Compressed sensing: notes and remarks

This short note is about the tutorial [1] on compressed sensing (CS) recently published in Optical Engineering journal. The tutorial introduces a mathematical framework that provide insight into how a high resolution image can be inferred from a relatively small number of measurements. Among other applications, such as IR imaging and compressing video sequences, astronomical applications [2] of CS are very attractive.


The idea of Compressed Sensing
The basic idea of CS [1] is that when the image of interest is very sparse or highly compressible in some basis, relatively few well-chosen observations suffice to reconstruct the most significant nonzero components. It can be also considered as projecting onto incoherent measurement ensembles [2]. Such an approach should be directly applied in the design of the detector. Devising an optical system that directly “measures” incoherent projections of the input image would provide a compression system that encodes in the analog domain.

Rather than measuring each pixel and then computing a compressed representation, CS suggests that we can measure a “compressed” representation directly.

The paper [1] provides a very illustrative example of searching the bright dot on a black background: instead of full comparison (N possible locations), the CS allows to do it in M=log2(N) binary measurements using binary masks.

The picture from the paper [1]

The key insight of CS is that, with slightly more than K well-chosen measurements, we can determine which coefficients of some basis are significant and accurately estimate their values.

A hardware example of Compressed sensing
An example of a CS imager is the rice single-pixel camera developed by Duarte et al [3,4]. This architecture uses only a single detector element to image a scene. A digital micromirror array is used to represent a pseudorandom binary array, and the scene of interest is then projected onto that array before the aggregate intensity of the projection is measured with a single detector.


Using Compressed Sensing in Astronomy
Astronomical images in many ways represent a good example of highly compressible data. An example provided in [2] is Joint Recovery of Multiple Observations. In [2], they considered a case that the data are made of N=100 images such that each image is a noise-less observation of the same sky area. The goal is to propose the decompression the set of observations in a joint recovery scheme. As the paper [2] shows, CS provides better visual and quantitative results: the recover SNR for CS is 46.8 dB, while for the JPEG2000 it is only 9.77 dB.

Remarks on using the Compressed Sensing in Adaptive optics
The possible application of the CS in AO can be for centroiding estimation. Indeed, the centroid image occupies only a small portion of the sensor. The multiple observations of the same centroid can lead to increased resolution in centroiding and, therefore, better overall performance of the AO system.

References:
[1] Rebecca M. Willett, Roummel F. Marcia, Jonathan M. Nichols, Compressed sensing for practical optical imaging systems: a tutorial. Optical Engineering 50(7), 072601 (July 2011).

[2] Jérôme Bobin, Jean-Luc Starck, and Roland Ottensamer, Compressed Sensing in Astronomy, IEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING, VOL. 2, NO. 5, OCTOBER 2008.

[3] M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).

[4] W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).