Enhanced image approximation using shifted rank-1 reconstruction

TL;DR

This paper introduces a shifted rank-1 matrix approximation method that generalizes low rank approximation to better model data with shifted structures, demonstrated through theoretical analysis and practical applications.

Contribution

It proposes a novel shifted rank-1 approximation framework, including an efficient algorithm and theoretical insights, for improved modeling of shifted data structures.

Findings

Efficient $O(NM \,\log M)$ algorithm for shifted rank-1 approximation.

Demonstrated improved approximation in numerical experiments.

Application to real-world data like seismic and video signals.

Abstract

Low rank approximation has been extensively studied in the past. It is most suitable to reproduce rectangular like structures in the data. In this work we introduce a generalization using shifted rank-1 matrices to approximate $A\in\mathbb{C}^{M\times N}$. These matrices are of the form $S_{\lambda}(uv^*)$ where $u\in\mathbb{C}^M$, $v\in\mathbb{C}^N$ and $\lambda\in\mathbb{Z}^N$.The operator $S_{\lambda}$ circularly shifts the k-th column of $uv^*$ by $\lambda_k$. These kind of shifts naturally appear in applications, where an object $u$ is observed in $N$ measurements at different positions indicated by the shift $\lambda$. The vector $v$ gives the observation intensity. Exemplary, a seismic wave can be recorded at $N$ sensors with different time of arrival $\lambda$; Or a car moves through a video changing its position in every frame. We present theoretical results as well as an efficient algorithm to calculate a shifted rank-1 approximation in $O(NM \log M)$. The benefit of the proposed method is demonstrated in numerical experiments. A comparison to other sparse approximation methods is given. Finally, we illustrate the utility of the extracted parameters for direct information extraction in several applications including video processing or non-destructive testing.