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This paper introduces a deep shape regression model tailored for open planar curves, effectively integrating multimodal and high-dimensional covariates. By representing curves as complex-valued functions and utilizing a novel deep conditional covariance smoother, the model achieves invariance to translation, rotation, and scaling, while adeptly managing sparsely and irregularly sampled curves. The approach is validated through simulations and applied to hippocampal outlines from the ADNI cohort, demonstrating covariate effects that align with existing literature.
A novel deep shape regression model reveals covariate effects in hippocampal outlines that align with established neuroimaging findings, showcasing its potential in health applications.
The shape of a planar curve is the geometric information that remains once translation, rotation, scale and reparametrisation are removed and is of interest in many health applications, e.g. in neuroimaging. We propose a deep shape regression model for open planar curves that admits multimodal and high-dimensional covariates. Representing curves as complex-valued functions, we show that the conditional full Procrustes mean is the leading eigenfunction of the conditional covariance. To estimate this covariance surface, we propose a novel deep conditional covariance smoother with modality-specific encoders - e.g. splines for scalar covariates and convolutional networks for images, which classical spline smoothers cannot accommodate. Our model is by construction invariant to the translation, rotation and scaling of the input curves and handles sparsely and irregularly sampled curves. We further provide an algorithm for elastic mean estimation that also removes parametrisation by iterating covariance smoothing, rotational alignment and parametrisation alignment. We illustrate the method on simulated outlines with known conditional mean and multimodal covariates, and give a first application to hippocampal outlines from the ADNI cohort, recovering covariate effects consistent with the literature. Code is available at https://github.com/mpff/dnn-shapes.
