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Max-Normalized Radon Cumulative Distribution Transform for Limited Data Classification

Matthias Beckmann, Robert Beinert, Jonas Bresch

TL;DR

The aim of this paper is to make the R-CDT and the related sliced Wasserstein distance invariant under affine transformations and to prove that the novel transform allows linear separation of affinely transformed image classes.

Abstract

The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates classification tasks, especially in the small data regime, like the recognition of watermarks in filigranology. Here, a typical issue is that the given data may be subject to affine transformations caused by the measuring process. The aim of this paper is to make the R-CDT and the related sliced Wasserstein distance invariant under affine transformations. For this, we propose a two-step normalization of the R-CDT and prove that our novel transform allows linear separation of affinely transformed image classes. The theoretical results are supported by numerical experiments showing a significant increase of the classification accuracy compared to the original R-CDT.

Max-Normalized Radon Cumulative Distribution Transform for Limited Data Classification

TL;DR

The aim of this paper is to make the R-CDT and the related sliced Wasserstein distance invariant under affine transformations and to prove that the novel transform allows linear separation of affinely transformed image classes.

Abstract

The Radon cumulative distribution transform (R-CDT) exploits one-dimensional Wasserstein transport and the Radon transform to represent prominent features in images. It is closely related to the sliced Wasserstein distance and facilitates classification tasks, especially in the small data regime, like the recognition of watermarks in filigranology. Here, a typical issue is that the given data may be subject to affine transformations caused by the measuring process. The aim of this paper is to make the R-CDT and the related sliced Wasserstein distance invariant under affine transformations. For this, we propose a two-step normalization of the R-CDT and prove that our novel transform allows linear separation of affinely transformed image classes. The theoretical results are supported by numerical experiments showing a significant increase of the classification accuracy compared to the original R-CDT.

Paper Structure

This paper contains 15 sections, 9 theorems, 38 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Contribution.
  3. Radon Transform
  4. Radon Transform of Functions
  5. Radon Transform of Measures
  6. Radon Transform of Affine Transformations
  7. Optimal Transport-Based Transforms
  8. R-CDT for Measures
  9. Normalized R-CDT
  10. Max-Normalized R-CDT
  11. Numerical experiments
  12. Datasets.
  13. Nearest Neighbour Classification
  14. Support Vector Machine Classification
  15. Conclusion

Key Result

proposition 1

Let $\mu \in \mathcal{M}(\mathbb R^2)$. Then, $\mathcal{R} [\mu]$ can be disintegrated into the family $\mathcal{R}_{\boldsymbol{\theta}} [\mu]$ with respect to $u_{\mathbb{S}_1}$, i.e., for all continuous $g \in C_0(\mathbb R \times \mathbb{S}_1)$ vanishing at infinity, we have

Figures (3)

  • Figure 1: Samples of the academic dataset consisting of randomly affine-transformed synthetic template images.
  • Figure 2: Accuracy of nearest neighbor classification for the academic dataset with 10 images per class and the LinMNIST dataset with 50 images per class.
  • Figure 3: Visualization of mNR-CDT for the academic dataset and 128 angles in $[0,\pi)$.

Theorems & Definitions (18)

  • proposition 1
  • proof
  • proposition 2
  • proof
  • proposition 3
  • proof
  • proposition 4
  • lemma 1
  • proof
  • proof : Proposition \ref{['prop:sigma_bounded']}
  • ...and 8 more