medical images by decomposing structures into joint embeddings and preserving smoothness properties. The proposed computational framework would incorporate highly innovative theoretical developments, namely: (1) an algorithm to reconstruct both manifold and non-manifold surfaces from high-dimensional training sets independently of the inherent data structure using joint Grassmannian embeddings, (2) a parameterization model to perform mappings between ambient and manifold spaces and (3) a discrete optimization framework using higher-order Markov Random Fields to infer new models from the underlying space. This innovative platform can be of great value for a number of applications, such as in medical imaging and in computer vision, by providing the required tools to process high volume and multi- parametric data. It has the potential to contribute new insights and indications for possible early predictors of neurological disorders and open new leads in computational anatomy, tumor growth and organ modeling.