CS: Fourier-transform Ghost Imaging based on Compressive Sampling algorithm, A First-Order Augmented Lagrangian Method for Compressed Sensing

Dick Gordon , one of the discoverer of the ART algorithm, sent me the following:

Compressive Sensing Hardware: Compressive Sensing Scanners

1. If you’ve ever had to scan an old fashioned paper document with a halftone image, you are aware of the difficulty in selecting dots per inch that does not produce an aliasing effect. It seems to me that this is an ideal application for compressive sensing, which by its very nature should not emphasize the high frequency components. So I offer it as a challenge for both algorithm and hardware development. It would probably compete well with general purpose line scanners.

2. Regarding [the item in the Compressive Sensing Hardware page]
1.1.12 The Heriot-Watt University/SELEX Galileo single-pixel, galvo-mirror-based scanning system

This could be further improved through the use of turnstile photons, which are emitted on command. Reflection of emitted photons by lasers involves the convolution of two Poisson processes: emission and reflection. With turnstile photons only one Poisson process occurs: reflection. Therefore the statistics are significantly improved:

Melvin, C., K. Abdel-Hadi, S. Cenzano & R. Gordon (2002). A simulated comparison of turnstile and poisson photons for x-ray imaging. In: Canadian Conference on Electrical and Computer Engineering, 2002. IEEE CCECE 2002. Eds., IEEE. 2: 165-1170. (attached)


Yours, -Dick Gordon

I am not sure about 1) and 2) looks interesting. I need to read that paper. In the meantime, here are two papers that just came out:

Fourier-transfrom Ghost Imaging based on Compressive Sampling algorithm by Hui Wang, Shensheng Han. The abstract reads:

A special algorithm for the Fourier-transform Ghost Imaging (GI) scheme is discussed based on the Compressive Sampling (CS) theory. The CS algorithm could also be used for the Fourier spectrum reconstruction of pure phase object by setting a proper sensing matrix. This could find its application in diffraction imaging of X-ray, neutron and electron with higher efficiency and resolution. Experiment results are also presented to prove the feasibility.

A First-Order Augmented Lagrangian Method for Compressed Sensing by Necdet Serhat Aybat, Garud Iyengar. The abstract reads:

In this paper, we propose a first-order augmented Lagrangian algorithm (FAL) that solves the basis pursuit problem min{||x||_1 : Ax=b} by inexactly solving a sequence of problems of the form min{lambda_k ||x||_1 + (1/2)||Ax-b-lambda_k theta_k||_2^2}, for an appropriately chosen sequence of multipliers {(lambda_k, theta_k)}. Each of these subproblems are solved using a proximal gradient algorithm proposed by Tseng wherein each update reduces to ``shrinkage" or constrained ``shrinkage". We show that FAL converges to an optimal solution x* of the basis pursuit problem, i.e. x*=argmin{||x||1 : Ax=b} and that there exists a priori fixed sequence {lambda_k} such that for all e \gt 0, iterates x_k computed by FAL are e-feasible, i.e. ||A x_k -b||_2 \lt e, after O(1/e) iterations, where the complexity of each iteration is O(n log(n)). We also report the results of numerical experiments comparing the performance of FAL with SPA, NESTA, FPC, FPC-AS and a Bregman-regularized solver. A very striking result that we observed in our numerical experiments was that FAL always correctly identifies the zero-set of the target signal without any thresholding or post-processing for all reasonably small error tolerance values.

Finally, from this Press Release "Arce named first Fulbright-Nokia Distinguished Chair", I get that Nokia is interested in Compressive Sensing. Congratulations Gonzalo.