Intro to Fourier Optics and the 4F correlator 

In the mid 1980s a colleague (now deceased) and myself tried this in my basement lab, with a HeNe laser, and 35mm film snapshots of various silhouette objects, same general setup. The concept was for target object acquisition and recognition within a 2D "search" region . . . it sort of worked. Then I went to the CLEO conference in Baltimore (1986 ?), and first thing I saw, walking into the tradeshow demo area, was the Perkin Elmer high tech version of the same concept, sorting out planes and tanks silhouette targets, from anywhere within an addressable 2D vision plane. Well, anyway, gave up on trying to compete with Perkin Elmer . . . but it was interesting. You video brought all that back to life . . . thanks for posting.

A nice quick way to make a small pinhole for the spatial filter is to get a stack of aluminum foil sheets, and take a sharp, thin sewing needle and lightly poke the stack. Separate the foil sheets and you will get pinholes of smaller and smaller diameter. One of these will be what you are looking for. Then mount the foil. I use this method all the time rather than worrying about purchasing a pinhole disk, and is a good replacement for when someone in your lab burns out your spatial filter with too much power! ;)

Great presentation! My physics professor showed me this in the lab one day, more than 50 years ago now. He used black and white microfilm and microfiche. He showed me newspapers on microfiche that could be searched for specific things. Your clearer explanation of how to use the "amplitude only" part of the spectrum to make the information location independent was helpful. There will be geometric issues I think, but that helped clean up an old memory. I have over the years kept an eye on this, and now there are micromirror arrays, and many controlled transparency materials to generate the spatial filters using the computer. Nothing was ever cheap enough for me to try it. It seems everything that is interesting quickly becomes costly. But your videos are a good "buy" because you are always trying to show how things work and how to build them and understand them. I can think of a few things to try, but there is no way to interact on RU-vid, except by quickly disappearing text notes. Thank you for your work. I just watched an old Raman spectroscopy video of yours where you said you were going to get back to it, but I could not find it. I really liked what you showed with the pinhole. I wish you had spent more time. I want to find a way to spread out the intense points in a better way. Bessel beams and optics seems a good path. I am getting older and there are billions of things to look at and try to calculate. I can't build anything, mostly too costly or too hard. But I can calculate most anything. Go figure. Richard Collins, Director, The Internet Foundation

If it works with polarized light, it would be nice to project an opaque monochromatic LCD. You could plot an image in the LCD that represents an electric signal and get it's Fourier transform. Today I found out about your channel and I watched all your videos. I don't remember having a better day. Thank You, people like you are rare.

When I heard about the 4F correlator, I knew that I had to do it. Thanks for the compliment. I'm pretty out-of-my-league on the mathematics, but it was frustrating to search around and not see anyone else at least try to explain it plainly, so I gave it a shot.

That is awesome. There's zero chance I would understand any of fourier optics without this genius, simple explanation.

This was such a great explanation - I have been working for weeks to intuitively understand what the spatial domain represents - this finally put everything together for me! Thank you so much for your continually excellent insights

About 15 years ago, I had designed several optical stages that performed multiplexing of 3 beams of different lambda, into a single fiber. Normally this is an expensive set of complicated optics. Moreover, I needed a tophat beam profile, meaning instead of a gaussian concentric beam energy density like you showed, the brightness of the beam as a spot on a target had to be uniform strength. This beam forming is a fundamentally desirable optical function, yet the methods to achieve such a profile are extremely complicated and require several optical stages to perform. To top it off, I am not an optical engineer. Anyway, the mux ended well with my design being good enough and very inexpensive. As for the beam former, I searched so many resources and found an obscure optical schematic for tophat beam formation using only to biconvex lenses and a spatial filter. But there was absolutely no explanation. I had theories, but no understanding of how or why it worked. Your video 15 years later made it clear before you got to your aparatus. Just the description of the spatial filter, combined with the spatial FFT intro. Effectively, there's two ways to look at it. The cheesy way, is how I thought it must work: The rough beam profile is compressed to the focal point and the spatial filter just takes the prime, central cut of the center of the beam and that's all that gets expanded back through the output objective stage, meaning you lose energy, but it's evenly distributed. But now, from this video, I know better, I think: The rough beam has some spatial features in it's profile. The distribution of the energy density of the beam cross section itself could be argued as a monochromatic photograph in complexity. That's why your setup needs to clean the beam first, by taking the prime cut and collimating it. Now you might thing the theory I had come up with is exactly the same only no collimation required. But I believe it's all about filtering the FFT, Such that the spatial filter in the middle is taking only the lowest spatial frequencies and filtering out the higher frequencies, just by taking the center brightest point of the FFT, and the objective rebuilds a spatial distribution of energy density of only near-zero spatial frequencies, thus creating a flat beam profile. So it's a lowpass filtering of a rough beam profile's spatial FFT, reverse spatially FFT'd back into a more uniform beam profile. The high frequency spatial noise is filtered out. Extremely cool! But now, I understand that the aperture of the spatial filter and quality of the lenses are paramount. But since my image is just to the beam diameter, I don't have the complication of collimation. Thank you very much for this!!

8:59 - Nice to see Ben "upcycling" redundant equipment 😉. The best on RU-vid. I always learn something new from every video. Top man.

Very cool. Brought back memories of doing feature finding independent of location and rotation using Kohonen neural networks. Thanks for providing such a clear explanation. I've had the same problem with wikipedia many times.

the projects that you work on never cease to amaze me! never stop being awesome!

Also, under certain circumstances you CAN recover the object from only the magnitude of the Fourier transform. The problem is called phase retrieval and it’s important in fields like astronomical imaging, x-ray diffraction, and holography. In the past, I’ve used iterative phase retrieval to generate phase-only holograms for a phase only, liquid crystal spatial light modulator given only the desired far-field irradiance pattern (i.e. a picture). The result was a holographic video projector.

Just listening to you explain how you set up the columnizing lens had me in awe. You are a genius!

Fourier optics is wonderful stuff! A few comments: Originally, the reconstruction process for synthetic aperture radar was accomplished using an analog, optical computer. Take a look at Goodman (THE Fourier Optics book) section 8.9. Your spatial filter is also a filter in the Fourier plane. The pinhole cuts out the high frequencies to clean up the beam. Making your mask objects smaller will make your Fourier plane bigger. Also, you can magnify the Fourier plane to make it easier to see.

Completely agree with you: optics has such a ridiculous learning curve and everyone seems incapable of talking down. Your setup is 100% legit. However, I would caution against hot gluing objectives; the optics inside are always effected (eventually) by temperature contrasts. I noticed you have an EO container; objective adapters are easily available from them (or cheaper, Thorlabs). They should also have pinholes which I remember being not too bad, they should get down to around the 1um mark, especially if you can get them used

A webcam with the lens removed works great for this. The chip is often nice and small, excellent for viewing the small image. If the features on the slide are small enough, the pattern in the fourier plane will be millimeter sized and you can easily view the effects of using a mask to remove the zeroth order, the higher frequency components or components in certain directions, for instance filtering out all the diagonal parts of characters in a text. Works really well.

Thank you Ben! In my university I made a setup to make optics experiments for other students (simple diffraction from slit) and my professor told me to prepare some experiments in the fourier optics field. I haven't studied this subject in detail (very time spending) so you gave me a very good clue! Thank you :D

Great video. This video helped me realize the potential of optical computing where fixed function optical elements are arranged to perform compute operations. There would be no clock frequency, it would simply be instant computation.

the first 50 seconds of this video explained fourier transforms more clearly than 2 weeks of university lectures.

Even your explanation of "gaussian distribution" was easy to understand. Thanks so much for sharing your knowledge. Please keep the videos coming.

Have you tried scanning the image into a computer and taking a 2d fft of the image just to see if you are getting similar results?

Ben, there is a great book, a bit old now, by Keigo Iizuka called "Engineering Optics". It goes through a lot of the math in very intuitive style, with lots of pictures and awful puns, and ties fourier optics to stuff a lot of EEs have seen. I see there is a new edition just a few years old, but it costs bigtime, and mine from the 80s is still great for figuring things out.

Ben, this is great! I refurbished a few devices that used Fourier Optics to detect droplet size, and spray plume density in a delivery device. The key was to keep the lenses clean with pure acetone and lens tissues, spiral cleaning pattern from the center.

At 11:03, I start describing what you can do with the 4f correlator. Notably, it was used for image shape search before computers were able to do so. For example, finding airplane silhouettes in film negatives. It can also be used for image processing, such as edge-detection, blurring, etc. Often, I will start a tutorial video with the real-world application and then describe how it works, but in this case, I wanted to describe Fourier space in general first.

Thank you for this! As you say, both wikipedia and most of my subject literature has a tendency to make everything extremely complicated, without pausing to also do the simple explanation. I could not wrap my head around why we got a projection of the fourier plane just from the lenses

Excellent video, would love a Part 2 with more images and examples shown

Hi, thank you for your very informative videos! I am preparing for my thesis about diffractive optics and your videos are helping me very much to get more into these topic!

So nice to see someone finally explain Fourier Optics without resorting to mathematics. I'm making holographic microscopes using a $0.30 laser and a $7 image sensor. Denis Gabor deserved his Nobel Prize!

Would love to see you come back to this and show it in action, optics physics is so interesting

You just couldn't resist trying it could you? :) Best explanation available on the 'net, hats off mate!

I love your pragmatic approach to physics.

Really cool. I really like your way of explaining things in a practical context. I am a little similar. I like to do things practically before I can understand it theoretically.

I see some kind of futuristic application of this Fourier Optics to hologram image generation. Thanks for the video.

I'm learning so much from you! I'm glad I subscribed!

The pinhole used to "clean up" the laser beam is in fact the same thing as the filter you talk about later. Higher order Fourier components in the laser beam itself due to imperfections of the laser cavity, optics, cleanliness of the cavity + optics, etc are "filtered out" by the pinhole. When an image is "created" in the colimated beam section, you are simply addging "higher order" Fourier compoents back into the "cleaned up" laser beam.

I want to see the followup video!

Excited for the follow up!

Thanks for your research and sharing your experiments. This has sparked the thought that maybe I could understand more about the characterization of DNA by Watson, Wilson and Crick. I've always wanted to grasp thet

Very good explination! thnks so much! I needed this for the Uni. Greetings from Germany

"I took a helium-neon laser and I hot-glued a microscope objective to the front of it". Ben, you're awesome! Great video, very interesting. :)

Spatial filtering of super high intensity laser pulses was the savior of laser driven nuclear fusion research in the 70s. In beams in the gigawatt scale, certain self-focusing effects begin to become apparent due to the intensity of the electric field component causing a change in refractive index in air and causing the beam to locally collapse into a filament. Image relaying and filtering after every amplification stage cleans the beam and prevents the laser medium from blowing itself apart.

Wow! I have so many questions. I won’t ask here but I’ll be reading into this for a month for sure. I’m now in “have lenses will laser them” mode. Thank you for posting this PLEASE FOLLOW UP THIS VIDEO!

i would really love to see you apply a low pass or high pass optically to an image on another video.

@Applied Science Now if you just cut of the central frequency(which is constant) by placing a tiny sphere in the focal point between 2nd and 3rd lenses, only light that is slightly deflected will get through. Now if you put for example a hot air between 1st and 2nd lens, some rays will get bent due to differential in refraction coefficient and won't be cut by the sphere. This system is called Schlieren method and serves for visualizing slightly bent rays (such as hot or condensed air flows) and I'm really surprised you haven't shown it. Here's the image of how it looks: nptel.ac.in/courses/101103004/module7/lec7/images/24.png I did try building this system myself back in the day, however, I did not have any lens of suitable size which led the system being ineffective. Your lens are fairly large and should work really well, you don't even need a coherent light source for that.

The FT/DFT is actually pretty straight forward if you think of it as the result of cross-correlations between your signal and a basis set of complex sinusoids (each of which is just two real sinusoids 90 degrees out of phase - using complex numbers just lets you stuff the phase information resolved from the quadrature sinusoids into a single number). The x-correlation finds the amount of each component sinusoid in your signal, decomposing it into its magnitude and phase spectra.

Cool. Early (pre-1980s) synthetic aperture radar (SAR) systems used optical processing. Optical have been revisited fairly recently as the processing can easily be done in real-time and is very power efficient.

Hi Ben, scottrharris is right. With large objects, like the letters you showed, the pattern in the fourier plane is only a couple of tens of microns across, making it hard to see and manipulate. The lenses you use are fine, I use similar ones when I show this setup to physics students in an optics class. For the object, try using a slide projector slide with very small features on it. The image in the exit plane will also be small, but an easy way to view it is to project it onto a bare ccd.

This is simply put a Bertrand lens. In microscopes it allows you to see the back focal plane, i.e. shifting the conjugate planes of a microscope, and obtaining an image that can be described as a Fourier transform. The same result can be achieved in a microscope by removing the oculars. Of course one huge problem is that in the conjugate plane of an image you also see the emission source itself, which can be mitigated through Koehler illumination. A cathode tube is a microscope by all means so all tricks in optical microscopy can be applied.

Awesome video and explanation

Great explanation. Looking for follow-up videos.

How about a 'double slit' experiment for your next video? Would love to see this in action! :)

This is also the way to do pattern recognition in real time. To calculate the cross correlation you put the transformed image of the pattern to be recognized in the focal point between the two lenses. you will get a bright dot if the two images correlate.

Very nice coherent (pun intended) explanation! Thanks!

This helped. A bit. I still appreciate the little that I walked away with that I didn't have before.

super cool. with a laser illuminated LCD as your source and a line scanner you could do hybrid optical computing with this setup

In order to take care of scaling, couldn't you take the collimated portion through which you shine the light through your transparency and replace a portion of that with a non-collimated region that gets re-collimated and then move the image parallel to the direction of light travel? Effectively increasing and decreasing the relative diameter of the collimated portion with respect to the image?

I learned that Optalysys is using this (with some LCDs in line) to do high speed computations akin to deep learning.

wonderfully made.it has been explained so well.i was searching all sort of data on net to find mtf experiment this made me back interested in optics.how to carry out calculation for college level experiments if any........or mtf can be calculated using matlab

Ben, you say you have to use black on clear to allow the diffraction pattern to form. What would happen if you used translucent photograph negatives?? Would that produce a photo quality diffraction image which you could then FFT??

How can you be a scientist student and not find the drive to do something just because it is interesting? I am not even there yet, but I find science so interesting, I can't avoid trying stuff I know it's useless but is plain amazing! Scientists do science because it's cool and amazing. I think that's the real drive behind the great minds.

knowledge should always be free ! tnx for vids !

I didn't understand most of it, but damn is it great to listen to you.

Thanks, that was very helpful

cool job!!!really helpful

About printing your own transparencies, last night I tried using a laser printer on transparency slides, then using TRF to seal it and make it significantly more opaque and had varying success. If you are interested, I could print you a slide with the black part in copper and that way it can be totally opaque with my glass PCB process... I'd have to see what I could about completely cleaning off glue rather than sealing it with polyurethane.

What happens if you modify this setup to make a hologram, either of the fourier plane or of the reconstructed filtered image plane?

What if you have a divergent beam input but smaller than the aperture of the first lens?

Well you can try the image comparison with your eyes. If you have 2 identical pictures and one has a verry small diference tan the other if you manage to look at each image with the coresponding eye yourbrain will fuse them in one and the difference will be flickering. I have tryed it and you can basicaly solve the "find the difference" puzles in seconds, no matter how hard they are.

Combine this with your x-ray experiment to get 4F analysis of your x-rays. You can do crystallography experiments. You see patterns like this in everyday life when your driving by double chain linked fences.

Thank u so much sir..

is there a more intuitive explanation of why, when focusing diffraction pattern to a single point, the higher frequencies component of the original light gets focused in the center ? i understand that that high frequency component shows up in the center in a fourier space but not quite sure why it does that intuitively..thank you

A whole new definition of the scarlet letter.

I only understood about a fifth of that, but that sounds awesome.

There's a Fourier plugin for GIMP that does forward and reverse transforms a bit more interactively than ImageMagick, fwiw.

This guy knows his sh&t, no question about it.

Now that you have this setup, you should take a look at digital holographic microscopy. I think all you would need to add is another microscope objective, a beam splitter and a web cam.

Can you put a LCD in the fourier plane to customize the filter? You can put a LCD on the image plane too, to customize the image to be processed! 😀

Wow that link is great ..so basically you can use spacial filters in the fourier domain to block out frequencies in the visual domain . essentially like digital filtering in audio only in visual context . I guess this completely makes sense in the way that pictures are frequencies of light ..

It's only a transform if it's inverse exists. That is, transform the image first and then recover the original image from the transform.

Really intereesting...

You said that you are working for Valve software which is in Seattle WA. But that looks like your shop there in Redwood City CA. You don't have some kind of teleporter system to go to and from work do you? I just can't imagine you driving 1600 miles to and from work everyday?

NICE! THANKS for ALL the different kinds of math. Of course, I speak in the name of the movement of the of 2000, and no redundancy. Praise "Secret name of higher Power and certainly YOU too!

Do an inverse transform of the captured Fourier image in Matlab.

Is a rainbow the Fourier transform of the Sun in the frequency domain? Or a... radial histogram? I actually thought of this today, not while looking at a conventional rainbow, but while observing a cloud near the Sun (in terms of angle of course) that was exhibiting iridescence. Totally separate thought: when a spacecraft launches shortly after sunset or shortly before sunrise, as its altitude increases its exhaust can be lit directly by sunlight against what we see as the night sky, exhibiting a spectacular contrast in brightness. Would there be any way to create the conditions for a "rainbow" phenomenon at such an altitude, so that a bright, colorful rainbow would appear across a dark sky? It's my understanding that small ice crystals in the upper atmosphere are responsible for the large multi-colored rings that sometimes appear around the Moon at night. I feel that this doesn't count.

Please Do the FTIR setup it might be easier light path. Since it is similar to Raman setup.

This is an awesome video. Yep, a phase modulated reference hologram grating is great for using finite conjugate optics to create classifier gratings for real-time object recognition.

The pinhole reminds me of the double-slit experiment. Perhaps the laser interferes with itself, such that it produces the gaussian result

I had a lab very similar to this but a few weeks ago if only this had come out then :(

i like when the lens disappeared 5:32

Modify it to reflect the object such as a picture that you can compare the know Fourier transform from file and the tested object for noise and distortion result that if both are equal, the display would be consistent. If it's a fake, noise and distortion would be seen.

Make a loud sound near the beam about 3/4 of the way down = optical microphone. aka the optophone.

I'm kind of a n00b, but couldn't phase information be inferred from the interference patterns? Lasers are pretty consistent. Which reminds me: you should do something cool/different with holography! I wish I had your workshop. I would never leave.

Have you considered doing any hologram work?

FT-IR spectrometer for organic chemistry next???

Hmm, could we use photographic negatives for this? Instead of letters, numbers or other simple things.

Im a lay person but curious nevertheless, where is the fourier data on a holographic film, or in other words how is the information non local, meaning any segment of the film contains all information regardless how small. Even if we cut the film any peace regardless how small will contain all information. Can we extrapolate the data and do we know how this actually works or is it a known unknown? I do know fourier mathematics are involved in holograms and Im curious to know if there is a clear way of presenting this to a lay person like myself as recorded holographic data seem rather counterintuitive. Meaning if you remove a peace of a holographic film one would think information would be lost. I do know that the image is recorded using 2 beams, a reference beam and an interference beam. How this actually works, Im clueless.

would it work with a photo negative?

Is that the basic principle for x-ray crystallography

follow up video!

0:45 for squre you see a each odd harmonic, not every harmonic.