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$k$-means on Positive Definite Matrices, and an Application to Clustering in Radar Image Sequences

Daniel Fryer, Hien Nguyen, Pascal Castellazzi

TL;DR

A novel application is provided, to time-series clustering of pixels in a sequence of Synthetic Aperture Radar images, via their finite-lag autocovariance matrices.

Abstract

We state theoretical properties for $k$-means clustering of Symmetric Positive Definite (SPD) matrices, in a non-Euclidean space, that provides a natural and favourable representation of these data. We then provide a novel application for this method, to time-series clustering of pixels in a sequence of Synthetic Aperture Radar images, via their finite-lag autocovariance matrices.

$k$-means on Positive Definite Matrices, and an Application to Clustering in Radar Image Sequences

TL;DR

A novel application is provided, to time-series clustering of pixels in a sequence of Synthetic Aperture Radar images, via their finite-lag autocovariance matrices.

Abstract

We state theoretical properties for -means clustering of Symmetric Positive Definite (SPD) matrices, in a non-Euclidean space, that provides a natural and favourable representation of these data. We then provide a novel application for this method, to time-series clustering of pixels in a sequence of Synthetic Aperture Radar images, via their finite-lag autocovariance matrices.

Paper Structure

This paper contains 13 sections, 4 theorems, 21 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of $k$-means and Fréchet mean
  3. Symmetric Positive Definite Matrices
  4. Log-Cholesky $k$-means
  5. Log-Cholesky distance and mean
  6. Reduction to Euclidean mean
  7. Theoretical results
  8. Application
  9. Methodology
  10. Data pre-processing
  11. Results
  12. Discussion
  13. Future work

Key Result

Proposition 1

Assume that $S_{1},\dots,S_{n}$ are IID and arise from a data generating process with probability measure $P_{S}$, with $\mathbb{E}_{P_{S}}\left\Vert \mathcal{V}\left(S\right)\right\Vert ^{2}<\infty$, and that for each $j\in\left[k\right]$, there exists a unique set $A^{j}$, such that $\mathcal{Q}\l

Figures (4)

  • Figure 1: The left panel gives the adjusted Rand index for lags $\ell \in [5]$, clusters $k \in [8]$ and each of the products CC, VH and VV. The right panel gives that for $k=2$, in the CC product, and patch sizes $p \in \{4,5,\ldots,10\}$.
  • Figure 2: Cross-sectional projection scatter plots of the $n$ time-series (pixels), represented in the 3-dimensional space of autocovariance matrices transformed via \ref{['eq:log-chol-vec']}. Pixels in $k$-means clusters $1,2,5$ and $12$ (clusters with greater than 5% empirical probability of GDV) are coloured black, and all others are red.
  • Figure 3: Cross-sectional projection scatter plots of the $n$ time-series (pixels), represented in the 3-dimensional space of autocovariance matrices transformed via \ref{['eq:log-chol-vec']}. Pixels whose $\text{SARGDE}_{v1}$ is in the lower 25% sample quartile are coloured black, the middle 50% red, and the upper 25% green.
  • Figure 4: (A) The 15 $k$-means classes, coloured by empirical probability $P$ of overlap with GDV; (B) the Bureau of Meteorology GDV atlas ground truth labels; (C) the $\text{SARGDE}_{v1}$ index.

Theorems & Definitions (4)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4