The proliferation and ubiquity of temporal data across many disciplines has
sparked interest for similarity, classification and clustering methods
specifically designed to handle time series data. A core issue when dealing
with time series is determining their pairwise similarity, i.e., the degree to
which a given time series resembles another. Traditional distance measures such
as the Euclidean are not well-suited due to the time-dependent nature of the
data. Elastic metrics such as dynamic time warping (DTW) offer a promising
approach, but are limited by their computational complexity,
non-differentiability and sensitivity to noise and outliers. This thesis
proposes novel elastic alignment methods that use parametric \& diffeomorphic
warping transformations as a means of overcoming the shortcomings of DTW-based
metrics. The proposed method is differentiable \& invertible, well-suited for
deep learning architectures, robust to noise and outliers, computationally
efficient, and is expressive and flexible enough to capture complex patterns.
Furthermore, a closed-form solution was developed for the gradient of these
diffeomorphic transformations, which allows an efficient search in the
parameter space, leading to better solutions at convergence. Leveraging the
benefits of these closed-form diffeomorphic transformations, this thesis
proposes a suite of advancements that include: (a) an enhanced temporal
transformer network for time series alignment and averaging, (b) a
deep-learning based time series classification model to simultaneously align
and classify signals with high accuracy, (c) an incremental time series
clustering algorithm that is warping-invariant, scalable and can operate under
limited computational and time resources, and finally, (d) a normalizing flow
model that enhances the flexibility of affine transformations in coupling and
autoregressive layers.

这篇论文提出了一种新颖的弹性对齐方法，采用参数化和可微分的扭曲变换来克服动态时间规整（DTW）度量的不足，该方法适用于深度学习架构，鲁棒性强，具有计算效率高、表达和灵活性强等优点。

时间序列分析的同胚变换：非线性扭曲的高效方法

Diffeomorphic Transformations for Time Series Analysis: An Efficient  Approach to Nonlinear Warping

This work proposes $\texttt{NePhi}$, a neural deformation model which results
in approximately diffeomorphic transformations. In contrast to the predominant
voxel-based approaches, $\texttt{NePhi}$ represents deformations functionally
which allows for memory-efficient training and inference. This is of particular
importance for large volumetric registrations. Further, while medical image
registration approaches representing transformation maps via multi-layer
perceptrons have been proposed, $\texttt{NePhi}$ facilitates both pairwise
optimization-based registration $\textit{as well as}$ learning-based
registration via predicted or optimized global and local latent codes. Lastly,
as deformation regularity is a highly desirable property for most medical image
registration tasks, $\texttt{NePhi}$ makes use of gradient inverse consistency
regularization which empirically results in approximately diffeomorphic
transformations. We show the performance of $\texttt{NePhi}$ on two 2D
synthetic datasets as well as on real 3D lung registration. Our results show
that $\texttt{NePhi}$ can achieve similar accuracies as voxel-based
representations in a single-resolution registration setting while using less
memory and allowing for faster instance-optimization.

本研究提出了一种神经变形模型 NePhi，该模型能够产生近似微分同胚转换，并通过功能性表示变形实现了高效的训练和推断；NePhi 在医学图像配准中展示了出色的性能，可以达到基于体素表示的相关准确性，在单分辨率配准设置中占用更少的内存并允许更快的实例优化。