title

Convolutions in image processing | Week 1 | MIT 18.S191 Fall 2020 | Grant Sanderson

description

The basics of convolutions in the context of image processing.
For full course information, visit https://github.com/mitmath/computational-thinking/tree/Fall20
Course website: https://computationalthinking.mit.edu/Fall20/
To learn more about Julia, head to https://julialang.org
Contents
00:00 Introduction
01:12 Box blur as an average
03:00 Dealing with the edges
04:31 Gaussian blur
05:30 Visualizing gaussian blur
06:04 Convolution
06:40 Kernels and the gaussian kernel
07:26 Looking at the convolution in Julia
08:45 Julia: `ImageFiltering` package and Kernels
09:08 Julia: `OffsetArray` with different indices
10:15 Visualizing a kernel
11:25 Computational complexity
12:00 Julia: `prod` function for a product
13:00 Example of a non-blurring kernel
16:00 Sharpening edges in an image
17:13 Edge detection with Sobel filters
21:25 Relation to polynomial multiplication
25:00 Convolution in polynomial multiplication
26:08 Relation to Fourier transforms
28:50 Fourier transform of an image
31:50 Convolution via Fourier transform is faster
34:00 Final thoughts

detail

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'duration': 2.021}, {'end': 17.968, 'text': "So if you check the link in the description, there's a new official course website.", 'start': 14.467, 'duration': 3.501}, {'end': 23.371, 'text': 'And again, this is something that anybody in the world is free to attend the lectures for if you want to.', 'start': 18.328, 'duration': 5.043}, {'end': 27.735, 'text': "Second, we're going to be doing the discussion during this lecture through Discord.", 'start': 24.211, 'duration': 3.524}, {'end': 34.762, 'text': 'So, if you want to answer questions or ask questions or interact with the other professors while I give this lecture,', 'start': 28.135, 'duration': 6.627}, {'end': 35.923, 'text': 'the Discord is the place to do it.', 'start': 34.762, 'duration': 1.161}, {'end': 39.526, 'text': 'Questions will be fed to me and we can pull them up on screen here.', 'start': 37.104, 'duration': 2.422}], 'summary': 'First live lecture with new website and discord for global audience', 'duration': 34.904, 'max_score': 4.622, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA4622.jpg'}], 'start': 4.622, 'title': 'Course live lecture logistics', 'summary': 'Covers the logistics of the first live lecture, including the announcement of a new official course website and the use of discord for discussions and interactions with professors.', 'chapters': [{'end': 39.526, 'start': 4.622, 'title': 'Course live lecture logistics', 'summary': 'Covers the logistics of the first live lecture, including the announcement of a new official course website and the use of discord for discussions and interactions with professors.', 'duration': 34.904, 'highlights': ['The chapter discusses the launch of a new official course website, inviting anyone worldwide to attend the lectures.', 'The use of Discord for discussions and interactions with professors during the lecture is highlighted.']}], 'duration': 34.904, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA4622.jpg', 'highlights': ['The launch of a new official course website invites worldwide attendance.', 'Discord is used for discussions and interactions with professors during the lecture.']}, {'end': 499.294, 'segs': [{'end': 110.624, 'src': 'embed', 'start': 79.822, 'weight': 3, 'content': [{'end': 81.503, 'text': "And if you're thinking about blurring an image,", 'start': 79.822, 'duration': 1.681}, {'end': 89.348, 'text': 'maybe the first thing that you could think to do would be to look at each pixel and kind of average it with all of its neighbors and do this sort of moving average.', 'start': 81.503, 'duration': 7.845}, {'end': 91.829, 'text': "So the way I'm doing it here we've got this little grid,", 'start': 89.948, 'duration': 1.881}, {'end': 98.313, 'text': "3 by 3 grid of values that's marching across the image and slowly creating a new image on the right.", 'start': 91.829, 'duration': 6.484}, {'end': 106.42, 'text': "And to be concrete, what's going on here is each of those values is 1, 9th, and we're sort of thinking of lining them up with some pixels,", 'start': 98.993, 'duration': 7.427}, {'end': 110.624, 'text': 'multiplying each of those values by the corresponding pixel and then adding them together.', 'start': 106.42, 'duration': 4.204}], 'summary': 'Blurring an image involves averaging pixels with neighbors using a 3x3 grid, each value being 1/9th, to create a new image.', 'duration': 30.802, 'max_score': 79.822, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA79822.jpg'}, {'end': 205.03, 'src': 'embed', 'start': 176.737, 'weight': 2, 'content': [{'end': 179.92, 'text': "In this case it's a literal edge case, like what's going on at the edge?", 'start': 176.737, 'duration': 3.183}, {'end': 189.067, 'text': "Because if we just let this grid of values fall off the side and maybe we either don't count them or we're multiplying them by zero,", 'start': 180.6, 'duration': 8.467}, {'end': 196.193, 'text': "what ends up happening is we would only be taking 4, 9ths of the given corner value and it's going to end up disproportionately dark.", 'start': 189.067, 'duration': 7.126}, {'end': 199.987, 'text': "which in some mathematical contexts, maybe that's what you want.", 'start': 196.966, 'duration': 3.021}, {'end': 205.03, 'text': "You would want to treat the image as being something where outside of a certain domain, it's entirely zero.", 'start': 200.027, 'duration': 5.003}], 'summary': 'Handling edge cases is important to avoid disproportionate effects on image values.', 'duration': 28.293, 'max_score': 176.737, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA176737.jpg'}, {'end': 440.907, 'src': 'embed', 'start': 414.167, 'weight': 0, 'content': [{'end': 418.071, 'text': "And I guess there's some use cases for blurring images, like blurring out faces.", 'start': 414.167, 'duration': 3.904}, {'end': 420.493, 'text': 'But it goes way beyond that, actually.', 'start': 419.032, 'duration': 1.461}, {'end': 426.598, 'text': 'First of all, smoothing out an image with a Gaussian kernel is useful in a lot of contexts beyond what you would expect.', 'start': 420.713, 'duration': 5.885}, {'end': 434.543, 'text': "But the important thing I want you to keep in mind here is not the specific example, but the general framework that's being developed, where,", 'start': 427.319, 'duration': 7.224}, {'end': 440.907, 'text': 'if we tweak what the values inside that kernel are, we can start to do some more nuanced and interesting things with images.', 'start': 434.543, 'duration': 6.364}], 'summary': 'Blurring images has diverse use cases beyond faces, with a gaussian kernel enabling nuanced image manipulations.', 'duration': 26.74, 'max_score': 414.167, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA414167.jpg'}], 'start': 39.947, 'title': 'Image blurring with convolution', 'summary': 'Discusses the concept of image blurring using convolution with a 3x3 grid, addressing edge cases, gaussian kernels, and broader applications, offering potential for more nuanced image manipulation.', 'chapters': [{'end': 499.294, 'start': 39.947, 'title': 'Image blurring with convolution', 'summary': 'Discusses the concept of image blurring using convolution, demonstrating how a 3x3 grid with weighted pixel values creates a blurred effect, while addressing nuances like edge cases and the use of gaussian kernels. it also emphasizes the broader applications and the potential for more nuanced image manipulation.', 'duration': 459.347, 'highlights': ['Image blurring is achieved through convolution using a 3x3 grid with weighted pixel values, creating a blurred effect. The speaker demonstrates the process of blurring an 8-bit Mario image by using a 3x3 grid of weighted pixel values, resulting in a gradual blur effect.', 'Nuances like edge cases are addressed, ensuring proper handling and avoiding disproportionately dark outcomes. The speaker discusses the literal edge cases in image blurring, emphasizing the importance of handling edge pixels to avoid disproportionately dark outcomes.', 'The use of Gaussian kernels for weighted pixel averaging is explained, highlighting its effectiveness in achieving a more natural blur effect. The speaker explains the use of Gaussian kernels for weighted pixel averaging, emphasizing its effectiveness in achieving a more natural blur effect due to its mathematical properties and nuanced averaging.', 'The broader applications and potential for nuanced image manipulation using convolution are emphasized. The lecture emphasizes the broader applications of convolution, highlighting the potential for more nuanced and interesting image manipulations by tweaking the values inside the kernel.']}], 'duration': 459.347, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA39947.jpg', 'highlights': ['The broader applications and potential for nuanced image manipulation using convolution are emphasized.', 'The use of Gaussian kernels for weighted pixel averaging is explained, highlighting its effectiveness in achieving a more natural blur effect.', 'Nuances like edge cases are addressed, ensuring proper handling and avoiding disproportionately dark outcomes.', 'Image blurring is achieved through convolution using a 3x3 grid with weighted pixel values, creating a blurred effect.']}, {'end': 815.252, 'segs': [{'end': 535.976, 'src': 'embed', 'start': 501.837, 'weight': 0, 'content': [{'end': 504.84, 'text': 'we want to actually do this convolving operation.', 'start': 501.837, 'duration': 3.003}, {'end': 507.784, 'text': 'We want to actually see what happens when we mess around with different kernels.', 'start': 505.001, 'duration': 2.783}, {'end': 512.789, 'text': "So first of all, to get a given kernel, there's this library that I'm going to use.", 'start': 508.444, 'duration': 4.345}, {'end': 516.554, 'text': "We've got all of our definitions down here for various functions I'm going to have.", 'start': 513.71, 'duration': 2.844}, {'end': 518.895, 'text': "But I'm going to be using one called image filtering.", 'start': 516.974, 'duration': 1.921}, {'end': 525.168, 'text': 'And in it It has this module called kernel, and we can pull up the documentation for it.', 'start': 519.717, 'duration': 5.451}, {'end': 529.631, 'text': 'you know just using the live docs here, where it tells us the various different functions it has in there.', 'start': 525.168, 'duration': 4.463}, {'end': 533.013, 'text': 'And among them is this Gaussian, which is exactly what you want.', 'start': 529.991, 'duration': 3.022}, {'end': 535.976, 'text': "But it already tells you that there's more interesting kernels that can be played with.", 'start': 533.154, 'duration': 2.822}], 'summary': 'Exploring convolving operations with different kernels and libraries for image filtering, including gaussian kernel.', 'duration': 34.139, 'max_score': 501.837, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA501837.jpg'}, {'end': 653.903, 'src': 'embed', 'start': 630.473, 'weight': 2, 'content': [{'end': 637.918, 'text': "It'll take in an image, it'll take in this kernel of values, and it'll produce a new image, which in this context is a little bit blurrier.", 'start': 630.473, 'duration': 7.445}, {'end': 640.239, 'text': 'Now, because this is a small kernel.', 'start': 638.678, 'duration': 1.561}, {'end': 645.34, 'text': "you know, maybe it's not as visible, but if I changed it to be a Gaussian with standard deviation 3,", 'start': 640.239, 'duration': 5.101}, {'end': 648.681, 'text': 'the library is smart and it knows to make it a little bit bigger if I want it to.', 'start': 645.34, 'duration': 3.341}, {'end': 653.903, 'text': "so now it's a 13 by 13 array and we see that it indeed does make it notably blurrier.", 'start': 648.681, 'duration': 5.222}], 'summary': 'Using a gaussian kernel with standard deviation 3 makes the image notably blurrier.', 'duration': 23.43, 'max_score': 630.473, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA630473.jpg'}, {'end': 774.074, 'src': 'embed', 'start': 750.553, 'weight': 1, 'content': [{'end': 758.198, 'text': 'as the kernel gets larger unless you do something clever it will by default take a lot longer to process images,', 'start': 750.553, 'duration': 7.645}, {'end': 759.599, 'text': "especially if they're higher resolution.", 'start': 758.198, 'duration': 1.401}, {'end': 767.957, 'text': "But before we talk about how to diminish that kind of complexity, let's just play around with what happens if I change this kernel.", 'start': 760.58, 'duration': 7.377}, {'end': 774.074, 'text': 'So actually, before I even show you, I think I want to do this as a live quiz.', 'start': 768.931, 'duration': 5.143}], 'summary': 'Larger kernel size leads to longer image processing time, especially for higher resolution images. experimentation with kernel changes to be demonstrated.', 'duration': 23.521, 'max_score': 750.553, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA750553.jpg'}], 'start': 501.837, 'title': 'Image convolution and gaussian kernels', 'summary': 'Explores image convolution and kernel manipulation, focusing on gaussian kernels and their impact on image blurriness. it also discusses computational complexity, demonstrating a 9.5-second processing time for a small image, and introduces a live quiz for audience engagement.', 'chapters': [{'end': 535.976, 'start': 501.837, 'title': 'Image convolution and kernel manipulation', 'summary': 'Explores image convolution and kernel manipulation using a library that provides functions for various kernels, including the gaussian kernel, which is of particular interest for experimentation.', 'duration': 34.139, 'highlights': ['The module used for image filtering includes a variety of functions, with the Gaussian kernel being highlighted for experimentation.', 'The library provides access to documentation for the available functions and emphasizes the potential for manipulating interesting kernels.', 'The chapter delves into the convolving operation and explores the impact of different kernels, with a focus on the Gaussian kernel.']}, {'end': 815.252, 'start': 536.756, 'title': 'Image convolution and gaussian kernels', 'summary': 'Discusses image convolution using gaussian kernels, explaining how different standard deviations affect the blurriness of the image. it also delves into computational complexity, highlighting the impact of kernel size on processing time, with an example showing a 9.5-second processing time for a small image. additionally, it introduces a live quiz to engage the audience in understanding the effects of a given kernel on the image.', 'duration': 278.496, 'highlights': ['The convolve function takes an image and a kernel of values, producing a new, slightly blurrier image, with the blurriness varying based on the standard deviation of the Gaussian kernel.', 'The processing time for image convolution can increase significantly with larger kernels, with an example showing a 9.5-second processing time for a small image, highlighting the impact of computational complexity.', 'The introduction of a live quiz to engage the audience in understanding the effects of a given kernel on the image, promoting interactive learning and participation.']}], 'duration': 313.415, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA501837.jpg', 'highlights': ['The convolve function applies a kernel to an image, producing a blurrier image.', 'Larger kernels can significantly increase image convolution processing time.', "The Gaussian kernel's impact on image blurriness is a key focus of the chapter.", 'The library provides access to documentation for available functions and kernel manipulation potential.', 'Introduction of a live quiz promotes audience engagement and interactive learning.']}, {'end': 1346.79, 'segs': [{'end': 920.566, 'src': 'embed', 'start': 887.065, 'weight': 1, 'content': [{'end': 887.805, 'text': "That's an interesting question.", 'start': 887.065, 'duration': 0.74}, {'end': 888.946, 'text': "We'll have to ask the other profs.", 'start': 887.825, 'duration': 1.121}, {'end': 897.06, 'text': "I definitely don't think so, because you can take this to an extreme where It feels like you're definitely losing out on information.", 'start': 889.046, 'duration': 8.014}, {'end': 900.504, 'text': "There's other things that you can do to it, though.", 'start': 898.322, 'duration': 2.182}, {'end': 903.748, 'text': 'Fast Fourier transform would be an example where you can recover the image exactly.', 'start': 900.544, 'duration': 3.204}, {'end': 905.511, 'text': 'But those feel very rare.', 'start': 903.989, 'duration': 1.522}, {'end': 909.535, 'text': "I don't want to say that offhandedly, because I don't know quite off the top of my head.", 'start': 905.531, 'duration': 4.004}, {'end': 912.679, 'text': "One of the other props who's smarter than I will might come in to answer.", 'start': 910.416, 'duration': 2.263}, {'end': 920.566, 'text': 'And the second one, how does convolving an image with a kernel directly relate to convolving two functions? Glad that you asked that.', 'start': 914.261, 'duration': 6.305}], 'summary': 'Discussion on information loss in image recovery and relating image convolution to function convolution.', 'duration': 33.501, 'max_score': 887.065, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA887065.jpg'}, {'end': 978.554, 'src': 'heatmap', 'start': 930.454, 'weight': 0.844, 'content': [{'end': 934.077, 'text': "It's essentially the same idea, though, that you have one thing, that's marching across another.", 'start': 930.454, 'duration': 3.623}, {'end': 936.719, 'text': 'you multiply corresponding values and then you add them up.', 'start': 934.077, 'duration': 2.642}, {'end': 941.042, 'text': "It's really the same operation, modulo one or two small conventions.", 'start': 937.119, 'duration': 3.923}, {'end': 946.666, 'text': "All right, so if we go back to our quiz, I'll go ahead and see what the answers turn out to be here.", 'start': 941.783, 'duration': 4.883}, {'end': 949.528, 'text': 'And fascinating.', 'start': 947.607, 'duration': 1.921}, {'end': 952.731, 'text': 'So it looks like most people thought that it would detect edges.', 'start': 949.789, 'duration': 2.942}, {'end': 959.555, 'text': "But if we actually play with this, if we go to our notebook here and I've gone ahead and written in the kernel,", 'start': 953.571, 'duration': 5.984}, {'end': 965.181, 'text': 'So that same grid of values that I gave you were seven in the center and then some negatives around.', 'start': 960.496, 'duration': 4.685}, {'end': 970.126, 'text': 'The effect that it has on the picture of the cat here is it actually kind of sharpens it up.', 'start': 965.621, 'duration': 4.505}, {'end': 974.01, 'text': 'So, if we look previously, his chest is very fluffy, very soft.', 'start': 970.446, 'duration': 3.564}, {'end': 978.554, 'text': "but over here what it's done is really accentuated the differences between one piece of fur and the next.", 'start': 974.01, 'duration': 4.544}], 'summary': 'Multiplying corresponding values and adding them up, sharpening the cat picture by accentuating fur differences.', 'duration': 48.1, 'max_score': 930.454, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA930454.jpg'}, {'end': 1030.483, 'src': 'embed', 'start': 1000.757, 'weight': 2, 'content': [{'end': 1003.099, 'text': "If you have all the weights, they've got to be 1.", 'start': 1000.757, 'duration': 2.342}, {'end': 1010.047, 'text': "And what that means is that if you're in a region of the image without any edges, you're in a totally monochromatic spot.", 'start': 1003.099, 'duration': 6.948}, {'end': 1015.299, 'text': "if you add up all of the values and you're multiplying them and the sum for the kernel is 1,", 'start': 1010.047, 'duration': 5.252}, {'end': 1017.8, 'text': "Well that means that you're just going to get the original color back.", 'start': 1015.299, 'duration': 2.501}, {'end': 1023.521, 'text': "The circumstances under which this does something non-obvious are when there's a lot of variation near a point.", 'start': 1018.16, 'duration': 5.361}, {'end': 1030.483, 'text': "So if you have one point, that's, let's say, a very bright pixel, and then it's surrounded by very dark pixels that will end up looking quite bright,", 'start': 1023.881, 'duration': 6.602}], 'summary': 'Using weights of 1 ensures original color in monochromatic areas.', 'duration': 29.726, 'max_score': 1000.757, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1000757.jpg'}, {'end': 1270.742, 'src': 'embed', 'start': 1233.3, 'weight': 0, 'content': [{'end': 1237.344, 'text': 'And you can really get more and more sophisticated as you tweak the specific values of this kernel.', 'start': 1233.3, 'duration': 4.044}, {'end': 1240.587, 'text': "It's a really powerful tool beyond just a moving average.", 'start': 1237.404, 'duration': 3.183}, {'end': 1244.511, 'text': 'Many of you may have heard of the idea of a convolutional neural network.', 'start': 1241.568, 'duration': 2.943}, {'end': 1249.796, 'text': 'This is in a context where what you want is for a machine to learn specific kernels.', 'start': 1245.091, 'duration': 4.705}, {'end': 1254.281, 'text': 'So, rather than hand baking one, like handing it a Gaussian or handing it a Sobel,', 'start': 1250.077, 'duration': 4.204}, {'end': 1258.966, 'text': 'instead you try to have it learn which kernels are going to pull out some meaningful information from an image.', 'start': 1254.281, 'duration': 4.685}, {'end': 1263.111, 'text': "So it's a very good building block to add up on top of all of that.", 'start': 1259.407, 'duration': 3.704}, {'end': 1270.742, 'text': "Now, one thing to point out while we're looking at this Sobel filter is, unlike the other ones, it's asymmetric.", 'start': 1264.076, 'duration': 6.666}], 'summary': 'Using convolutional neural networks to learn specific kernels for extracting meaningful information from images can be a powerful tool.', 'duration': 37.442, 'max_score': 1233.3, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1233300.jpg'}, {'end': 1356.063, 'src': 'embed', 'start': 1330.215, 'weight': 4, 'content': [{'end': 1334.719, 'text': 'And indeed, in some image processing ones, they just define convolutions to do the thing that you would expect.', 'start': 1330.215, 'duration': 4.504}, {'end': 1342.466, 'text': 'But I do want to take a little moment to explain why you sometimes see it this way and why it actually gives you these really nice mathematical properties.', 'start': 1335.54, 'duration': 6.926}, {'end': 1346.79, 'text': "I don't know if I'll have full time to delve into the details here,", 'start': 1342.967, 'duration': 3.823}, {'end': 1353.739, 'text': "but We're going to start by talking about something that seems totally unrelated to image processing,", 'start': 1346.79, 'duration': 6.949}, {'end': 1356.063, 'text': 'which is to multiply two different polynomials together.', 'start': 1353.739, 'duration': 2.324}], 'summary': 'Convolution in image processing provides nice mathematical properties.', 'duration': 25.848, 'max_score': 1330.215, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1330215.jpg'}], 'start': 815.292, 'title': 'Image convolution and kernel computation', 'summary': 'Explores the computation of image convolution using kernels, the impact of different kernels on image properties, and the challenge of reconstructing original images from blurred ones. it delves into the properties and applications of convolution kernels, emphasizing their sum equals 1, their impact on edge detection, and the role of convolutional neural networks in learning specific kernels for information extraction.', 'chapters': [{'end': 978.554, 'start': 815.292, 'title': 'Image convolution and kernel computation', 'summary': 'Discusses the computation of image convolution using kernels, exploring the impact of different kernels on image properties and clarifying the relationship with mathematical convolution. it also addresses the challenge of reconstructing original images from blurred ones and highlights the misconception regarding the outcome of convolution with kernels.', 'duration': 163.262, 'highlights': ['The majority believed the kernel would detect edges, but it actually sharpens the image, accentuating differences between fur textures. Most people thought the kernel would detect edges, but it actually sharpens the image, accentuating differences between fur textures.', 'The challenge of reconstructing the original image from a blurred one is addressed, with emphasis on potential loss of information and the rare possibility of exact image recovery through methods like Fast Fourier transform. The challenge of reconstructing the original image from a blurred one is addressed, with emphasis on potential loss of information and the rare possibility of exact image recovery through methods like Fast Fourier transform.', 'The misconception regarding the outcome of convolution with kernels is highlighted, emphasizing the similarity with mathematical convolution and the impact on image properties. The misconception regarding the outcome of convolution with kernels is highlighted, emphasizing the similarity with mathematical convolution and the impact on image properties.']}, {'end': 1346.79, 'start': 980.217, 'title': 'Understanding convolution kernels', 'summary': 'Delves into the properties and applications of convolution kernels, emphasizing how their sum equals 1 and their impact on edge detection, demonstrating how specific kernels like the sobel kernel can be used for edge detection and the implications of asymmetry, as well as discussing the convention of flipping kernels before convolving and the role of convolutional neural networks in learning specific kernels for meaningful information extraction from images.', 'duration': 366.573, 'highlights': ['The sum of convolution kernel values equals 1, ensuring a weighted average effect and preservation of the original color in monochromatic areas. The sum of all kernel values equals 1, leading to a weighted average effect and preservation of the original color in monochromatic areas.', 'Explaining the behavior of convolution when there is a lot of variation near a point and the impact of positive and negative values on brightness and darkness in the image. Discussion on the behavior of convolution in the presence of variation near a point and the impact of positive and negative values on brightness and darkness in the image.', 'Demonstrating the properties and visualization of the Sobel kernel for edge detection, showcasing its asymmetry and its ability to highlight micro variations in images. Explanation and visualization of the properties of the Sobel kernel for edge detection, highlighting its asymmetry and ability to capture micro variations in images.', 'Discussion on the convention of flipping kernels before convolving and the rationale behind this unconventional approach, highlighting its mathematical properties. Explanation of the convention of flipping kernels before convolving and the rationale behind this unconventional approach, emphasizing its mathematical properties.', 'Insight into the role of convolutional neural networks in learning specific kernels for extracting meaningful information from images, highlighting its importance as a building block in image processing. Explanation of the role of convolutional neural networks in learning specific kernels for extracting meaningful information from images, emphasizing its importance as a building block in image processing.']}], 'duration': 531.498, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA815292.jpg', 'highlights': ['The role of convolutional neural networks in learning specific kernels for extracting meaningful information from images, emphasizing its importance as a building block in image processing.', 'The challenge of reconstructing the original image from a blurred one is addressed, with emphasis on potential loss of information and the rare possibility of exact image recovery through methods like Fast Fourier transform.', 'The sum of all kernel values equals 1, leading to a weighted average effect and preservation of the original color in monochromatic areas.', 'Explanation and visualization of the properties of the Sobel kernel for edge detection, highlighting its asymmetry and ability to capture micro variations in images.', 'The misconception regarding the outcome of convolution with kernels is highlighted, emphasizing the similarity with mathematical convolution and the impact on image properties.']}, {'end': 1771.985, 'segs': [{'end': 1373.124, 'src': 'embed', 'start': 1346.79, 'weight': 3, 'content': [{'end': 1353.739, 'text': "but We're going to start by talking about something that seems totally unrelated to image processing,", 'start': 1346.79, 'duration': 6.949}, {'end': 1356.063, 'text': 'which is to multiply two different polynomials together.', 'start': 1353.739, 'duration': 2.324}, {'end': 1361.711, 'text': 'So for this one, let me actually pop back to our live quiz and just have you do a little bit of active thinking.', 'start': 1356.704, 'duration': 5.007}, {'end': 1367.979, 'text': 'So the question asks that you consider the following expression.', 'start': 1363.354, 'duration': 4.625}, {'end': 1373.124, 'text': "You've got a certain polynomial term whose coefficients are k0, k1, k2,", 'start': 1368.099, 'duration': 5.025}], 'summary': 'Discussion on multiplying polynomials in image processing.', 'duration': 26.334, 'max_score': 1346.79, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1346790.jpg'}, {'end': 1480.501, 'src': 'embed', 'start': 1454.574, 'weight': 0, 'content': [{'end': 1463.756, 'text': "So it looks like we've got 92, 93 people answering correctly that it's A, which basically is just marching through all of the pairs,", 'start': 1454.574, 'duration': 9.182}, {'end': 1468.298, 'text': 'so that the index on the K added together with the index on the A is equal to three.', 'start': 1463.756, 'duration': 4.542}, {'end': 1473.359, 'text': "And those of you who answered B, it's like completely in the right thought.", 'start': 1469.598, 'duration': 3.761}, {'end': 1480.501, 'text': "It's just that we're multiplying by a quadratic polynomial instead of multiplying by something that goes on forever as well.", 'start': 1474.219, 'duration': 6.282}], 'summary': '92, 93 people answered correctly that the index on k added with the index on a equals three. those who answered b had the right thought but were multiplying by a quadratic polynomial instead of something that goes on forever.', 'duration': 25.927, 'max_score': 1454.574, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1454574.jpg'}, {'end': 1689.053, 'src': 'embed', 'start': 1649.026, 'weight': 2, 'content': [{'end': 1652.348, 'text': "So all we're doing is taking these values and multiplying them by something.", 'start': 1649.026, 'duration': 3.322}, {'end': 1655.429, 'text': "It's not x, but it's something that we have successive powers of.", 'start': 1652.428, 'duration': 3.001}, {'end': 1663.253, 'text': 'And what that means is that if we do that to two different arrays, to two different collections of values,', 'start': 1656.09, 'duration': 7.163}, {'end': 1667.876, 'text': 'multiplying the two results corresponds to convolving the original terms.', 'start': 1663.253, 'duration': 4.623}, {'end': 1676.925, 'text': 'So, for example, right when we take the polynomial that was generated by k and we multiply it by the polynomial that was generated by these a terms,', 'start': 1668.84, 'duration': 8.085}, {'end': 1682.168, 'text': 'we get a new polynomial where each term is a convolution between a and k.', 'start': 1676.925, 'duration': 5.243}, {'end': 1687.151, 'text': 'Same thing would happen if we replaced x with something like this e to the two pi i stuff.', 'start': 1682.168, 'duration': 4.983}, {'end': 1689.053, 'text': 'All of the terms from one.', 'start': 1687.612, 'duration': 1.441}], 'summary': 'Multiplying values by successive powers corresponds to convolving original terms.', 'duration': 40.027, 'max_score': 1649.026, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1649026.jpg'}, {'end': 1759.095, 'src': 'embed', 'start': 1730.803, 'weight': 1, 'content': [{'end': 1734.806, 'text': "but at a high level I want to be able to describe what information about the image it's pulling out.", 'start': 1730.803, 'duration': 4.003}, {'end': 1740.67, 'text': 'So Tom, the cat, is a rather soft image with not a lot of variation happening from left to right.', 'start': 1735.527, 'duration': 5.143}, {'end': 1749.736, 'text': 'So if I go through and I try to ask is there very fine detail, a lot of high frequency changes happening as we move from left to right?', 'start': 1741.17, 'duration': 8.566}, {'end': 1753.259, 'text': "there's not necessarily going to be a lot of information there.", 'start': 1749.736, 'duration': 3.523}, {'end': 1759.095, 'text': "Now, I don't know why this is taking as long as it is to process.", 'start': 1753.991, 'duration': 5.104}], 'summary': 'Describing low variation in the image, lacking fine detail. processing delay.', 'duration': 28.292, 'max_score': 1730.803, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1730803.jpg'}], 'start': 1346.79, 'title': 'Multiplying polynomials, convolutions, and fourier transforms in image processing', 'summary': 'Introduces the concept of multiplying polynomials and its relevance to image processing, along with the relationship between convolutions and fourier transforms. a live quiz outcome showed 92-93 people answered correctly. the demonstration using a cat image showcases the usage of these concepts.', 'chapters': [{'end': 1473.359, 'start': 1346.79, 'title': 'Multiplying polynomials and image processing connection', 'summary': 'Introduces the concept of multiplying polynomials and its relevance to image processing, with a brief explanation of the process and a live quiz outcome, where 92-93 people answered correctly.', 'duration': 126.569, 'highlights': ['The chapter introduces the concept of multiplying polynomials and its relevance to image processing The chapter starts by discussing the seemingly unrelated topic of multiplying polynomials and how it relates to image processing.', 'Live quiz outcome: 92-93 people answered correctly During a live quiz, 92-93 people correctly answered a question related to the multiplication of polynomials and infinite power series.', 'Explanation of the process of multiplying polynomials The process of multiplying polynomials is briefly explained, including the consideration of specific terms and the use of foil method.']}, {'end': 1771.985, 'start': 1474.219, 'title': 'Understanding convolutions and fourier transforms', 'summary': 'Explains the process of multiplying polynomials and the relationship between convolutions and fourier transforms, showing how they are utilized to extract frequency information from images, with a demonstration using a cat image.', 'duration': 297.766, 'highlights': ['Convolution involves multiplying polynomials and can be expressed as a convolution between the terms in the original polynomials, requiring one of the polynomials to be flipped around before the marching process, heavily related to the concept of convolutions and Fourier transforms. Explanation of the process of multiplying polynomials and the relationship between convolutions and Fourier transforms.', 'Using Fourier transforms to extract frequency information from an image, demonstrated by analyzing the Fourier spectrum of a cat image to determine the level of variation and high-frequency changes present. Demonstration of utilizing Fourier transforms to extract frequency information from images, with an analysis of a cat image to understand the level of variation and high-frequency changes.', 'Multiplying two arrays corresponds to convolving the original terms, and replacing x with a term like e to the two pi i results in convolving the terms from one with the terms from the other, showing the relationship between multiplication and convolution. Explanation of the relationship between multiplication and convolution, showcasing how multiplying two arrays corresponds to convolving the original terms, and the impact of replacing x with a term like e to the two pi i.']}], 'duration': 425.195, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1346790.jpg', 'highlights': ['Live quiz outcome: 92-93 people answered correctly', 'Demonstration of utilizing Fourier transforms to extract frequency information from images, with an analysis of a cat image to understand the level of variation and high-frequency changes', 'Explanation of the relationship between multiplication and convolution, showcasing how multiplying two arrays corresponds to convolving the original terms, and the impact of replacing x with a term like e to the two pi i', 'The chapter introduces the concept of multiplying polynomials and its relevance to image processing, discussing the seemingly unrelated topic of multiplying polynomials and how it relates to image processing', 'Convolution involves multiplying polynomials and can be expressed as a convolution between the terms in the original polynomials, requiring one of the polynomials to be flipped around before the marching process, heavily related to the concept of convolutions and Fourier transforms']}, {'end': 2122.655, 'segs': [{'end': 1797.311, 'src': 'embed', 'start': 1772.486, 'weight': 1, 'content': [{'end': 1779.651, 'text': "So what we could do instead, I'll go ahead and get rid of his convolution term there, is take a more interesting image.", 'start': 1772.486, 'duration': 7.165}, {'end': 1782.994, 'text': "So I've prepared one that's just a herd of zebra.", 'start': 1780.632, 'duration': 2.362}, {'end': 1789.263, 'text': "So this herd of zebra, we've got lots of stripes as you move from the left to the right.", 'start': 1785.199, 'duration': 4.064}, {'end': 1793.267, 'text': "Kernel, by the way, I'm just gonna prepare ourselves a Gaussian kernel here.", 'start': 1789.904, 'duration': 3.363}, {'end': 1797.311, 'text': "Let's get a Gaussian, it's maybe this nine by nine thing.", 'start': 1793.787, 'duration': 3.524}], 'summary': 'Removing the convolution term, using a gaussian kernel for an interesting image of a herd of zebra with lots of stripes.', 'duration': 24.825, 'max_score': 1772.486, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1772486.jpg'}, {'end': 1865.08, 'src': 'embed', 'start': 1839.45, 'weight': 3, 'content': [{'end': 1846.62, 'text': 'But now, if we do the same thing, we plot this 1D Fourier spectrum, but we do it on the convolved version of the image.', 'start': 1839.45, 'duration': 7.17}, {'end': 1851.305, 'text': "what we get looks very similar to the original, but notice how what's happening around the edges have been pushed down.", 'start': 1846.62, 'duration': 4.685}, {'end': 1857.832, 'text': "And that's a way of saying all the high frequency stuff happening with the image, all that finer detail, is no longer there.", 'start': 1852.006, 'duration': 5.826}, {'end': 1861.976, 'text': "So now it's behaving a lot more like the image of the cat that was soft all around.", 'start': 1858.473, 'duration': 3.503}, {'end': 1865.08, 'text': "And there's actually something much more specific.", 'start': 1863.338, 'duration': 1.742}], 'summary': 'Convolution of image reduces high frequency details, making it softer.', 'duration': 25.63, 'max_score': 1839.45, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1839450.jpg'}, {'end': 1920.119, 'src': 'embed', 'start': 1895.836, 'weight': 0, 'content': [{'end': 1901.722, 'text': 'And remember how, a little bit earlier I was talking about the computational complexity here how, as we have a really big kernel,', 'start': 1895.836, 'duration': 5.886}, {'end': 1903.183, 'text': 'it requires a lot of multiplications?', 'start': 1901.722, 'duration': 1.461}, {'end': 1909.629, 'text': 'It turns out if instead we first translate things to describing it in terms of the frequencies of the image,', 'start': 1903.844, 'duration': 5.785}, {'end': 1911.971, 'text': 'where convolutions just like multiplication.', 'start': 1909.629, 'duration': 2.342}, {'end': 1918.057, 'text': 'so we can just multiply the two functions as functions, just pixel by pixel, and then we take it back.', 'start': 1911.971, 'duration': 6.086}, {'end': 1920.119, 'text': 'it gives us a much faster way to do this operation.', 'start': 1918.057, 'duration': 2.062}], 'summary': 'Translating to image frequencies speeds up operations by reducing computational complexity.', 'duration': 24.283, 'max_score': 1895.836, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1895836.jpg'}, {'end': 2096.049, 'src': 'embed', 'start': 2066.784, 'weight': 4, 'content': [{'end': 2071.888, 'text': 'and we tweak some data, we tweak the kernel that lets you do qualitatively different things with your data.', 'start': 2066.784, 'duration': 5.104}, {'end': 2076.27, 'text': 'So in this case, it let us do things like blurring, like sharpening, like edge detecting.', 'start': 2072.348, 'duration': 3.922}, {'end': 2080.411, 'text': 'And then the other thing is that when you represent your data in distinct ways.', 'start': 2076.29, 'duration': 4.121}, {'end': 2086.899, 'text': 'so in this context, if we were taking the Fourier information, taking this frequency information sometimes,', 'start': 2080.411, 'duration': 6.488}, {'end': 2090.983, 'text': "that actually lets you do things much faster, even though you're doing under the hood.", 'start': 2086.899, 'duration': 4.084}, {'end': 2096.049, 'text': "well, not under the hood, like in the abstract sense, you're doing the same operations on the data.", 'start': 2090.983, 'duration': 5.066}], 'summary': 'Tweaking data and kernel enables qualitative changes; fourier info allows faster operations.', 'duration': 29.265, 'max_score': 2066.784, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA2066784.jpg'}], 'start': 1772.486, 'title': 'Image processing and convolution', 'summary': 'Discusses the use of gaussian kernel for fourier analysis and image convolution, demonstrating computational efficiency and faster operations, and explains the basics of image convolutions for operations like blurring, sharpening, and edge detection.', 'chapters': [{'end': 1817.172, 'start': 1772.486, 'title': 'Image processing with gaussian kernel', 'summary': 'Discusses using a gaussian kernel for fourier analysis of an image to identify the frequency of stripes in a zebra herd image.', 'duration': 44.686, 'highlights': ['The Fourier analysis of the image shows higher values as you move away from 0, peaking at almost 50, corresponding to the frequency of the stripes in the zebra herd.', 'Preparing a Gaussian kernel with a nine by nine size for further image analysis.', 'Discussing the use of an interesting image, a herd of zebras, with lots of stripes for Fourier analysis.']}, {'end': 2013.541, 'start': 1818.865, 'title': 'Image convolution and fourier transform', 'summary': 'Discusses the impact of convolving an image with a gaussian kernel, demonstrating how it affects the 1d and 2d fourier spectra, and the computational efficiency gained by representing information in terms of frequencies, ultimately leading to a faster operation.', 'duration': 194.676, 'highlights': ['The Fourier transform of the Gaussian kernel results in a Gaussian curve, where a narrow Gaussian yields a broad Fourier transform and vice versa. The relationship between the Gaussian kernel and its Fourier transform is highlighted, showing how the width of the Gaussian kernel affects the resulting Fourier transform.', 'Describing images in terms of frequencies allows for faster convolutions through simple pixel-by-pixel multiplication, leading to computational efficiency. The advantage of representing image information in frequency terms is explained, emphasizing the computational efficiency gained by using this approach.', 'The impact of convolving an image with a Gaussian kernel is demonstrated through the changes in the 1D and 2D Fourier spectra, showcasing the reduction of high-frequency details and the concentration of values towards the center. The effect of convolving an image with a Gaussian kernel on the Fourier spectra is illustrated, highlighting the reduction of high-frequency details and the concentration of values towards the center.']}, {'end': 2122.655, 'start': 2013.541, 'title': 'Understanding image convolutions', 'summary': 'Explains the basics of image convolutions, highlighting the use of general convolution principles to perform operations like blurring, sharpening, and edge detecting, and the potential for faster processing when utilizing different data representations.', 'duration': 109.114, 'highlights': ['The general convolution principle allows for qualitative changes in data, such as blurring, sharpening, and edge detection.', 'Utilizing different data representations, like Fourier information, can lead to faster processing of operations on the data.', 'The chapter emphasizes the importance of understanding the basics of image convolutions and the potential for applying general convolution principles to achieve various image processing operations.']}], 'duration': 350.169, 'thumbnail': 'https://coursnap.oss-ap-southeast-1.aliyuncs.com/video-capture/8rrHTtUzyZA/pics/8rrHTtUzyZA1772486.jpg', 'highlights': ['Demonstrating computational efficiency through frequency-based image convolutions.', 'Preparing a Gaussian kernel with a nine by nine size for further image analysis.', 'Describing images in terms of frequencies allows for faster convolutions.', 'The impact of convolving an image with a Gaussian kernel is demonstrated through changes in the 1D and 2D Fourier spectra.', 'The general convolution principle allows for qualitative changes in data, such as blurring, sharpening, and edge detection.']}], 'highlights': ['The launch of a new official course website invites worldwide attendance.', 'Discord is used for discussions and interactions with professors during the lecture.', 'The broader applications and potential for nuanced image manipulation using convolution are emphasized.', 'The use of Gaussian kernels for weighted pixel averaging is explained, highlighting its effectiveness in achieving a more natural blur effect.', 'Nuances like edge cases are addressed, ensuring proper handling and avoiding disproportionately dark outcomes.', 'Image blurring is achieved through convolution using a 3x3 grid with weighted pixel values, creating a blurred effect.', 'The convolve function applies a kernel to an image, producing a blurrier image.', 'Larger kernels can significantly increase image convolution processing time.', "The Gaussian kernel's impact on image blurriness is a key focus of the chapter.", 'The library provides access to documentation for available functions and kernel manipulation potential.', 'Introduction of a live quiz promotes audience engagement and interactive learning.', 'The role of convolutional neural networks in learning specific kernels for extracting meaningful information from images, emphasizing its importance as a building block in image processing.', 'The challenge of reconstructing the original image from a blurred one is addressed, with emphasis on potential loss of information and the rare possibility of exact image recovery through methods like Fast Fourier transform.', 'The sum of all kernel values equals 1, leading to a weighted average effect and preservation of the original color in monochromatic areas.', 'Explanation and visualization of the properties of the Sobel kernel for edge detection, highlighting its asymmetry and ability to capture micro variations in images.', 'The misconception regarding the outcome of convolution with kernels is highlighted, emphasizing the similarity with mathematical convolution and the impact on image properties.', 'Live quiz outcome: 92-93 people answered correctly', 'Demonstration of utilizing Fourier transforms to extract frequency information from images, with an analysis of a cat image to understand the level of variation and high-frequency changes', 'Explanation of the relationship between multiplication and convolution, showcasing how multiplying two arrays corresponds to convolving the original terms, and the impact of replacing x with a term like e to the two pi i', 'The chapter introduces the concept of multiplying polynomials and its relevance to image processing, discussing the seemingly unrelated topic of multiplying polynomials and how it relates to image processing', 'Convolution involves multiplying polynomials and can be expressed as a convolution between the terms in the original polynomials, requiring one of the polynomials to be flipped around before the marching process, heavily related to the concept of convolutions and Fourier transforms', 'Demonstrating computational efficiency through frequency-based image convolutions.', 'Preparing a Gaussian kernel with a nine by nine size for further image analysis.', 'Describing images in terms of frequencies allows for faster convolutions.', 'The impact of convolving an image with a Gaussian kernel is demonstrated through changes in the 1D and 2D Fourier spectra.', 'The general convolution principle allows for qualitative changes in data, such as blurring, sharpening, and edge detection.']}