- Researcher: Bogdan Toader
- Academic Supervisors: Jared Tanner and Andrew Thompson
- Industrial Supervisors: Peter Harris and Stephane Chretien

## Background

Given a number of moving point sources emitting sound in a specific region, we want to estimate the overall sound and the exact locations and paths of the sources by taking only a small number of measurements of the sound in the region. We model the problem using the framework of super-resolution, where the (unknown) input signal is a discrete measure (sum of Diracs) and the measurements consist of samples of the convolution of the signal with a known kernel, for example a Gaussian. In Figure 1 we show an example of such a signal in one dimension; the aim is to recover the discrete signal (blue spikes) from the noisy measurements (black circles). These techniques represent an equivalent of compressed sensing in a continuous setting.

*Figure 1. Example of discrete input signal and noisy samples.*

The main goal of the project is to develop theory and algorithms for solving the deconvolution problem in a two-dimensional and dynamic setting, where we can provably recover the source locations and the paths they follow. We will then apply these techniques to the problem of recovering the locations of a number of ships in a shipping lane based on measurements of the sound at different locations in the shipping lane.

## Progress

In most of the literature in this area, the focus is on solving the total variation norm minimisation problem, where we seek a 'sparse' non-negative measure which matches the observed measurements. This is a continuous analog of the 1-norm minimisation problem for vectors, common in compressed sensing. However, in our work we have considered the feasibility problem for non-negative measures; that is, we simply seek any measure which matches the observed measurements. This is motivated by the existing literature in compressed sensing, where it has been shown that under the non-negativity assumption, the true signal is the unique signal consistent with the measurements.

By relying on the machinery of the Chebyshev systems, we have shown that the true signal is the only solution of the feasibility problem in the noise-free, one-dimensional case. In the more difficult case when the measurements have additive noise, we want to control the magnitude of the error between the true signal and the solution to the feasibility problem on intervals around each source and away from the sources. In [1], we have shown that if, for each source, there exist two nearby samples, the error depends linearly on the level of noise.

## Future Work

The next step is to work on algorithms that solve the super-resolution problem, first in one dimension and then in two dimensions with extension to the dynamic case (reconstruction of paths). Non-linear least-squares approaches or a continuous version of the iterative hard thresholding algorithm, using ideas from optimal transport, are two directions that we would like to explore. A more ambitious objective for the project would be to extend our theory for the feasibility problem to the two-dimensional dynamic setting.

## Publications

[1] Armin Eftekhari, Jared Tanner, Andrew Thompson, Bogdan Toader and Hemant Tyagi. Sparse non-negative super-resolution - simplified and stabilised. *arXiv preprint arXiv:1804.01490* (2018).