Adventures in Signal Processing and Open Science

Tag: compressed sensing

Magni: A Python Package for Compressive Sampling and Reconstruction of Atomic Force Microscopy Images

Our new software metapaper Magni: A Python Package for Compressive Sampling and Reconstruction of Atomic Force Microscopy Images has just been published in Journal of Open Research Software. The paper describes our new software package Magni:

Magni is an open source Python package that embraces compressed sensing and Atomic Force Microscopy (AFM) imaging techniques. It provides AFM-specific functionality for undersampling and reconstructing images from AFM equipment and thereby accelerating the acquisition of AFM images. Magni also provides researchers in compressed sensing with a selection of algorithms for reconstructing undersampled general images, and offers a consistent and rigorous way to efficiently evaluate the researchers own developed reconstruction algorithms in terms of phase transitions. The package also serves as a convenient platform for researchers in compressed sensing aiming at obtaining a high degree of reproducibility of their research.

The software itself is on GitHub as well as on Aalborg University’s repository: DOI 10.5278/VBN/MISC/Magni

Go ahead and check it out if you are into compressed sensing or atomic force microscopy. Pull requests welcome if you have ideas.

Compressed Sensing – and more – in Python

Compressed Sensing – and more – in Python

The availability of compressed sensing reconstruction algorithms for Python has so far been quite scarce. A new software package improves on this situation. The package PyUnLocBox from the LTS2 lab at EPFL is a convex optimisation toolbox using proximal splitting methods. It can, among other things, be used to solve the regularised version of the LASSO/BPDN optimisation problem used for reconstruction in compressed sensing:

\underset{x}{\mathrm{argmin}} \| Ax - y \|_2 + \tau \| x \|_1

See http://pyunlocbox.readthedocs.org/en/latest/tutorials/compressed_sensing_1.html

Heard through Pierre Vandergheynst.

I have yet to find out if it also solves the constrained version. Update: Pierre Vandergheynst informed me that the package does not yet solve the constrained version of the above optimisation problem, but it is coming:

\underset{x}{\mathrm{argmin}} \quad \| x \|_1 \\ \text{s.t.} \quad \| Ax - y \|_2 < \epsilon

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.

Reconstruction MSE

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

 

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