# Picture Perfect: The Mathematics of JPEG Compression

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1 Picture Perfect: The Mathematics of JPEG Compression May 19, 2011

2 1 2 3 in 2D Sampling and the DCT in 2D 2D Compression

3 Images Outline A typical color image might be 600 by 800 pixels.

4 Images Outline A typical color image might be 600 by 800 pixels. Each pixel has a red (R), green (G) and blue (B) value associated to it.

5 Images Outline A typical color image might be 600 by 800 pixels. Each pixel has a red (R), green (G) and blue (B) value associated to it. Each pixel needs 3 bytes of storage (one for each color).

6 Images Outline A typical color image might be 600 by 800 pixels. Each pixel has a red (R), green (G) and blue (B) value associated to it. Each pixel needs 3 bytes of storage (one for each color). That s = 1.44 megabytes.

7 Images Outline A typical color image might be 600 by 800 pixels. Each pixel has a red (R), green (G) and blue (B) value associated to it. Each pixel needs 3 bytes of storage (one for each color). That s = 1.44 megabytes. But a typical 600 by 800 JPEG image takes only about 120 kilobytes, less than 1/10 the expected amount!

8 Images Outline A typical color image might be 600 by 800 pixels. Each pixel has a red (R), green (G) and blue (B) value associated to it. Each pixel needs 3 bytes of storage (one for each color). That s = 1.44 megabytes. But a typical 600 by 800 JPEG image takes only about 120 kilobytes, less than 1/10 the expected amount! How can we eliminate 90 percent of the required storage?

9 Compression Outline An interesting image:

10 Bad Compression Image compressed 10-fold:

11 The Basis for Compression JPEG compression (and a good chunk of applied mathematics) is based on one of the great ideas of 19th century mathematics: Functions can be decomposed into sums of sines and cosines of various frequencies.

12 Fourier Cosine Series Outline Fourier series come in many flavors. We ll be interested in Fourier Cosine Series: Any reasonable function f (t) defined on [0, π] can be well-approximated as a sum of cosines, f (t) a 0 + a 1 cos(t) + a 2 cos(2t) + + a N cos(nt) if we pick the a k correctly (and take N large enough).

13 A Function to Approximate f (t)

14 Cosine Series Example f (t) 4.70

15 Cosine Series Example f (t) cos(t)

16 Cosine Series Example f (t) cos(t) cos(2t)

17 The Cosine Series Outline f (t) cos(t) cos(2t) 5.88 cos(3t)

18 The Cosine Series Outline f (t) cos(t) cos(2t) 5.88 cos(3t) 9.92 cos(4t)

19 The Cosine Series Outline cos(5t) cos(6t)

20 The Cosine Series Outline cos(7t) 0.53 cos(8t)

21 The Cosine Coefficients A piecewise continuously-differentiable function f (t) defined on [0, π] can be approximated with a Fourier sum where f (t) a a 1 cos(t) + a 2 cos(2t) + + a N cos(nt) a k = 2 π π 0 f (t) cos(kt) dt As N the sum converges to f (t) for any t at which f is continuous.

22 Periodicity Outline Outside the interval [0, π] the cosine series extends f as an even period 2π function to the real line:

23 Discontinuities Outline Discontinuities are difficult to represent, for they take a lot of cosine terms to synthesize accurately. Here N = 50:

24 Compressing a Function So a function f (t) on [0, π] is determined by its Fourier cosine coefficients a 0, a 1, a 2..., an infinite amount of information. We could compress this information by

25 Compressing a Function So a function f (t) on [0, π] is determined by its Fourier cosine coefficients a 0, a 1, a 2..., an infinite amount of information. We could compress this information by Throwing out all small a k, e.g., a k < δ for some δ, and

26 Compressing a Function So a function f (t) on [0, π] is determined by its Fourier cosine coefficients a 0, a 1, a 2..., an infinite amount of information. We could compress this information by Throwing out all small a k, e.g., a k < δ for some δ, and Quantizing the remaining a k by, for example, rounding them to the nearest integer, or more generally,

27 Compressing a Function So a function f (t) on [0, π] is determined by its Fourier cosine coefficients a 0, a 1, a 2..., an infinite amount of information. We could compress this information by Throwing out all small a k, e.g., a k < δ for some δ, and Quantizing the remaining a k by, for example, rounding them to the nearest integer, or more generally, Rounding them to the nearest multiple of r for some fixed r.

28 Function Compression Example The function f (t) above has as its first 12 Fourier cosine coefficients 9.403, 19.09, 18.97, 5.883, 9.919, 12.38, 2.967, 1.705, , ,

29 Function Compression Example The function f (t) above has as its first 12 Fourier cosine coefficients 9.403, 19.09, 18.97, 5.883, 9.919, 12.38, 2.967, 1.705, , , If we round each to the nearest integer we obtain 9, 19, 19, 6, 10, 12, 3, 2, 1, 0, 0, 0 We can reconstruct f from these approximate cosine coefficients.

30 Function Compression Example The original function (black) and compressed/reconstructed version (red)

31 Sampling and Discretization Signals and images aren t presented as functions, but as sampled function values:

32 Sampling and Discretization We might break [0, π] into N subintervals, sample f at each midpoint t 0, t 1,..., t N 1 :

33 Sampling and Discretization We replace f (t) with the N-vector f = (f (t 0 ), f (t 1 ),..., f (t N 1 )). Each basis function cos(kt) also gets replaced with a vector 2 v k = N (cos(kt 0), cos(kt 1 ),..., cos(kt N 1 )) except v 0 = 1 (1, 1,..., 1) N (The factor in front makes the vectors have length one.)

34 Symmetries and Discretization The vectors v 0,..., v N 1 are orthonormal, and so form a basis for R N! We can thus write any signal vector f as f = c 0 v 0 + c 1 v c N 1 v N 1 for certain constants c k, analogous to the cosine series f (t) = a a 1 cos(t) + a 2 cos(2t) + How do we compute the c k?

35 Sampling and Discretization Dot each side of f = c 0 v 0 + c 1 v c N 1 v N 1 with v k and use v j v k = 0 for j k (while v k v k = 1) to find c k = f v k.

36 Sampling and Discretization The formula is analogous to a k = 2 π c k = f v k π 0 f (t) cos(kt) dt. In fact, the former is just the midpoint rule for evaluating the latter integral!

37 Then any signal vector f = c 0 v 0 + c 1 v c N 1 v N 1 where c k = f v k. The map f c = (c 0,..., c N 1 ) is the Discrete Cosine Transform (DCT).

38 DCT Example Outline Consider a signal vector in R 8 : f =< 36.0, 22.3, 24.2, 1.55, 40.4, 9.90, 10.3, 3.99 >

39 DCT Example Outline We can decompose f = c 0 v 0 + c 1 v c 6 v 6 + c 7 v 7, f = Here c =< 19.0, 42.5, 31.7, 14.0, 14.9, 23.0, 11.7, 4.70 >.

40 Matrix Formulation of the DCT The DCT maps a vector f = (f 0,..., f N 1 ) to a vector c = (c 0,..., c N 1 ) as c k = v k f, or in matrix terms, c = C N f where C N is the N N matrix with the v k as rows. The inverse DCT is just f = C T N c since C N turns out to be orthogonal.

41 Discrete Compression To compress a discretized signal (vector) f R N :

42 Discrete Compression To compress a discretized signal (vector) f R N : 1 Perform a DCT c = C N f.

43 Discrete Compression To compress a discretized signal (vector) f R N : 1 Perform a DCT c = C N f. 2 Zero out all c k below a certain threshold, quantize the rest (round to nearest integer, near tenth, etc.) This is where we get compression.

44 Discrete Compression To compress a discretized signal (vector) f R N : 1 Perform a DCT c = C N f. 2 Zero out all c k below a certain threshold, quantize the rest (round to nearest integer, near tenth, etc.) This is where we get compression. 3 The signal can be (approximately) reconstituted using the thresholded quantized c k and an inverse DCT.

45 Audio Compression Example The file gong1.wav consists of 50,000 sampled audio values, so f R

46 Audio Compression Example The file gong1.wav consists of 50,000 sampled audio values, so f R With Matlab, we can perform a DCT, round each coefficient to the nearest integer (or multiple of r). This is where compression happens (most c k will be zero).

47 Audio Compression Example The file gong1.wav consists of 50,000 sampled audio values, so f R With Matlab, we can perform a DCT, round each coefficient to the nearest integer (or multiple of r). This is where compression happens (most c k will be zero). We can approximately reconstitute the original audio signal with an inverse DCT.

48 Blocking Outline 1 Discontinuities cause many DCT coefficients to be large they don t decay to zero very fast and so the file is hard to compress.

49 Blocking Outline 1 Discontinuities cause many DCT coefficients to be large they don t decay to zero very fast and so the file is hard to compress. 2 It can help to break the signal into blocks and compress each individually, to limit the effect of any discontinuity to one block.

50 Compression Example 2 A signal with discontinuities, and its compressed version (eight-bit) 62 percent of the DCT coefficients remain non-zero.

51 Compression Example 2 A signal with discontinuities, compressed in blocks of 50 samples: 38 percent of the DCT coefficients remain non-zero.

52 Grayscale Images Outline in 2D Sampling and the DCT in 2D 2D Compression 1 A grayscale image on a region (x, y) [0, A] [0, B] is modeled by a function f (x, y), with (for example) f = 0 as black, f = 255 as white.

53 Grayscale Images Outline in 2D Sampling and the DCT in 2D 2D Compression 1 A grayscale image on a region (x, y) [0, A] [0, B] is modeled by a function f (x, y), with (for example) f = 0 as black, f = 255 as white. 2 A color image might be modeled by three functions, R(x, y), G(x, y), and B(x, y), assigning a red, green, and blue value to each point.

54 Grayscale Images Outline in 2D Sampling and the DCT in 2D 2D Compression 1 A grayscale image on a region (x, y) [0, A] [0, B] is modeled by a function f (x, y), with (for example) f = 0 as black, f = 255 as white. 2 A color image might be modeled by three functions, R(x, y), G(x, y), and B(x, y), assigning a red, green, and blue value to each point. 3 We ll only be concerned with grayscale images.

55 in 2D Outline in 2D Sampling and the DCT in 2D 2D Compression A function f (x, y) defined on (x, y) [0, π] [0, π] can be expanded into a Fourier cosine series f (x, y) = a 0, j=1 k=1 (a j,0 cos(jx) + a 0,j cos(jy)) j=1 a jk cos(jx) cos(ky) where a jk = 4 π 2 π π 0 0 f (x, y) cos(jx) cos(ky) dx dy.

56 2D Example Outline in 2D Sampling and the DCT in 2D 2D Compression f (x, y) = e ((x 2)2 +(y 1) 2 )/5 + sin(x 2 y)/20, a 0,0 term only:

57 2D Example Outline in 2D Sampling and the DCT in 2D 2D Compression All terms up to j = 2, k = 2:

58 2D Example Outline in 2D Sampling and the DCT in 2D 2D Compression All terms up to j = 10, k = 10:

59 Sampling in 2D Outline in 2D Sampling and the DCT in 2D 2D Compression A grayscale image on a square 0 x, y π can be considered as a function f (x, y). The sampled version is the M N matrix q with entries q jk = f (x k, y j ), sampled on a rectangular grid. Each sample value q jk is the gray value of the corresponding pixel.

60 The 2D DCT Outline in 2D Sampling and the DCT in 2D 2D Compression

61 The 2D DCT Outline in 2D Sampling and the DCT in 2D 2D Compression Replace image function f with sampled version q, an M N matrix.

62 The 2D DCT Outline in 2D Sampling and the DCT in 2D 2D Compression Replace image function f with sampled version q, an M N matrix. Replace cos(jx) cos(ky) basis function with sampled version, M N basis matrix E jk, 0 j N 1, 0 k M 1.

63 The 2D DCT Outline in 2D Sampling and the DCT in 2D 2D Compression Replace image function f with sampled version q, an M N matrix. Replace cos(jx) cos(ky) basis function with sampled version, M N basis matrix E jk, 0 j N 1, 0 k M 1. We can write q = M j=0 k=0 N q jk E jk for certain q jk, components of an M N matrix q.

64 The 2D DCT Outline in 2D Sampling and the DCT in 2D 2D Compression Replace image function f with sampled version q, an M N matrix. Replace cos(jx) cos(ky) basis function with sampled version, M N basis matrix E jk, 0 j N 1, 0 k M 1. We can write q = M j=0 k=0 N q jk E jk for certain q jk, components of an M N matrix q. It turns out that q = C M qc T N.

65 The 2D DCT Outline in 2D Sampling and the DCT in 2D 2D Compression As grayscale images, the matrices E jk look like (cases (j, k) = (0, 0), (0, 1), (3, 7), (4, 21)): The 2D DCT expresses any image as a sum of these basic images.

66 2D Compression Example in 2D Sampling and the DCT in 2D 2D Compression A grayscale image to compress:

67 2D Compression Example in 2D Sampling and the DCT in 2D 2D Compression The DCT, displayed as a grayscale image (log rescaling):

68 2D Compression Example in 2D Sampling and the DCT in 2D 2D Compression The strategy: 1 Perform a DCT on the image, to produce M N array q of DCT coefficients.

69 2D Compression Example in 2D Sampling and the DCT in 2D 2D Compression The strategy: 1 Perform a DCT on the image, to produce M N array q of DCT coefficients. 2 Quantize the array by rounding each q jk, zeroing out small q jk (this is where compression occurs).

70 2D Compression Example in 2D Sampling and the DCT in 2D 2D Compression The strategy: 1 Perform a DCT on the image, to produce M N array q of DCT coefficients. 2 Quantize the array by rounding each q jk, zeroing out small q jk (this is where compression occurs). 3 The image can be reconstructed by using the quantized array and an inverse DCT (2D).

71 2D Compression Example in 2D Sampling and the DCT in 2D 2D Compression Round DCT coefficients to near multiple of 100 (yields 90 percent compression):

72 Block Compression Outline in 2D Sampling and the DCT in 2D 2D Compression As in 1D, discontinuities (edges) cause trouble. It can help to divide the image into blocks and compress each individually.

73 Block Compression Outline in 2D Sampling and the DCT in 2D 2D Compression As in 1D, discontinuities (edges) cause trouble. It can help to divide the image into blocks and compress each individually. The JPEG standard uses 8 8 pixel blocks.

74 2D JPEG Example Outline in 2D Sampling and the DCT in 2D 2D Compression The satellite image compressed in 8 8 blocks, 90 percent DCT coefficients zeroed out:

75 Final Remarks Outline in 2D Sampling and the DCT in 2D 2D Compression The ideas of Fourier analysis appears in many other areas of image processing, e.g., noise removal, edge detection.

76 Final Remarks Outline in 2D Sampling and the DCT in 2D 2D Compression The ideas of Fourier analysis appears in many other areas of image processing, e.g., noise removal, edge detection. The new JPEG 2000 standard replaces the cosine basis functions with wavelet basis functions.

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