WEBVTT 00:00.660 --> 00:03.150 -: Hello, welcome back to the course on Deep Learning. 00:03.150 --> 00:04.620 In the previous tutorial, we found out 00:04.620 --> 00:07.320 what convolution neural networks are all about. 00:07.320 --> 00:11.070 And today we're going to dive into step one convolution. 00:11.070 --> 00:15.000 So this is the convolution function. 00:15.000 --> 00:17.130 I know we tried to survey from mathematics 00:17.130 --> 00:19.689 and keep things intuitive, but I couldn't help 00:19.689 --> 00:23.160 but share this formula for you because it is so simple. 00:23.160 --> 00:26.250 A convolution is basically a combined integration 00:26.250 --> 00:27.300 of two functions 00:27.300 --> 00:31.320 and it shows you how one function modifies the other 00:31.320 --> 00:32.730 or modifies the shape of the other. 00:32.730 --> 00:35.250 And if you've done any signal processing 00:35.250 --> 00:36.390 or electrical engineering 00:36.390 --> 00:39.570 or profession where signal processing is required 00:39.570 --> 00:41.160 you would've inevitably come 00:41.160 --> 00:42.420 across the convolution function. 00:42.420 --> 00:43.830 It is quite popular. 00:43.830 --> 00:48.000 Now, once again, we're going to keep the mathematics light 00:48.000 --> 00:49.500 or keep them separate. 00:49.500 --> 00:51.090 And if you'd like to get 00:51.090 --> 00:54.930 into the math behind the convolution neural networks 00:54.930 --> 00:57.277 a great additional read is 00:57.277 --> 01:02.277 "Introduction to Convolution Neural Networks," by Jianxin Wu 01:02.340 --> 01:05.790 who is a professor at the Nanjing University in China. 01:05.790 --> 01:08.820 This paper was published literally days ago 01:08.820 --> 01:12.780 like five or six days ago, and it is oriented specifically 01:12.780 --> 01:14.670 at people who are starting out at beginners 01:14.670 --> 01:17.430 who are getting to know convolutional neural networks. 01:17.430 --> 01:20.160 So the mathematics there should be accessible. 01:20.160 --> 01:23.700 I actually emailed Professor Jianxin Wu 01:23.700 --> 01:28.260 and yeah he said his whole goal is to make 01:28.260 --> 01:30.630 break the complex things down 01:30.630 --> 01:33.360 so that people who are new to this field can understand. 01:33.360 --> 01:34.950 And also he mentioned 01:34.950 --> 01:38.970 that he's got some materials available on his homepage. 01:38.970 --> 01:40.560 So if you, in the URL 01:40.560 --> 01:42.900 if you just remove the last two parts and you just go 01:42.900 --> 01:47.900 to like /wjx to that part, that's his homepage 01:47.970 --> 01:50.820 and you'll be able to find more additional tutorials 01:50.820 --> 01:53.400 and materials which haven't been published as papers 01:53.400 --> 01:55.920 but he uses them in his tutorials. 01:55.920 --> 01:57.930 So you might find those useful. 01:57.930 --> 02:01.530 So browser around there if you'd like to get an introduction 02:01.530 --> 02:04.800 into the mathematics behind convolutional neural networks 02:04.800 --> 02:08.550 and kind of build a solid base around that area. 02:08.550 --> 02:09.990 But we're going to move on 02:09.990 --> 02:12.540 and we're gonna talk about the convolution. 02:12.540 --> 02:16.500 So what is the convolution in intuitive terms? 02:16.500 --> 02:19.020 Here on the left we've got an input image as we discussed. 02:19.020 --> 02:20.970 That's how we're going to look at images 02:20.970 --> 02:22.890 just ones and zeros to simplify things. 02:22.890 --> 02:25.050 And you can see the smiley face there. 02:25.050 --> 02:26.370 There we've got a feature detector. 02:26.370 --> 02:28.800 So a feature detectors are three by three matrix. 02:28.800 --> 02:30.120 Does it have to be three by three? 02:30.120 --> 02:31.890 No, it doesn't. 02:31.890 --> 02:34.923 Alex Net I think uses seven by seven. 02:35.850 --> 02:38.700 And then some other, one of those other famous ones 02:38.700 --> 02:40.750 uses like five by five feature detectors. 02:41.640 --> 02:44.280 They can be different, but usually you'll see 02:44.280 --> 02:46.020 that they are three by three 02:46.020 --> 02:49.380 and they are reasons to make them three by three. 02:49.380 --> 02:51.780 So we eventually stick to the convention 02:51.780 --> 02:54.600 of a having a three by three feature detector. 02:54.600 --> 02:57.540 Also, the feature detectors called these are important terms 02:57.540 --> 02:58.740 because you might come across them. 02:58.740 --> 03:01.800 They're many different terms for the feature detector 03:01.800 --> 03:04.140 but the most common ones are feature detector. 03:04.140 --> 03:07.050 Or you might hear it being called kernel 03:07.050 --> 03:09.510 or you might hear it being called filter. 03:09.510 --> 03:11.520 So in this course we're gonna be using 03:11.520 --> 03:14.430 either filter or feature detector interchangeably 03:14.430 --> 03:17.040 but just bear in mind that it has those names. 03:17.040 --> 03:20.040 And a convolution operation is signified 03:20.040 --> 03:25.040 by an X in a circle, just as you saw in the formulas before. 03:25.770 --> 03:29.820 And here what happens is on an intuitive level 03:29.820 --> 03:31.770 or just to think of it 03:31.770 --> 03:33.750 in terms of what is actually happening in the background 03:33.750 --> 03:35.010 rather than the mathematics. 03:35.010 --> 03:38.340 Well, you take this feature detector or filter 03:38.340 --> 03:40.680 and you put it on your image like you see on the left. 03:40.680 --> 03:44.490 So you cover the, for instance, in this case 03:44.490 --> 03:48.420 the top left corner, the nine pixels in the top left corner 03:48.420 --> 03:53.420 and you basically multiply each value by each value. 03:54.120 --> 03:54.990 So respective value. 03:54.990 --> 03:58.770 So the top zero by the top left value 03:58.770 --> 04:01.680 by the top left value, then basically 04:01.680 --> 04:03.750 position number one one by position number one, one 04:03.750 --> 04:08.670 position by number or 01 by 01, 02 by 02 and so on. 04:08.670 --> 04:13.230 So just it's element wise, multiplication of these matrices. 04:13.230 --> 04:14.490 And then you add up the results. 04:14.490 --> 04:16.710 So in this case, nothing matches up. 04:16.710 --> 04:19.980 So always it's always either a zero by zero, zero by one. 04:19.980 --> 04:21.540 So the result is zero. 04:21.540 --> 04:23.940 Here you can see that one of them matched up, 04:23.940 --> 04:26.550 the one on the left matched up 04:26.550 --> 04:28.140 and therefore we got a one here. 04:28.140 --> 04:30.603 Nothing matched up, nothing matched up, nothing matched up. 04:30.603 --> 04:32.130 Then we move on to the next row. 04:32.130 --> 04:35.610 So, and the step at which we're moving 04:35.610 --> 04:38.610 this whole filter is called the stride. 04:38.610 --> 04:40.500 So here we have a stride of one pixel. 04:40.500 --> 04:42.270 So here you can see again something matched up, 04:42.270 --> 04:44.130 the bottom right corner matched up. 04:44.130 --> 04:48.300 Again, stride, bottom one in the middle, matched up 04:48.300 --> 04:50.100 here top right one matched up. 04:50.100 --> 04:50.970 Then nothing matched up. 04:50.970 --> 04:52.170 The stride is one. 04:52.170 --> 04:56.340 You can change the stride, you can make it one, two 04:56.340 --> 04:58.920 you can make it three, whatever you like. 04:58.920 --> 05:01.740 The conventionally, the one that works well 05:01.740 --> 05:02.790 is usually at two. 05:02.790 --> 05:04.560 So that's what people stick to. 05:04.560 --> 05:07.590 And we'll talk about what the stride is 05:07.590 --> 05:09.510 towards the end of the tutorial. 05:09.510 --> 05:12.450 So here we've got, we're matching up, so we just keep, oh 05:12.450 --> 05:13.590 here you can see we've got a two 05:13.590 --> 05:15.750 because two of them matched up 05:15.750 --> 05:17.730 and so on and so on and so on. 05:17.730 --> 05:18.563 And there we go. 05:18.563 --> 05:20.070 There's another one that matched up. 05:21.390 --> 05:22.503 There we go. 05:23.700 --> 05:24.840 And there we're done. 05:24.840 --> 05:28.800 So what's have we created, right? 05:28.800 --> 05:32.010 Couple of important things here. 05:32.010 --> 05:35.280 The image on the right is called a feature map 05:35.280 --> 05:36.660 also has several terms. 05:36.660 --> 05:40.980 It also can be called, sometimes it can bold feature. 05:40.980 --> 05:44.190 So when you apply convolution operator to something, 05:44.190 --> 05:47.070 it doesn't become convoluted it becomes convolved. 05:47.070 --> 05:51.120 And yes, sometimes I think to myself 05:51.120 --> 05:54.936 in the wrong way, but it's the correct term it's convolved 05:54.936 --> 05:56.760 it's a convolved feature, or it can also be called 05:56.760 --> 05:58.050 activation map. 05:58.050 --> 06:01.080 But we are gonna be calling it a feature map in this course. 06:01.080 --> 06:03.480 So it can be called any one of those things. 06:03.480 --> 06:06.330 And what have we done here? 06:06.330 --> 06:09.900 Well, as you can see, we've reduced the size of the image. 06:09.900 --> 06:10.733 That's number one. 06:10.733 --> 06:12.720 And that's the important thing I wanted to mention 06:12.720 --> 06:15.087 about your input image 06:15.087 --> 06:17.250 and the featured detect and the stride, right? 06:17.250 --> 06:18.690 If you have a stride of one 06:18.690 --> 06:20.040 you can see the image reduced a bit. 06:20.040 --> 06:21.330 But if you have a stride of two 06:21.330 --> 06:23.190 the image is going to reduce more. 06:23.190 --> 06:25.590 So the feature app is gonna be even smaller. 06:25.590 --> 06:29.730 And that's an very important function 06:29.730 --> 06:30.960 of the feature detector 06:30.960 --> 06:35.960 of this whole convolution step is to make the image smaller 06:36.090 --> 06:39.070 because it'll be easier to process it 06:40.140 --> 06:45.140 and it'll be just faster and it'll be just faster. 06:46.890 --> 06:49.200 Because imagine like here we've got a what 06:49.200 --> 06:51.870 a seven by seven image. 06:51.870 --> 06:55.710 But imagine if you have a proper photo, right? 06:55.710 --> 06:59.280 Or you have like a 256 by 256 pixel image. 06:59.280 --> 07:04.280 That's a huge number of pixel 256 squared. 07:04.410 --> 07:07.230 Or like, let's say you have a 300 by 300 pixels, 07:07.230 --> 07:09.930 so we don't get confused with the RGB 256 07:09.930 --> 07:12.960 let's just say like we have a 300 by 300 image 07:12.960 --> 07:14.760 in terms of size in pixels 07:14.760 --> 07:17.400 then you have 300 squared number of pixels. 07:17.400 --> 07:19.170 That's a huge number. 07:19.170 --> 07:20.890 And therefore feature detectors 07:22.020 --> 07:24.540 will reduce the size of the image, 07:24.540 --> 07:27.750 and therefore stride of two is actually beneficial. 07:27.750 --> 07:29.970 But then the question is, do we lose information? 07:29.970 --> 07:31.950 Are we losing information 07:31.950 --> 07:34.500 when we're applying the feature detector? 07:34.500 --> 07:37.320 Well, some information we are losing, of course 07:37.320 --> 07:40.710 because we have less values in our resulting matrix. 07:40.710 --> 07:43.350 But at the same time, the purpose of the feature detector 07:43.350 --> 07:44.820 is to detect certain features 07:44.820 --> 07:48.600 certain parts of the image that are integral. 07:48.600 --> 07:51.030 And so for instance, if you think about it this way 07:51.030 --> 07:53.970 like the feature detector has a certain pattern on it 07:53.970 --> 07:56.340 the highest number in your feature map 07:56.340 --> 07:57.960 is when that pattern matches up. 07:57.960 --> 07:59.820 In fact, the highest number you can get 07:59.820 --> 08:02.220 is in an now simplified example, 08:02.220 --> 08:05.580 is when the feature matches exactly. 08:05.580 --> 08:07.950 And you can see with that number four, 08:07.950 --> 08:10.560 we have in our feature map, that's exactly, 08:10.560 --> 08:12.150 so if you look over here, 08:12.150 --> 08:15.780 that's exactly where this feature detector, 08:15.780 --> 08:19.110 because there's only four ones in it matched perfectly. 08:19.110 --> 08:21.480 So you can see this part over here. 08:21.480 --> 08:23.430 So the feature was detected here. 08:23.430 --> 08:27.280 And as we discuss at the very start of this section 08:28.350 --> 08:31.530 that is features is how we see things, 08:31.530 --> 08:33.090 is how we recognize things. 08:33.090 --> 08:37.410 We don't look at every single pixel, so to speak 08:37.410 --> 08:40.380 in what we see on an image or in real life. 08:40.380 --> 08:41.820 We don't look at every single pic. 08:41.820 --> 08:45.298 We look at features, we look at the nose, the hats 08:45.298 --> 08:50.298 the feather, the the eyes under or the little black marks 08:51.180 --> 08:54.210 under the cheaters eyes to distinguish it 08:54.210 --> 08:57.648 between a cheater and a leopard, or the shape of the train, 08:57.648 --> 09:00.390 to distinguish it between a bullet train and a normal train 09:00.390 --> 09:01.223 and so on. 09:01.223 --> 09:03.240 So we don't look at everything, we look at features 09:03.240 --> 09:04.650 and that's what we are preserving 09:04.650 --> 09:08.520 and that's what the feature map helps us preserve actually. 09:08.520 --> 09:12.600 That's what it allows us to bring forward 09:12.600 --> 09:15.270 and get rid of all of the unnecessary things 09:15.270 --> 09:17.520 that even as humans, we don't process this. 09:17.520 --> 09:20.873 So much information going into your eyes 09:20.873 --> 09:24.270 at any given time, like gigabytes of information 09:24.270 --> 09:27.870 if you look at every single dot, if not terabytes 09:27.870 --> 09:30.420 of information going into your eyes per second. 09:30.420 --> 09:32.730 And still we're able to process that 09:32.730 --> 09:36.000 because we get rid of what is unnecessary only focus 09:36.000 --> 09:37.020 on the important features. 09:37.020 --> 09:38.940 Features are important to us. 09:38.940 --> 09:42.240 And that is exactly what the feature map does. 09:42.240 --> 09:44.670 So now moving on 09:44.670 --> 09:49.500 this is our input image and you, we create a feature map. 09:49.500 --> 09:50.730 So the front one 09:50.730 --> 09:52.740 let's say the front one is the one we just created 09:52.740 --> 09:54.300 but then how come there is many of them? 09:54.300 --> 09:56.980 But we create multiple feature maps 09:58.230 --> 10:00.600 because we use different filters, right? 10:00.600 --> 10:03.000 And that's another way that we preserve 10:03.000 --> 10:03.840 lots of the information. 10:03.840 --> 10:05.850 So we don't just have one feature map 10:05.850 --> 10:08.320 we look for certain features and then 10:09.300 --> 10:12.330 or basically the network decides through its training. 10:12.330 --> 10:13.410 And this is something we'll discuss 10:13.410 --> 10:14.460 towards the end of this section. 10:14.460 --> 10:15.570 Through its training 10:15.570 --> 10:18.930 it decides which features are important 10:18.930 --> 10:21.660 for certain types or certain categories, 10:21.660 --> 10:22.620 and it looks for them. 10:22.620 --> 10:24.690 And therefore it'll have different filters. 10:24.690 --> 10:26.160 And we'll talk about filters just now 10:26.160 --> 10:27.810 but basically it'll apply these filters. 10:27.810 --> 10:29.550 So to get this feature map 10:29.550 --> 10:32.550 it applied a filter like the one we saw. 10:32.550 --> 10:33.550 But then to get this feature map it 10:33.550 --> 10:35.107 apply a different filter to get this feature map 10:35.107 --> 10:37.533 and apply a different filter and so on. 10:38.370 --> 10:43.370 And so basically it just creates these feature maps. 10:43.620 --> 10:45.330 And actually that's why personally 10:45.330 --> 10:49.590 I think the term feature detector is better than filter. 10:49.590 --> 10:52.950 So remember over here we have this filter, 10:52.950 --> 10:55.080 which we also can call a feature detector. 10:55.080 --> 10:57.780 Well actually the word feature detector 10:57.780 --> 10:59.460 I think is better suited. 10:59.460 --> 11:03.334 And the reason for that is that's what the purpose is. 11:03.334 --> 11:06.510 We don't wanna just filter out our image, 11:06.510 --> 11:08.895 but even though that's the whole 11:08.895 --> 11:10.230 that's the same just question of terminology. 11:10.230 --> 11:12.503 But basically we want to detect features. 11:12.503 --> 11:14.980 In this layer, we're going to, 11:14.980 --> 11:18.090 or in this feature map we've detected 11:18.090 --> 11:20.180 where certain features are in the image. 11:20.180 --> 11:21.810 In this feature map we've detected 11:21.810 --> 11:23.100 where certain other features are, 11:23.100 --> 11:25.380 where a certain specific feature is located. 11:25.380 --> 11:27.840 And this feature map we've detected 11:27.840 --> 11:31.410 where a certain other feature is located on the image. 11:31.410 --> 11:33.390 So that's what we are doing. 11:33.390 --> 11:34.650 And let's have a look at a couple of examples. 11:34.650 --> 11:39.650 So here we're using, and this is from gib.org 11:40.530 --> 11:44.430 their documentation, it's a free like of kind of tool 11:44.430 --> 11:48.690 like paint and you can use it to adjust your images 11:48.690 --> 11:49.560 or work with your images. 11:49.560 --> 11:52.500 But basically they have some valuable examples 11:52.500 --> 11:53.580 in their documentation. 11:53.580 --> 11:57.150 And here they have a picture of the Taj Mahal 11:57.150 --> 11:59.850 and you can choose which filter you want to apply. 11:59.850 --> 12:01.560 So if you download this program 12:01.560 --> 12:02.790 and you upload a photo into it 12:02.790 --> 12:06.660 and then you can actually start a convolution matrix 12:06.660 --> 12:07.590 and apply filters 12:07.590 --> 12:11.730 and you will see that these things, these convolution matrix 12:11.730 --> 12:15.240 I actually applied in image processing and design and so on. 12:15.240 --> 12:16.830 So let's have a look at what we get. 12:16.830 --> 12:19.470 So if we apply this filter five in the middle 12:19.470 --> 12:21.660 minus one, minus one, minus one minus one 12:21.660 --> 12:23.790 you can see that it sharpens the image. 12:23.790 --> 12:26.880 And yeah, so this is 12:26.880 --> 12:28.980 it's quite intuitive if you think of it. 12:28.980 --> 12:31.830 So five is the pixel, the main pixel, 12:31.830 --> 12:36.570 like in the middle of the filter or the feature detector. 12:36.570 --> 12:38.280 And then minus one, minus one, minus one 12:38.280 --> 12:42.390 just you kind of like reduces the pixels around it. 12:42.390 --> 12:47.010 In an intuitive sense then blur. 12:47.010 --> 12:50.400 So basically it takes equal significance 12:50.400 --> 12:53.340 gives equal significance to all of the pixels 12:53.340 --> 12:54.600 around the one in the center 12:54.600 --> 12:57.060 and therefore it combines them together 12:57.060 --> 12:59.070 and you get a blur edge in hands. 12:59.070 --> 13:02.610 So here you can see that minus one and one 13:02.610 --> 13:03.900 and then you get zeros, right? 13:03.900 --> 13:08.900 So you delete, remove the pixels around the main one 13:09.300 --> 13:12.120 in the middle and you only keep this one at a minus one. 13:12.120 --> 13:13.020 And it gives you an edge. 13:13.020 --> 13:16.320 This one's a bit harder to understand how it works. 13:16.320 --> 13:18.360 Like probably harder just to think 13:18.360 --> 13:20.940 of it intuitively edge detect, right? 13:20.940 --> 13:23.550 So this one probably makes more sense, right? 13:23.550 --> 13:27.580 You take the middle one, you reduce the middle one 13:29.196 --> 13:32.520 probably like the strength of the middle pixel. 13:32.520 --> 13:37.110 And then you look for the ones, you look for these ones 13:37.110 --> 13:42.090 you increase the strength of the ones around them. 13:42.090 --> 13:43.440 So you have the ones there. 13:44.670 --> 13:47.190 Yeah, so that gives you like an edge text 13:47.190 --> 13:48.990 and you can see what you get there. 13:48.990 --> 13:50.700 And boss another one. 13:50.700 --> 13:55.700 So the key here is that it's asymmetrical 13:55.710 --> 13:58.170 and you can see the image becomes asymmetrical as well. 13:58.170 --> 14:01.410 So you've got like that kind of feeling 14:01.410 --> 14:03.750 that it's standing out towards you. 14:03.750 --> 14:06.570 And that's what you get when you have like minuses here 14:06.570 --> 14:08.460 And pluses here, again, this is very 14:08.460 --> 14:10.050 this is getting a bit technical now, 14:10.050 --> 14:11.460 but at least we can get 14:11.460 --> 14:12.810 some kind of intuitive understanding. 14:12.810 --> 14:14.190 Lets just go quickly through them again. 14:14.190 --> 14:18.360 So there's sharpen, there's blur, there's edge enhance 14:18.360 --> 14:20.880 there's edge detect, there's emboss. 14:20.880 --> 14:23.010 And so as you can see, these are great examples 14:23.010 --> 14:27.300 of the same image, but we are getting feature maps. 14:27.300 --> 14:29.220 So we use different feature detectors 14:29.220 --> 14:31.860 to get different feature maps of the same image. 14:31.860 --> 14:35.610 And therefore now we have lots of this versions 14:35.610 --> 14:39.840 of this image where in each one we've tried 14:39.840 --> 14:41.160 to detect certain things. 14:41.160 --> 14:45.049 These terms, they're not applicable to us. 14:45.049 --> 14:47.460 Like I said, like in boss is probably not applicable 14:47.460 --> 14:50.100 to us in terms of convolutional neural networks. 14:50.100 --> 14:51.630 But edge detect that's important. 14:51.630 --> 14:53.280 We want to detect the edges, 14:53.280 --> 14:56.520 edge enhance probably not, blur, sharpen. 14:56.520 --> 14:58.530 So certain things like edge detect 14:58.530 --> 15:02.460 is probably the most important one for our type of work. 15:02.460 --> 15:04.140 And in terms of understanding, 15:04.140 --> 15:06.360 like computers they will decide for themselves 15:06.360 --> 15:07.980 their neural network will decide for itself 15:07.980 --> 15:09.030 what's important, what's not. 15:09.030 --> 15:12.900 And it probably won't be even recognizable to the human eye. 15:12.900 --> 15:14.910 You won't be able to understand what those features mean 15:14.910 --> 15:16.800 but the computer will decide. 15:16.800 --> 15:19.830 And that's the beauty that of neural networks 15:19.830 --> 15:23.190 that they can process so many different things 15:23.190 --> 15:26.640 and understand without even having that intuition 15:26.640 --> 15:29.430 without having that explanation why 15:29.430 --> 15:32.400 they will understand which features are important to them 15:32.400 --> 15:35.258 whether we have a name for them or not, 15:35.258 --> 15:37.740 that's an irrelevant question 15:37.740 --> 15:39.960 for their artificial neural network. 15:39.960 --> 15:44.070 And my favorite one, here's a image of Jeffrey Hinton 15:44.070 --> 15:49.003 photo of Jeffrey Hinton passed through one of these filters. 15:50.910 --> 15:53.070 All right, so that brings us to the end of today's tutorial. 15:53.070 --> 15:55.440 I hope you enjoyed learning about convolution. 15:55.440 --> 15:58.680 The key takeaway is that a convolution, 15:58.680 --> 16:00.450 the primary purpose of a convolution 16:00.450 --> 16:04.590 is to find features in your image using the feature detector 16:04.590 --> 16:06.300 put them into a feature map. 16:06.300 --> 16:08.310 And by having them in a feature map, 16:08.310 --> 16:13.050 it still preserves the spatial relationships between pixels 16:13.050 --> 16:15.690 which is very important for us to, 16:15.690 --> 16:17.160 because if they're completely jumbled up, 16:17.160 --> 16:19.320 then we've lost the pattern. 16:19.320 --> 16:21.600 And at the same time, it's important to understand 16:21.600 --> 16:25.080 that most of the time the features a neural network 16:25.080 --> 16:29.400 will detect and use to recognize certain images 16:29.400 --> 16:31.800 and classes will mean nothing to humans. 16:31.800 --> 16:33.090 But nevertheless, they work. 16:33.090 --> 16:34.410 And that's what convolution is. 16:34.410 --> 16:36.270 And I look forward to seeing you on the tutorial. 16:36.270 --> 16:38.163 Until then, enjoy Deep Learning.