Computable elastic distances between shapes
SIAM Journal on Applied Mathematics
Analysis of Planar Shapes Using Geodesic Paths on Shape Spaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
Computing Large Deformation Metric Mappings via Geodesic Flows of Diffeomorphisms
International Journal of Computer Vision
Statistical Shape Analysis: Clustering, Learning, and Testing
IEEE Transactions on Pattern Analysis and Machine Intelligence
Symmetric Non-rigid Registration: A Geometric Theory and Some Numerical Techniques
Journal of Mathematical Imaging and Vision
Rate-invariant recognition of humans and their activities
IEEE Transactions on Image Processing
Shape Analysis of Elastic Curves in Euclidean Spaces
IEEE Transactions on Pattern Analysis and Machine Intelligence
An information-geometric framework for statistical inferences in the neural spike train space
Journal of Computational Neuroscience
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Constructing generative models for functional observations is an important task in statistical functional analysis. In general, functional data contains both phase (or x or horizontal) and amplitude (or y or vertical) variability. Traditional methods often ignore the phase variability and focus solely on the amplitude variation, using cross-sectional techniques such as fPCA for dimensional reduction and data modeling. Ignoring phase variability leads to a loss of structure in the data and inefficiency in data models. This paper presents an approach that relies on separating the phase (x-axis) and amplitude (y-axis), then modeling these components using joint distributions. This separation, in turn, is performed using a technique called elastic shape analysis of curves that involves a new mathematical representation of functional data. Then, using individual fPCAs, one each for phase and amplitude components, it imposes joint probability models on principal coefficients of these components while respecting the nonlinear geometry of the phase representation space. These ideas are demonstrated using random sampling, for models estimated from simulated and real datasets, and show their superiority over models that ignore phase-amplitude separation. Furthermore, the generative models are applied to classification of functional data and achieve high performance in applications involving SONAR signals of underwater objects, handwritten signatures, and periodic body movements recorded by smart phones.