The Bold Blueprint Podcast
The Bold Blueprint Avideh Zakhor Successful people aren’t defined
Hey Amazon Prime members, why pay more for groceries when you can save big on thousands of items at Amazon Fresh? Shop Prime exclusive deals and save up to 50% on weekly grocery favorites. Plus save 10% on Amazon brands like our new brand Amazon Saver, 365 by Whole Foods Market, Aplenty and more. Come back for new deals rotating every week. Don't miss out on savings. Shop Prime exclusive deals at Amazon Fresh. Select varieties. A message paid for by Veterans for all voters. Listen to this message from Ted Delacath, former army infantryman and Ranger qualified platoon leader active in the army reserves. When I enlist in the army, I swore an oath to this country, not any political party. That's why I'm interested in citizens ballot measures around the country to reduce the power of political parties. Colorado votes on one too. Right now, election rules allow political insiders to hand pick party nominees. It's the reason we're usually stuck voting for the lesser of two evils. Colorado's plan creates an open primary where all candidates appear on one primary ballot. Every voter has the freedom to vote for any candidate, no matter which party. The Colorado plan advances four candidates to the general election, not two. That means more choices for voters in the primary and general election. Get the facts. Elections belong to the voters, not political parties. Paid for by Veterans for all voters, Anthony Haas registered agent. The use of military rank and job titles does not imply endorsement by the Department of the Army or the Department of Defense of this ballot measure. Okay, just so that not that you've all done the actual homework, just a quick show of hands, how many people took more than 10 hours to do this last homework? Almost none. Five and ten? Less than five? Everyone? Okay, good. That was an easy homework. Okay. So as I mentioned last time for Wednesday's lecture, most likely, I will come up with a list of projects. It was nearly impossible to do that today. I will do that on Wednesday of next week. This is the last homework where you do problems off of Jay Lim's book for the semester. So if you're kind of tired of that, then that's the end of that. There's at least two more lab assignments that you'll do on image enhancement and image restoration. It won't be off of the book, it'll be just design, kind of custom design lab assignment from our own description. So what I'm going to talk about today is wrap up a little bit of the MacLillan or transformation filter design technique that we started off on Wednesday. And then start talking about some basics of image processing in particular. The primary additive and subtractive color system, the very simple models of the human eye, a little bit of the anatomy of the human eye, how it works. And talk a little bit about Weber's law and talk a little bit about the electromagnetic spectrum in general. So that's kind of the outline of what we're doing today. So before I start continuing from last time's lecture, I'm going to just briefly tell you where we were. So remember, we have a four-step transformation filter design technique where you start with translating the specs from 2D to 1D, you design your one-dimensional filter using some kind of a remains exchange algorithm, etc. And then you then combine the, first of all you pick some random transformation filter like T of N1 and N2, then you translate the specs using that T of N1 and N2, you translate your specs from 2D to 1D, you design your 1D filter and then you combine the 1D filter with the transformation filter T of N1 and N2 at the end get your final big two-dimensional filter H of N1 and N2. That was the basic flow and what we were questioning was step one, which was pick a transformation filter. And one way to do that is to just say, "Okay, well I'm going to pick whatever my clella gave me and that's certainly one way to do it." But the isocontours of that transformation filter, the closer they are to the final filter that you have the specs for, final two-dimensional filter that you're trying to build, the better it is, the lower the number of taps will be, etc. So the thing that we talked about last time was what if we want to design, we don't want to stick to my clella transformation as our T filter, as our transformation filter, or if we want to design that filter ourselves. And these were the four steps, if you can put the camera on here, and rotate and zoom, and thank you. Great. Okay, so these were the four steps straight from last time that we used in order to design this filter T of N1 and N2, and the steps where you assume some sort of a shape and size, you import some constraint, and then you just kind of tentatively choose Omega P and Omega S for your one-dimensional filter. Based on that tentative assumption, design T such that these conditions are satisfied, and we decided to do that, we would kind of come up with an error function, E sub P and E sub S, and then solve the least square problems, which involve solving linear equations. And then once we have a tentative design for T, we massage it a little bit, we refine or tune it up a little bit, we change T to T prime, such that these conditions here are satisfied, such that between for Omega, between 0 and Omega P in one dimension, you want those range of frequencies to get mapped into the region inside C sub P, the passband contour for our final filter, and we also want the range of Omega's between Omega S and P to get mapped outside of C sub S, which is our stopband contour. So, we kind of went ahead with an example, this is the example of elliptic filter, and can you zoom out just a tiny bit, so that, thank you, the whole picture. Okay, so this was our elliptic filter, then we imposed a bunch of constraints, this is our T filter, we imposed a bunch of constraints, and I at the end, after we imposed constraint, we came up with three unknowns, A prime, C prime, D prime, and then we came up with this error criteria, which says, along the passband contour C sub P, we wanted the error between cosine Omega P and T to be small, because we want cosine Omega P and T of, for all values of Omega 1 and Omega 2 in the passband contour, we want these two quantities to be almost equal to each other. So, we minimized this error, and the same thing for stopping, we minimized this error, take the derivative, and in doing that, we'd realize that we end up having linear system of equations, with six linear system equations and three unknowns, you apply least squares, and boom, you get your unknowns, A prime, C prime, B prime, E prime, B prime, and that gives us the T. Now, where we left things last time, was that that T wasn't necessarily a good, a good filter, because, if you look at figure, I think it was 4.20. All right, if you, if you zoom in here, so this shows the contours, of course, on omega equals 2 omega 1, omega 2, and for, for all the, for all the regions in which, so these are the isocontours, for, when magnitude of T of omega 1, omega 2 is less than 1, and these costs, this, whatever you want to call them, shaded regions are where magnitude of T of omega 1 and omega 2 is larger than 1. So the fundamental, so these are problematic, the reason they're problematic is because cosine omega, for real values of omega, is always between plus 1 and minus 1, and the transformation you have is cosine omega equals T of omega 1, omega 2, and the way things work, if you remember from last time, is really this picture, right? You go to your H of omega, you pick a particular omega, let's say omega naught prime, for all the contour points corresponding to cosine omega naught prime equals T of omega 1, omega 2, which is like this one, the, the height of this whole contour is equal to the height of H of omega at omega naught prime. So if in the process of doing that transformation, you've come up with a T that's larger than 1, it would not correspond to any real values of omega between 0 and pi, okay? In fact, what, what omega would it correspond to? If cosine of omega larger than 1, what, what values of omega would correspond to cosine omega larger than 1? Exactly, that's what they teach you in complex variable theory, imagine values of omega, and, and, and, well, we never design H of omega for imagine values of omega, we, in fact, we don't give a hoot about the imagine values of omega when we design H of omega, and so, so all, essentially, all hell breaks loose because then, yeah, for, then, then, there was values of omega 1 and omega 2 for these regions are going to correspond to some random value of H of omega for omega being imaginary. That's not such a good thing, and so we want to avoid, so, so, why did this happen? Well, it happened because T of omega 1 and omega 2, its magnitude exceeded 1, and you see, you're, you're gonna ask, well, did we do anything specifically to prevent that from happening? No, well, then it's not surprising that it happened. If you don't, if you don't actively constrain or enforce some function being in a specific range, it can do whatever it wants. So this guy T of omega 1 and omega 2, go back to this picture, did whatever it wanted, it exceeded 1. Hey Amazon Prime members, why pay more for groceries when you can save big on thousands of items at Amazon Fresh? Shop Prime exclusive deals and save up to 50% on weekly grocery favorites. Plus, save 10% on Amazon brands, like our new brand Amazon Saver, 365 by Whole Foods Market, Aplenty, and more. Come back for new deals rotating every week. Don't miss out on savings. Shop Prime exclusive deals at Amazon Fresh. Select varieties. We wear our work day by day, stitch by stitch. At Dickies, we believe work is what we're made of. So, whether you're gearing up for a new project, or looking to add some tried and true workware to your collection, remember that Dickies has been standing the test of time for a reason. Their workware isn't just about looking good, it's about performing under pressure and lasting through the toughest jobs. Head over to Dickies.com and use the promo code Workware20 at checkout to save 20% on your purchase. It's the perfect time to experience the quality and reliability that has made Dickies a trusted name for over a century. If you plot it here, you can see that in Part B, what you have is the frequency response of the filter designed using this. If you don't do anything more, this is the frequency response that you end up getting. In these four corners, things are all messed up. And then, believe in these four corners are high frequencies. The pi comma pi minus pi pi minus pi minus pi comma minus pi. Does this look like a low-pass filter with elliptic contours? No, not at all. So, all hell did actually break, broke loose. And now, the question is how do we fix that? You need not worry too much, there's a really simple fix to it. And that's the step four of our design. Right? If you go back to step four in this design that we enumerated here, was to modify this t to t prime in order for these two conditions to hold true. In other words, for the range omega to omega p, you wanted to get mapped inside c sub p. And the problem here is that omega s to omega pi in the omega side didn't get mapped outside c sub s. In other words, there are some omega ones and omega twos here that did not get mapped into the outside of c sub s. Sorry, there's no corresponding value of omega in one dimension that got mapped to these four values. And so, this condition here, this condition didn't quite get satisfied. So, if we go back and tune t, we're going to be all okay. And so, how would we go about doing that? Okay. So, let me, that's the beginning of today's lecture. That's why it's, it's always problematic when you don't finish a topic in a whole nugget in one lecture to pick it up the next lecture. You have to spend like 10 minutes bringing back the context in everybody's mind. But let's just, let's just do that. So, so problems occur. So, this is 2D filter design using transformation. That's the topic of today's lecture. And basically, summary of all of the things that I just said was that, was that if magnitude of t of omega one and omega two is larger than one for some omega ones and omega twos in between minus pi to plus pi times minus pi to plus pi. Then, resulting h of omega one and omega two is, I would just call it ill-behaved. That means it can do anything it wants. It's uncontrolled, unspecified, or ill-behaved for those, for those omega ones and omega twos, i.e. for the ones that, that, that this guy t got to be larger than one. Okay. So, so this is problematic. We can't live with this. So, instead, how do we fix this problem? We go back to step 1.4, and we make sure that the regions inside C sub p, i.e. the passband in the two-dimensional filter correspond to omega in the one-dimensional case in between zero and omega p in the 1D filter. And zero, I'm not going to write all of this. Well, let me, let me expand on that just, just a little bit more. Well, can you roll up please? So, what does, what do I mean? I want h of omega one and omega two, which is h of omega evaluated at cosine omega equals t. Because this guy cosine omega is smaller than one, we want to make sure that t of omega one and omega two magnitude is also smaller than one. And, and you want this to be true for C sub p, but also ditto for C sub s. I don't want to write all of it. Okay. And so, how do we, how do we get around this problem? Here's my proposal to fix this problem. Renormalize t such that it is in between range plus one and minus one. C, renormalize t of omega one and omega two, such that it is in the range magnitude of t smaller than one. In other words, try to do that with minimal perturbation of t. So, minimally perturbed t to get a new transformation, which we call t prime. And, and this guy t prime is going to be t prime of omega one and omega two. Here's my proposal of minimally perturbing t is some constant k one times t of omega one and omega two plus some other constant k two. And, and, and why, intuitively, why do you think that minimally perturbs things? It, it preserves the shape of the contours, right? If, if, if t of omega one and omega two had nine elliptical kind of a contour, when, when you do this transformation, the contours of this guy are kind of also, you can convince yourself, are kind of elliptical. And, and now you could say, well, how do I find k one and k two? Well, our, our approach is to say find k one and k two such that magnitude of t prime of omega one and omega two is smaller than one for all omega one and omega two in the range of interest, which is minus pi to pi in minus pi to plus pi times minus pi to plus pi. Okay, so how do we do that? Well, that's not probably difficult either. What you do is you, you find, you call the maximum of t one, t of omega one and omega two over all omega ones and omega twos in this range of minus pi to plus pi. Let's just call that t max, call the minimum of t of omega one and omega two, or omega one and omega two, call this t min. And, essentially what you want to do is you want to find k one and k two such that maximum of t prime is one and minimum of t prime is, is minus one. So that's what we mean by renormalizing. Just come up with a linear transformation of t, find the old maximum and minimums and then find k one and k two such that the new maximum and minimums are, this time is strictly speaking between plus one and minus one. We don't want, we don't want magnitude of t to exceed one. And so what does this, what does this imply? Well, if you can roll up a little bit, what it would imply is that you want t prime of omega one and omega two, which is given by k one times t of omega one and omega two plus k two, you want it to be equal to plus one at k one t max plus k two. That's what this means. And you want it to be minus one at k one t min plus k two, right? You want to find k one and k two such that when it's at its maximum, it becomes plus one and it's at minimum, it becomes minus one. And using these two things, that's a linear system of equations. You can find k one and k two. And now this, it gives us a new transformation t prime. And using this analytically, k one ends up being two over t max minus t min. And k two ends up being t max plus t min over t max minus t min. Hey Amazon Prime members, why pay more for groceries when you can save big on thousands of items at Amazon Fresh? Shop Prime exclusive deals and save up to 50% on weekly grocery favorites. Plus save 10% on Amazon brands, like our new brand Amazon Saver, 365 by Whole Foods Market, a plenty and more. Come back for new deals rotating every week. Don't miss out on savings. Shop Prime exclusive deals at Amazon Fresh. Select varieties. We wear our work day by day, stitch by stitch. At Dickies, we believe work is what we're made of. So whether you're gearing up for a new project or looking to add some tried and true work where to your collection, remember that Dickies has been standing the test of time for a reason. The work where isn't just about looking good. It's about performing under pressure and lasting through the toughest jobs. Head over to Dickies.com and use the promo code work where 20 at checkout to save 20% on your purchase. It's the perfect time to experience the quality and reliability that has made Dickies a trusted name for over a century. So that's our new transformation. And now that we have a new transformation, going back to the flow diagram that we have, step one, two, three, four from two lectures ago, now we have to redo everything again. Now we have a new transformation, now we have to come up with a set of specific, now we have to translate the specs in 2D to the specs in 1D. Okay. So we have to design a new H of Omega. We have to design a new one-dimensional filter to marry it with this transformation in order to design the big filter. And as it turns out, this new transformation results in Omega S and Omega P in the one-dimensional case. That's just the linear transformation of the Omega S and Omega P for the tentative filter that we had just designed. We had that T, T of N1 and N2, and that resulted in some Omega S and Omega P, or we picked some Omega and S, Omega P associated with it. So now that we minimally perturbed T to get T prime, what you can show is that we just need to modify the old Omega P and Omega S that we had for T as follows. So we had chosen Omega S to be 0.5 pi, Omega, sorry, Omega P to be 0.5, and Omega S to be 0.6 pi. This was for our old transformation. We had T of Omega 1, Omega 2 is approximately cosine Omega P, and then for all Omega ones and Omega twos in CP. And we also had T of Omega 1 and Omega 2 for all Omega 1 and Omega 2 in CS to be approximately cosine Omega S. Not that we have a new T, we have a new thing called T prime, so T has been changed to T prime, and in particular, T prime is just K1 times T plus K2. Well, the new Omega P and Omega S is going to be cosine Omega P prime is going to be approximately equal to T prime of Omega 1 and Omega 2 for Omega 1, Omega 2 in CP, which means that cosine Omega P prime, because this thing is already true, is just K1 times cosine Omega P plus K2. And cosine Omega S prime is equal to K1 cosine Omega S plus K2. Because already we had ensured that we had picked some Omega P and Omega S out of the hat, it was 0.5 and 0.6, and we knew it's not going to be our final choice. But using that 0.5 and 0.6, we designed the T. So that T that we designed is contour on C sub P, approximate cosine Omega P. And it's contour on C sub S, approximate Omega S. So not that we change T to T prime using this relationship, the contours of the new transformation, corresponding cosine Omega P prime, the Omega P prime is just the old Omega P modified by this to get the new one. Is that clear to everyone? Okay, so now we have these new values of Omega P prime and Omega S prime. And these are, so plug in Omega new Omega P prime, Omega S prime, and the Delta P and Delta S. You notice Delta P and Delta S hardly ever cause any trouble. They get translated straight from 2D to 1D, from 1D to 1D, they remain the same, but plug all of this thing into the remains exchange algorithm to design an optimal 1D filter, a new optimal 1D filter, we call that H prime. And now combine H prime with T prime and out comes the new H of Omega 1 and Omega 2. And this filter is the final thing that we end up getting. So it's kind of an iterative, it's a two-step kind of approach. You first make a guess with some random Omega S, Omega P's, that results in the T, that T is not exactly what you wanted, modify the T again, and then give you, that gives you a new Omega S and an Omega P, design the 1D filter again, and this time it is going to work. And so just so that you can see this in action, that's figure 421 shows you the details of this, if you can zoom in please. So these are the ISO contours and now everything is between plus 1 and minus 1. So now Omega equals 0.9 pi corresponds to here, whereas before Omega equals 0.9 pi would of corresponded here, and this would pretty much be corresponding to cosine equals pi. Here now all the way, 0.9 pi is here and pi is right there. And these are elliptical contours, and my ellipses were much taller and narrower than this because I didn't draw them to the right proportion. And here's the resulting 2-dimensional filter. Yeah, okay. So this kind of wraps up the discussion on 2D filter design using transformation. This is a good point for me to stop and see if there's any questions. Is this methodology clear? Any confusion? Any questions? Yeah, please. Why do you have to go to 1-1 to 1-4 and then go back game? Why can't you just go get T and then normalize it and then everybody who's normalized when we get cos? Is that what we should do? Oh, I see. In other words, you may seem like we have to go twice, is there? Oh, I see. Can we do a shortcut? Like 1-1 or determine T and then do the normalization numerator? That's essentially. Oh, I see. You said to complete the 1D filter twice, basically. Oh, I see. The only reason I says that is for illustrative purposes, so you know why I'd renormalized it. Because if you design, I see what you mean. You said you don't design all the way the big H of omega 1, omega 2 because, yeah, in real life you only do it once, but I did it twice just so that you see why we have to do it twice. I mean, why we need to renormalize? In real life, you get T, you normalize it and then you design the filter. Any other questions? Okay, so next, what I want to do is talk a little bit about image processing. Start the discussion on image processing. And in particular, what I'm going to talk about today is just basics of light and image processing systems and various other things. So just to give you a layer of the land kind of picture as to what are the topics that we'll cover, we'll start with some basic stuff, like today's lecture, essentially, and perhaps a little bit next Wednesday. Like what's the electric magnetic lights and how does the eye work and additive color system, subtractive color system, that kind of stuff. Then we move on to image enhancement. And image enhancement, basically, is the notion of you have an image and you want to change it in such a way that something in it is enhanced so you can see better. It could mean that, I don't know, it could even mean that you have, you want to just attract the viewer's eyes to enhance it. Let's say you have a, there's the famous painting, for example, where there's a watermelon that, and a pineapple, and the pineapple is painted red the same color as watermelon with textures, the watermelon, and the watermelon is painted. Hey Amazon Prime members, why pay more for groceries when you can save big on thousands of items at Amazon Fresh? Shop prime exclusive deals and save up to 50% on weekly grocery favorites. Plus, save 10% on Amazon brands, like our new brand Amazon Saver, 365 by Whole Foods Market, Aplenty, and more. Come back for new deals rotating every week. Don't miss out on savings. Shop prime exclusive deals at Amazon Fresh. Select varieties. We wear our work, day by day, stitch by stitch. At Dickies, we believe work is what we're made of. So, whether you're gearing up for a new project or looking to add some tried and true work wear to your collection, remember that Dickies has been standing the test of time for a reason. Their work wear isn't just about looking good. It's about performing under pressure and lasting through the toughest jobs. Head over to Dickies.com and use the promo code Workwear20 at checkout to save 20% on your purchase. It's the perfect time to experience the quality and reliability that has made Dickies a trusted name for over a century. The inside of it with yellow, the same color as the pineapple. That's an example of an enhancement. So, you're just enhancing things to drive a point home or something like that or make the picture more intelligible. Then, in contrast with enhancement, there's a thing called image restoration. And what restoration means is that something went wrong with the original picture. And now, you're trying to model that to undo the damage, the undo the degradation that the picture went through. So, you can see the difference between enhancement and restoration, right? And then, after we talk about restoration and enhancement, in some of these things, we talk a little bit about motion blur or the degradation due to, for example, the signal got blurred, either due to motion or due to the lens not being in focus or whatever. And by the way, the undoing the motion blur is a very hot topic. Can everybody say why? Because there's lots of images being captured by your cell phones. And you keep seeing these advertisements, you know, buy it. Three megapixel cell phone camera or five megapixel. It's good. There's a lot of pixels, but the pictures that come out of it, those are very blurred. Can anyone even say why? Because your hands doesn't stay still. When you take the picture, there's an infinitesimal amount of motion. And so, when you get out, even though the manufacturers are not lying, there was three million sensors inside the cell phone, but there's a huge amount of motion blur. And if you actually want to see what I mean by that, I'm sure all of you have already done it, but I can show you the background image of my cell phone, which is my husband and my son in Hawaii. And I can't even see the eyes of my son. And this is like a one megapixel. It's supposed to be a thousand by thousand pixels. And my husband's face is slapped with singular logos. So, I can't see him either. But yeah. How is that different than if you buy like a regular digital camera and take a picture? Like what software or what stuff does that have at the cell? There's a lot more processing and there's a lot more... There's a lot... There's electronic stabilizers, there's optical stabilizers in an electronic... In a real camera that you buy. There's a lot more hardware and software to undo these effects. The problem with this is... This is given away for free if you sign up for two years. So, right? It's zero. The cost is zero. Okay. If I sign up for one year, it's forty nine dollars. Right? The electronic camera you buy. I mean, I didn't buy one recently, but what is it? It's a couple of hundred bucks, right? It's three, four hundred. It has a lot of sophistication. This is supposed to be very light, right? This you can throw and you turn it on, it still works. You won't do that with your camera. This is a mass market consumer electronics kind of a thing. It's forty nine dollars. It's kind of the price. That is much more higher and it has a lot more sophisticated processing. Okay. So, we talk a little bit about image restoration and then we talk about image compression, the basics of it, cosine transforms and talk about wavelets when we get to compression. And then we move on to talk about video, motion estimation and challenges that video poses. And then from then on, we can talk about a variety of topics. We can talk a little bit about half toning, which is the process of printing pictures on paper. I don't know if you've read the article in New York Times yesterday. HP is getting into the HP is the biggest color printer in the world. They make a lot of their money off of that. And they're in competition, obviously, with Lexmark and other car printing. But they're introducing kiosks for printing your pictures, digital pictures, into longs drugs and other chains. Already, if you go to a lot of other chains like Walmart, you can go to Kodak kiosks and put in your digital pictures and get printouts. But these guys are coming up with a faster method. But supposedly the way Kodak does it, you need to have a store employee kind of babysit you through it. And they have to make sure enough chemicals are poured in there. And this limited number of prints you can get out of a machine. But HP is trying to do that in a much more automated way. So printing color pictures is big business. And initially, HP and Kodak and other companies were banking on the fact that we all buy the little HP printers, the color printers. What they call it? Smart something? What's the, they use a funny name to advertise for it, smart photo smart. Thank you. Yeah. Yeah, the world banking, we spent a hundred bucks on buy a printer. And that didn't quite happen. I mean, they still make a lot of money off of that. But the truth is that if you take a picture and you don't have a photo smart thing to print it at home, you just take it to longs or Walmart and you want to get a hard copy, or maybe you send it over the internet to snap fish or photo and get a hard copy that way. But for the rest of us, you just go to longs drugs and you want to get a hard copy. So that's big business. And how do you print? How is the half toning process? You have to modulate the inks and combine the colors and stuff like that. So we might talk a little bit about that. We might talk a little bit about understanding images, although that's more computer vision. And for those of you interested, that should probably take CS 280 in the fall. And whatever other topic that you guys are interested in. And if you have time at the end of the semester, and if you divide you up into groups that did projects, we want to have a presentation of the projects. Actually, that's one thing I should say. As I said, on Wednesday, I'm going to come up with a list of projects. But it would be great if you guys teamed up in groups of five or smaller or even 10 to do projects that would otherwise be difficult to do individually. Don't worry too much about your grade. And oh, what if my partner doesn't behave and doesn't perform? That's for undergraduates. I get a lot of undergraduates knocking at my door saying, oh, my CS 150 partner didn't do his job. And now I'm getting a C. I won't get into grad school. You're already in grad school. There's no more things to get into, right? This is it. Right. I mean, it's not that you won't get your PhD. If your grade becomes A to A minus or anything like that. Pardon? How? Oh, you won't. No. No. So anyway, my message to you is be bold. And I encourage you, like at the end of the class, especially actually, today we have a very good turnout. And I can't really understand why, but that's good. So at the end of the class, talk to each other and see if you can form teams that if you're interested in projects or after I declare the topic. So people who have similar interests can kind of team up. Okay. So any questions? Just as a quick show of hand, how many of you roughly have an idea what project or term paper you want to do? Wow, very good. Excellent. Okay. So let me get started here. So what we have, what I want to talk about is really light as a as an electron. So light as a electromagnetic wave. So you thought that, you know, because you're now doing image processing, you can forget about your E and M theory, but you really never can't. So what is, so you can talk about this function C of x comma y comma t comma lambda as the energy density of an electromagnetic wave. And the notation is pretty obvious. x is x and y are location or positions. T is time and lambda is the wavelength. Okay. So if you have C of lambda as a function of lambda. So most of the times we forget about x, y and t. And for this discussion, I'm just showing you C of lambda, for example. Okay. As a function of lambda. Okay. Then there's a, let's say this is a distribution of the different wavelengths within a particular light. And there's a center frequency called lambda center. Okay. The most important thing that you need to understand is that is what, what range of frequencies can you observe? Can you see things? What's the, what the range of wavelengths for which the eye can see things? Hey Amazon Prime members, why pay more for groceries when you can save big on thousands of items at Amazon Fresh? Shop Prime exclusive deals and save up to 50% on weekly grocery favorites. Plus save 10% on Amazon brands, like our new brand Amazon Saver, 365 by Whole Foods Market, Aplenty and more. Come back for new deals rotating every week. Don't miss out on savings. Shop Prime exclusive deals at Amazon Fresh. Select varieties. We wear our work day by day, stitch by stitch. At Dickies, we believe work is what we're made of. So whether you're gearing up for a new project or looking to add some tried and true work wear to your collection, remember that Dickies has been standing the test of time for a reason. Their work wear isn't just about looking good. It's about performing under pressure and lasting through the toughest jobs. Head over to Dickies.com and use the promo code WorkWear20 at checkout to save 20% on your purchase. It's the perfect time to experience the quality and reliability that has made Dickies a trusted name for over a century. That's exactly right. Okay. And I forgot. Are you guys the bioengineering crowd here? Optics. Okay. Okay. So it's approximately 350 nanometers to about 780 nanometers. I'm stretching it a little bit, right? But roughly speaking, that's the range. Okay. And beyond here is called ultraviolet. And beyond this direction is called infrared. And so this thing here that I'm plotting is essentially it has units of energy per area times time times wavelength. It's the amount of energy. And so in the metric system, energy is in joules. So it would be joules per meter squared second. And then wavelength is one over meter. So it would be meter cube. So the units of C is this. Okay. And now what I can do is I can talk about approximate physical properties of this sea guy. Okay. So if I can you roll up just a little bit. So we can we can think about physical domain in one side and perceptual domain in the other side. Okay. And here I want to emphasize very it's very approximately we can relate some properties in the physical domain to three things in the perceptual domain, namely brightness, hue, and saturation. So in the perceptual domain you talk about brightness, hue, and saturation. And roughly speaking, we want to relate the characteristics of the C function to these three quantities. So if I integrate C of lambda with respect to lambda from some from the lambda mean, which in this case is this 350 nanometers to lambda max, roughly speaking up very approximately that ends up corresponding to brightness. So brightness it's in common term or in English if you were to define is how bright the color is that you're looking at. Okay. Hue refers to approximately the color. It tells you that lambda subcenter. So if so it's approximately color, this is how bright things are. So if lambda center is approximately 700 nanometer, you get the illusion of what? What color is 700 nanometer? Red. Okay. If you're about 550 nanometers, then approximately green. And if you're about 450 nanometers, it's approximately blue. Okay. And there's actually some theories that you don't want to be looking at red stuff too much because it irritates you or something. It could be because I don't know how true that is, but it could be because of this wavelength is too high. And if I come back to this picture, so Hue is the hardest one to intuitively define is saturation. Okay. Or coma. In English, it's kind of you can think of it as how vivid or how dull the color is. Is it vivid or dull? But in terms of this thing here, approximately, if I define lambda sub w to be the approximate width of this function, saturation corresponds to lambda sub w. Okay. So if lambda sub w is narrow or small, if that c of the lambda is narrower, then it's kind of the color is more vivid. If it is broader, it's more dull. Okay. Since you guys are from vision science, do you have a better definition of it like intuitively, like from a perceptual point of view with people called saturation? I've always had a hard time kind of visually explaining the saturation portion of the thing. Yeah, but suppose you're talking to a layman who doesn't know energy densities and doesn't know width. Kind of, you know, how would you say, ah, this is highly saturated. This is a lot of coma. To me, it's a measure of the purity of the color that you're looking at. But how do you translate that kind of to a layman? That's, that's what I'm trying to. Okay. Since we're all puritans, right? Okay. No, no, we're descendants of the puritans. Okay. Yeah, sure. The other way around. The other way around. Right. All right. Okay. So let me now flash at you. The, um, two pictures. I don't want to plot these. So if you zoom in here, right, this, this, these two curves show the spectral contents of sun's radiation. So as you know, there's a, there's a lot of discussion about what ozone hole in, and, and the ozone global warning is causing the ozone hole. And that, that, that, what's, what's so bad about that? That let's ultraviolet light come right down to earth, right? And so we get the more ultraviolet, like we get, the sicker we get, and cancer, and all those bad issues. But at some point you, you, you could ask yourself, what's the distribution of the visible light that's hitting the ground? So it, this plot kind of is showing that going from 400 to 700 nanometers, the dark line here, the solid line, sorry, shows the, the spectral contents up in the earth's atmosphere. And this is on the ground at a particular point in a particular time. So you can see that there's, um, there's a little, at this point in time, anyway, there was a little ultraviolet, and the, the infrared content is kind of a little bit higher. So zooming into the spectrum, to the, to the, um, wavelength axis. So you might say, okay, well, 400 to 700 nanometers is roughly speaking the, the wavelengths of the visible light, and nanometer is 10 to the nine meters, we all know that. So there's violet, blue, green, yellow, orange, red, etc. As you move on in this, in this axis. But in the bigger scheme of things, so if visible light sits here, in the bigger scheme of things, what are the other frequencies or other wavelengths correspond to? Well, there, there, there are electromagnetic waves that have a shorter wavelength than the visible light, and those are x-rays and gamma rays and, and other things. And then there, there's waves that have a longer wavelength. For example, radar, microwave, uh, VHF, UHF, radio broadcast band, that's gonna be, uh, AM, FM kind of band, etc. So again, this refers back to the talk, the FCC guy was giving about a week and a half ago. The, the spectrum is a very scarce resource for any country and how it's allocated between different users. You know, not on top, this book is kind of old, but on top of this, there's, there's the cellular spectrum that the carriers are using, for example, in the United States and others. Um, and in the US, it's around, um, um, it depends what system you're talking to, but the analog system was around in 900 megahertz. Now, the digital systems are around much higher frequencies, like 1800, 1900 megahertz, etc. So, um, what, what I'm trying to emphasize is the visible light is a small portion of the entire spectrum that, um, that we all deal with. Okay. Um, so next, I want to talk about a little bit about the additive and the subtractive color system. Okay. So, you zoom in, please. So additive and subtractive color system. Okay. So, in general, um, if I, I'll start with additive explaining what additive is. So, if I have, Hey Amazon Prime members, why pay more for groceries when you can save big on thousands of items at Amazon Fresh? Shop prime exclusive deals and save up to 50% on weekly grocery favorites. Plus save 10% on Amazon brands, like our new brand Amazon Saver, 365 by Whole Foods Market, Uplenty and more. Come back for new deals rotating every week. Don't miss out on savings. Shop prime exclusive deals at Amazon Fresh. Select varieties. We wear our work day by day, stitch by stitch. At Dickies, we believe work is what we're made of. So, whether you're gearing up for a new project, or looking to add some tried and true workware to your collection, remember that Dickies has been standing the test of time for a reason. The workware isn't just about looking good. It's about performing under pressure and lasting through the toughest jobs. Head over to Dickies.com and use the promo code Workware 20 at checkout to save 20% on your purchase. It's the perfect time to experience the quality and reliability that has made Dickies a trusted name for over a century. Two C of lambdas. C1 of Lambda and C2 of Lambda. And if I combine those electromagnetic waves, this is one, this is another one, then I get a new wave called C of Lambda, which is C1 plus C2. In this case, the energy of this guy and the energy of the other wave have been combined with each other to result in a new shape in this. So, this is an example of an additive system. So, you add the light at different wavelengths and you get a new color. And what's an example of an additive system in real life? An example of a system we build in which colors are added. There's one spectrum, one C of Lambda combined with another one resulting in new colors. The TV, right? You have the red gun, the green gun, and the blue gun, and essentially what the TV signal over the air is doing, it's modulating each of those guns, so they get combined in the right proportions with each other so that what you end up seeing on the screen is the weighted sum of a red blob plus a green blob plus a blue blob. So, if you wanted to make a particular color, you await those things appropriately and boom, you result in a desired color. So, an example of an additive system is the TV. You have the red gun, the blue gun, and the green gun. And the additive system has three primary colors, and there are red, green, and blue. And you can talk about the combination of these two things. If I combine equal amount of red and green, I get a color that's called yellow. I combine blue and green, I get cyan, combine blue and red, get magenta. What if I combine equal amounts of red, green, and blue? You get white, exactly. And so, from a spectrum point of view, that's kind of exactly what's happening. So, if you roll up, please. So, for example, if I have as a function of lambda, if I have a blob here around 400 nanometers, this is blue, then as a function of lambda, if I have a blob here around 500 nanometer, that's green. And if I have a blob here around 700 nanometer, that's red. And now if I combine red and green from a spectral point of view, I get a blob that's this much. So, this is red plus green that covers both of them. And that is essentially yellow. That results. What I'm saying is that if you had a device that measured the C guy, and each of these individual C's were, had this kind of a spectrum, when you add them up, you get this kind of a spectrum, it would look to the human eye as what we call yellow. So, essentially here, you're adding up C1 of lambda. You have no light, or black light, start with here, and you add C1 of lambda. You add C2 of lambda together, and what comes out is a mixture. And the mixture is going to have a spectrum. There's the sum of the spectrums of the other two things. So, that's why it's called the additive color systems. The spectrums are added in order to result in the spectrum of the final thing you look at. And, besides TV, what other color system has additive? The sensors in a camera. The CMOS sensors in today's camera, in fact, there's three of them, red, green, and blue, all packed together. And so, you tile the plane with red, green, blue sensors, and then combine them together to result in a final color. So, another example of additive system, let me redraw that thing here. So, you got start with black, and you get a mixture, which is C1 of lambda plus C2 of lambda. So, another example is the camera sensors. Now, not every instance of a coloring system in nature is additive. Sometimes nature works in a subtractive way. And you might say, "Oh, what does that mean?" Well, there's this molecules that are called pigments. Like in the red apple has red pigments, right? The word pigment in that means or whatever language it means color, right? So, when a white light has wavelengths in all, has wavelengths all the way from 400 to 700. It has a mixture of all possible lights. You can also think about white noise in electrical engineering. What does that mean? It has a flat spectrum. So, let me just show that. So, white light, before we even move forward, is it goes all the way from 400 to 700. It's essentially what I showed here. When you say the sunlight is white, it means it's supposed to have all equal amounts of frequencies of all different colors, right? So, when you have an apple, which has red pigments, the white light hits it, and what happens is that the apple absorbs all the other frequencies, except for red, and it reflects red. And that's the color that you end up seeing. That's an example of the subtractive color system, okay? So, let's talk about that for just a second. For just a second. Okay, so, in essentially white light comes in, and each time it goes through a system, it goes through a filter that you can think of it as, it goes through a band pass filter with certain frequencies, and what comes out is the process of this linear operator applied to this, right? So, what do I mean by that? Well, let's say that C1 of lambda was a cyan color, and how does cyan work? What's the spectrum of cyan? Well, coming back to here, okay, cyan is the mixture of red and blue, and now I'm oversimplifying this to the max, right? Sorry, blue and green, thank you. Okay, so if I were to plot the spectrum of cyan, and again, massively oversimplified, it's going to have blue and let's see. Hold on, this is 400, this is 700, and right. So, blue and green, okay, so it goes, remember, blue was centered around 400, green was about 500, and red was about 700, and we're kind of assuming they're contiguous under equal bands. So, blue plus green starts at 400 and ends somewhere around 550 or so, right? So, that's cyan, and then suppose I passed this, start with the white light, and I passed this through another filter magenta, and the spectrum of magenta is, if you go back here, is red and blue, and come back here, here's red, and here's blue, so there's a hole in the middle, which is green, so it does like this, it does like that. 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It's the perfect time to experience the quality and reliability that has made Dickies a trusted name for over a century. That comes back up. This is the spectrum for my gentle. So, if I think of the white signal, white signal ideally has, and how does the white look like? It's a flat all the way from here to here ideally, right? So, think of a signal with the white spectrum, all these frequencies going in, right? And it passes through a filter that only allows this piece to go through and blocks from 550 to 700. So, it keeps this portion. So, now at the output here, I only have this portion of the signal. And then I pass it through the magenta, which is a filter shown here, and magenta blocks from here to here. So, now, this part gets knocked out, and all I'm left with is from here to here. Is that clear? I'm just doing straightforward filtering. I start with the white signal, which is here. It has all the frequencies from 400 to 700, right? I pass it through cyan filter. I pass it through a filter with this kind of characteristic, and it keeps this part all the way from here to here and gets rid of this part. So, at this point, I have a signal that goes from here to here. And then I pass it through a magenta filter with magenta, knocks out from here to here. So, I get rid of these as well. And I'm only left between here and here. So, what comes out of, at the output here, is something that has just, I ran out of space now, but only from 400, the beginning part, this region. And what color would that be? Looking at this picture, what's the, what's the color that's within 400 and here? Blue. Okay. So, you can think of a subtractive color, in the additive color system, we add these things, these spectrons in parallel with each other. We, we mix them up to get C1 lambda plus C2 lambda. Here, we're thinking, if you think of a linear time and variance systems, you're cascading these systems one after the other. And, and what comes out is, is the, the remaining spectrum that, that, that comes out this way. Okay. So, what's, what's a typical, so what are the primary colors of the, why did I draw all of these things? Well, the three primary colors of subtractive system, or what? Cyan, yellow, and magenta. So, here's cyan, here's yellow, and magenta. So, notice that the, the primary colors of a subtractive color system are the secondary colors of the additive color system. Remember, this picture we had, here. Okay. So, for, for primary color system, this is red, this is green, this is blue, this were the primary ones, let's just circle the primary ones this way. And then the secondary ones where yellow, magenta, and cyan, here, it's the opposite. So, yellow, yellow, cyan, and magenta are the primary colors, and the secondary are the intersections. And, how do I find out the intersections of, for example, cyan and magenta? Cyan and magenta. It's exactly this process that I just went through. You start with the white color, you pass it through a cyan filter, pass it through a magenta, whatever comes out, that's the intersection. So, it's blue. And then, how about the intersection of yellow and magenta? Well, you go through that same process, and you realize it's red. How about the intersection of yellow and cyan? It's green. And the intersection of everything, here, is black. It means that you filter once through cyan and yellow, and then you filter again with magenta, then nothing is left out. You start with white, you go through a cyan filter, then through your yellow filter, then your magenta filter, all the light has been limited, and then you end up with black. So, now, these are the secondary colors of the subtractive color system, and these are the primary colors. So, what would be an example of a real life engineering system that uses a subtractive color system? The printers, right? How do printers work? Let's say, how does a color printer work? I'm sure you've bought ink, right? When you buy ink, do you buy red, green, blue ink? Or do you buy cyan, yellow, magenta? Exactly. Cyan, yellow, magenta. When you open up San Francisco Chronicle to read it in the morning, at the bottom of it, it wasn't colored like 10 years ago. The fact that we have colored pictures in the newspapers is a fairly recent phenomenon. Anyway, at the bottom of it, you always see the cyan, yellow, magenta, and black. So, how does it work? Well, essentially, you have these repositories of ink in these three colors, and you throw droplets onto the paper in order to mix them up to create the colors that you really want. So, on a white paper, by superimposing these different colors, you're able to use the, you're using, essentially, subtract the color system to generate the colors that you intend to do to come out with. Yeah, question. So, are some image formats in the cyan, you know, magenta just for dealing with printers then, or is there anything? It's not an image format. It's a, it's a, it's a, it's a, you can think of it more as a system rather than a format, right? See, in your TV, when the blue gun and the green gun interact with each other, essentially, they're, they're getting added up. The spectrum of one is getting added on top of the spectrum of the other one, right? So, if you want to have different shades of, I don't know, yellow, you mix them up with different proportions to generate mixed, different proportions of yellow, different colors of yellow, right? Here, it's, again, a system, right? Light hits. So, if I put, you know, red dot here on this white paper, if you can put a red dot here, you know, the white lights hit it, the red pigments on the color absorb everything except for red, they send the rest of it back. So, this system of a red ink on a, on a, on a white paper is a subtractive color system. It's a mechanism, not a format. Does that make sense? Yeah, so then, but then are the pixels stored differently? And if you save them like in Photoshop, are they under RGB or CMYK? Oh, those are, those are entirely different concepts. Usually, I don't, I don't think people, I mean, the image formats, there's like hundreds of them. There's YIQ, there's RGB, there's C, C, I, C, I, look at this. There's YIQ, and then C, C, C, R, C, B, et cetera. Those are all linear transformations of each other. You apply it. So, you have a, you have a three dimensional vector of RGB, for example, or YIQ, you apply three by three matrix to it and then get a new one. That, that's a different concept than this one, okay? So, and then of course, you can, you're right. You can also express an image in terms of the linear combinations of this, this dionial or magenta. So, that would be a different format. But, but what I'm talking about here is, is more than just a formatting thing. It's, that's right. It's, it's a physical system type thing, right? Physically, the, the red and green gun get mixed up with each other, their spectrums get added here. Physically, things are subtracted from each other. Depends on what system you're in. And so, one of the important things when you, when you're printing with, with tionial or magenta is what? I mean, have you looked at the pictures on the chronicle? Have you ever looked at it and said, oh, this looks awful. What, what artifacts have you seen in them when you look at like a color printed picture? It's not in your, in your photo smart printer at home because you paid a bunch of money for that. And then, if you think about it, the newspaper is a mass production thing. I don't know. They, they, they, they print like a million or five million at every morning, right? They can't use photos for chronicle, unfortunately. But when, so, Cora called it, you know, it's, it's a cheap printing system. So, what could go wrong when you use a cheap printing system? Exactly. Listen, are you going to say the same thing? Yeah. Sometimes you see people with like two eyebrows, two mouths. I don't know if you've noticed that, right? If, if the, so, I mean, why is there misalignment? Because you have a, you have some sort of a mechanical system that essentially it shoots a drop out of cyan and it shoots a drop out of yellow and it shoots a drop out of magenta onto a piece of paper. And if this mechanical system is not extremely fine-tuned and I think we have mechanical engineers in this class as well, I hope we do, then, then things get out of whack and, and, and you get this misalignment problems causes a lot of, a lot of distortion. And how do you, and what's the other big problem? Okay. So, let me ask you a slightly different question. Why is it that when you go buy your, your cartridge printers, why is it that you, that you always have to buy black as well? Why is it that it's not enough to buy cyan and yellow magenta? And you have to buy black. Exactly. To generate black in this system, you have to superimpose, say, cyan and yellow and, and, and magenta dot or ink on all on top of each other. And, and what's the problem with that is that there's so much black in all of our pictures, right? The background is dark. I don't know, people's hairs are black. I'll do not all of you have black hair, but there's a lot of black. So each time you want to generate black, if you did that, you run out of color for all your other things very quickly. So as a result, and on top of that is the alignment problem. If, if they're slightly off, you never get real black. So as a result, well, they, they just sell you a black, a black cartridge, so that you, so that you can, you can, you can just, when there's, when, when the printer sees there's a black thing, you just print black and that, that's the end of it. Okay. And there's this beautiful pictures I'll probably bring it next time is, it's called more patterns showing that, that if these things are misaligned, you get this funny patterns and people have done a lot of analysis on this, in this kind of subtractive color systems. Okay. Any, any questions on this? Okay. So next I'm going to talk a little bit about the eye. And again, I won't even attempt to draw the picture of the, of the, the, the eye, horizontal cross section. But if you can zoom in here, please, as much as you can. So this is a, this is a, kind of a cross section, a horizontal cross section of the, of a, of a right human eye. Okay. So you, you've just cut it like, like this. Okay. And, and what are the components that, that we see here? Okay. Let's start from outside and kind of go in. First of all, there's the, the cornea. Okay. And its job is to just direct as much light as, as possible inside, inside the eye. And then this, this aqueous humor, there's nothing humorous about it, but it's basically the liquid. Your eye is made of a liquid. That's the liquid kind of behind it. Okay. Then behind that, there's the iris, which is this thing here and this thing here. And iris basically controls the amount of light that goes into your eye. So at, at night, actually the human eye has an incredible dynamic range. You can see things in extreme dark and you can see them in extreme brightness. This is something that still, we cannot reproduce it on a single camera in a single vision system to have such a big dynamic range as the human eye does. But anyway, how do we accomplish that? Well, the iris, when, when you're an extreme bright light, it, it, it closes into the, limits the amount of eye that gets into the eye, inside, just so that you don't damage your eyes. And when you're, when you're at night in a very dark situation, it opens up to, to let more light get in. Okay. And then you have the lens and what does the lens do, anybody? It becomes thinner or it becomes thicker and thereby allowing you to focus. So if you're looking at close by things, then it, it, or further away things by becoming thicker and thinner, it makes sure that the image is projected onto the right spot in your retina so that you can see it. Okay. And if you become near-sided or far-sided, essentially you've lost, the lens has lost its elasticity or the ability to do this becoming thicker and thinner, therefore the image gets formed here or here. And that's when you, and, and it, it, it's, at the sharpest, when I say it's formed, I mean, it's the sharpest here and here, and therefore this part, which is where you actually see, this is where your sensors are, your rods and cones and your cells are, or the highest concentration of your cells is, is missed. So the lens, what it does is by, by thickening and thinning, it can, it can form the, it tries to form the image right here where your eye is the most sensitive. And by the way, this is called the fovea. It's a piece of, so retina here is a, is a, is a, which is all over here is the, has all kinds of light sensitive cells that creates the signal that goes to the brain. And right in the middle of it here, fovea is the piece of retina that has the highest concentration of these, of these cells. That's, that's where the image gets formed. Okay. So what kinds of cells are there in the retina? Anybody knows? Right, what are they called? There are two kinds of cells, rods and cones, exactly. So this is kind of the distributions. So it, this, as a function of angle, per metric angle in degree, this is a distribution of rods and cones in, in your eyes. Okay. These are the light receptive cells, both of them. And there's about 7 million cones and 120 million rods. And what's the difference between cones and rods? Well cones are less sensitive to light than rods are. Okay. And they're primarily responsible for, for day vision and they're responsible for seeing color. And the distribution of cones is, is, for example, shown here, the solid line in this picture. Okay. There's a lot of them near the zero degrees where the fovea is. And the distribution goes very close to zero in the middle at, at the two ends. Rods on the other hand, there's more oven, there's 120 million. They're more sensitive to light and they're primarily responsible for, for night color, for, for, for our vision at night. Sorry, for not night color, for our vision at night. That's exactly what, and they're not the ones that deal with color. That's exactly why at night, you don't see as much color. So the human visual system is a lot poorer for, in terms of color, perception is a lot poorer at night than it is at daytime. Simply because at night, you want to use the more sensitive cells, i.e. the, the rods, and the rods are, are, are not very color sensitive. Most of the color is done by the cones and the cones mostly work in, in the, in the regime where there's a lot of daylight. Okay. And as you can see, there's, there's a blind spot here, which is where there's, there's, there's no rods and there's no coins. And so the eye has a, uh, has a blind spot. It's, it's quite narrow. That's why it's not problematic. If it was wide, then they have a big blind spot and God knows what the evolution consequences of that would have been. Actually, I shouldn't say God and evolution won't sentence, but hey, it just came out. So, you know, my bad, just scratch that. Okay. Um, so, so few words about the human visual system. So these, these electrical signals kind of, so the, the retina has kind of, um, layers of cells, the light hits it and then, uh, it goes through layers of cells and, and it hits the cones and stuff like that. And, and finally, at the end of the day, these electrical signals here that, the light gets converted into electrical signals and then it gets sent on our optical nerve. And unfortunately that's, that's really all about what we know. We know something's about the physics of the eye and we know that it gets converted into some signal that goes to the nerve, but we don't really know or completely understand how the human brain makes sense of that signal. In fact, we, as I've mentioned, I think in this course before, the physical signal that gets sent is upside down, but somehow your brain interprets it as the right side up. And even if you make the person look at upside down images for, for a while, for three weeks, at the end of, I think, previous was the time at, at, at which the person just collected it again. So, so there's a, there's a lot of interpretive kind of processing going on in the brain with that electrical signal that we don't really understand. How does, how does the brain know that it has to invert the image? It does. It, it's, I think my, I mean one possibility is that you, you experience the world with your other senses and you kind of train it. Hey Amazon Prime members, why pay more for groceries when you can save big on thousands of items at Amazon Fresh. Shop Prime exclusive deals and save up to 50% on weekly grocery favorites. Plus save 10% on Amazon brands, like our new brand Amazon Saver, 365 by Whole Foods Market, Aplenty and more. Come back for new deals rotating every week. Don't miss out on savings. Shop Prime exclusive deals at Amazon Fresh. Select varieties. We wear our work day by day, stitch by stitch. At Dickies, we believe work is what we're made of. So, whether you're gearing up for a new project or looking to add some tried and true work wear to your collection, remember that Dickies has been standing the test of time for a reason. Their work wear isn't just about looking good. It's about performing under pressure and lasting through the toughest jobs. Head over to Dickies.com and use the promo code WorkWear20 at checkout to save 20% on your purchase. It's the perfect time to experience the quality and reliability that has made Dickies a trusted name for over a century.