Deep Shape Regression for Planar Curves with Multimodal Covariates

Area of Science:

• Computational geometry and topology
• Medical image analysis
• Machine learning and deep learning

Background:

• Shape analysis of planar curves is vital in health applications, particularly neuroimaging.
• Existing methods struggle with multimodal, high-dimensional data and irregular curve sampling.
• Invariance to transformations (translation, rotation, scale, reparametrization) is essential for shape analysis.

Purpose of the Study:

• To propose a novel deep shape regression model for open planar curves.
• To handle multimodal and high-dimensional covariates effectively.
• To develop a method invariant to curve transformations and robust to sparse sampling.

Main Methods:

• Representing curves as complex-valued functions.
• Utilizing a deep conditional covariance smoother with modality-specific encoders (splines, CNNs).
• Implementing an elastic mean estimation algorithm for shape and parametrization alignment.

Main Results:

• The conditional full Procrustes mean is identified as the leading eigenfunction of the conditional covariance.
• The proposed deep smoother accommodates diverse covariate types, outperforming classical methods.
• The model demonstrates invariance to translation, rotation, and scaling, and handles irregular data.

Conclusions:

• The deep shape regression model offers a robust and flexible framework for analyzing planar curve shapes.
• The method successfully applied to hippocampal outlines, yielding clinically relevant covariate effects.
• This approach advances shape analysis in medical imaging and other fields requiring geometric interpretation.