Wavelets and Signal Processing

Transcription

1 Wavelets and Signal Processing John E. Gilbert Mathematics in Science Lecture April 30, 2002.

2 Publicity Mathematics In Science* A LECTURE SERIES FOR UNDERGRADUATES Wavelets Professor John Gilbert Mathematics and TICAM Compression of 3D images for the web Motivation for wavelets and contributions to the theory of wavelets came from seismology, physics, and signal processing as much as from mathematics itself. In turn, this means that there are many varied applications of the mathematics underlying wavelets. This talk will describe some basic wavelet ideas and how they can be used in signal analysis and data detection. Compression of fingerprints for the FBI database Tuesday, April 30, 4pm RLM Version: April 29, 2002; Typeset on May 25, 2002,17:41 2

3 Misconceptions Math isn t about real things in daily life versus: way to hear music or speech; read EKG s; transmit data; image enhancement. Version: April 29, 2002; Typeset on May 25, 2002,17:41 3

4 Misconceptions Math isn t about real things in daily life versus: way to hear music or speech; read EKG s; transmit data; image enhancement. Math results are abstract versus: can see results, e.g. compression ; hear them, e.g. de-noising; reconstruct images, e.g. extrapolation. Version: April 29, 2002; Typeset on May 25, 2002,17:41 3

5 Misconceptions Math isn t about real things in daily life versus: way to hear music or speech; read EKG s; transmit data; image enhancement. Math results are abstract versus: can see results, e.g. compression ; hear them, e.g. de-noising; reconstruct images, e.g. extrapolation. Math amounts to finding right formula or right theorem. versus: iterative series of explorations, asking questions, getting back answers, refining techniques, approximations. Version: April 29, 2002; Typeset on May 25, 2002,17:41 3

6 Signal = Function Mathematically, a signal is a function of continuous variable(s) = analogue of discrete variable(s) = digital Version: April 29, 2002; Typeset on May 25, 2002,17:41 4

7 Signal = Function Mathematically, a signal is a function of continuous variable(s) = analogue of discrete variable(s) = digital one variable: audio signals, sensor outputs, heart beats,... two variables: video images (still, moving), MRI,... Version: April 29, 2002; Typeset on May 25, 2002,17:41 4

8 Signal = Function Mathematically, a signal is a function of continuous variable(s) = analogue of discrete variable(s) = digital one variable: audio signals, sensor outputs, heart beats,... two variables: video images (still, moving), MRI,... Fundamental concerns: Compression for storage or transmission, de-noising Version: April 29, 2002; Typeset on May 25, 2002,17:41 4

9 Signal = Function Mathematically, a signal is a function of continuous variable(s) = analogue of discrete variable(s) = digital one variable: audio signals, sensor outputs, heart beats,... two variables: video images (still, moving), MRI,... Fundamental concerns: Compression for storage or transmission, de-noising Data-mining, Edge-detection in (possibly) noisy signals Version: April 29, 2002; Typeset on May 25, 2002,17:41 4

10 Audio signals Caruso is just a few mouse clicks away original denoised noise Version: April 29, 2002; Typeset on May 25, 2002,17:41 5

11 Audio signals Caruso is just a few mouse clicks away original denoised noise Represent noisy signal graphically: original signal = clear signal + noise Version: April 29, 2002; Typeset on May 25, 2002,17:41 5

12 Images: test cases Some basic features of images are shown in Barbara Version: April 29, 2002; Typeset on May 25, 2002,17:41 6

13 Images: test cases Some basic features of images are shown in Barbara ridges, edges. Same for fingerprints In images transients may depend on direction (think fingerprints). Version: April 29, 2002; Typeset on May 25, 2002,17:41 6

14 Compress fingerprints each one 1 MB. storage, transmission, security Version: April 29, 2002; Typeset on May 25, 2002,17:41 7

15 Compress fingerprints each one 1 MB. storage, transmission, security original image % compression Can probably do much better with specialized software. Version: April 29, 2002; Typeset on May 25, 2002,17:41 7

16 To your good health real data. Need to analyze distance between peaks. Monitor patients; immediate detection onset heart irregularities = transients. edge detec- Basic problem: tion Version: April 29, 2002; Typeset on May 25, 2002,17:41 8

17 To your good health real data. Need to analyze distance between peaks. Monitor patients; immediate detection onset heart irregularities = transients. edge detec- Basic problem: tion tumor Tumor malignant when spikes present. Basic problem: (directional) edge detection. Version: April 29, 2002; Typeset on May 25, 2002,17:41 8

18 Bases, Representations Standard idea: given mathematical object, represent it with respect to carefully chosen basis. Version: April 29, 2002; Typeset on May 25, 2002,17:41 9

19 Bases, Representations Standard idea: given mathematical object, represent it with respect to carefully chosen basis. Vectors in R 3 : usual Cartesian basis i, j, k v = a 1 i + a 2 j + a 3 k (a 1, a 2, a 3 ). Measure size ( = energy ) by dot product of vectors Version: April 29, 2002; Typeset on May 25, 2002,17:41 9

20 Bases, Representations Standard idea: given mathematical object, represent it with respect to carefully chosen basis. Vectors in R 3 : usual Cartesian basis i, j, k v = a 1 i + a 2 j + a 3 k (a 1, a 2, a 3 ). Measure size ( = energy ) by dot product of vectors energy(v) = length(v) 2 = a a2 2 + a2 3 = v.v Version: April 29, 2002; Typeset on May 25, 2002,17:41 9

21 Bases, Representations Standard idea: given mathematical object, represent it with respect to carefully chosen basis. Vectors in R 3 : usual Cartesian basis i, j, k v = a 1 i + a 2 j + a 3 k (a 1, a 2, a 3 ). Measure size ( = energy ) by dot product of vectors energy(v) = length(v) 2 = a a2 2 + a2 3 = v.v In high dimensions achieve compression by projecting onto low-dimensional subspace. Loss of energy? Version: April 29, 2002; Typeset on May 25, 2002,17:41 9

22 Bases, Representations Standard idea: given mathematical object, represent it with respect to carefully chosen basis. Vectors in R 3 : usual Cartesian basis i, j, k v = a 1 i + a 2 j + a 3 k (a 1, a 2, a 3 ). Measure size ( = energy ) by dot product of vectors energy(v) = length(v) 2 = a a2 2 + a2 3 = v.v In high dimensions achieve compression by projecting onto low-dimensional subspace. Loss of energy? Maybe need to change basis depending on how subspace lies. Version: April 29, 2002; Typeset on May 25, 2002,17:41 9

23 Bases from Calculus Need bases having good diff and integral properties. Choose monomials. Version: April 29, 2002; Typeset on May 25, 2002,17:41 10

24 Bases from Calculus Need bases having good diff and integral properties. Choose monomials. Taylor series representations: f smooth near origin. 1 f(x) = n! f (n) (0) x n. n = 0 Version: April 29, 2002; Typeset on May 25, 2002,17:41 10

25 Bases from Calculus Need bases having good diff and integral properties. Choose monomials. Taylor series representations: f smooth near origin. 1 f(x) = n! f (n) (0) x n. n = 0 Dot product ( d ) f.g = f dx g(x) x = 0 Orthonormal basis 1 n! x n, n = 0, 1, 2,.... Version: April 29, 2002; Typeset on May 25, 2002,17:41 10

26 Bases from Calculus Need bases having good diff and integral properties. Choose monomials. Taylor series representations: f smooth near origin. 1 f(x) = n! f (n) (0) x n. n = 0 Dot product ( d ) f.g = f dx g(x) x = 0 Orthonormal basis 1 n! x n, n = 0, 1, 2,.... Energy(f) = n = 0 1 n! (f (n) (0)) 2. Version: April 29, 2002; Typeset on May 25, 2002,17:41 10

27 Compression in Calculus Taylor polynomials P N f(x) = N n = 0 1 n! f (n) (0) x n provide compression with little loss of energy if tail 1 ( f (n) (0) ) 2 n! of energy is small. n = N+1 Version: April 29, 2002; Typeset on May 25, 2002,17:41 11

28 Compression in Calculus Taylor polynomials P N f(x) = N n = 0 1 n! f (n) (0) x n provide compression with little loss of energy if tail 1 ( f (n) (0) ) 2 n! of energy is small. n = N+1 Usually don t measure error f(x) P N f(x) = n = N+1 1 n! f (n) (0) x n by loss of energy, but by pointwise maximum. Version: April 29, 2002; Typeset on May 25, 2002,17:41 11

29 Compression of matrices Simple linear transformation T : R 2 R 2. In matrix form with respect to usual basis of plane [ ] [ a b 1 T, T : c d 2 ] [ a + 2b c + 2d ]. Version: April 29, 2002; Typeset on May 25, 2002,17:41 12

30 Compression of matrices Simple linear transformation T : R 2 R 2. In matrix form with respect to usual basis of plane [ ] [ a b 1 T, T : c d 2 ] [ a + 2b c + 2d Energy(T ) = a 2 + b 2 + c 2 + d 2 = trace (T T trans ). (independent of basis). ]. Version: April 29, 2002; Typeset on May 25, 2002,17:41 12

31 Compression of matrices Simple linear transformation T : R 2 R 2. In matrix form with respect to usual basis of plane [ ] [ a b 1 T, T : c d 2 ] [ a + 2b c + 2d Energy(T ) = a 2 + b 2 + c 2 + d 2 = trace (T T trans ). (independent of basis). compression when diagonalized by orthogonal matrix [ ] λ1 0 T 0 λ 2 by changing to new basis of eigenvectors. No loss of energy!! ]. Version: April 29, 2002; Typeset on May 25, 2002,17:41 12

32 Example T Usual basis: [ ] 5 4 T 4 5 New basis: [ ] 9 0 T 0 1 Compress: [ ] 9 0 T 0 0 compression of 75% with less than 1.25% loss of energy. Version: April 29, 2002; Typeset on May 25, 2002,17:41 13

33 Example T Usual basis: [ ] 5 4 T 4 5 New basis: [ ] 9 0 T 0 1 Compress: [ ] 9 0 T 0 0 compression of 75% with less than 1.25% loss of energy. T on (1, 2) compressed T on (1, 2) Version: April 29, 2002; Typeset on May 25, 2002,17:41 13

34 Basic wavelet ideas Most data sets have correlation in time (space) and frequency. Version: April 29, 2002; Typeset on May 25, 2002,17:41 14

35 Basic wavelet ideas Most data sets have correlation in time (space) and frequency. Wavelets: building blocks to quickly decorrelate data. Version: April 29, 2002; Typeset on May 25, 2002,17:41 14

36 Basic wavelet ideas Most data sets have correlation in time (space) and frequency. Wavelets: building blocks to quickly decorrelate data. Building blocks = bases: f = n γ n ψ n Version: April 29, 2002; Typeset on May 25, 2002,17:41 14

37 Basic wavelet ideas Most data sets have correlation in time (space) and frequency. Wavelets: building blocks to quickly decorrelate data. Building blocks = bases: f = n γ n ψ n Decorrelate data: wavelets to resemble given data. Version: April 29, 2002; Typeset on May 25, 2002,17:41 14

38 Basic wavelet ideas Most data sets have correlation in time (space) and frequency. Wavelets: building blocks to quickly decorrelate data. Building blocks = bases: f = n γ n ψ n Decorrelate data: wavelets to resemble given data. 3 2 simple + transients signal: { f ( n ) } n = sample values Version: April 29, 2002; Typeset on May 25, 2002,17:41 14

39 Wavelets = little waves Simplest wavelet functions introduced by: Haar (1910) φ(x) dx = 1, ψ(x) dx = 0. scaling function φ wavelet function ψ Version: April 29, 2002; Typeset on May 25, 2002,17:41 15

40 Wavelets = little waves Simplest wavelet functions introduced by: Haar (1910) φ(x) dx = 1, ψ(x) dx = 0. scaling function φ wavelet function ψ Basic relations: φ(x) = φ(2x) + φ(2x 1), ψ(x) = φ(2x) φ(2x 1) Version: April 29, 2002; Typeset on May 25, 2002,17:41 15

41 Wavelets = little waves Simplest wavelet functions introduced by: Haar (1910) φ(x) dx = 1, ψ(x) dx = 0. scaling function φ wavelet function ψ Basic relations: φ(x) = φ(2x) + φ(2x 1), ψ(x) = φ(2x) φ(2x 1) φ(2x) = 1 2 ( ) φ(x) + ψ(x), φ(2x 1) = 1 2 ( ) φ(x) ψ(x) Version: April 29, 2002; Typeset on May 25, 2002,17:41 15

42 wavelet analysis, synthesis Decompose into hierarchichal set of approximations and details Version: April 29, 2002; Typeset on May 25, 2002,17:41 16

43 wavelet analysis, synthesis Decompose into hierarchichal set of approximations and details Hierarchichal: decreasing resolution = increasing coarseness. Version: April 29, 2002; Typeset on May 25, 2002,17:41 16

44 wavelet analysis, synthesis Decompose into hierarchichal set of approximations and details Hierarchichal: decreasing resolution = increasing coarseness. digitized signal n = 0 f ( n 256 ) φ(2 8 x n) level 8 resolution Version: April 29, 2002; Typeset on May 25, 2002,17:41 16

45 Coarse + detail idea Use relation φ(2x) = 1 2 algebraically: graphically: ( ) φ(x) + ψ(x), φ(2x 1) = 1 ( ) φ(x) ψ(x) 2 aφ(2x) + bφ(2x 1) = 1 2 ( ) a + b φ(x) + 1 ( ) a b ψ(x) 2 + original coarse detail + fine detail Version: April 29, 2002; Typeset on May 25, 2002,17:41 17

46 Hunt the Transients simple signal transients noisy signal transients in noise Version: April 29, 2002; Typeset on May 25, 2002,17:41 18

47 Digitized signal { ( f n 256 ) } 255 n = 0 Split signal: at resolution coarse detail level fine detail level 7 Now split level 7 coarse detail; then split level 6 coarse detail. Version: April 29, 2002; Typeset on May 25, 2002,17:41 19

48 From level 8 down to level coarse detail level fine detail level fine detail level 7 fine detail level 5 Version: April 29, 2002; Typeset on May 25, 2002,17:41 20

49 Source of hierachical idea wavelet functions are related by scaling equations. In general: filter coefficients h 0, h 1,..., h N N N φ(x) = h n φ(2x k), ψ(x) = ( 1) k h 1 2k ψ(2x k). k = 0 k = 0 Version: April 29, 2002; Typeset on May 25, 2002,17:41 21

50 Source of hierachical idea wavelet functions are related by scaling equations. In general: filter coefficients h 0, h 1,..., h N N N φ(x) = h n φ(2x k), ψ(x) = ( 1) k h 1 2k ψ(2x k). k = 0 k = 0 Coarse + detail: f(x) = n a (1) n φ(2x n) = n a (0) n φ(x n)+ n d (0) n ψ(x n) Version: April 29, 2002; Typeset on May 25, 2002,17:41 21

51 Source of hierachical idea wavelet functions are related by scaling equations. In general: filter coefficients h 0, h 1,..., h N N N φ(x) = h n φ(2x k), ψ(x) = ( 1) k h 1 2k ψ(2x k). k = 0 k = 0 Coarse + detail: f(x) = n a (1) n φ(2x n) = n a (0) n φ(x n)+ n d (0) n ψ(x n) where a (0) n running average, d (0) n running difference, of a (1) n using h 0, h 1,..., h N. Version: April 29, 2002; Typeset on May 25, 2002,17:41 21

52 Ingrid Daubechies (1988) Constructed smooth, compactly supported wavelets with N = 3, 5, 7,.... (Haar N = 1) Version: April 29, 2002; Typeset on May 25, 2002,17:41 22

53 Ingrid Daubechies (1988) Constructed smooth, compactly supported wavelets with N = 3, 5, 7,.... (Haar N = 1) db2 scaling function db2 wavelet h 0 = , h 1 = , h 2 = 3 3 4, h 3 = Version: April 29, 2002; Typeset on May 25, 2002,17:41 22

54 Ingrid Daubechies (1988) Constructed smooth, compactly supported wavelets with N = 3, 5, 7,.... (Haar N = 1) db2 scaling function db2 wavelet h 0 = , h 1 = , h 2 = 3 3 4, h 3 = How does one arrive at these numbers? Version: April 29, 2002; Typeset on May 25, 2002,17:41 22

55 Poor Ingrid Image restoration Received Signal (s) Reconstructed Signal (r) Done by Maia Croft as Honors Thesis at McQuarie University. Version: April 29, 2002; Typeset on May 25, 2002,17:41 23

56 Poor Ingrid Image restoration Received Signal (s) Reconstructed Signal (r) Done by Maia Croft as Honors Thesis at McQuarie University. Wavelets can be used to add to, fill-in images. Extend ideas to image fusion, morphing? Version: April 29, 2002; Typeset on May 25, 2002,17:41 23

57 Complex signals; speech General Fourier advises that excessive use of wavelets is dangerous to your success: need more sophisticated decompositions. At each stage choose to split coarse detail or fine detail. Need cost function to determine which is cheaper. Version: April 29, 2002; Typeset on May 25, 2002,17:41 24

58 Complex signals; speech General Fourier advises that excessive use of wavelets is dangerous to your success: need more sophisticated decompositions. At each stage choose to split coarse detail or fine detail. Need cost function to determine which is cheaper. Get wavelet packets: Dictionary of bases. original 10%wavelet 10%wavepacket Version: April 29, 2002; Typeset on May 25, 2002,17:41 24

59 Complex signals; speech General Fourier advises that excessive use of wavelets is dangerous to your success: need more sophisticated decompositions. At each stage choose to split coarse detail or fine detail. Need cost function to determine which is cheaper. Get wavelet packets: Dictionary of bases. original 10%wavelet 10%wavepacket Wavelets exploit: scaling (resolution), translation. Version: April 29, 2002; Typeset on May 25, 2002,17:41 24

60 Complex signals; speech General Fourier advises that excessive use of wavelets is dangerous to your success: need more sophisticated decompositions. At each stage choose to split coarse detail or fine detail. Need cost function to determine which is cheaper. Get wavelet packets: Dictionary of bases. original 10%wavelet 10%wavepacket Wavelets exploit: scaling (resolution), translation. Wavelet packets add: frequency. Version: April 29, 2002; Typeset on May 25, 2002,17:41 24

61 Want to learn more? R E U Research Experience for Undergraduates In Summer 2002, Professor John Gilbert will present a sixweek intensive course on Fourier series, discrete Fourier transforms, spectrograms and wavelets. The course will discuss Haar wavelets, and use them as a means of analyzing and also compressing data. The course will also introduce the Daubechies wavelets, and apply them to reducing noise or extracting music from LP recordings. We also give applications to image compression (the FBI fingerprint databank) and image reconstruction. The course is open to all UT Austin undergraduates. In addition, funds are available to support 13 undergraduates for the six weeks, at $2400 per person. Applications, topics, and reading lists are all online at: Cross section of the organ of Corti, performing frequency analysis in the ear. Power spectrum from the song Malaika, by Angelique Kidjo. Spectrogram from the song Malaika, by Angelique Kidjo. Wavelet transform from the song Malaika, by Angelique Kidjo. Version: April 29, 2002; Typeset on May 25, 2002,17:41 25