[FRIAM] [WedTech] Diffracton: minding the gap

Sun Aug 15 11:14:49 EDT 2021

One possible answer to your final observation is that your model is a model of a physical phenomena and if it is a good model it should exhibit much of the same behavior that we see in the physical world. The Fourier analysis I posed should yield results close to what we observe for the continuous case which should be echoed in a good computational model.

If you have any of my graphics books, they all have an appendix on sampling and aliasing which shows how the sinc and rect functions arise as fundamental to sampling/reconstruction. I don’t use Fourier analysis there. 

Here’s some further connections and a tie to Owen.

I joined the EE Dept  U Rochester in 1973. I was doing a lot digital image processing then. UR had one of the two Institutes of Optics (the other is at Arizona) and I started working with Brian Thompson, the head of the Institute, on hybrid image processing which got me to learn some about Fourier optics. We taught a joint Optics/EE course on hybrid processing. The optics students didn’t know anything about digital image processing and the EE students knew almost nothing about Fourier analysis in the complex domain. I gave a final exam to the optics students and Brian gave a final to the EE students. We also taught a number of short courses on hybrid processing. We also worked with the image processing group at Xerox Research in Webster. At the time, Owen was in that facility and we worked with the same people but never met. That’s really coming full circle.

A couple of comments about the sinc function. It comes up in sampling and in the continuous world whenever you have a window. There are many consequences of it having negative side lobes. In signal processing, it means that you can’t make a perfect low-pass filter. In optics it comes up in lens design because of the finite size of a lens. Alvy Ray glossed over it in his pixel talk. When you have discrete value and want to produce a finite pixel on an analog device, the value modulates a beam that covers a finite area. Sampling theory tells us that this beam should have the shape of a sinc function. But this is not physically possible because the negative side lobes of the sinc would require the beam to extract energy from the projection surface. So in physical systems such as CRTs, the beam has a Gaussian profile which has no negative values. In digital systems, the equivalent is a profile such as the one Alvy Ray showed that has a finite extent and can be described by polygons. Similar results hold for designing lens coatings, In all cases, these designs are trying minimize the errors, which can be seen in the frequency domain, caused by our inability to a perfect reconstruction.

Ed
__________

Ed Angel

Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
Professor Emeritus of Computer Science, University of New Mexico

1017 Sierra Pinon
Santa Fe, NM 87501
505-984-0136 (home)		 	edward.angel at gmail.com
505-453-4944 (cell) 				http://www.cs.unm.edu/~angel

> On Aug 14, 2021, at 9:56 PM, Stephen Guerin <stephen.guerin at simtable.com> wrote:
> 
>  Ed, 
> 
> Yes, that's how I'm seeing it. 
> 
> For others, Ed's Step function is what I was calling Rect pulse function which is Fourier dual of the Sinc function. Attached is an .mp4 recording of Ed description of the relationship of gap width, frequency, and the amount of spread of the Sinc function which is the diffusion pattern observed. I recorded the mp4 from the illustration/animation"  I linked to earlier <https://www.olympus-lifescience.com/en/microscope-resource/primer/java/diffraction/>.
> 
> And I think it's pretty cool to think of the gap as a sampler. I suspect this is a well-known idea in optics/physics and old hat to many of you but it's exciting to come onto ideas like this for oneself :-)   I can almost hear John Zingale saying, "of course, it's just the convolution theorem <https://en.wikipedia.org/wiki/Convolution_theorem> applied to a square wave". In the past I would nod and frankly, my eyes glaze if I can't ground it in a microscopic understanding that guides my intuition.
> 
> Or if given this amazingly deep statement I came across as I'm searching for connecting sampling and diffraction -  "the diffraction pattern of an object is the Fourier Transform of the object" from here <http://www.sci.sdsu.edu/TFrey/Bio750/FourierTransforms.html>, it finally makes sense to me. 
> 
> And I can practically hear Steve Smith and our dear and late Fred Untersher calling out, "that's how we've been describing holograms to you for 20 years and you always nodded like you understood". 
> 
> Or Ed who brought Pradeep Sen into our world saying what do you think I was showing you with Dual Photography <https://graphics.stanford.edu/papers/dual_photography/>, you idiot?
> 
> And Alvy Ray Smith, again Ed bringing into our office,  saying that's what a Pixel is <https://youtu.be/dvHDXUV7hmQ>! it's not a little square nor gaussian point sample, it's Kotelnikov Sampling (Nyquist-Shannon Sampling Theorem),
> 
> or potentially worse is Eric Smith and Roger Critchlow shaking their heads saying "you're just confused and making connections that aren't there". :-)
> 
> --------------------------
> 
> Now even after having said this, I *still* want to know how the diffraction is happening using only the interaction rules in the model. Obviously, there are no Sinc or Rect functions in the code, nor Fourier transforms explicitly coded. All these wonderful explanations above are emergent properties from the model I would call a macroscopic explanation and description. If nothing else perhaps I learn a better phrase for the level of explanation I'm asking for when you trace an algorithm and understand where the emergent property comes from. (BTW, I think I have a micro answer and will put it in my response to Alex).
> 
> -S
> 
> _______________________________________________________________________
> Stephen.Guerin at Simtable.com <mailto:stephen.guerin at simtable.com>
> CEO, Simtable  http://www.simtable.com <http://www.simtable.com/>
> 1600 Lena St #D1, Santa Fe, NM 87505
> office: (505)995-0206 mobile: (505)577-5828
> twitter: @simtable
> z <http://zoom.com/j/5055775828>oom.simtable.com <http://oom.simtable.com/>
> 
> 
> On Sat, Aug 14, 2021 at 3:57 PM Angel Edward <edward.angel at gmail.com <mailto:edward.angel at gmail.com>> wrote:
> I hope someone can check out the analysis below. 
> 
> If you look at the gap as a sampler, you can do the following analysis using Fourier methods:
> 
> A gap is a window on a continuous function. A perfect gap is a step function multiplying the continuous function. 
> 
> In the Fourier domain, the Fourier transform of the continuous function on the input side of the gap is convolved with the Fourier transform of gap (the step function).
> 
> The Fourier transform of a step function is a sinc (sin(ax)/(ax)) function.
> 
> The width of the main lobe of the sinc is inversely proportional to the width of the gap.
> 
> Consequently, the smaller the width of the gap, the more a given frequency is distorted because the sinc is wider. Convolution applies the sinc at each frequency of the input function.
> 
> I think it gets more complicated when we add in sampling. If we take a number of samples that is proportional to the width of the gap, then as we make the gap smaller there are fewer samples, hence more reconstruction issues which is the second, often overlooked, part of the sampling theorem.
> 
> In the limit as the gap goes to zero width, there is no distortion to the continuous function but in the digital world you could have only a single sample.
> 
> Ed
> __________
> 
> Ed Angel
> 
> Founding Director, Art, Research, Technology and Science Laboratory (ARTS Lab)
> Professor Emeritus of Computer Science, University of New Mexico
> 
> 1017 Sierra Pinon
> Santa Fe, NM 87501
> 505-984-0136 (home)		 	edward.angel at gmail.com <mailto:edward.angel at gmail.com>
> 505-453-4944 (cell) 				http://www.cs.unm.edu/~angel <http://www.cs.unm.edu/~angel>
> 
>> 
>> On Sat, Aug 14, 2021 at 10:17 AM Stephen Guerin <stephen.guerin at simtable.com <mailto:stephen.guerin at simtable.com>> wrote:
>> At yesterday's Virtual Friam I asked a question on diffraction and said I would send more background.
>> 
>> The gist of my question is: 
>> 
>> Even though I completely understand the micro-level rules that generate diffraction in the wave model described below,  I still don't have an intuition **how** the gaps in an obstacle have the emergent effect of diffracting waves when wavelengths >= gap width. Can anyone help?
>> 
>> 
>> Background:
>> The question arose from my mentoring NM School for the Arts high school students in the NM Supercomputing Challenge <http://nmsupercomputingchallenge.org/> where the students simulated spatial acoustics by appropriating Saint-Venant equations used for shallow water waves to instead model acoustic pressure waves. We wrote a Netogo agent-based model with Python extension for reading / writing the sound files and simulating spatial acoustics.
>> 
>> <image.png>
>> 
>> 
>> The students explored the effects of different room configurations on acoustics.  
>> 
>> One configuration of interest was a wall gap illustrated below in the top right under Madelyn's video below. The wall gap is hard to see on right side.
>> 
>> <image.png>
>> 
>> They simulated microphones in Netlogo by recording amplitudes at a patch (red dot below in top-right visualization of room) and simulated speakers (hard-to-see blue dot below red dot on other side of wall) by driving amplitudes at a patch from the time series of amplitudes in  .wav files (recordings of a singer and viola performance). They could hear, and through Fourier analysis, see the gap acting as a low-pass filter on the acoustic signal. ie, only the low frequencies were "bending" around the wall to reach the microphone. 
>> 
>> You can see and listen to this effect and the spectrogram visualization at time 33:11 in their presentation <https://youtu.be/61p97NWJiQ8?t=2117>.
>> 
>> <image.png>
>> 
>> It took me a few weeks after their presentation in the NM Supercomputing Challenge - they got second place - to connect the low pass filter behavior to the concept of diffraction. Had this been a light model and I saw the rainbow effects I would have clued in much faster.  Their presentation was a month after finals and they added this epilogue in the presentation above to identify the effect as diffraction. <https://youtu.be/61p97NWJiQ8?t=2761>
>> 
>> Their presentation included this physical wavepool video demonstration <https://youtu.be/BH0NfVUTWG4> which was helpful to me to begin to understand the diffraction relationship with frequency and gap width.
>> 
>> Note: my question is not about "describing" the behavior with macroscopic equations or geometric models but fundamentally how does the gap become a point source ala Huygens Principle at the micro-level of the patches interacting with the emergent waves. To help with the distinction, I consider this interactive model  <https://www.olympus-lifescience.com/en/microscope-resource/primer/java/diffraction/> a great macroscopic description of the phenomenon that nicely illustrates the relationship of frequency and gap width but doesn't help me interpret the micro-level interactions giving rise to the diffraction effect in our simple shallow-water model.
>> 
>> The students describe the details of the shallow water model at this point in their presentation <https://youtu.be/61p97NWJiQ8?t=870>:
>> <image.png>
>> 
>> 
>> Here is my simplified Netlogo wave model <https://anysurface.com/sguerin/models/shallowWaterDoubleSlit.html> of the same shallow water equations without the acoustics. It's set up to explore double slit but you can change it to single slit and mess with frequency and gap and watch the wave propagations, diffractions and interference patterns
>> https://anysurface.com/sguerin/models/shallowWaterDoubleSlit.html <https://anysurface.com/sguerin/models/shallowWaterDoubleSlit.html>
>> <image.png>
>> 
>> As a related aside, with some follow-up discussions with Ed Angel and Steve Smith I am also trying to understand how the gap might be considered a sampling function on the signal. My intuition is that the diffraction of the wave creates a spreader Sinc function and the gap is Rect function which are Fourier duals. In some way, i see Nyquist-Shannon Sampling Theorem <https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem> related to the gap. Note that diffraction creates a spreader function on the back wall in single gap experiments and the gap may be considered a Rect pulse when smaller than the wavelength.
>> 
>> <image.png>
>> 
>> 
>> 
>> _______________________________________________________________________
>> Stephen.Guerin at Simtable.com <mailto:stephen.guerin at simtable.com>
>> CEO, Simtable  http://www.simtable.com <http://www.simtable.com/>
>> 1600 Lena St #D1, Santa Fe, NM 87505
>> office: (505)995-0206 mobile: (505)577-5828
>> twitter: @simtable
>> z <http://zoom.com/j/5055775828>oom.simtable.com <http://oom.simtable.com/>
> 
> <gapFrequency.mp4>_______________________________________________
> Wedtech mailing list
> Wedtech at redfish.com
> http://redfish.com/mailman/listinfo/wedtech_redfish.com

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://redfish.com/pipermail/friam_redfish.com/attachments/20210815/80ac78b7/attachment.html>