Imaging with Nature (part 3)

In the comment section of the previous entry on Imaging with Nature, I had the following interaction with one of the reader. The reason I am putting this exchange is that often, when trying to explain compressive sensing, the issue of random sampling as understood in a compressive sensing setting is often misunderstood:

Anonymous said...

Imagine a Demon (a relative of Maxwell's demon?) with complete knowledge of the entire scene, cameras, mathematics, optics, and s tiny paint brush.

Could such a demon touch up the scene in a subpixel manner such that the normal imaging system sees the same image as before, while the random imager sees something wildly different? It seems to me that it could. The touched up scene would be highly improbable I suppose.

I find this fascinating, as much as I understand this.

To what I said:

I don't think it would but being random there would be little control about the capability for superresolution.

Anonymous then said...

Is this demon interesting? I don't want to waste your time!

It seems to me that such a demon could defeat superresolution. To the demon, nothing is random, he knows everything, past and future. His only power is his ability to paint very fine detail in such a way that the detail looks like one thing at one resolution and another completely differnt thing at another resolution.

Salvador Dali made paintings like this years ago.

I really want to understand this.

Thanks,

Michael

I then responded:

Michael,

Thanks for your comment, I am sure a lot a people are asking themselves similar questions, so this is a good exercise.

In effect, while the random medium is indeed random, the calibration step I am mentioning is really about having a full characterization of it. So, in effect, we are building the demon. Once we know everything about the random medium we can use the medium to perform the imaging.

Compressed sensing tells you that if the scene of interest is sparse, the random medium (known after being characterized) and a certain number of measurement, done through this medium, allow one to know perfectly the scene of interest. Only the physics will put some constraints on what you can see.

Hope this answer your question, if not let me know.

Thank you Michael!

As Jacques Laurent rightly pointed out recently this type of approach parallels in-situ measurements found in the work of Lawrence Carin, Dehong Liu, and Ya Xue ( In Situ Compressive Sensing (short version) ).