### Compressed sensing with linear correlation between signal and measurement noise

Torben Larsen and I have recently published a paper, “Compressed sensing with linear correlation between signal and measurement noise” in EURASIP Signal Processing. This post is an attempt and a sort of experiment to provide a front page summarizing the paper’s contributions and providing an overview of available versions of the paper and its accompanying code.

We considered compressed sensing with measurement noise in the case where the measurement noise is linearly correlated with the signal of interest. So we have the typical compressed sensing model with measurement noise:

$\mathbf y = \mathbf{Ax} + \mathbf n$

where the noise $\mathbf n$ is now correlated with $\mathbf x$. This can be modelled as a scaling by some factor $\alpha$ of the measured signal in addition to additive random noise:

$\mathbf y = \alpha \mathbf{Ax} + \mathbf w$

The difference in the measurement between the original and scaled signals constitutes the part of the resulting measurement noise that is correlated with the input signal:

$\mathbf n = \alpha \mathbf{Ax} + \mathbf w - \mathbf{Ax} = (\alpha - 1) \mathbf{Ax} + \mathbf w$

We show that in the case of reconstruction of the measured signal by basis pursuit de-noising (BPDN), the correlation between the measurement noise and the measured signal can be compensated simply by scaling the BPDN solution by $1/\alpha$.

It turns out that this simple correlated noise model models the error introduced by low-resolution quantisation quite well. We have tested the proposed reconstruction approach on compressed measurements quantised to 1, 3, and 5 bits, respectively. Especially in the extreme case of 1 bit quantisation we see substantial improvements in reconstruction error, reducing the error by up to around 7dB. This simple modification of BPDN performs better than BIHT (which is specifically designed for 1 bit quantisation) in a large portion of the undersampling/sparsity phase space.

Relative reconstruction MSE of the proposed approach. The fat contour line marks the region (above and left of it) where the error is below that of BIHT reconstruction.

Below, you can find links to both the official published version of the paper, all versions from the review process on arXiv, and the code for running the numerical simulations.

Paper versions and simulation code