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).

SPIE Optical Engineering and Applications 2011 - presentations from Astromentry section

Some interesting papers from the Astrometry section that held on Wednesday. This is not about the Adaptive optics, but still contains some interesting points.


 

1. Differention Tip-Tilt Jitter.

It is well known fact that the Tip/Tilt is the main source of distrubance in atmospherical seeing. Other distortions to consider are geometrical ones, like cushion/barrel.

Atmosphere is like a prism - it can displace the star position. Advantages of large telescopes are therefore reduced by CDAR noise.

Dynamic distortion calibration using a diffracting pupil: high-precision astrometry laboratory demonstration for exoplanet detection, . . . . . . [8151-29]

They want to create diffraction spikes. 

Interesting astronomical papers from SPIE Optical Engineering and Applications conference 2011

More about papers from the SPIE conference; main section about the adatpvie optics was on Sunday, but some other interesting posters were in other days as well. 

Advancements in laser tomography implementation at the 6.5m MMT,  . . . .[8149-07]

The system on the MMT uses 5 LGS stars, 336 voice-coil actuators and they trying to use dynamics focus. The LGS they use is sodium beacon, and, as it is well known fact, the sodium LGS tends to elongate.

They capture everything on one CCD - this means that all of LGS on one CCD. They also use the WFS instrument for the NGS light from tip-tilt star (to sense the tip/tilt distortion).

Least-squares LTAO implementation uses SVD decompostition (modal decomposition) for tomographical reconstruction. Wind can be detected from multiple LGS beacons. They obtain then a tomographic matrix.

However, the problem with the SVD is computationally intensive algorithms.

The further challenges are presented on the slide above.

Wavefront control with SCExAO: concepts and first on-sky results,
Olivier Guyon, Frantz Martinache, Christophe Clergeon, Robert Russell, Subaru Telescope, National Astronomical Observatory of Japan (United States); . . . . . . . . . . . .[8149-08]

The paper presents a wavefront control on the Subaru telescope. They use phase induced amplitude apodizer (PIAA) - a novel concept that can be used for the coronography.

The PIAA is used for the redistribution of light without loss. They try to decrease the speackles using the PIAA.

A sensitivity comparison between the non-linear curvature wavefront
sensor and the Shack-Hartmann wavefront sensor in broadband, Mala Mateen,  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .[8149-09]

This was a really strange presentation. The promising title was ruined by poor presentation: out of slides it was impossible to understand the point.

They tried to compare Curvature WFS that measures:

\[ C = \frac{W_+ - W_-}{W_+ + W_-}

They observed Talbot effect:

Talbot imaging is a well-known effect that causes sinusoidal patterns to be reimaged by diffraction with characteristic period that varies inversely with both wavelength and the square of the spatial frequency. This effect is treated using the Fresnel diffraction integral for fields with sinusoidal ripples in amplitude or phase. The periodic nature is demonstrated and explained, and a sinusoidal approximation is made for the case where the phase or amplitude ripples are small, which allows direct determination of the field for arbitrary propagation distance.

[from the paper: Analysis of wavefront propagation using the Talbot effect
Ping Zhou and James H. Burge, Applied Optics, Vol. 49, Issue 28, pp. 5351-5359 (2010)       doi:10.1364/AO.49.005351 » View Full Text: Acrobat PDF (785 KB) ]

Image plane phase-shifting wavefront sensor for giant telescope
active and adaptive optics, François Hénault, Univ. de Nice Sophia Antipolis (France) . . . . . . . . . . . . . . . . . . . . . . . . . . . .[8149-10]

The paper is about phase shifting WFS, although the speaker was not very detailed in descriptions.

The thing is, they use it for making a cross-spectram measurements.

Finding a problem for the Research

In the July issue of IEEE Spectrum journal, there were a short but interesting article entitled "In Research, the Problem is the Problem". The author of it reflects about some issues of problem finding - that is, how to find a really worthy research question. There are no definite solution, of cource, but the article itself (one page only) is worth to read and think about. Here are some quotes from it (OCRed version is here).

A problem well stated is a problem half solved. -- Inventor Charles Franklin Kettering (1876-1958)

The solution of problem is not difficult; but finding a problem -- there's the rub. Engineering education is based on the presumption that there exists a predefined problem worthy of a solution.



Internet pioneer Craig Partridge recently sent around a list of open research problems in communications and networking, as well as a set of criteria for what constitutes a good problem. He offers some sensible guidelines for choosing research problems:

• having a reasonable expectation of results
• believing that someone will care about your results
• others will be able to build upon them
• ensuring that the problem is indeed open and under-explored.

All of this is easier said than done, however. Given any prospective problem, a search may reveal a plethora of previous work, but much of it will be hard to retrieve. On the other hand, if there is little or no previous work, maybe there's a reason no one is interested in this problem.



Real progress usually comes from a succession of incremental and progressive results, as opposed to those that feature only variations on a problem's theme.



At Bell Labs, the mathematician Richard Hamming used to divide his fellow researchers into two groups: those who worked behind closed doors and those whose doors were always open. The closed-door people were more focused and worked harder to produce good immediate results, but they failed in the long term.



Today I think we can take the open or closed door as a metaphor for researchers who are actively connected and those who are not. And just as there may be a right amount of networking, there may also be a right amount of reading, as opposed to writing. Hamming observed that some people spent all their time in the library but never produced any original results, while others wrote furiously but were relatively ignorant of the relevant literature.

Highlights of the SPIE Optical Engineering + Applications Conference at San Diego, CA, 2011

Most of interesting oral presentations was on Sunday, where the astronomical adaptive optics was discussed. Here are some remarks on them from the section Astronomical Adaptive Optics Systems and Applications V.


Integration and test of the Gemini Planet Imager . . . . . . .[8149-01]
For the extreme AO, they plan to achieve 2-4 arcseconds of angular resolution. Since it is a Cassegrain focus, the instruments must be located under the focus and move with the telescope.
There are some interesting lessons they learned from the WFS:

• small subapertures make it hard to align;
• mount of the camera is hard to align;

The lenslet size is 63 micrometers that introduces a lot of diffraction effects.

The controller they use is commercial closed black-box (Fourier, predictive controller).
The WFS used is Shack-Hartmann quadcell. WFS noise is 4-5 e- at 1 KHz speed.



The TMTracer: a modeling tool for the TMT alignment and phasing system, Piotr K. Piatrou, Gary A. Chanan, Univ. of California, Irvine (United States) . . . .[8149-03]

This is about the simulator of the TMT parts written by Piotr K. Piatrou on FORTRAN 95. No diffraction effects, only ray tracing.

This is for alignment and phase sensing of the telescope mirrors. The control of the wavefront is LS tomography - filtration of commands directly to DM. This is due to huge amount of data.
Observability must be computed theoretically, then the SVD decomposition is used.



Athermal design of the optical tube assemblies for the ESO VLT Four Laser Guide Star Facility, Rens Henselmans, David Nijkerk, Martin Lemmen, Fred Kamphues, TNO Science and Industry (Netherlands) . . . .[8149-04]

Interesting speech about the design of LGS tube, they actually use it for VLT with 4 GS for lambda=589 nm and power 25 W.
Optical design of the LGS tube is classical Gallilean 20x beamer expander. Athermal design was a primary goal. Laser absorption is estimated as 0.1 ... 1% for 25 W which is pretty much. They are making a big tube from invar (steel alloy) to expand a light beam - not from caron since it is very expensive. L2 Lense in their design is 38 cm in diameter.


Overview of the control strategies for the TMT alignment and phasing system, Piotr K. Piatrou, Gary A. Chanan, Univ. of California, Irvine (United States) . . . . . . . . .[8149-05]

The main goal here is to automatically control alignment of AO parts on the TMT. The multidirectional tomography is the mainstream approach for TMT alignment.The PAS (alignment and phasing system) is n open-loop system without accounting for the dynamics.

This is for the alignment only. The control is brute-force LS tomography because of huge amount of data. They have 33GB for SVD, and therefore use complexity reduction methods. For instance, projection mehtod like Oa = s -> \[ P\dagger O a = P\dagger a \]

In the case of TMT, I think, it is possbile to nglect the dynamics of the system completely and just assume that the system is static. For low and medium frequencies it will probbly work well. That cruel algorithm (just throw the command to DM) explains the 2*opd coefficient 2.

The quasi-continuous part for the control.

Manufacturing and using the continuous-facesheet deformable mirrors

There are some notes from SPIE Optical Engineering + Applications poster discussions and presentations about deformable mirrors.



It was interesting to know that Xinetics still uses very strong actuators that can actually break the mirror even in the presence of electrical protection. Some say they just do not want to upgrade their technology as NASA buys from them. Therefore, if there are very powerful actuators, the beackage of a mirror is possible. Other, like Gleb Vdovin from OKO Technologies, use soft actuators (piezoelectric ones) and the force of them is not enough to break the mirror.





How to attach an actuator to a Deformable Mirror?

Simple answer: using epoxy glue. Surprisingly, even after many days of work epoxy glue still holds actuators very well. Moreover, there were special tests: for instance, one can write a program (DMKILL) that throws random "0" and "+max" voltages on the mirror. It was reported that the first actuator broken down after at least 10 days (!) of constant work in such a regimen.



In the case of broken actuator, many people suggested just to throw it away as a loss of one actuator does not effects the rest too badly.





Main problem is Tip\Tilt

Tip and Tilt are the major sources of problems caused by the atmospheric turbulence. An impact of the higher aberrations falls very quickly. The correction is therefore required for the tip/tilt.

For this, a WFS with Zernike fitting can be helpful.



However, as Guan Ming Dai notes in his paper "Comparison of wavefront reconstructions with Zernike polynomials and Fourier transforms." [J Refract Surg. 2006 Nov;22(9):943-8.],

Fourier full reconstruction was more accurate than Zernike reconstruction from the 6th to the 10th orders for low-to-moderate noise levels. Fourier reconstruction was found to be approximately 100 times faster than Zernike reconstruction. Fourier reconstruction always makes optimal use of information. For Zernike reconstruction, however, the optimal number of orders must be chosen manually.

Interesting posters from SPIE Optical Engineering + Applications San Diego, CA, 2011

1. Na variability and LGS elongation: impact on wavefront error, Katharine J. Jones, WBAO Consultant Group (United States). . . . . . . . . . . . . . . . . . . .[8149-14]
2. MT_RAYOR: a versatile raytracing tool for x-ray telescopes, Niels Jørgen S.
   
   Westergaard, Technical Univ. of Denmark (Denmark) . . . . . . . . . . . . . [8147-64] <---- this simulator is actually written on Yorick
3. A hardware implementation of nonlinear correlation filters, Saul Martinez-
   
   Diaz, Hugo Castañeda Giron, Instituto Tecnológico de La Paz (Mexico)[8135-49] <--- the poster actually is about morphological filtres implemented in hardware.
4. Development of the visual encryption device using higher-order
   
   birefringence, Hiroyuki Kowa, Takanori Murana, Kentaro Iwami, Norihiro
   
   Umeda, Tokyo Univ. of Agriculture and Technology (Japan); Mitsuo Tsukiji,
   
   Uniopt Co. Ltd. (Japan); Atsuo Takayanagi, Tokyo Univ. of Agriculture and
   
   Technology (Japan) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [8134-32]
5. Enhancement of the accuracy of the astronomical measurements carried
   
   on the wide-field astronomical image data, Martin Rerábek, Petr Páta, Czech
   
   Technical Univ. in Prague (Czech Republic) . . . . . . . . . . . . . . . . . . . . . [8135-58]
6. Astronomical telescope with holographic primary objective, Thomas D.
   
   Ditto, 3DeWitt LLC (United States) . . . . . . . . . . . . . . . . . . . . . . . . . . . . [8146-40]
7. Calibration of the AVHRR near-infrared (0.86 μm) channel at the Dome
   
   C site, Sirish Uprety, Changyong Cao, National Oceanic and Atmospheric
   
   Administration (United States). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . [8153-74]