[SciPy-User] 2D phase unwrapping

[Friedrich Romstedt]

Hi Robert,

This post contains reasoning which is OT to the problem at hand. It is related, but it does not solve anything about this "unwrapping" process. Rather it tries to understand what we are doing here and where it might lead to, what the aiming is. :-)

If the reader is only interested in getting his data done, he better does not read it. I've done it because I found there was be something to it (for me), and I wanted to know what it was. This means to be able to formulate it. I wanted to know what thing it is that I apparently did not take into account, and how it relates to the problem of the OP, and what the whole story looks like. If the reader finds this an interesting attitude, he might want to eventually read on. Otherwise, storage is cheap these days. I won't waste much. :-)

I know this is not a philosophy mailing list. It's just the best I can do at the moment. I found the thing interesting and gave my contribution. If someone complains enough that thinking like this does absolutely not fit here, I might want to unsubscribe. :-) Just be open. :-)

Am 12.05.2012 um 12:49 schrieb Robert Kern <robert.kern at gmail.com>:

> On Sat, May 12, 2012 at 11:30 AM, Friedrich Romstedt
> <friedrichromstedt at gmail.com> wrote:
> 
>> I worked pretty hard on an unwrapping algorithm for Fourier tranform results. By this I noticed the following points:
>> 
>> • Phases are angles, so they are just identical, if their angle "value" is identical modulus 2 pi. There's nothing to tell them apart. The numerical value is just an insufficient model and we fight these insufficiencies when trying to "unwrap". They are already unwrapped. We just don't see it anymore.
> 
> Usually, the point of phase unwrapping is to try to recover an
> underlying linear variable in the range (-inf, inf) that we are
> observing through the lens of a complex phase. For example, when doing
> Interferometric Synthetic Aperture RADAR (InSAR), the setup is to fly
> a satellite doing SAR twice over a given area. SAR gives you a complex
> image of the ground surface. The complex phase is related to the
> two-way travel time of the RADAR signal. The phase of a single SAR
> image is essentially meaningless because it depends on every little
> detail on the surface, but if the surface did not change much in its
> fine details (seriously, plants growing on just the order of
> centimeters is a problem) but moved in the direction of the RADAR
> signal due to seismic events, the difference in the phases
> (interferometry!) is linearly related to the distance that the ground
> moved. Except that the phase difference, the only thing we can
> directly observe, gets wrapped. There *is* an underlying non-phase
> linear variable here that we are trying to estimate via phase
> unwrapping. When we're unwrapping phase, we're not really interested
> in the phase itself. It's just the only thing that we can observe.
> There are better and worse ways to unwrap phase in order to estimate
> these underlying variables, but it's not a theoretically doomed
> effort.

Yes. 

[I concentrate in the following on InSAR]

This is a descriptive data analysis problem. I'm not really interested in that kind of problems, although I'm always tempted to be. Two different people with two different interests met here. 

As I see it, the theory is just about modeling how height change and lateral displacement (when moving along the map), as well as any other effect, comprise the net interferometric phase. Since the lateral displacement (when moving along the map) is encoded in the pixel coordinate (or the spacial extent of the image), it can be resolved. Now all the other effects remain. 

The theory now does not make any judgement on the probability of all the different explanations which are possible due to the 2 pi "ambiguity" of the phase (more precise, of the fact that this ambiguity arises when representing phase as a number). So the theory would be happy with all these rather rough height maps which incorporate jumps of a kilometer from pixel to pixel. Because the theory yields the sum of all this different height maps, all of them at the same time. 

It is now hence a matter of a-priori information which continuous interpretation of these contraint done by the measurement we prefer or like best. It's no longer up to the theory to give this. It's a judgement on the constraint done by the measurement, taking into account that it should be continuous, s.t. jumps of a kilometer are not that likely. We shrink the theoretically possible explanations down using a-priori information. 

If there was a jump of one kilometer from one scanline to the next it would still be there on earth, but the interpretation would fail to find this out. Because it has not an eye for it. It would prefer the explanation where the jump is the remaining 2 cm downwards or so (just an example). 

As I said, this is not the kind of problems I'm interested in. I'm not a believer of realism. :-). It's fine if the interpretation yields this 2 cm jump from one scanline to the next. It will create a world for you where you can build on. You might want to refine it by going down there, finding that 1 km jump, or not. Although it might have been preferrable for an alien doing InSAR to just agree on that he would not know what he will see. To be precise, the alien [having InSAR eyes] would have no interest in assuming that there would be a continuous surface. :-)

So it's rather "space unwrapping" or "how to choose the continuous surface" than "phase unwrapping". And I'm not going into that here, I just have not studied what others did about it so I'll better hide my childish ideas how it might be done. :-)

Friedrich

[some content deleted :-)]

P.S.: And if course, a-priori information involves theoretical notions. With "theory" I meant the theory of InSAR, implying phases starting from height and other information. The InSAR process is interesting because it brings together two different theories: First, that of how spacial coordinates etc. relate to phase, and second, the model of a continuous surface. It's an assumption that they hold at the same time. But if they hold at the same time, it won't create anything new on the basis of other things, and that's why I'm not interested in it. So much of text just to find that out :-/. 

And the process of choosing the surface is just the attempt to see if the two theories mentioned can hold at the same time.