Y Combinator Startup Podcast - #76 - Carola Schönlieb

Episode Date: May 9, 2018

Carola Schönlieb is an applied mathematician at the University of Cambridge. She’s also a Turing Fellow at the Alan Turing Institute and the head of the Image Analysis group at Cambridge’s Department of Applied Mathematics and Theoretical Physics.In this episode we cover mathematical approaches to image processing.The YC podcast is hosted by Craig Cannon.

Hey, how's it going? This is Craig Cannon, and you're listening to Y Combinators podcast. Today's episode is with Karola Shonlebe. Karola is an applied mathematician at the University of Cambridge. She's also a Turing Fellow at the Allen Turing Institute and the head of the Image Analysis Group at Cambridge's Department of Applied Mathematics and Theoretical Physics. In this episode, we cover mathematical approaches to image processing. All right, here we go. We ought to start with a little bit of your background.

So what did you start researching and then what are you researching now? Okay. So I started out my research in mathematics in Austria, in Vienna, where I actually didn't look at image processing or imaging at all. I started out with so-called partial differential equations, which are equations of a function and its derivatives. So they can express change over time or space. and they are models for various natural phenomena in physics and biology.

Lots of things are explained via these differential equations. And my first paper, again, had nothing to do with image processing. It was actually on the conhealate equation, which is an equation that describes phase separation and coarsening in alloys, in metallic alloys, for instance. So when you cool them down to a certain temperature, you have a mixture of two. and if you cool them down to a certain temperature, they are starting to separate from each other and coarsen out and build these larger areas.

And so there is an equation that models this kind of phenomenon, which is the conhealate equation. And my first paper was on the stability analysis of a certain type of solutions to this conhalid equation. Stability analysis, meaning that if you perturb your initial condition a little bit, how much is your stationary solution that is when you let time evolve infinitely? Okay.

How, you know, when a stationary state is a state where the system is in no change anymore, how much do these stationary states differ from each other when you just perturb the initial condition a little bit? And this is in the context of creating alloys or building structures, allies for structures, or was there any particular purpose? Well, the purpose is a lot with these differential equations to similarly, certain phenomena. And so if you understand how stable these stationary states are, so if you are at a stationary state and then you perturb the stationary state a little bit, is it going back to

the same stationary state? Okay. Or is it going somewhere completely different? So you kind of understand how these systems, how these systems react to perturbations that are naturally occurring because we are in real life and things happen. Gotcha. Okay. Yeah.

So it's more an understanding of the physical processes. involved in, you know, mixture of alloys, for instance, or things like that. And were you at a technical university where you would be, like, focusing on all? Or this was a personal interest? Not at all. So, actually, you know, a lot of applied mathematics on the continent, which is everything else in the UK, basically here in Europe, is applied mathematics very much means that what you're doing is inspired by applications, but eventually you end up with a

mathematical problems. So it was really the driving factor was, well, we were interested in analyzing this equation and there were techniques coming up that they're kind of cool. Yeah, so it was just a kind of intellectual interest in this equation. That was the driving factor for this particular paper. But then during writing this paper, research at UCLA, researchers at UCLA, in particular the group of Andrea Bertoczi, used this same equation to do image restoration. And image restoration, meaning you have a digital image and there are parts of this image which are damaged for some reason

or where you have objects which are occluding some other object of interest that you want to get rid of the occlusion or something like this. So you have one part in the image. that you somehow want to replace by something that is suggested by the surrounding area of this region. So is this similar to like content aware fill in Photoshop? Exactly. Okay. Exactly.

Okay. But this predates the Photoshop development, I assume. It actually does. And I mean, also the content aware feel is actually very much based on some of the things that have been initiated by people like Andrea Batotsi. So, I mean, the technique is different. in what Photoshop is using, but it's still based on research in mathematics, in fact.

It's a differential equation. Maybe if you wish that it's more, is not a connealate equation, but it's a different type of differential equation that is non-local. It's taking patches in images and kind of copy and pasting them into the region that you want to replace.

Yep. But anyway, so she used the canylaid equation to do that. And that was a kind of eye-opening moment. And then I moved into image processing, still sticking to differential equations at the time. And actually looking at image restoration, so at this Photoshop content-aware fill type problem. And yeah, and that was basically my PhD. My PhD was about image restoration.

Okay. And during my postdoc, then I moved more and more into what is called inverse imaging problems, where what you are observing or what you're measuring in the first place is not an image. Like when you take a photo, you know, the digital image is an image. So. But there are certain applications like in biomedical imaging where what you're observing is not an image directly, but there's some transform of this image, like an image tomography, for instance. Okay.

Think about CT, for instance, computer tomography. What you are, what the CT, what the tomograph is measuring, are projections of your three-dimensional object, which is whatever you have in your body. And from that you want to reconstruct the object. So projections, meaning in the CT sense, a particular sense, which is that you send x-rays through the body. And what you're measuring, so you're sending them through, what you're measuring at the other

end is the attenuation that they feel when they travel through the body, depending on which type of tissues they hit. And so that's what you're measuring on the other hand. And you can model that by saying what you're measuring is a line, is an integral along the line that the x-ray takes through your body, where you're integrating over the attenuation that it feels. Yep. And so from that, and that is a very old problem, it goes back to radon.

It's called the radon transform. What you're measuring is not an image, but it's the radon transform of your image, which are line integrals over the image density that you want to reconstruct, right? That consists and where the density is different in different parts of your body, and then you can see organs in your body and stuff like that. Right, and so the likelihood of there to be some amount of it missing that you need to fill or recreate or do. D noise is much higher than an image.

Yeah. That's obvious. That's quite obvious because, well, first of all, we are in a finite dimensional world. So, you know, you don't have all possible infinitely many line integrals of your body measured. Yeah. And then it's not even, you know, it's not even, that would be still okay. If you're measuring as many line integrals as you're corresponding to the resolution of the image

that you then want to compute from these line integrals. But then very often it's not like that because you don't want to, you want a very high resolution image because you want to look at all the details in the body. Right. But you don't want to measure so many line integrals because you don't want to radiate the patient so much. You don't want to send tons of x-rays through the patient. So you have a lack of data. You don't have as much data as you want.

the you know for a high resolution image to reconstruct and then there is noise because these are measurements right and there is always noise and measurements and so were you doing denoising work as well at the same time it's it's uh it's it's integrated in the reconstruction approach so in in the in the mathematical algorithm that reconstructs an image or you know the three dimensional inside of your body from these line measurements. There is the denoising

is integrated into this reconstruction step coming from these line integrals reconstructing a 3E object. Okay, and so what I know about denoising mostly through audio like a 4 transform and that kind of thing. So how are you doing it with an image?

How are you denoising in the algorithm? So with images it depends. it depends what you think is important in an image. That will determine how you're going to denoise it, let's say. A very successful assumption that has been made for designing image denoising approaches is and has been and still is, that the most important information that visually guides you of what this image is showing you,

but also that helps you, if you later want to quantify something in the image, are the edges in the image. This is the most important thing. Where are boundaries between different objects? Okay. When you think about it, what really makes an impression on you of what this image shows are colors, you know, and the boundary between these colors. Where are the colors changing?

And these are the edges in the image. Interesting. And to preserve those and not make them blurry, blurred, blurred out is something that a lot of research in image denoising has gone into. So image denoising methods which can preserve edges in an image. And so the Fourier, you know, Fourier type techniques are good. They can smoothen out the noise by taking away the high. frequencies. Yeah.

But they will take away the high frequencies everywhere, which means they will also take away the high frequencies that correspond to edges where the image function is changing rapidly. Yeah. So you're looking for the delta. This is a very high frequency component of your image. But this is a component you would like to keep. Yeah. And so you want to differentiate between the high frequency components in the image, which are

just noise and the high frequency components which correspond to these very characteristic features that you want to keep. And so, you know, there are various techniques, but one very successful one is total variation regularization, for instance, which is a technique that has been used a lot by people in image denoising to, you know, that models this assumption that you have sharp discontinuities. Medium filtering is a maybe simpler thing to understand or that people might have heard about, which is not exactly total variation denoising, but it's related. So median filtering instead of Gaussian filtering, maybe where Gaussian filtering corresponds to your Fourier taking away to high frequencies type of that. Oh, okay, got you. You know, it's so funny when I was doing Photoshop of the Onion, we were always actually interested in blurring edges because one of the most obvious things to spot a Photoshop is a sharp edge and a soft edge in the same photo. So for instance, like if I were to cut you out and then put you in front of the White House, if the photo has a slight blur so like,

The depth of field in the photo is, like, say, like a 1.4 aperture, which creates a very, very, like, shallow depth of field. So there's a lot of blur. Yeah. But if you're crispy, someone can immediately spot that you were dropped into the photo. So it was all about blurring the edges to trick someone into thinking that it was in the same photo. Yeah, yeah. Okay. So in your context, these algorithms that will handle the edge sharpness, are they hand-coded or are you using machine learning to create them?

How does that work? So they are classically hand-coded. And this is maybe something that is now, you know, more and more being replaced by other things where image denoising, nowadays I think the best image denoising approaches are actually coming from deep neural networks. Okay. So, you know, these handcrafted methods get more and more beaten in terms of performance by some of these. neural network approaches. They get beaten in certain scenarios, though. They get beaten on the type of examples they have seen already or similar type of

images that they have seen already, right? If you present them with something completely different, right, if you only train them on photographs of animals or whatever, and then you present them with a CT image or with the CT scan, they will not be able to handle that. So that is one of the things I think we're still handcrafted models have a certain justification of existence in a sense because there is not there is still, you know, although we can do GPU programming and everything, there is still not enough computational power to train a machine to know everything, to learn everything about the world. Right. And so I think a lot, so while, you know, in certain scenarios, if you know what you want to apply your image denoising approach to. Well, it was like the ImageNet thing from like almost 10 years ago.

Exactly. Yeah. If you know that, then it's fine. And that's good. But if you want, you know, think about, for instance, one big thing in CT, let's say. Or in different types of biometric imaging, let's say, MRI. let's say MRI, the nettegrisance tomography.

The type of image that you get, the resolution, the contrast and everything very much depends on how you do the acquisition, how many, let's say, in the CT case, how many x-rays you have been shooting through the patient. But also, and that is actually

connected to what I just said, also the type of scanner you're using. Are you using a G or a Siemens or Cheshiba or whatever? They have different settings and they have different ways of going from the measurements to an image. And so, you know, if you train an algorithm, for instance, a neural network on one of these scanners, it doesn't mean that it works on images of another scanner. Really? So they're producing entirely different data.

I thought they were just like basically the same tools inside with a different logo. Well, so this is the other interesting thing. It's not entirely different, right? You might not spot also visually what the difference is. But this is one of the things that also people start, you know, more and more hopefully start to, you know, do some research and understanding this, that even small perturbances that are consistent in small differences that are consistent between the different scanners

might contribute to your algorithm then failing. I don't know if you have seen these adversarial errors where you do a little perturbation and then all of a sudden it classifies the image into something completely different. So yeah. So I think the really very exciting and for mathematicians, in particular,

the exciting opportunity that neural networks are now offering, in contrast to these handcrafted models that they can go beyond just saying, I want an algorithm that preserves edges. Right. Which is a very simplistic view of the world. But on the other hand, that there are lots of unknowns in these algorithms,

on the one hand, that mathematicians, I think, should be exploring and try to bring some of the analysis and some of the methodologies that help us to understand why these handcrafted models work because we can prove properties about the denoising abilities of these methods of how stable they are for instance to perturbations in the images we know we know how that works so we can prove things about that we have error estimates and things like this and to bring those over to neural networks, I think, is very exciting. But for that, bringing some structure into these neural networks is also important.

And that might, on the other hand, when you think about these neural networks having these 100, millions of parameters that are adapting themselves to the data, maybe in some cases it would be better to not have a million parameters, but have an intelligent, structural way of reducing the search space. And as such, bring some structure into the problem which helps you make statements about stability and things like that. And then also statements about what the algorithm is actually doing. And statements about what the algorithm is doing.

Yeah. Because, so that is another thing, right? Because since when you look at these handcrafted models, you have started with hypotheses, right? you have started with a hypothesis of edges are important in images. And then you come up with a mathematical algorithm that is exactly doing what you wanted to do, right? Or, you know, then you have to make sure that it is actually doing what you want it to do. And if it doesn't, then that code is bad. Then it's not thinking.

The code is bad or your model is bad. Maybe you have to change your model in a certain way. Okay. But you understand why things are happening. Yeah. if you have millions of parameters and then, you know, you train this algorithm to do something and then you get a parameterization that is a one million different parameters, how are you ever going to interpret that? There are ways, you know, where machine learning people are trying to interpret classification results, for instance.

You have these salient features that you can detect in the image what was important for the classification to do this or this. but it's still limited. And I think, yeah, there are lots of very, very cool opportunities. And so are you guys working on hand stitching the two together at this point? What's the status of the current research? Yeah. So there are different people are trying to do different things. So I can first tell you what I've been doing over the last couple of years.

So the last couple of years, what I've been doing is I've been, and trying to, starting with these more handcrafted models, nothing to do yet with neural networks. I started with the handcrafted models. And then for certain parts in these models, where I wasn't quite sure about, our edge is really the only thing I'm looking for. For instance, I've tried to parametrize them in a certain way.

Okay. But not with a million parameters, but maybe with 10 parameters or something like this. And then learn these parameters from actual examples that I would like my handcrafted model to spit out. And this is what we call bi-level optimization or parameter estimation. I mean, people have been doing this for a long time, but now I think the motivation comes more from, you know,

there is a certain interpretation in terms of machine learning that is kind of exciting where people are, yeah, more interested in. So this is one way. And levels of parametization vary in this context. But the good thing is you have an handcrafted model in the end that you still understand. Right. And that you can still prove things about. You still have guarantees on your solution.

You know, you have guarantees that if you, you don't have these adversarial errors that if you perturb a little bit, you get a completely different result. This is really something you don't want. Right. The other thing is, and this is more blue sky, and this actually goes a little bit against what I said before, which is we have been starting to use deep neural networks for problems in computer tomography, for instance. And there at the moment we cannot prove a lot of things, but we can see some ways of how to combine these more handcrafts. models with neural networks in the sense of what you feed them with, for instance, the prior information you feed them with, the data, maybe not just the measurements, but maybe also the information that the measurements are actually line integrals of the 3D

object that you want to reconstruct. Yep. And doing this in a kind of iterated fashion where you always go back to the fact that, actually remember neural network, these are line measurements that I'm feeding you with. remember this and then you do another sweep through a neural network. But then how does that work in the context of building out a model around, say, like, I mean, I don't even know in an MRI how many images are created or lines are monitored, but like say you have 10,000 images,

but you want to create a combination of a hand-coded algorithm and machine learning system. How do you go about tagging all that stuff? What do you mean exactly? How are you going to... So what I understand you're saying is like you're giving it more data than just like the original source material. Yes. And so how do you do that with a more like at larger scale? Ah, computationally you mean?

Yeah. Okay. So computationally, we are doing this in a sequential manner. Okay. So we're not... So you can do it in different ways, but in a sequential manner means that you're not feeding it to 10,000 images at the same time. but you're doing it bit by bit and you're adapting your objective towards this. Another thing about computational performance is also, of course, that the optimization

that is underlying, but this is not just the problem that we have. This is a problem that neural networks have in general is that you do not necessarily need to solve your optimization problem, your training exactly, and maybe sometimes, or most of the time you actually don't want it, want to save it exactly, because you're only have a finite amount of training examples. And so when you think about what these neural networks are doing, they are trying to minimize a loss over the training examples that you have. But this loss is only an approximation of many, many, many, many more images that you want

your neural network to work for. And so very often you do not want to solve that exactly. You don't want to minimize your loss exactly for this training set. Okay. And so there are different types of optimization methods that people are using, but the main thing in machine learning is stochastic optimization. So you don't minimize, you know, exactly for all the variables that you have, but you randomly pick a certain amount in every sweep through the network that you're optimizing for,

and then you randomly change which ones you're optimizing the next sweep and so on. And just so I understand, minimizing loss, why don't you want to do that? So what you're minimizing, so the loss, let's say, could be the least squares error. Yeah. Let's go back to denoising. Let's say you want to train your neural network to optimally de-noise images by saying, for this training set where I have both noisy and clean images, I want that if I sum over the difference between the de-noised image, so you feed your neural network with a noisy image,

it gives you a de-noised image. You want that this de-noised image is closest in a least square sense to the clean image that you know in this case, because you have a training set. You have a label, you have a true label for this noisy image. image, which the label in this case is your ground truth image. Okay. And you want your denoising method, which is this neural network, to produce denoiced images

such that all of them are in the least square sense closest to the original label, to the ground label, which is the clean image. And you want that to work over all the images in the training set. Got you. Okay. Okay, so, but let's say you have 10,000 of these images that you both know the clean and the noisy image. If you would perfectly fit to this training set, if you would perfectly minimize this loss function, you could think, and again, you know, people are not really understanding this. And I also don't really understand this.

But conceptually, the idea is what you actually want to minimize is not the loss just over the training set, but is the loss over an infinite. amount of images, which you then want to denoise. Gotcha. Okay. But you don't have all these infinite amount of images. So why would you want to very accurately minimize the loss over this finite amount of images? Maybe you don't. Maybe you only approximately want such that you still have freedom.

Right. Such that it could be optimal also for more images that you don't have. So in other words, you could train it on the wrong thing. And it could only work for, you know, like denoising photos of apple trees. Exactly. And then you're in the same place that you were in the beginning. Yeah, exactly. With a hand-coded.

Okay. Gotcha. So the idea is if you only do it approximately, you might be able to generalize it more. But all of this, really, I mean, there are some attempts to understand this, but all of this is not really, so I'm hand-waving here because I can't really say anything mathematically about that. But have you pushed your research into practical applications at this point? Like are you working with, you know, companies or student groups or anyone else? So my main collaborations are actually with people in academia, but from other disciplines.

So we have been collaborating a lot in recent years with people in the hospital and the university hospital in Cambridge. So with clinicians and medical physicists. Different types of applications. You know, one of the things I said before is that I got more and more interested in these. problems where you don't measure an image directly but only indirectly via these x-rays, for instance. So developing algorithms which can get the most out of very limited amount of data, the most out in terms of very high-resolution images, is something we have been collaborating a lot with people in magnetic resonance tomography, in particular in the Edinburgh's Hospital,

which is the local Cambridge Hospital here, but also with people in chemical engineering, where one of the driving factors for people in chemical engineering is, for instance, there is a group here, which is the Magnetic Resonance Research Center, where they look in particular at processes which are dynamic. So they have these tubes filled with water,

and then they pump certain things through, and they want to understand what the dynamics of this process are. So now if you think about not, just having a static 3D object, but having something that changes over time as well. And now thinking back about how many x-rays, not in magnetic resonance tomography, these are not x-rays, but just going back to an example of what we had before. So not sending through so many x-rays means you don't have a lot of data to reconstruct. Right. Which now, if you want to track something dynamically, also means you're not measuring a lot

per time step. If you want to have a very high resolution over time, it means per time stamp, you can't acquire as much data as if you would have, you know, if you just have one second for reconstructing your organ inside the body at this particular timestamp,

and then the organ is moving again and you need to go to the next time stamp and so on. You have less data for reconstructing each time stamp, as if you would have, have a static object and you would have 10 seconds to acquire this instead of one second. You can measure much more, right? Right.

And then you reconstruct just one image. But now we have maybe one to reconstruct not just one image in 10 seconds, but 10 images because we want to see something evolving over time. So here also the challenges are along these lines of getting high resolution out of limited data. Another thing which is

not connected to indirect measurement so much than then these applications in magnetic resonance tomography is that we have collaborations with people in plant sciences, for instance. So they are interested in monitoring forest health or forest constituencies, let's say, from airborne imaging data. So they fly, mostly in my collaboration, they fly. So not so much satellite, but more flying. They fly over forest regions.

and then they acquire different types of imaging data. They acquire just photographs, aerial photographs, hyperspectral imaging data or multi-spectoral imaging data, which means you do not only have RGB, but you have a broader range, you cover a broader range over the light spectrum, so also the invisible light. So you don't have just three channels,

but you have 200 channels or something like this in your image. Yeah. And hyperspectual imaging. is interesting. So the spectral component that you get from these measurements gives you an idea of what the material properties are of these trees. So it tells you something about what... Really? Yeah. So this is... So the spectral component tells you something about the material that you are looking at. So in other words, like, the... So different materials have a different signature in the light spectrum of how they reflect light back.

They have a different signature in the light spectrum. Okay. And so the intent would be to figure out, you know, say for instance, like an invasive tree that was taking over an area, they could figure that out by just by flying right over it. Gotcha. Okay. And then the other thing, so this is one and then, or two aerial photographs and hyper spectral imaging. And then the third thing that they are often acquiring are lighter measurements. Yep.

where you do not just get kind of a planar picture of the trees, but you actually get a 3D model of the trees. So this is also nice. I was just watching a documentary about that, about searching for Mayan ruins with Lidar, flying over the Yucatan Peninsula or something, essentially like saying like, hey, we could take 20 years for an archaeologist to like dig around in the dirt

or we could just fly over it and look for the hard stuff and see what happens. Yeah. Yeah. Very interesting. And are people also looking to this in the context of, you know, for instance, like denoising, like camera footage from anything, you know, like security on one hand. Yeah, I haven't done so much work in that myself, but there are, of course, you know, the, I mean, CCTV cameras are everywhere. Yeah. I mean, it's like, it's kind of the terrifying output of figuring out this research, right, like being tracked everywhere, like in the UK in particular.

Like, I imagine people are looking to do this, right? you know um it's quite funny because uh when you think about uh these um crime tv shows c s i whatever miami or whatever they are always these these enhance funny things right so you don't you have a very pixelated image and uh you press a magic button and then you can zoom in and you can see everything so um when you think it so this is really ridiculous, yeah? Of course, you can't do that, but you can't do it now. Maybe, you know, if you have all these machine learning methods, which have learned to look at just pixels and then know what, what is a very probable match in terms of high resolution, maybe at some

point you can do it. But then you don't know if you're right or wrong, right? Just by chance, I was reading a New York article from, I think, 2010 about this guy in Montreal, allegedly finding 500-year-old fingerprints using different kinds of spectral photography. Okay, cool. I haven't heard about that. But tell me more about it. So I don't want to give away the whole thing. And then there was an ensuing lawsuit, actually, from him to the New Yorker saying they, like,

it was libel. But basically what happens is like he was accused of faking these fingerprints that may or may not have existed. Oh, man. Okay. Yeah. And like copying them from a real one, duplicating them onto the back, using like proprietary methods to find them out. But you are interested in doing it whether or whether or not it's legit. Like you want to work with art.

I hope so. I mean, I'm going to tell people that it's fake. Yes. Okay. Yeah, that's the whole idea. Yeah. Is it like, yeah, what direction are you going with art? So it kind of, in Cambridge, it's, well, okay, let me say a bit more.

So when I, again, during my PhD, in Vienna, there was a collaboration that we had with physical conservators, so with conservatives who are, we're looking at particular wall frescoes, at frescoes in an, in a, in a, an old apartment in the city center of Vienna, which are called the Nighthut frescoes. I'm not going more into detail, but they were in the process of restoring these frescoes. And so that was my first hand experience there. And there, the idea was that, you know,

it takes them a long time to physically restore these wall paintings. And once you have restored it, there is no way back, right? you need to decide what to do. Yeah. Because then it sticks. And so our idea was to help them by creating a virtual template of how the restoration could look if they do this or this or this. Right.

Yeah. So because the important part is a fresco is actually part of the wall chemically. It's not paint. Exactly. Yeah. Exactly. But even with paintings, you know, if you do something, if you do, if you manually, really, you know, physically restore them. Yeah.

You've done it. I mean, you can still maybe, you know, try to do that. But, I mean, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, you're, uh, uh, so, uh, so, um, so, so coming here to came here to, uh, uh, I got, uh, uh, uh, uh, uh, uh, uh, which is a, which is a, which is a museum here in Cambridge. And they're interested in illuminated manuscripts. So I met a very good colleague

of mine who is the keeper of manuscripts in the Fitzwilliam got interested in this idea of virtual restoration because illuminated manuscripts are so fragile that the culture is you never physically restore them. You never physically restore them.

They, you know, if they get damaged or altered over time, you leave it. Wow. Okay. You leave them like this. And so there, the idea was, couldn't we create a virtual restoration and, you know, kind of exhibit the original manuscript and the virtual restoration next to each other? And so last year, there was an exhibition in the Fitzhilim Museum, which was called color. And in this exhibition, we had one piece, which was in a,

a page of an illuminated manuscript, which had been altered over time, actually manually overpainted. Okay. And what we did was that we exhibited the manuscript and next to it, the virtual restoration, where we took off the overpaint. And, yeah, and that has led to other things. But, I mean, this is, so this is kind of the idea that you don't physically change something, but you virtually do it, which is, you know, nothing damaged. You just virtually create a digital copy of this manuscript and you play around with it.

So you're not only going like back in time to see maybe like restoring it to its original, you know, vitality, like its original color, but you're actually like going deeper into the layers. Like this has been painted over. Yeah. So you can go further in with imaging. And then you kind of like apply everything you might already. Wow. That's super cool.

Yeah. what would you point them to? Where should they get started? Depends what their background is. Okay. Yeah, yeah. So they have like, you know, they have a CS degree. They're interested in imaging. So they're like technical, but they haven't done anything in particular, like in this field. Okay.

So what I would advise is to look. So I think in particular when you think about the US, I think some of the cool things that came out of image processing in the last couple of years were from UCLA. So if you look at some of the applied math faculty there and some of the online lecture material or YouTube videos of some of their talks, I think that would be a good source to look at. So I mean, very classical names are Stan Osher, Andrea Bette.

Totzea mentioned, Malik, Perona, Stefano Soato, there are lots of people there is, now the name is case me, I can tell you a bit of a few more things afterwards, but I think just to look for mathematical approaches to image processing, I think it would be the first thing I would do. there are very good introductory books to look at that explain a bit of the basics. Great. But, yeah, I would first start reading a little bit in these more general foundational books. And then I think just starting from that, you immediately come, go to the more modern

recent years research. I think that would be a good way to start. I can catch up to you, maybe. Yeah, maybe. Or apply here. Awesome. Well, thank you so much. Thanks for making time.

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