Heriot-Watt University

Edinburgh EH14 4ASColin Maclaurin Building CMT.19A
a.repetti@hw.ac.uk
Associate Professor
Department of Actuarial Mathematics and Statistics School of Mathematical and Computer Sciences
Institute of Sensors, Signals, and Systems School of Engineering and Physical Sciences
Maxwell Institute for Mathematical Sciences - Edinburgh 

MATLAB codes

Approximated forward-backward for reweighting procedures

Reweighted procedures have shown high efficiency in computational imaging. They aim to handle non-convex composite penalization functions by iteratively solving multiple approximated sub-problems. Although the asymptotic behaviour of these methods has recently been investigated in several works, they all necessitate the sub-problems to be solved accurately, which can be sub-optimal in practice. In this work we present a reweighted forward-backward algorithm designed to handle non-convex composite functions. Unlike existing convergence studies in the literature, the weighting procedure is directly included within the iterations, avoiding the need for solving any sub-problem. We show that the obtained reweighted forward-backward algorithm converges to a critical point of the initial objective function. 
The associated code illustrates the good behaviour of the proposed approach on a Fourier imaging application, specific to radio-astronomical imaging.

Related articles 


- A. Repetti and Y. Wiaux. A forward-backward algorithm for reweighted procedures: Application to radio-astronomical imaging. In Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Barcelona, Spain, 4-8 Mai 2020. [pdf] 
- A. Repetti, and  Y. Wiaux. Variable Metric Forward-Backward Algorithm for Composite Minimization Problems.  SIAM Journal on Optimization, vol. 31, no. 2, pp. 1215-1241, May 2021. [pdf]  

SARA-COIL: Compressive optical imaging with a photonic lantern

The thin and flexible nature of optical fibres often makes them the ideal technology to view biological processes in-vivo, but current microendoscopic approaches are limited in spatial resolution. Here, we demonstrate a new route to high resolution microendoscopy using a multicore fibre (MCF) with an adiabatic multimode-to-singlemode photonic lantern transition formed at the distal end by tapering. We show that distinct multimode patterns of light can be projected from the output of the lantern by individually exciting the single-mode MCF cores, and that these patterns are highly stable to fibre movement. This capability is then exploited to demonstrate a form of single-pixel imaging, where a single pixel detector is used to detect the fraction of light transmitted through the object for each multimode pattern. A custom compressive imaging algorithm we call SARA-COIL is used to reconstruct the object using only the pre-measured multimode patterns themselves and the detector signals.

Related articles 


- D. Choudhury, D. K. McNicholl, A. Repetti, I. Gris-Sánchez, S. Li, D. B. Phillips, G. Whyte,  T. A. Birks,  Y. Wiaux, and R. R. Thomson. Computational optical imaging with a photonic lantern.  Nature Communications, vol. 11, no. 1, pp. 5217,Oct. 2020. [pdf]
- D. McNicholl, S. Li, D. Choudhury, A. Repetti, I. Gris-Sauchez, G. Whyte, T. A. Birks, D. B. Phillips, Y. Wiaux, and R. R. Thomson. Towards Photonic Lantern-Based Microendoscopy. In 14th Pacific Rim Conference on Lasers and Electro-Optics (CLEO PR 2020), OSA Technical Digest (Optical Society of America, 2020), paper C10D_5, Sydney, Australia, 3-5 August 2020. [pdf]

BUQO: Bayesian Uncertainty Quantification by Optimization


We propose a Bayesian uncertainty quantification method for large-scale imaging inverse problems. Our method applies to all Bayesian models that are log-concave, where maximum-a-posteriori (MAP) estimation is a convex optimization problem. The method is a framework to analyse the confidence in specific structures observed in MAP estimates (e.g., lesions in medical imaging, celestial sources in astronomical imaging), to enable using them as evidence to inform decisions and conclusions. Precisely, following Bayesian decision theory, we seek to assert the structures under scrutiny by performing a Bayesian hypothesis test that proceeds as follows: firstly, it postulates that the structures are not present in the true image, and then seeks to use the data and prior knowledge to reject this null hypothesis with high probability. Computing such tests for imaging problems is generally very difficult because of the high dimensionality involved. A main feature of this work is to leverage probability concentration phenomena and the underlying convex geometry to formulate the Bayesian hypothesis test as a convex problem, that we then efficiently solve by using scalable optimization algorithms. This allows scaling to high-resolution and high-sensitivity imaging problems that are computationally unaffordable for other Bayesian computation approaches. We illustrate our methodology, dubbed BUQO (Bayesian Uncertainty Quantification by Optimization), on a range of challenging Fourier imaging problems arising in astronomy and medicine.

Related articles 


- A. Repetti, M. Pereyra, and Y. Wiaux - Scalable Bayesian uncertainty quantification in imaging inverse problems via convex optimization, SIAM Journal on Imaging Sciences, vol. 12, no. 1, pp. 87-118, 2019.  [pdf]
- A. Repetti, M. Pereyra and Y. Wiaux. Uncertainty Quantification in Imaging: When Convex Optimization Meets Bayesian Analysis. In Proceedings of the 26th European Signal Processing Conference (EUSIPCO 2018), Rome, Italy, 3-8 Sept. 2018. [pdf] 

CALIM in astronomy: Self direction-dependent effect calibration and imaging in radio-interferometry

Description


The code represents a proof of concept MATLAB implementation of the proposed algorithm, for self-calibration of direction-dependent effects (DDEs) in radio interferometric imaging. Radio interferometric imaging aims to estimate an unknown sky intensity image from degraded observations, acquired through an antenna array. In the theoretical case of a perfectly calibrated array, it has been shown that solving the corresponding imaging problem by iterative algorithms based on convex optimization and compressive sensing theory can be competitive with classical algorithms such as CLEAN. However, in practice, antenna-based gains are unknown and have to be calibrated. Future radio telescopes, such as the SKA, aim at improving imaging resolution and sensitivity by orders of magnitude. At this precision level, the direction-dependency of the gains must be accounted for, and radio interferometric imaging can be understood as a blind deconvolution problem. In this context, the underlying minimization problem is non-convex, and adapted techniques have to be designed. In this toolbox, leveraging recent developments in non-convex optimization, we propose the first joint calibration and imaging method in radio interferometry, with proven convergence guarantees. Our approach, based on a block-coordinate forward-backward algorithm, jointly accounts for visibilities and suitable priors on both the image and the DDEs. As demonstrated in recent works, sparsity remains the prior of choice for the image, while DDEs are modelled as smooth functions of the sky, i.e. spatially band-limited.

Related articles 


- P. A. Thouvenin, A. Repetti, A. Dabbech and Y. Wiaux. Time-Regularized Blind Deconvolution Approach for Radio Interferometry. In Proceedings of the IEEE SAM 2018 workshop, Sheffield, UK, pp. 475-479, 8-11 July 2018. [pdf] 
- A. Repetti, J. Birdi, A. Dabbech, and Y. Wiaux, Non-convex optimization for self-calibration of direction-dependent effects in radio interferometric imaging. Monthly Notices of the Royal Astronomical Society, vol. 470, no. 4, pp. 3981-4006, Oct. 2017.  [pdf]
- A. Repetti and Y. Wiaux, A non-convex perspective on calibration and imaging in radio interferometry. In Proceedings of the conference on Wavelets and Sparsity XVII, part of the SPIE Optical Engineering + Applications, San Diego, California, United States, 6-9 August 2017. [pdf]
- A. Repetti, J. Birdi, and Y. Wiaux, Non-convex blind deconvolution approach for sparse radio-interferometric imaging. In Proceedings of the Signal Processing with Adaptive Sparse Structured Representations (SPARS 2017), Lisbon, Portugal, 5-8 June 2017. [pdf] 
- A. Repetti, J. Birdi, and Y. Wiaux, Joint imaging and DDEs calibration for radio interferometry. In Proceedings of international Biomedical and Astronomical Signal Processing (BASP) Frontiers workshop, page 25, Villars-sur-Ollon, Suisse, 29 Janvier-3 Février 2017. [pdf] 
- E. Chouzenoux, J.-C. Pesquet and A. Repetti, A Block Coordinate Variable Metric Forward-Backward Algorithm. J. Global Optim, vol. 66, no. 3, pp. 457-485, Nov. 2016.  [pdf]

SOOT algorithm: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization

Description


This Matlab toolbox is designed to reconstruct a sparse seismic signal x, from the degraded observation model y=h*x+w, where h is an unknown convolution kernel and w ia a realization of a white Gaussian noise. In the context of blind deconvolution problems, the regularization term l1/l2 is known to be particularly suitable to reconstruct sparse signals. 
In the article associated with this toolbox, a smooth approximation of this regularization term is proposed. The resulting non-convex minimization problem is then solved leveraging a variable block-coordinate forward-backward algorithm.

Related articles 



- E. Chouzenoux, J.-C. Pesquet and A. Repetti, A Block Coordinate Variable Metric Forward-Backward Algorithm. J. Global Optim, vol. 66, no. 3, pp. 457-485, Nov. 2016.  [pdf]
- A. Repetti, M. Q. Pham, L. Duval, E. Chouzenoux and J.-C. Pesquet,  Euclid in a Taxicab: Sparse Blind Deconvolution with Smoothed l1/l2 Regularization. Signal Processing Letters., vol. 22, no. 5, pp. 539-543, May 2015.  [pdf] 

Variable Metric Forward-Backward algorithm for image reconstruction

Description


This Matlab toolbox can be used to estimate an image degraded by a linear operator, and corrupted by an additive signal-dependent Gaussian noise. 
Using a maximum a posteriori approach, this problem is solved by minimizing a penalized criterion G = F + R, where F is the data-fidelity term corresponding to the forward model (corresponding to the negative log-likelihood associated with the noise), and R is the regularization term incorporating prior information on the target solution (in the toolbox corresponding to the indicator function of a convex - to constraint the amplitude of the estimated image pixels - and an isotropic total variation term). To minimize the function G, a variable-metric forward-backward algorithm is leveraged. The variable metric is used to accelerate the convergence of the usual forward-backward algorithm, and is chosen using a majorization-minimization strategy.

Related articles 



- E. Chouzenoux, J.-C. Pesquet and A. Repetti, Variable Metric Forward-Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function. J. Optim. Theory and Appl., vol.162, no. 1, pp. 107-132, Jul. 2014.  [pdf]
- A. Repetti, E. Chouzenoux and J.-C. Pesquet, Reconstruction d'image en présence de bruit gaussien dépendant par un algorithme Explicite-Implicite à métrique variable. In Actes du 24e colloque GRETSI, Brest, France, 3-6 septembre 2013.  [abstract] [pdf]