WEBVTT 00:00:07.560 --> 00:00:08.640 OK. 00:00:08.640 --> 00:00:12.290 So, coming nearer the end of the course, 00:00:12.290 --> 00:00:20.340 this lecture will be a mixture of the linear algebra that 00:00:20.340 --> 00:00:22.610 comes with a change of basis. 00:00:22.610 --> 00:00:26.340 And a change of basis from one basis to another basis 00:00:26.340 --> 00:00:30.270 is something you really do in applications. 00:00:30.270 --> 00:00:33.710 And, I would like to talk about those applications. 00:00:33.710 --> 00:00:36.920 I got a little bit involved with compression. 00:00:36.920 --> 00:00:39.820 Compressing a signal, compressing an image. 00:00:39.820 --> 00:00:43.910 And that's exactly change-of-basis. 00:00:43.910 --> 00:00:48.510 And then, the main theme in this chapter 00:00:48.510 --> 00:00:54.320 is th- the connection between a linear transformation, which 00:00:54.320 --> 00:00:56.870 doesn't have to have coordinates, 00:00:56.870 --> 00:01:01.610 and the matrix that tells us that transformation 00:01:01.610 --> 00:01:04.069 with respect to coordinates. 00:01:04.069 --> 00:01:08.420 So the matrix is the coordinate-based description 00:01:08.420 --> 00:01:11.090 of the linear transformation. 00:01:11.090 --> 00:01:14.480 Let me start out with the nice part, which 00:01:14.480 --> 00:01:18.770 is just to tell you something about image compression. 00:01:18.770 --> 00:01:21.070 Those of you -- well, everybody's going to meet 00:01:21.070 --> 00:01:25.950 compression, because you know that the amount of data that 00:01:25.950 --> 00:01:27.730 we're getting -- 00:01:27.730 --> 00:01:31.640 well, these lectures are compressed. 00:01:31.640 --> 00:01:37.500 So that, actually, probably you see my motion as 00:01:37.500 --> 00:01:38.030 jerky? 00:01:38.030 --> 00:01:39.050 Shall I use that word? 00:01:39.050 --> 00:01:41.760 Have you looked on the web? 00:01:41.760 --> 00:01:46.470 I should like to find a better word. 00:01:46.470 --> 00:01:47.770 Compressed, let's say. 00:01:47.770 --> 00:01:53.900 So the complete signal is, of course, in those video cameras, 00:01:53.900 --> 00:01:56.360 and in the videotape, but that goes 00:01:56.360 --> 00:01:59.540 to the bottom of building nine, and out of that 00:01:59.540 --> 00:02:06.240 comes a jumpy motion because it uses a standard system 00:02:06.240 --> 00:02:09.020 for compressing images. 00:02:09.020 --> 00:02:13.990 And, you'll notice that the stuff that sits on the board 00:02:13.990 --> 00:02:20.200 comes very clearly, but it's my motion that needs 00:02:20.200 --> 00:02:22.150 a whole lot of bits, right? 00:02:22.150 --> 00:02:26.060 So, and if I were to run up and back up there and back, 00:02:26.060 --> 00:02:29.380 that would need too many bits, and I'd 00:02:29.380 --> 00:02:32.860 be compressed even more. 00:02:32.860 --> 00:02:36.180 So, what does compression mean? 00:02:36.180 --> 00:02:39.100 Let me just think of a still image. 00:02:39.100 --> 00:02:44.570 And of course, satellites, and computations 00:02:44.570 --> 00:02:47.420 of the climate, computations of combustion, 00:02:47.420 --> 00:02:52.410 the computers and sensors of all kinds 00:02:52.410 --> 00:02:55.910 are just giving us overwhelming amounts of data. 00:02:55.910 --> 00:02:58.720 The Web is, too. 00:02:58.720 --> 00:03:02.910 Now, some compression can be done with no loss. 00:03:02.910 --> 00:03:07.150 Lossless compression is possible just using, sort of, the fact 00:03:07.150 --> 00:03:10.410 that there are redundancies. 00:03:10.410 --> 00:03:13.550 But I'm talking here about lossy compression. 00:03:13.550 --> 00:03:18.730 So I'm talking about -- here's an image. 00:03:18.730 --> 00:03:20.960 And what does an image consist of? 00:03:20.960 --> 00:03:24.770 It consists of a lot of little pixels, right? 00:03:24.770 --> 00:03:28.870 Maybe five hundred and twelve by five hundred and twelve. 00:03:28.870 --> 00:03:31.930 Two to the ninth by two to the ninth pixels, 00:03:31.930 --> 00:03:40.200 and so this is pixel number one, one, so that's a pixel. 00:03:40.200 --> 00:03:45.860 And if we're in black and white, the typical pixel 00:03:45.860 --> 00:03:52.640 would tell us a gray-scale, from zero to two fifty five. 00:03:52.640 --> 00:03:59.250 So a pixel is usually a value of one of the xi, 00:03:59.250 --> 00:04:02.190 so this would be the i-th pixel, is -- 00:04:02.190 --> 00:04:05.260 it's usually a real number on a scale 00:04:05.260 --> 00:04:06.860 from zero to two fifty five. 00:04:06.860 --> 00:04:10.260 In other words, two to the eighth possibilities. 00:04:10.260 --> 00:04:19.720 So usually, that's the standard, so that's eight -- eight bits. 00:04:22.620 --> 00:04:26.610 But then we have that for every pixel, 00:04:26.610 --> 00:04:29.490 so we have five hundred and twelve squared pixels, 00:04:29.490 --> 00:04:36.640 we're really operating x is a vector in R^n, but what is n? 00:04:36.640 --> 00:04:41.080 n is five hundred and twelve squared. 00:04:41.080 --> 00:04:45.470 That's our problem, right there. 00:04:45.470 --> 00:04:49.040 A pixel is a vector that gives us 00:04:49.040 --> 00:04:53.010 the information about the image. 00:04:53.010 --> 00:04:55.240 I'm sorry. 00:04:55.240 --> 00:05:02.530 The image that comes through is a vector of that length that -- 00:05:02.530 --> 00:05:04.800 that's the information that we have about the image, 00:05:04.800 --> 00:05:08.550 if it's a color image, we would have three times that length, 00:05:08.550 --> 00:05:12.920 because we'd need three coordinates to get color. 00:05:12.920 --> 00:05:16.830 So it would be three times five hundred and twelve squared. 00:05:16.830 --> 00:05:18.830 It's an enormous amount of information, 00:05:18.830 --> 00:05:22.810 and we couldn't send out the image for these lectures 00:05:22.810 --> 00:05:25.510 without compressing it. 00:05:25.510 --> 00:05:29.070 It would overload the system. 00:05:29.070 --> 00:05:31.100 So it has to be compressed. 00:05:31.100 --> 00:05:37.920 The standard compression, and still used with lectures 00:05:37.920 --> 00:05:39.810 is, called JPEG. 00:05:42.950 --> 00:05:46.870 I think that stands for Joint Photographic Experts Group. 00:05:46.870 --> 00:05:50.390 They established a system of compression. 00:05:50.390 --> 00:05:53.080 And I just want to tell you what it's about. 00:05:53.080 --> 00:05:55.190 It's a change-of-basis. 00:05:55.190 --> 00:05:58.360 What basis do we have? 00:05:58.360 --> 00:06:00.860 The current basis we have is, you 00:06:00.860 --> 00:06:07.050 could say, the standard basis is, every pixel, give a value. 00:06:07.050 --> 00:06:12.440 So that's like we have a vector x which is five hundred 00:06:12.440 --> 00:06:16.380 and twelve squared long and, in the i-th position, 00:06:16.380 --> 00:06:21.420 we get a number like one twenty one or something. 00:06:21.420 --> 00:06:27.720 The pixel next to it might be one twenty four, maybe 00:06:27.720 --> 00:06:34.470 where my tie begins to enter, so if it was mostly blue shirt, 00:06:34.470 --> 00:06:37.240 this would be a slight difference in shading, 00:06:37.240 --> 00:06:41.430 but pretty close, then the tie would be a different color, 00:06:41.430 --> 00:06:45.180 so we might have quite a few pixels 00:06:45.180 --> 00:06:47.820 for the blue shirt, and a whole lot 00:06:47.820 --> 00:06:52.450 more for the blackboard, that are very close. 00:06:52.450 --> 00:06:55.430 And that's what are very correlated. 00:06:55.430 --> 00:06:59.250 And that's what gives us the possibility of compression. 00:06:59.250 --> 00:07:02.770 For example, before the lecture starts, 00:07:02.770 --> 00:07:07.660 if we had a blank blackboard, then there's an image, 00:07:07.660 --> 00:07:12.250 but it would make no sense to take that image 00:07:12.250 --> 00:07:17.030 and tell you what it is pixel by pixel. 00:07:17.030 --> 00:07:21.720 I mean, there's a case in which all pixel values, 00:07:21.720 --> 00:07:24.180 all gray levels are the same -- 00:07:24.180 --> 00:07:26.950 or practically the same, depending on the erasing 00:07:26.950 --> 00:07:29.240 of the board, but extremely close -- 00:07:29.240 --> 00:07:35.260 and, so that's an image where the standard basis is lousy. 00:07:35.260 --> 00:07:40.490 That's the basic fact, that the standard basis which gives 00:07:40.490 --> 00:07:44.640 the value of every pixel makes no use of the fact that 00:07:44.640 --> 00:07:49.250 we're getting a whole lot of pixels whose gray levels -- 00:07:49.250 --> 00:07:54.450 the neighboring pixels tend to have the same gray level 00:07:54.450 --> 00:07:57.260 as their neighbors. 00:07:57.260 --> 00:07:59.950 So how do we take advantage of that fact? 00:07:59.950 --> 00:08:04.820 Well, one basis vector that would 00:08:04.820 --> 00:08:07.810 be extremely nice to include in the basis 00:08:07.810 --> 00:08:12.830 would be a vector of all ones. 00:08:12.830 --> 00:08:14.830 That's not in our standard basis, 00:08:14.830 --> 00:08:23.890 so let me just write again, the standard basis is our one, 00:08:23.890 --> 00:08:29.440 and all the rest zeroes, zero, one, and all the rest, zeroes, 00:08:29.440 --> 00:08:33.820 everybody knows what these standard basis is. 00:08:33.820 --> 00:08:39.320 Now, any other basis for R -- so this is -- 00:08:39.320 --> 00:08:43.100 for this very high-dimensional space -- 00:08:43.100 --> 00:08:46.180 now I'm going to speak about a better basis. 00:08:46.180 --> 00:08:51.810 Better basis -- and let me just emphasize, 00:08:51.810 --> 00:08:54.580 one vector that would be extremely nice to have in that 00:08:54.580 --> 00:08:56.770 basis is the vector of all ones. 00:08:59.790 --> 00:09:00.540 Why is that? 00:09:00.540 --> 00:09:04.730 Let me just say again, because that vector of all ones, 00:09:04.730 --> 00:09:11.180 by itself, one vector is able to completely give 00:09:11.180 --> 00:09:15.940 the information on a solid image. 00:09:15.940 --> 00:09:17.650 Of course, our image won't be solid, 00:09:17.650 --> 00:09:26.070 it will have a mix of solid and signal. 00:09:26.070 --> 00:09:30.470 So having that one vector in the basis 00:09:30.470 --> 00:09:33.260 is going to save us a whole lot. 00:09:33.260 --> 00:09:36.400 Now, the question is, what other vectors should be in the basis? 00:09:36.400 --> 00:09:41.050 The extreme vector in the basis might 00:09:41.050 --> 00:09:47.050 be a vector of one minus one, one minus one, one minus one. 00:09:47.050 --> 00:09:49.420 That would be a vector that shows -- 00:09:49.420 --> 00:09:52.560 I mean, that's like a checkerboard vector, right? 00:09:52.560 --> 00:09:58.630 That's a vector that would, if the image was 00:09:58.630 --> 00:10:01.040 like a huge checkerboard of plus, minus, 00:10:01.040 --> 00:10:03.720 plus, minus, plus, minus, that vector 00:10:03.720 --> 00:10:06.260 would carry the whole signal. 00:10:06.260 --> 00:10:08.740 But much more common would be maybe 00:10:08.740 --> 00:10:14.760 to have half the image, darker and the other half lighter. 00:10:14.760 --> 00:10:18.320 So another vector that might be quite useful in here 00:10:18.320 --> 00:10:21.480 would be half ones and half minus ones. 00:10:24.950 --> 00:10:31.780 I'm just trying to get across the idea of that a basis could 00:10:31.780 --> 00:10:33.950 be where, that first of all, we've 00:10:33.950 --> 00:10:38.330 got the bases at our disposal. 00:10:38.330 --> 00:10:39.960 Like, we're free to choose that. 00:10:39.960 --> 00:10:45.330 And it's a billion-dollar decision what we choose. 00:10:45.330 --> 00:10:50.270 So, and TV people would rather pre- 00:10:50.270 --> 00:10:55.480 would prefer one basis based on the way the signal is scanned, 00:10:55.480 --> 00:10:58.040 and movie people would prefer another, 00:10:58.040 --> 00:11:02.540 I mean, there's giant politics in this question that 00:11:02.540 --> 00:11:05.710 really reduces to a linear algebra problem, 00:11:05.710 --> 00:11:07.900 what basis to choose. 00:11:07.900 --> 00:11:15.540 I'll just mention the best known basis, which JPEG uses, -- 00:11:15.540 --> 00:11:17.760 let me put that here -- 00:11:17.760 --> 00:11:21.490 is the Fourier basis. 00:11:21.490 --> 00:11:27.730 So when you use the Fourier basis, that includes -- 00:11:27.730 --> 00:11:31.820 this is the constant vector, the D C vector if we're electrical 00:11:31.820 --> 00:11:37.170 engineers, the l- vector of all ones, so it would include one, 00:11:37.170 --> 00:11:39.070 one, one, one. 00:11:39.070 --> 00:11:41.370 Often eight by eight is a good choice. 00:11:44.770 --> 00:11:46.580 Eight by eight is a good choice. 00:11:46.580 --> 00:11:50.340 So, what do I mean by this eight by eight? 00:11:50.340 --> 00:11:55.430 I mean that the big signal, which is five twelve by five 00:11:55.430 --> 00:12:00.480 twelve, gets broken down, and JPEG does this, into eight 00:12:00.480 --> 00:12:02.540 by eight blocks. 00:12:02.540 --> 00:12:06.800 And we -- sort of, this is too much to deal with at once. 00:12:06.800 --> 00:12:10.090 So what JPEG does is take this eight by eight block, 00:12:10.090 --> 00:12:16.250 which is sixty four coefficients, sixty four, 00:12:16.250 --> 00:12:21.030 pixels, and changes the basis on that 00:12:21.030 --> 00:12:22.090 piece. 00:12:22.090 --> 00:12:26.260 And then, now, let's see, I was going to write down Fourier, 00:12:26.260 --> 00:12:31.120 so you remember Fourier as this vector of all ones, and then, 00:12:31.120 --> 00:12:35.100 the vector -- oh, well, actually, 00:12:35.100 --> 00:12:39.480 I gave a lecture earlier about the Fourier matrix, 00:12:39.480 --> 00:12:50.040 this matrix whose columns are powers of a complex number w. 00:12:50.040 --> 00:12:52.820 I won't repeat that, because I don't 00:12:52.820 --> 00:12:56.600 want to go into the details of the Fourier basis, 00:12:56.600 --> 00:13:00.640 just to tell you how compression works. 00:13:00.640 --> 00:13:02.880 So what happens in JPEG? 00:13:02.880 --> 00:13:11.640 What happens to the video, to each image, of these lectures? 00:13:11.640 --> 00:13:16.270 It gets broken into eight by eight blocks. 00:13:16.270 --> 00:13:16.910 OK. 00:13:16.910 --> 00:13:21.280 Within each block, we have sixty four coefficients, 00:13:21.280 --> 00:13:24.210 sixty four basis vectors, sixty four pixels, 00:13:24.210 --> 00:13:28.870 and we change basis in sixty four dimensional space 00:13:28.870 --> 00:13:31.662 using these Fourier vectors. 00:13:35.000 --> 00:13:38.140 Just note, that was a lossless step. 00:13:38.140 --> 00:13:40.240 Let me emphasize. 00:13:40.240 --> 00:13:46.150 In comes the signal x. 00:13:46.150 --> 00:13:47.460 We change basis. 00:13:47.460 --> 00:13:49.490 This is the basis change. 00:13:49.490 --> 00:13:50.800 Change basis. 00:13:50.800 --> 00:13:54.080 Choose a better basis. 00:13:54.080 --> 00:14:03.350 So it produces, the coefficients c. 00:14:03.350 --> 00:14:07.590 So sixty four pixels come in, sixty four coefficients 00:14:07.590 --> 00:14:08.400 come out. 00:14:08.400 --> 00:14:11.150 Now comes the compression. 00:14:11.150 --> 00:14:12.735 Now come -- this was lossless. 00:14:16.320 --> 00:14:18.860 It's just -- we know that R -- 00:14:18.860 --> 00:14:25.250 R sixty four has plenty of bases, and we've chosen one. 00:14:25.250 --> 00:14:32.040 Now, in that basis, we write the signal in that basis, 00:14:32.040 --> 00:14:34.350 and that's what my lecture -- that's the math part 00:14:34.350 --> 00:14:35.710 of my lecture. 00:14:35.710 --> 00:14:37.610 Now here's the application part. 00:14:37.610 --> 00:14:40.340 The next part is going to be the compression step. 00:14:43.790 --> 00:14:46.030 And that's lossy. 00:14:46.030 --> 00:14:47.840 We're going to lose information. 00:14:47.840 --> 00:14:52.220 And what will actually happen at that step? 00:14:52.220 --> 00:14:56.500 Well, one thing we could do is just throw away 00:14:56.500 --> 00:14:58.680 the small coefficients. 00:14:58.680 --> 00:15:03.520 So that's called thresholding, we set some threshold. 00:15:03.520 --> 00:15:07.150 Every coefficient, every basis vector that's 00:15:07.150 --> 00:15:10.510 not in there more than the threshold value, 00:15:10.510 --> 00:15:14.240 and we set them threshold so that our eye can't 00:15:14.240 --> 00:15:19.650 see the difference, or can hardly see the difference, 00:15:19.650 --> 00:15:23.230 whether we throw away that little bit of that basis vector 00:15:23.230 --> 00:15:24.220 or keep it. 00:15:24.220 --> 00:15:28.004 So this compression step produces a compressed set of 00:15:28.004 --> 00:15:29.670 I'll just keep going here. coefficients. 00:15:32.900 --> 00:15:36.220 So it keeps going, this compression step 00:15:36.220 --> 00:15:40.190 produces some coefficient c hat. 00:15:40.190 --> 00:15:42.560 And with many zeroes. 00:15:45.220 --> 00:15:48.160 So that's where the compression came. 00:15:48.160 --> 00:15:52.480 Probably, there is enough of this vector of all ones -- 00:15:52.480 --> 00:15:55.240 we very seldom throw that away. 00:15:55.240 --> 00:15:57.950 Usually, its coefficient will be large. 00:15:57.950 --> 00:16:01.110 But the coefficient of something like this, 00:16:01.110 --> 00:16:04.500 that quickly alternative vector, there's 00:16:04.500 --> 00:16:07.560 probably very little of that in any smooth signal. 00:16:07.560 --> 00:16:12.450 That's high-frequency -- this is low-frequency, zero frequency. 00:16:12.450 --> 00:16:15.350 This stuff is the highest frequency we could have, 00:16:15.350 --> 00:16:23.470 and if the noise, the jitter is producing that sort of output, 00:16:23.470 --> 00:16:26.470 but a smooth lecture like this one 00:16:26.470 --> 00:16:30.740 is, has very little of that highest frequency, 00:16:30.740 --> 00:16:32.700 very little noise in this lecture. 00:16:32.700 --> 00:16:37.150 OK, so we throw away whatever there is, 00:16:37.150 --> 00:16:40.860 and we're left with just a few coefficients, 00:16:40.860 --> 00:16:48.560 and then we reconstruct a signal using those coefficients. 00:16:48.560 --> 00:16:53.580 We take those coefficients, times their basis vectors, 00:16:53.580 --> 00:16:59.040 but this sum doesn't have sixty four terms any more. 00:16:59.040 --> 00:17:02.130 Probably, it has about two or three terms. 00:17:02.130 --> 00:17:03.230 So that would -- 00:17:03.230 --> 00:17:05.109 say it has three terms. 00:17:05.109 --> 00:17:07.000 From sixty four down to three, that's 00:17:07.000 --> 00:17:09.670 compression of twenty one to one. 00:17:09.670 --> 00:17:12.500 That's the kind of compression you're looking for. 00:17:12.500 --> 00:17:17.480 And everybody is looking for that sort of compression. 00:17:17.480 --> 00:17:19.890 Let's see, I guess I met the problem 00:17:19.890 --> 00:17:23.990 with the FBI and fingerprints. 00:17:23.990 --> 00:17:26.390 So there's a whole lot of still images. 00:17:26.390 --> 00:17:30.590 You know, with your thumb, you make these inky marks 00:17:30.590 --> 00:17:34.300 which go somewhere. 00:17:34.300 --> 00:17:38.460 it used to go to Washington and get stored in a big file. 00:17:38.460 --> 00:17:44.240 So Washington had a file of thirty million murderers, 00:17:44.240 --> 00:17:50.670 cheaters on quizzes, other stuff, 00:17:50.670 --> 00:17:56.280 and actually, there was no way to retrieve them in time. 00:17:56.280 --> 00:17:59.740 So suppose you're at the police station, they say, OK, 00:17:59.740 --> 00:18:03.650 this person may have done this, check with Washington, 00:18:03.650 --> 00:18:07.410 have they got -- are his or her fingerprints on file? 00:18:07.410 --> 00:18:12.220 Well, Washington won't know the answer within a week 00:18:12.220 --> 00:18:15.870 if it's got filing cabinets full of fingerprints. 00:18:15.870 --> 00:18:21.190 So of course, the natural step is digitizing. 00:18:21.190 --> 00:18:24.380 So all fingerprints are now digitized, 00:18:24.380 --> 00:18:30.090 so now it's at least electronic, but still there's too much 00:18:30.090 --> 00:18:33.580 information in each one. 00:18:33.580 --> 00:18:39.260 I mean, you can't search through that many, fingerprints 00:18:39.260 --> 00:18:45.040 if the digital image is five twelve squared by five twelve 00:18:45.040 --> 00:18:47.280 squared, if it's that many pixels. 00:18:47.280 --> 00:18:49.110 So you get compressed. 00:18:49.110 --> 00:18:54.970 So the FBI had to decide what basis to choose for compression 00:18:54.970 --> 00:18:55.900 of fingerprints. 00:18:55.900 --> 00:18:59.510 And then they built a big new facility in West Virginia, 00:18:59.510 --> 00:19:03.220 and that's where fingerprints now are sent. 00:19:03.220 --> 00:19:06.110 So I think, if you get your fingerprints done now 00:19:06.110 --> 00:19:09.610 at the police station, if it's an up-to-date police station, 00:19:09.610 --> 00:19:13.990 it happens digitally, and the signal is sent digitally, 00:19:13.990 --> 00:19:18.560 and then in West Virginia, it's compressed and indexed. 00:19:18.560 --> 00:19:21.960 And then, if they want to find you, 00:19:21.960 --> 00:19:26.710 they can do it within minutes instead of within a week. 00:19:26.710 --> 00:19:27.410 OK. 00:19:27.410 --> 00:19:31.860 So this compression comes up for signals, for images, 00:19:31.860 --> 00:19:36.020 for video -- which is, like these lectures -- 00:19:36.020 --> 00:19:38.400 there's another aspect. 00:19:38.400 --> 00:19:43.980 You could treat the video as one still image after another one, 00:19:43.980 --> 00:19:50.770 and compress each one, and then run them and make a video. 00:19:50.770 --> 00:19:52.950 But that misses -- 00:19:52.950 --> 00:19:55.990 well, you can see why that's not optimal. 00:19:55.990 --> 00:20:04.570 In a video thing, you have a sequence of images, 00:20:04.570 --> 00:20:14.340 so video is really a sequence of images but what about one 00:20:14.340 --> 00:20:16.800 image to the next image? 00:20:16.800 --> 00:20:18.780 They're extremely correlated. 00:20:18.780 --> 00:20:21.930 I mean that I'm getting an image every split-second, 00:20:21.930 --> 00:20:24.200 and also, I'm moving slightly. 00:20:24.200 --> 00:20:29.690 That's what's producing the, jumpy motion on the video. 00:20:29.690 --> 00:20:35.010 But I'm not, like, you know -- 00:20:35.010 --> 00:20:39.270 each image in the sequence is pretty close to the one before. 00:20:39.270 --> 00:20:44.480 So you have to use, like, prediction and correction. 00:20:44.480 --> 00:20:49.700 I mean, the image of me one instant -- one time-step later, 00:20:49.700 --> 00:20:51.830 you would assume would be the same, 00:20:51.830 --> 00:20:54.260 and then plus a small correction. 00:20:54.260 --> 00:20:58.620 And you would only code and digitize the correction, 00:20:58.620 --> 00:21:00.970 and compress the correction. 00:21:00.970 --> 00:21:05.350 So a sequence of images that's highly correlated 00:21:05.350 --> 00:21:11.250 and the problem in compression is always 00:21:11.250 --> 00:21:14.010 to use this correlation, this fact 00:21:14.010 --> 00:21:17.960 that, in time, or in space, things 00:21:17.960 --> 00:21:23.650 don't change instantly, they're very often smooth changes, 00:21:23.650 --> 00:21:29.170 and, you can predict one value from the previous value. 00:21:29.170 --> 00:21:29.670 OK. 00:21:29.670 --> 00:21:36.240 So those are applications which are pure linear algebra. 00:21:36.240 --> 00:21:42.680 I could, well, maybe you'll allow me to tell you, 00:21:42.680 --> 00:21:47.650 and the book describes, the new basis that's 00:21:47.650 --> 00:21:50.150 the competition for Fourier. 00:21:50.150 --> 00:21:54.680 So the competition for Fourier is called wavelets, 00:21:54.680 --> 00:21:58.460 and I can describe what that basis is like, 00:21:58.460 --> 00:22:00.680 say, in the eight by eight case. 00:22:00.680 --> 00:22:03.240 So the eight by eight wavelet basis 00:22:03.240 --> 00:22:08.410 is the vector of all ones, eight ones, then 00:22:08.410 --> 00:22:13.030 the vector of four ones and four minus ones, 00:22:13.030 --> 00:22:18.770 then the vector of two ones, and two minus ones, 00:22:18.770 --> 00:22:20.620 and four zeroes. 00:22:20.620 --> 00:22:24.930 And also the vector of four zeroes 00:22:24.930 --> 00:22:28.480 and two ones and two minus ones. 00:22:28.480 --> 00:22:31.630 So now I'm up to four, and I need four more, right? 00:22:31.630 --> 00:22:32.870 For R^8? 00:22:32.870 --> 00:22:39.690 The next basis vector will be one minus one and six zeroes, 00:22:39.690 --> 00:22:44.910 and then three more like that, with the one minus one 00:22:44.910 --> 00:22:47.400 there, and there, and there. 00:22:47.400 --> 00:22:52.490 So those are eight vectors in eight-dimensional space, 00:22:52.490 --> 00:22:59.830 those are called wavelets, and it's a very simple wavelet 00:22:59.830 --> 00:23:03.131 choice, it's a more sophisticated 00:23:03.131 --> 00:23:03.630 choice. 00:23:03.630 --> 00:23:09.320 This is a little jumpy, to jump between one and minus one. 00:23:09.320 --> 00:23:12.920 And, actually, you can see, now, suppose 00:23:12.920 --> 00:23:16.820 you compare the wavelet basis with the Fourier basis above. 00:23:16.820 --> 00:23:22.450 How could I write this guy, which is in the Fourier basis, 00:23:22.450 --> 00:23:24.780 it's an eight -- it's a vector in R^8. 00:23:24.780 --> 00:23:28.350 How would I write that as a combination 00:23:28.350 --> 00:23:30.920 of the wavelet basis? 00:23:30.920 --> 00:23:34.250 Have I told you enough about the wavelet basis that you can see, 00:23:34.250 --> 00:23:37.750 how does this very fast guy -- 00:23:37.750 --> 00:23:41.700 what combination of the wavelet basis is that very fast guy? 00:23:41.700 --> 00:23:45.020 It would be this one -- 00:23:45.020 --> 00:23:47.430 it would be the sum of these four, right? 00:23:47.430 --> 00:23:50.360 That very fast guy will be that one minus one, 00:23:50.360 --> 00:23:53.740 and the next one, and the next one, and the next one. 00:23:53.740 --> 00:23:59.350 So this is the sum of those last four wavelets. 00:23:59.350 --> 00:24:02.010 This one, we've kept, and so on. 00:24:02.010 --> 00:24:06.410 So, each -- well, every -- well, that's what a basis does. 00:24:06.410 --> 00:24:10.250 Every vector in R^8 is some combination of those, 00:24:10.250 --> 00:24:15.210 and for the linear algebra -- so the linear algebra is this 00:24:15.210 --> 00:24:18.860 step, find the coefficient. 00:24:18.860 --> 00:24:21.900 That's the step we want to take. 00:24:21.900 --> 00:24:28.010 What if I give you the basis, like this wavelet basis, 00:24:28.010 --> 00:24:34.360 and I give you the pixel -- so here are the pixel values, P1, 00:24:34.360 --> 00:24:35.660 P2, down to P8 -- 00:24:38.810 --> 00:24:39.930 what's the job? 00:24:39.930 --> 00:24:42.220 What's the linear algebra here? 00:24:42.220 --> 00:24:48.400 So these are the values, this is in the standard basis, right? 00:24:48.400 --> 00:24:53.990 Those are just the values at eight successive points. 00:24:53.990 --> 00:24:57.560 I guess I'm dropping down to one dimension, instead of eight 00:24:57.560 --> 00:25:01.180 by eight, I'm just going to take eight pixel values 00:25:01.180 --> 00:25:04.540 along that first top row. 00:25:04.540 --> 00:25:06.160 So what do I want to do? 00:25:06.160 --> 00:25:10.720 In standard basis, here are the pixel values. 00:25:10.720 --> 00:25:15.880 I want to write that as a combination of c1 times this 00:25:15.880 --> 00:25:21.220 guy, plus c2 times this guy, plus c3, 00:25:21.220 --> 00:25:25.590 these are the coefficients, plus c4 times this one -- 00:25:25.590 --> 00:25:26.720 do you see what I'm doing? 00:25:26.720 --> 00:25:32.010 I want to write this vector P as a combination of c1 times 00:25:32.010 --> 00:25:36.395 the first wavelet plus c8 times the eighth wavelet. 00:25:39.910 --> 00:25:42.640 That's the transform step. 00:25:42.640 --> 00:25:43.970 That's the lossless step. 00:25:43.970 --> 00:25:47.580 That's the step from P -- oh, I'm calling it P here, 00:25:47.580 --> 00:25:50.790 and I called it x there, so let me -- 00:25:50.790 --> 00:25:54.520 at the risk of moving, and therefore making this jumpy -- 00:25:54.520 --> 00:26:00.780 suppose the signal I'm now calling P, that a pixel values, 00:26:00.780 --> 00:26:02.860 and I'm looking for the coefficients. 00:26:02.860 --> 00:26:05.360 OK, tell me how to do it. 00:26:08.360 --> 00:26:12.960 If I give you eight basis vectors, 00:26:12.960 --> 00:26:17.930 and I give you the input signal, and I ask for the coefficients, 00:26:17.930 --> 00:26:19.950 what do I do? 00:26:19.950 --> 00:26:22.690 What's the step? 00:26:22.690 --> 00:26:28.650 I'm trying to solve this, I want to know the eight coefficients, 00:26:28.650 --> 00:26:32.200 so I'm changing from the standard basis, which is just 00:26:32.200 --> 00:26:37.220 the eight gray-scale values to the wavelet basis, where 00:26:37.220 --> 00:26:41.490 the same vector is represented by eight numbers. 00:26:41.490 --> 00:26:44.970 It's got to take eight numbers to tell you a vector in R^8, 00:26:44.970 --> 00:26:49.460 and those eight numbers are the coefficients of the basis. 00:26:49.460 --> 00:26:54.720 Look, we've done this thing before. 00:26:54.720 --> 00:26:58.510 There is the equation in vector notation, 00:26:58.510 --> 00:27:01.610 we want to see it as a matrix. 00:27:01.610 --> 00:27:06.870 This is a combination of columns of the wavelet matrix, 00:27:06.870 --> 00:27:07.430 right? 00:27:07.430 --> 00:27:14.440 This is P equals c1, c2, down to c8, 00:27:14.440 --> 00:27:17.370 and these guys are the columns. 00:27:17.370 --> 00:27:19.990 I mean, this is the step that we're constantly 00:27:19.990 --> 00:27:22.700 taking in this course, the first basis vector 00:27:22.700 --> 00:27:25.920 goes in the first column, the second basis vector 00:27:25.920 --> 00:27:29.170 goes in the second column, and so on, 00:27:29.170 --> 00:27:32.760 the eight columns of this wavelet matrix 00:27:32.760 --> 00:27:35.130 are the eight basis vectors. 00:27:35.130 --> 00:27:40.710 This is a wavelet matrix W. 00:27:40.710 --> 00:27:46.330 So, the step to change basis -- so now I'm finally coming 00:27:46.330 --> 00:27:51.780 to this change-of-basis, so the change of basis that, 00:27:51.780 --> 00:27:55.370 let me stay with this board, but -- well, 00:27:55.370 --> 00:27:56.790 let me just go above it, here. 00:28:00.500 --> 00:28:07.200 So the standard basis, we know, the wavelet basis we have here, 00:28:07.200 --> 00:28:14.300 and the transform is simply, solve the equations, 00:28:14.300 --> 00:28:23.300 P=W C. So the coefficients are W inverse P. Right. 00:28:23.300 --> 00:28:26.920 This shows a critical point. 00:28:26.920 --> 00:28:31.630 A good basis has a nice, fast, inverse. 00:28:31.630 --> 00:28:38.260 So good basis means what? 00:28:38.260 --> 00:28:40.470 So this is like the billion-dollar competition, 00:28:40.470 --> 00:28:41.710 Eh? and it's not over yet. 00:28:41.710 --> 00:28:46.360 People are going to come up with better bases than these. 00:28:46.360 --> 00:28:53.420 So a good basis will be, first good thing would be fast. 00:28:53.420 --> 00:28:58.600 I have to be able to multiply by W fast, and multiply by W -- 00:28:58.600 --> 00:29:00.930 by its inverse fast. 00:29:00.930 --> 00:29:04.630 That's -- if a basis doesn't allow you to do that fast, 00:29:04.630 --> 00:29:09.920 then it's going to take so much time that you can't afford it. 00:29:09.920 --> 00:29:12.210 So these bases -- 00:29:12.210 --> 00:29:15.900 the Fourier basis, everybody said, OK, I 00:29:15.900 --> 00:29:18.830 know how to deal quickly with the Fourier basis, 00:29:18.830 --> 00:29:23.690 because we have something called the Fast Fourier Transform. 00:29:23.690 --> 00:29:27.510 So there's a FFT that came in my earlier lecture, 00:29:27.510 --> 00:29:30.270 and comes in the last chapter of the book, 00:29:30.270 --> 00:29:34.400 so change-of-basis is done -- if, for the Fourier basis, 00:29:34.400 --> 00:29:39.380 it's done fast by the FFT and there's a fast wavelet 00:29:39.380 --> 00:29:40.420 transform. 00:29:40.420 --> 00:29:47.040 I can change, for this wavelet example, 00:29:47.040 --> 00:29:49.320 this matrix is easy to invert. 00:29:49.320 --> 00:29:52.720 It's just somebody had a smart idea 00:29:52.720 --> 00:29:56.930 in choosing that wavelet basis and inverting it, 00:29:56.930 --> 00:29:58.580 it has a nice inverse. 00:29:58.580 --> 00:30:01.290 Actually, you can see why it has a nice inverse. 00:30:01.290 --> 00:30:06.340 Do you see any property of these eight basis vectors? 00:30:06.340 --> 00:30:07.940 Well, I've only written five of them, 00:30:07.940 --> 00:30:11.220 but if you see that property for those five, 00:30:11.220 --> 00:30:13.220 you'll see it for the three 00:30:13.220 --> 00:30:14.030 remaining. 00:30:14.030 --> 00:30:16.890 Well, if I give you those eight vectors 00:30:16.890 --> 00:30:20.650 and ask, what's a nice property? 00:30:20.650 --> 00:30:23.770 Well, you would say, first, they're all ones and minus ones 00:30:23.770 --> 00:30:25.120 and zeroes. 00:30:25.120 --> 00:30:32.530 So every multiplication is very fast using -- just in binary. 00:30:32.530 --> 00:30:34.820 But what's the other great property of those vectors? 00:30:34.820 --> 00:30:37.750 Anybody see it? 00:30:37.750 --> 00:30:41.210 So, of course, when I think about a basis, 00:30:41.210 --> 00:30:42.480 one nice property -- 00:30:42.480 --> 00:30:45.320 I don't have to have it, but I'm happy if it's there -- 00:30:45.320 --> 00:30:48.830 is that they're orthogonal. 00:30:48.830 --> 00:30:51.070 If the basis vectors are orthogonal, 00:30:51.070 --> 00:30:52.780 then I'm in good shape. 00:30:52.780 --> 00:30:54.380 And these are... do you see? 00:30:54.380 --> 00:30:56.840 Take the dot product of that with that, 00:30:56.840 --> 00:31:00.910 you get four plus ones and four minus ones, you get zero. 00:31:00.910 --> 00:31:03.060 Take the dot product of that with that. 00:31:03.060 --> 00:31:05.880 You get two plus ones and two minus ones. 00:31:05.880 --> 00:31:07.660 Or the dot product of that with that. 00:31:07.660 --> 00:31:10.620 Two plus ones and two minus ones. 00:31:10.620 --> 00:31:14.060 You can easily check that that's an orthogonal basis. 00:31:14.060 --> 00:31:15.875 It's not orthonormal. 00:31:18.870 --> 00:31:22.470 To fix it up, I should divide by the length, 00:31:22.470 --> 00:31:23.920 to make them unit vectors. 00:31:23.920 --> 00:31:26.000 Let's suppose I do that. 00:31:26.000 --> 00:31:29.320 So somewhere in here, I've got to account for the fact 00:31:29.320 --> 00:31:32.590 that this has length square root of eight, that 00:31:32.590 --> 00:31:34.520 has length square root of four, that 00:31:34.520 --> 00:31:36.750 has length square root of two. 00:31:36.750 --> 00:31:40.750 But that's just a constant factor that's easy to -- 00:31:40.750 --> 00:31:42.620 so suppose we've done that. 00:31:42.620 --> 00:31:47.420 Then, tell me what's W inverse? 00:31:47.420 --> 00:31:52.270 That's what chapter four, section four point four 00:31:52.270 --> 00:31:53.630 was about. 00:31:53.630 --> 00:31:57.470 If we have orthonormal columns then 00:31:57.470 --> 00:32:02.330 the inverse is the same as the transpose. 00:32:02.330 --> 00:32:06.470 So if we have a fast way to multiply by W, which we do, 00:32:06.470 --> 00:32:09.010 the inverse is going to look just the same, 00:32:09.010 --> 00:32:10.450 and we'll have a fast way to do 00:32:10.450 --> 00:32:11.680 W inverse. 00:32:11.680 --> 00:32:17.760 So that's the wavelet basis passes this requirement 00:32:17.760 --> 00:32:19.020 for fast. 00:32:19.020 --> 00:32:21.250 We can use it fast. 00:32:21.250 --> 00:32:24.530 But there's a second requirement, is it any good? 00:32:24.530 --> 00:32:26.420 Because the the very fastest thing 00:32:26.420 --> 00:32:30.060 we could do is not to change basis at all. 00:32:30.060 --> 00:32:30.560 Right? 00:32:30.560 --> 00:32:33.660 The fastest thing would be, OK, stay with the standard basis, 00:32:33.660 --> 00:32:36.020 stay with eight pixel values. 00:32:36.020 --> 00:32:40.320 But that was poor from compression point of view, 00:32:40.320 --> 00:32:41.020 right? 00:32:41.020 --> 00:32:45.380 Those eight pixel values, if I just took those eight numbers, 00:32:45.380 --> 00:32:48.800 I can't throw some of those away. 00:32:48.800 --> 00:32:52.550 If I throw away ninety percent -- 00:32:52.550 --> 00:32:55.110 if I compress ten to one, and throw away 00:32:55.110 --> 00:32:57.610 ninety percent of my pixel values, 00:32:57.610 --> 00:33:00.660 well, my picture's just gone dark. 00:33:00.660 --> 00:33:03.470 Whereas, the basis that was good, 00:33:03.470 --> 00:33:05.960 the wavelet basis or the Fourier basis, 00:33:05.960 --> 00:33:12.910 if I throw away c5, c6, c7, and c8, all I'm throwing away 00:33:12.910 --> 00:33:16.060 is little blips that are probably there 00:33:16.060 --> 00:33:18.870 in very small amounts. 00:33:18.870 --> 00:33:23.210 So the second property that we need is good compression. 00:33:23.210 --> 00:33:32.470 So first, it has to be fast, and secondly, a few basis vectors 00:33:32.470 --> 00:33:36.270 should come close to the signal. 00:33:36.270 --> 00:33:40.140 So a few is enough. 00:33:40.140 --> 00:33:41.840 Can I write it that way? 00:33:41.840 --> 00:33:50.470 A few basis vectors are enough to reproduce the image just 00:33:50.470 --> 00:33:54.060 exactly as on a video of these 18.06 lectures. 00:33:54.060 --> 00:33:58.700 Uh, I don't know what the compression rate is, I'll ask, 00:33:58.700 --> 00:34:03.270 David, who does the compression -- and, by the way, 00:34:03.270 --> 00:34:09.639 I'll try to get the lectures, that are relevant for the quiz 00:34:09.639 --> 00:34:13.310 up onto the Web in time. 00:34:13.310 --> 00:34:15.980 So I'll send them a message today. 00:34:15.980 --> 00:34:21.679 So, he's using the Fourier basis because the JPEG -- 00:34:21.679 --> 00:34:26.219 so JPEG two thousand, which will be the next standard for image 00:34:26.219 --> 00:34:29.520 compression, will include wavelets. 00:34:29.520 --> 00:34:33.139 So, I mean, you're actually getting a kind of up-to-date, 00:34:33.139 --> 00:34:40.030 picture of where this big world of signal and image processing 00:34:40.030 --> 00:34:41.230 is. 00:34:41.230 --> 00:34:45.090 That Fourier is what everybody knew, 00:34:45.090 --> 00:34:47.949 and what people automatically used, 00:34:47.949 --> 00:34:53.460 and the new one is wavelets, where this is 00:34:53.460 --> 00:34:55.580 the simplest set of wavelets. 00:34:55.580 --> 00:34:59.650 And this isn't the one that the FBI uses, by the way, 00:34:59.650 --> 00:35:03.730 the FBI uses a smoother wavelet, instead 00:35:03.730 --> 00:35:08.690 of jumping from one to minus one, it's a smooth, Cutoff. 00:35:08.690 --> 00:35:15.010 and, that's what we'll be in in JPEG two thousand. 00:35:15.010 --> 00:35:17.130 OK, so that's that application. 00:35:17.130 --> 00:35:23.520 Now, let me come to the math, the linear algebra 00:35:23.520 --> 00:35:25.780 part of the lecture. 00:35:25.780 --> 00:35:29.470 Well, we've actually seen a change-of-basis. 00:35:29.470 --> 00:35:33.420 So let -- let me just review that eh-eh change-of-basis 00:35:33.420 --> 00:35:37.980 idea, and then the i- and then the transformation to a matrix. 00:35:37.980 --> 00:35:38.590 OK. 00:35:38.590 --> 00:35:43.630 So this, I hope you see that these applications are 00:35:43.630 --> 00:35:45.360 really big. 00:35:45.360 --> 00:35:49.100 Now, I have to talk a little about change-of-basis, 00:35:49.100 --> 00:35:50.960 and a little about that. 00:35:50.960 --> 00:35:52.380 The matrix. 00:35:52.380 --> 00:35:53.300 OK. 00:35:53.300 --> 00:35:55.880 OK. 00:35:55.880 --> 00:35:56.380 OK. 00:35:56.380 --> 00:35:57.171 So change-of-basis. 00:36:01.820 --> 00:36:10.390 Basically, forgive that put, OK, I have, 00:36:10.390 --> 00:36:15.250 I have my vector in one basis, and I want 00:36:15.250 --> 00:36:16.500 to change to a different one. 00:36:16.500 --> 00:36:19.590 Actually, you saw it for the wavelet case. 00:36:19.590 --> 00:36:22.680 So I need the -- 00:36:22.680 --> 00:36:31.300 let the matrix W, and the columns of W 00:36:31.300 --> 00:36:34.330 be the new basis vectors. 00:36:40.830 --> 00:36:43.820 Then the change-of-basis involves, just 00:36:43.820 --> 00:36:46.050 as it did there, W inverse. 00:36:46.050 --> 00:36:52.510 So we have the vector, say, x, in the old basis, 00:36:52.510 --> 00:37:04.050 and that converts to a vector, let's say, c, in the new basis, 00:37:04.050 --> 00:37:16.655 and the relation is exactly what we had there, that x is W c. 00:37:20.400 --> 00:37:22.680 That's the step we have to take. 00:37:22.680 --> 00:37:26.640 There's a matrix W that gives us a change-of-basis. 00:37:26.640 --> 00:37:28.650 OK. 00:37:28.650 --> 00:37:38.110 What I want to do is think about transformations on matrices. 00:37:38.110 --> 00:37:43.790 So here's the question to complete this lecture. 00:37:43.790 --> 00:37:50.930 Suppose I have a linear transformation T. 00:37:50.930 --> 00:37:57.390 So we would think of it as an eight -- as a n by n matrix. 00:37:57.390 --> 00:38:02.400 And it's computed with respect to a certain basis. 00:38:02.400 --> 00:38:04.960 So T -- no, I'm sorry. 00:38:04.960 --> 00:38:07.780 I've got the transformation T, period. 00:38:07.780 --> 00:38:09.760 That's taking eight-dimensional space 00:38:09.760 --> 00:38:12.240 to eight-dimensional space. 00:38:12.240 --> 00:38:15.340 Now, let's get matrices in there. 00:38:15.340 --> 00:38:16.650 OK. 00:38:16.650 --> 00:38:27.350 So, with respect to a first basis, say v1 up to v8, 00:38:27.350 --> 00:38:35.550 it has a matrix A. 00:38:35.550 --> 00:38:39.380 I'm just setting up letters here. 00:38:39.380 --> 00:38:48.000 With respect to a second basis, say, I'll make it u1 up to -- 00:38:48.000 --> 00:39:00.000 or w1, since I've used (w)s, w1 up to w8, it has a matrix B. 00:39:00.000 --> 00:39:05.830 And my question is, what's the connection between A 00:39:05.830 --> 00:39:09.890 How is the matrix -- the transformation T is settled. 00:39:09.890 --> 00:39:11.240 and B? 00:39:11.240 --> 00:39:16.660 We could say, it's a rotation, for example. 00:39:16.660 --> 00:39:18.490 So that would be one transformation 00:39:18.490 --> 00:39:21.640 of eight-dimensional space, just spin it a little. 00:39:24.740 --> 00:39:25.930 Or project it. 00:39:25.930 --> 00:39:32.070 Or whatever linear transformation we've got. 00:39:32.070 --> 00:39:35.240 Now, we have to remember -- 00:39:35.240 --> 00:39:41.420 my first step is to remind you how you create that matrix A. 00:39:41.420 --> 00:39:45.530 Then my second step is, we would use the same method to create 00:39:45.530 --> 00:39:49.520 B, but because it came from the same transformation, 00:39:49.520 --> 00:39:52.550 there's got to be a relation between A and B. 00:39:52.550 --> 00:39:55.660 What's the relation between A and B? 00:39:55.660 --> 00:40:02.230 And let me jump to the answer on that one. 00:40:02.230 --> 00:40:05.890 That if I have the same transformation, 00:40:05.890 --> 00:40:10.600 and I'm compute on its matrix in one basis, and then I computer 00:40:10.600 --> 00:40:16.910 it in another basis, those two matrices are similar. 00:40:16.910 --> 00:40:19.350 So these two matrices are similar. 00:40:19.350 --> 00:40:23.080 Now, do you remember what similar matrices meant? 00:40:23.080 --> 00:40:23.630 Similar. 00:40:23.630 --> 00:40:27.935 A is similar to -- the two matrices are similar. 00:40:27.935 --> 00:40:28.435 Similar. 00:40:31.280 --> 00:40:33.560 And what do I mean by that? 00:40:33.560 --> 00:40:40.700 I mean that I take the matrix B, and I can compute it 00:40:40.700 --> 00:40:47.320 from the matrix A using some similarity, some matrix 00:40:47.320 --> 00:40:52.180 M on one side, and M inverse on the other. 00:40:52.180 --> 00:40:55.530 And this M will be the change-of-basis matrix. 00:40:59.490 --> 00:41:03.740 This part of the lecture is, admittedly, compressed. 00:41:03.740 --> 00:41:05.820 What I wanted you to -- 00:41:05.820 --> 00:41:12.670 it's really the conclusion that I want you to spot. 00:41:12.670 --> 00:41:17.980 Now, I have to go back and say, what does it mean for A 00:41:17.980 --> 00:41:21.830 to be the matrix of this transformation T. 00:41:21.830 --> 00:41:23.710 So I have to remind you what that meant, 00:41:23.710 --> 00:41:26.280 that was in the last lecture. 00:41:26.280 --> 00:41:32.360 Then this is the conclusion that if I change to a different 00:41:32.360 --> 00:41:36.720 basis, we now know -- see, if I change to a different basis, 00:41:36.720 --> 00:41:38.340 two things happen. 00:41:38.340 --> 00:41:41.880 Every vector has new coordinates. 00:41:41.880 --> 00:41:45.190 There, the rule is this one, between the old coordinates 00:41:45.190 --> 00:41:46.660 and the new ones. 00:41:46.660 --> 00:41:50.570 Every matrix changes, every transformation has a new 00:41:50.570 --> 00:41:51.560 matrix. 00:41:51.560 --> 00:41:55.120 And the new matrix is related this way, 00:41:55.120 --> 00:41:57.710 the M could be the same as the W. 00:41:57.710 --> 00:42:00.300 The M there would be the W here. 00:42:00.300 --> 00:42:00.960 OK. 00:42:00.960 --> 00:42:04.790 So, can I, in the remaining minutes, 00:42:04.790 --> 00:42:09.620 recapture my lecture -- the end of my lecture that was just 00:42:09.620 --> 00:42:13.740 before Thanksgiving, about the matrix? 00:42:13.740 --> 00:42:15.660 OK. 00:42:15.660 --> 00:42:16.550 What's the matrix? 00:42:19.090 --> 00:42:21.040 And I'll just take one basis. 00:42:21.040 --> 00:42:26.830 So now this part is going to go onto this board here. 00:42:26.830 --> 00:42:27.970 What is the matrix? 00:42:27.970 --> 00:42:28.740 What is A? 00:42:32.830 --> 00:42:33.930 OK. 00:42:33.930 --> 00:42:42.280 Using a basis v1 up to v8. 00:42:42.280 --> 00:42:43.161 Mm. 00:42:43.161 --> 00:42:43.660 OK. 00:42:43.660 --> 00:42:46.350 What's the point? 00:42:46.350 --> 00:42:51.180 The point is, if I know what the transformation does 00:42:51.180 --> 00:42:56.990 to those eight basis vectors, I know it completely. 00:42:56.990 --> 00:42:59.140 I know T, I know everything about T, 00:42:59.140 --> 00:43:11.200 I know T completely from knowing T of V -- what T does to v1, 00:43:11.200 --> 00:43:15.040 what T does to v2, what T does to v8. 00:43:18.020 --> 00:43:19.760 Why is that? 00:43:19.760 --> 00:43:23.170 It's because T is a linear transformation. 00:43:23.170 --> 00:43:26.700 So that if I know what these outputs are -- 00:43:26.700 --> 00:43:29.360 so these are the inputs v1 up to v8, 00:43:29.360 --> 00:43:31.560 these are the outputs from the transformation, 00:43:31.560 --> 00:43:35.500 like everyone rotated, everyone projected, 00:43:35.500 --> 00:43:38.010 whatever transformation I've done, 00:43:38.010 --> 00:43:42.090 then why is it that I know everything? 00:43:42.090 --> 00:43:43.660 How does linearity work? 00:43:43.660 --> 00:43:45.020 Why? 00:43:45.020 --> 00:43:56.150 This is because every x is some combination of these basis 00:43:56.150 --> 00:43:57.870 vectors, right? 00:43:57.870 --> 00:44:05.570 c1v1, c2v2, c8v8, they were a basis. 00:44:05.570 --> 00:44:07.010 That's the whole point of a basis, 00:44:07.010 --> 00:44:09.790 that every vector is a combination of the basis 00:44:09.790 --> 00:44:12.210 vectors in exactly one way. 00:44:12.210 --> 00:44:14.737 And then, what is T of x? 00:44:18.670 --> 00:44:23.460 The point is, I claim that we know T of x completely 00:44:23.460 --> 00:44:27.770 for every x, because every x is a combination of those -- 00:44:27.770 --> 00:44:31.740 and now we use the linear transformation part to say that 00:44:31.740 --> 00:44:37.890 the output from x has to be c1 times the output from v1 plus 00:44:37.890 --> 00:44:43.330 v2 times the output from v2, and so on. 00:44:43.330 --> 00:44:46.360 Up through c8 times the output from v8. 00:44:46.360 --> 00:44:49.790 So this is like just saying, OK. 00:44:49.790 --> 00:44:54.900 We know everything when we know what 00:44:54.900 --> 00:44:57.790 T does to each basis vector. 00:44:57.790 --> 00:44:58.720 OK. 00:44:58.720 --> 00:45:02.000 So those are the eight things we need. 00:45:02.000 --> 00:45:08.330 Now -- but we need these answers in this basis. 00:45:08.330 --> 00:45:13.450 So this first output is some combination 00:45:13.450 --> 00:45:14.665 of the eight basis vectors. 00:45:17.480 --> 00:45:24.680 So write T acting on the first input -- in other words, 00:45:24.680 --> 00:45:27.880 write the first output as a combination of the basis 00:45:27.880 --> 00:45:38.040 vectors, say a11 v1 + a21 v2 and so on a81 v8. 00:45:41.740 --> 00:45:50.110 Write T of v2 as some combination a12 of v1, 00:45:50.110 --> 00:45:53.240 a22 of v2 and so on. 00:45:53.240 --> 00:45:57.270 I'm creating the matrix A, column by column. 00:45:57.270 --> 00:46:00.050 Those numbers go in the first column, 00:46:00.050 --> 00:46:06.200 these numbers go in the second column, the matrix A that thi- 00:46:06.200 --> 00:46:11.900 this -- this is our matrix that represents T in this basis is 00:46:11.900 --> 00:46:13.110 these numbers. 00:46:13.110 --> 00:46:22.290 a11 down to a18, a21 down to a28, and so on. 00:46:22.290 --> 00:46:25.060 OK. 00:46:25.060 --> 00:46:26.120 That's the recipe. 00:46:26.120 --> 00:46:31.090 In other words, if I give you a transformation, and a basis. 00:46:31.090 --> 00:46:32.720 So that's what I have to give you. 00:46:32.720 --> 00:46:37.550 The inputs are the basis and to tell you 00:46:37.550 --> 00:46:38.641 what the transformation 00:46:38.641 --> 00:46:39.140 is. 00:46:39.140 --> 00:46:42.840 And then, you tell me -- 00:46:42.840 --> 00:46:47.880 you compute T for each basis, expand 00:46:47.880 --> 00:46:51.270 that result in the basis, and that gives you 00:46:51.270 --> 00:46:55.900 the sixty four numbers that go into the matrix A. 00:46:55.900 --> 00:47:01.460 Let me suppose -- let's close with the best example of all. 00:47:01.460 --> 00:47:08.810 Suppose v1 to v8, this basis, is the eigenvectors. 00:47:08.810 --> 00:47:22.950 Suppose we have an eigenvector basis so that T(vi) 00:47:22.950 --> 00:47:27.660 is in the same direction of vi. 00:47:27.660 --> 00:47:29.790 Now, my question is, what is A? 00:47:35.370 --> 00:47:37.370 Can you carry through the steps? 00:47:37.370 --> 00:47:40.270 Let's do them together, because we can do it in one minute. 00:47:43.200 --> 00:47:47.120 So, we've chosen this perfect basis. 00:47:47.120 --> 00:47:52.010 And, actually, with signal image processing, 00:47:52.010 --> 00:47:55.520 they might look for the eigenvectors. 00:47:55.520 --> 00:47:57.790 But that would take more calculation 00:47:57.790 --> 00:48:02.040 time that just saying, OK, we'll use the wavelet basis. 00:48:02.040 --> 00:48:04.510 Or, OK, we'll use the Fourier basis. 00:48:04.510 --> 00:48:08.130 But the very best basis is the eigenvector basis. 00:48:08.130 --> 00:48:10.320 OK, what's the matrix? 00:48:10.320 --> 00:48:12.480 So, what's the first column of the matrix? 00:48:15.180 --> 00:48:17.130 How do I get the first column? 00:48:17.130 --> 00:48:19.185 I take the first basis vector v1. 00:48:22.000 --> 00:48:25.070 I opt -- I look to see, what does the transformation do 00:48:25.070 --> 00:48:26.940 to it? 00:48:26.940 --> 00:48:29.610 The output is lambda one v1. 00:48:29.610 --> 00:48:36.420 I express that output as a combination so the first input 00:48:36.420 --> 00:48:39.590 is v1. 00:48:39.590 --> 00:48:47.150 Its output is lambda one v1. 00:48:47.150 --> 00:48:51.900 Now write lambda one v1 as a combination of the basis 00:48:51.900 --> 00:48:54.330 vectors, well, it's already done. 00:48:54.330 --> 00:48:56.720 It's just lambda one times the first basis vector 00:48:56.720 --> 00:48:58.360 and zero times the others. 00:48:58.360 --> 00:49:03.490 So this first column will have lambda one and zeroes. 00:49:03.490 --> 00:49:05.060 OK. 00:49:05.060 --> 00:49:10.030 Second input is v2. 00:49:10.030 --> 00:49:15.220 Output is lambda two v2. 00:49:15.220 --> 00:49:19.060 OK, write that output as a combination of the (v)s. 00:49:19.060 --> 00:49:20.470 It's already done. 00:49:20.470 --> 00:49:23.120 It's just lambda two times the second v. 00:49:23.120 --> 00:49:25.190 So we need, in the second column, 00:49:25.190 --> 00:49:28.040 we have lambda two times the second v. 00:49:28.040 --> 00:49:31.410 Well, you see what's coming, that in that basis, 00:49:31.410 --> 00:49:37.470 in the eigenvector basis, the matrix is diagonal. 00:49:37.470 --> 00:49:40.950 So that's the perfect basis, that's 00:49:40.950 --> 00:49:43.840 the basis we'd love to have for image processing, 00:49:43.840 --> 00:49:47.770 but to find the eigenvectors of our pixel matrix 00:49:47.770 --> 00:49:49.640 would be too expensive. 00:49:49.640 --> 00:49:54.110 So we do something cheaper and close, 00:49:54.110 --> 00:49:58.200 which is to choose a good basis like wavelets. 00:49:58.200 --> 00:49:59.130 OK, thanks. 00:49:59.130 --> 00:50:04.490 So I'll -- quiz review on Wednesday, all day. 00:50:04.490 --> 00:50:06.040 Thanks.