Supervised deep learning of elastic SRV distances on the shape space of curves

TL;DR

This paper introduces a supervised deep learning framework for the direct computation of SRV distances between curves, which usually requires an optimization over the group of reparametrizations that act on the curves.

Abstract

Motivated by applications from computer vision to bioinformatics, the field of shape analysis deals with problems where one wants to analyze geometric objects, such as curves, while ignoring actions that preserve their shape, such as translations, rotations, or reparametrizations. Mathematical tools have been developed to define notions of distances, averages, and optimal deformations for geometric objects. One such framework, which has proven to be successful in many applications, is based on the square root velocity (SRV) transform, which allows one to define a computable distance between spatial curves regardless of how they are parametrized. This paper introduces a supervised deep learning framework for the direct computation of SRV distances between curves, which usually requires an optimization over the group of reparametrizations that act on the curves. The benefits of our approach in terms of computational speed and accuracy are illustrated via several numerical experiments.

Figures (7)

• Figure 1: Training step and network structure diagram for Shape Preserving Data Augmentation based training: Weights contained in the red blocks are trainable and the Siamese convolutional nodes have shared weights. Specific parameter details of the network architecture can be found in Section \ref{['secc:network_architecture']}. The green blocks perform shape preserving data augmentation as described in Section \ref{['sec:trainingmethod']}.
• Figure 2: Example of Shape Preserving Data Augmentation: The top curve is an example of a parameterization of a curve from the Swedish Leaf II dataset, see Section \ref{['sec:numerical_experiments']} for a description of this dataset. The three curves on the bottom represent parameterizations and rotations of this curve as produced by the shape preserving data augmentation described in Section \ref{['sec:trainingmethod']}.
• Figure 3: On both figures, the $x$-axis represents epochs, and on the $y$-axis, we plot the mean squared error of the network on the training data (blue), as well as on unseen testing data (red). Convergence curves for network trained on open, real-valued functions discretized at 90 points from our Synthetic I data set, trained for 500 epochs (left figure). Convergence curves for network trained on closed, 2-dimensional curves discretized at 100 points from the Kimia dataset, trained for 50 epochs (right figure). Descriptions of the datasets are given in Section \ref{['sec:numerical_experiments']}.
• Figure 4: Five examples from Synthetic I (top-left) and the CPC Precipitation dataset (top-right). Third and fourth figure: Comparison of DP (red) and Trained Network (blue). Scatter plot of relative errors for 1000 testing cases from the CPC Precipitation dataset, using a network trained on Synthetic I (bottom-left). Corresponding correlation plot for both methods, with exact distances on the $y$-axis, and estimated distances on the $x$-axis, and the line $y=x$ in green (bottom-right).
• Figure 5: Five examples from the MPEG-7 dataset (left). Five examples from the Swedish leaf dataset (right).

Theorems & Definitions (2)

• Theorem 1
• proof : Proof of Theorem \ref{['thm:existence']}