CS: Wavefront Coding for Random Lens Imagers ?

Do you recall that question I asked a while back to Rich Baraniuk annd Justin Romberg about the reason why the soccer ball image from the single pixel camera was Ok but still kind of blurry (this was a question initially asked by an anonymous commenter on the blog) ?

Do you recall the so-so quality of the reconstruction of the random lens imager ?

Maybe part of the issue is that, in either cases, the light wavefront phase was not considered in the calibration process. Indeed, if you recall the calibration process of the random lens imager, it does not into account for this parameter:

In a totally different area, Allard Mosk and Ivo Vellekoop [1] [2] are concerned, among other things, with light delivery in human tissues. One of the problems with human tissue is its extraordinary diffusivity. Some folks spent much time trying to compute light dispersion through these tissues in order to detect specific cancers or known markers and so forth. In effect, they either are solving an inverse problem with the diffusion equation and some are even heroic to the point of doing it with the full transport (or radiative transfer) equation. In the problem of delivering light to a specific area inside the body, one has to control the forward problem. In their recent publication [1][2] Allard Mosk and Ivo Vellekoop show that by modulating the phase of an incoming light beam through the use of an SLM they can rectify the direction of the light after it has gone through a random medium (also called 'opaque lens') as shown in the diagram below:




In their more recent paper, Ivo Vellekoop, A. Lagendijk and Allard Mosk have essentially achieved a higher focusing capability than what would be offered by a simple lens thanks to the use of a random medium.


This is explained in their recent Exploiting disorder for perfect focusing. The abstract reads:

We demonstrate experimentally that disordered scattering can be used to improve, rather than deteriorate, the focusing resolution of a lens. By using wavefront shaping to compensate for scattering, light was focused to a spot as small as one tenth of the diraction limit of the lens. We show both experimentally and theoretically that it is the scattering medium, rather than the lens, that determines the width of the focus. Despite the disordered propagation of the light, the prole of the focus was always exactly equal to the theoretical best focus that we derived.

One cannot escape the similarities between these figures and the ones above for the random lens imager. Figure 2.a is the same beam of light through a simple lens, Figure 2.b, is the random projection of the same single beam of light after it has gone through the random/opaque material. Figure 2.d is the configuration of the SLM that allows the initial beam of light to be focused. Figure 2.c is the resulting beam of light modulated with the SLM configuration in 2.d after it has gone through the random/opaque material. In other words, Figure 2.b is the random projection of a dirac like input in about the same way as in the Random Lens imager case.

What is very interesting from this paper is that they have an analog algorithm that allows them to go through a series of SLM configurations that eventually provides a single focused beam. For any random medium, they can find a certain SLM configuration so that a focused beam comes out on the outside. In effect, they are solving a calibration issue. In the random lens imager case, calibration is performed by sending several coded signals (not phase coded) and by gathering their responses. This collection of responses is then used to produce a dictionary. The dictionary is then used to build future images obtained from the random lens imager. What this paper shows is that the random medium provides phase modulation and that any random lens imager should need a calibration step that includes some phase information.

Other cameras designs that are either affected by this issue or are solving the phase coding problem include:

• Heterodyning Light Photography of Ashok Veeraraghavan, Ramesh Raskar, Amit Agrawal, Ankit Mohan and Jack Tumblin
• The Reference Structure work of David Brady and his team.
• Random Convolution Imager of Justin Romberg,

At my low level, my question is how can we build a dirt cheap random lens imager that integrates an SLM and can be calibrated in a simple (albeit slow) fashion ?

Thank you to Laurent Jacques for pointing me to this very nice work. We have to thank the Arxiv blog for raising, once again, our awareness on this issue.

References:

[1] I.M. Vellekoop and A.P. Mosk, Universal optimal transmission of light through disordered materials
[2] I. M. Vellekoop, and A. P. Mosk, Phase control algorithms for focusing light through turbid media
[3] Rice Compressive Sensing Single Pixel Camera using 40% samples.