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CMU SCS 15-826: Multimedia Databases and Data Mining Lecture #23: DSP tools – Fourier and Wavelets C. Faloutsos.

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Presentation on theme: "CMU SCS 15-826: Multimedia Databases and Data Mining Lecture #23: DSP tools – Fourier and Wavelets C. Faloutsos."— Presentation transcript:

1 CMU SCS 15-826: Multimedia Databases and Data Mining Lecture #23: DSP tools – Fourier and Wavelets C. Faloutsos

2 CMU SCS 15-826Copyright: C. Faloutsos (2012)2 Must-read Material DFT/DCT: In PTVF ch. 12.1, 12.3, 12.4; in Textbook Appendix B.PTVF Textbook Wavelets: In PTVF ch. 13.10; in MM Textbook Appendix CPTVFMM Textbook

3 CMU SCS 15-826Copyright: C. Faloutsos (2012)3 Outline Goal: ‘Find similar / interesting things’ Intro to DB Indexing - similarity search Data Mining

4 CMU SCS 15-826Copyright: C. Faloutsos (2012)4 Indexing - Detailed outline primary key indexing.. multimedia Digital Signal Processing (DSP) tools –Discrete Fourier Transform (DFT) –Discrete Wavelet Transform (DWT)

5 CMU SCS 15-826Copyright: C. Faloutsos (2012)5 DSP - Detailed outline DFT –what –why –how –Arithmetic examples –properties / observations –DCT –2-d DFT –Fast Fourier Transform (FFT)

6 CMU SCS 15-826Copyright: C. Faloutsos (2012)6 Introduction Goal: given a signal (eg., sales over time and/or space) Find: patterns and/or compress year count lynx caught per year

7 CMU SCS 15-826Copyright: C. Faloutsos (2012)7 What does DFT do? A: highlights the periodicities

8 CMU SCS 15-826Copyright: C. Faloutsos (2012)8 Why should we care? A: several real sequences are periodic Q: Such as?

9 CMU SCS 15-826Copyright: C. Faloutsos (2012)9 Why should we care? A: several real sequences are periodic Q: Such as? A: –sales patterns follow seasons; –economy follows 50-year cycle –temperature follows daily and yearly cycles Many real signals follow (multiple) cycles

10 CMU SCS 15-826Copyright: C. Faloutsos (2012)10 Why should we care? For example: human voice! Frequency analyzer http://www.relisoft.com/freeware/freq.html http://www.relisoft.com/freeware/freq.html speaker identification impulses/noise -> flat spectrum high pitch -> high frequency Freq.exe

11 CMU SCS 15-826Copyright: C. Faloutsos (2012)11 ‘Frequency Analyzer’

12 CMU SCS 15-826Copyright: C. Faloutsos (2012)12 DFT and stocks Dow Jones Industrial index, 6/18/2001- 12/21/2001

13 CMU SCS 15-826Copyright: C. Faloutsos (2012)13 DFT and stocks Dow Jones Industrial index, 6/18/2001- 12/21/2001 just 3 DFT coefficients give very good approximation freq Log(ampl)

14 CMU SCS 15-826Copyright: C. Faloutsos (2012)14 DFT: definition Discrete Fourier Transform (n-point): inverse DFT

15 CMU SCS 15-826Copyright: C. Faloutsos (2012)15 How does it work? Decomposes signal to a sum of sine (and cosine) waves. Q:How to assess ‘similarity’ of x with a wave? 0 1 n-1 time value x ={x 0, x 1,... x n-1 } Skip

16 CMU SCS 15-826Copyright: C. Faloutsos (2012)16 How does it work? A: consider the waves with frequency 0, 1,...; use the inner-product (~cosine similarity) 0 1 n-1 time value freq. f=0 0 1 n-1 time value freq. f=1 (sin(t * 2  n) ) Skip

17 CMU SCS 15-826Copyright: C. Faloutsos (2012)17 How does it work? A: consider the waves with frequency 0, 1,...; use the inner-product (~cosine similarity) 0 1 n-1 time value freq. f=2 Skip

18 CMU SCS 15-826Copyright: C. Faloutsos (2012)18 How does it work? ‘basis’ functions 0 1 n-1 0 1 0 1 sine, freq =1 sine, freq = 2 0 1 n-1 0 1 cosine, f=1 cosine, f=2 Skip

19 CMU SCS 15-826Copyright: C. Faloutsos (2012)19 How does it work? Basis functions are actually n-dim vectors, orthogonal to each other ‘similarity’ of x with each of them: inner product DFT: ~ all the similarities of x with the basis functions Skip

20 CMU SCS 15-826Copyright: C. Faloutsos (2012)20 How does it work? Since e jf = cos(f) + j sin(f) (j=sqrt(-1)), we finally have: Skip

21 CMU SCS 15-826Copyright: C. Faloutsos (2012)21 DFT: definition Discrete Fourier Transform (n-point): inverse DFT

22 CMU SCS 15-826Copyright: C. Faloutsos (2012)22 DFT: definition Good news: Available in all symbolic math packages, eg., in ‘mathematica’ x = [1,2,1,2]; X = Fourier[x]; Plot[ Abs[X] ];

23 CMU SCS 15-826Copyright: C. Faloutsos (2012)23 DFT: definition (variations: 1/n instead of 1/sqrt(n) exp(-...) instead of exp(+...)

24 CMU SCS 15-826Copyright: C. Faloutsos (2012)24 DFT: definition (variations: … or even

25 CMU SCS 15-826Copyright: C. Faloutsos (2012)25 DFT: definition Observations: X f : are complex numbers except –X 0, who is real Im (X f ): ~ amplitude of sine wave of frequency f Re (X f ): ~ amplitude of cosine wave of frequency f x: is the sum of the above sine/cosine waves

26 CMU SCS 15-826Copyright: C. Faloutsos (2012)26 DFT: definition Observation - SYMMETRY property: X f = (X n-f )* ( “*”: complex conjugate: (a + b j)* = a - b j ) Skip

27 CMU SCS 15-826Copyright: C. Faloutsos (2012)27 DFT: definition Definitions A f = |X f | : amplitude of frequency f |X f | 2 = Re(X f ) 2 + Im(X f ) 2 = energy of frequency f phase  f at frequency f XfXf Re Im AfAf ff

28 CMU SCS 15-826Copyright: C. Faloutsos (2012)28 DFT: definition Amplitude spectrum: | X f | vs f (f=0, 1,... n-1) 0 1n-1 SYMMETRIC (Thus, we plot the first half only) Skip

29 CMU SCS 15-826Copyright: C. Faloutsos (2012)29 DFT: definition Phase spectrum |  f | vs f (f=0, 1,... n-1): Anti-symmetric (Rarely used) Skip

30 CMU SCS 15-826Copyright: C. Faloutsos (2012)30 DFT: examples flat time freq Amplitude

31 CMU SCS 15-826Copyright: C. Faloutsos (2012)31 DFT: examples Low frequency sinusoid time freq

32 CMU SCS 15-826Copyright: C. Faloutsos (2012)32 DFT: examples Sinusoid - symmetry property: X f = X * n-f time freq

33 CMU SCS 15-826Copyright: C. Faloutsos (2012)33 DFT: examples Higher freq. sinusoid time freq

34 CMU SCS 15-826Copyright: C. Faloutsos (2012)34 DFT: examples examples = + +

35 CMU SCS 15-826Copyright: C. Faloutsos (2012)35 DFT: examples examples Freq. Ampl.

36 CMU SCS 15-826Copyright: C. Faloutsos (2012)36 DFT: Amplitude spectrum year count Freq. Ampl. freq=12 freq=0 Amplitude:

37 CMU SCS 15-826Copyright: C. Faloutsos (2012)37 DFT: Amplitude spectrum year count Freq. Ampl. freq=12 freq=0

38 CMU SCS 15-826Copyright: C. Faloutsos (2012)38 DFT: Amplitude spectrum year count Freq. Ampl. freq=12 freq=0

39 CMU SCS 15-826Copyright: C. Faloutsos (2012)39 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? Freq.

40 CMU SCS 15-826Copyright: C. Faloutsos (2012)40 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? A1: compression A2: pattern discovery (A3: forecasting)

41 CMU SCS 15-826Copyright: C. Faloutsos (2012)41 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? A1: (lossy) compression A2: pattern discovery

42 CMU SCS 15-826Copyright: C. Faloutsos (2012)42 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? A1: (lossy) compression A2: pattern discovery

43 CMU SCS 15-826Copyright: C. Faloutsos (2012)43 DFT: Amplitude spectrum Let’s see it in action (defunct now…) (http://www.dsptutor.freeuk.com/jsanalyser/FFTSpectrumAnalyser.html)(http://www.dsptutor.freeuk.com/jsanalyser/FFTSpectrumAnalyser.html plain sine phase shift two sine waves the ‘chirp’ function http://ion.researchsystems.com/

44 CMU SCS 15-826Copyright: C. Faloutsos (2012)44 Plain sine

45 CMU SCS 15-826Copyright: C. Faloutsos (2012)45 Plain sine

46 CMU SCS 15-826Copyright: C. Faloutsos (2012)46 Plain sine – phase shift

47 CMU SCS 15-826Copyright: C. Faloutsos (2012)47 Plain sine – phase shift

48 CMU SCS 15-826Copyright: C. Faloutsos (2012)48 Plain sine

49 CMU SCS 15-826Copyright: C. Faloutsos (2012)49 Two sines

50 CMU SCS 15-826Copyright: C. Faloutsos (2012)50 Two sines

51 CMU SCS 15-826Copyright: C. Faloutsos (2012)51 Chirp

52 CMU SCS 15-826Copyright: C. Faloutsos (2012)52 Chirp

53 CMU SCS 15-826Copyright: C. Faloutsos (2012)53 Another applet http://www.falstad.com/fourier/ (seems virus-free – but scan, before you install) FFT_applet/index.html

54 CMU SCS Properties Time shift sounds the same –Changes only phase, not amplitudes Sawtooth has almost all frequencies –With decreasing amplitude Spike has all frequencies 15-826Copyright: C. Faloutsos (2012)54

55 CMU SCS 15-826Copyright: C. Faloutsos (2012)55 DFT: Parseval’s theorem sum( x t 2 ) = sum ( | X f | 2 ) Ie., DFT preserves the ‘energy’ or, alternatively: it does an axis rotation: x0 x1 x = {x0, x1}

56 CMU SCS 15-826Copyright: C. Faloutsos (2012)56 DFT: Parseval’s theorem sum( x t 2 ) = sum ( | X f | 2 ) Ie., DFT preserves the ‘energy’ or, alternatively: it does an axis rotation: x0 x1 x = {x0, x1}

57 CMU SCS 15-826Copyright: C. Faloutsos (2012)57 DSP - Detailed outline DFT –what –why –how –Arithmetic examples –properties / observations –DCT –2-d DFT –Fast Fourier Transform (FFT)

58 CMU SCS 15-826Copyright: C. Faloutsos (2012)58 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? 0 1 time value 1 Skip

59 CMU SCS 15-826Copyright: C. Faloutsos (2012)59 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2  0 / n ) = 1/2 X 1 =? X 2 =? X 3 =? Skip

60 CMU SCS 15-826Copyright: C. Faloutsos (2012)60 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2  0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: does the ‘symmetry’ property hold? Skip

61 CMU SCS 15-826Copyright: C. Faloutsos (2012)61 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2  0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: does the ‘symmetry’ property hold? A: Yes (of course) Skip

62 CMU SCS 15-826Copyright: C. Faloutsos (2012)62 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2  0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: check Parseval’s theorem Skip

63 CMU SCS 15-826Copyright: C. Faloutsos (2012)63 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2  0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: (Amplitude) spectrum? Skip

64 CMU SCS 15-826Copyright: C. Faloutsos (2012)64 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2  0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: (Amplitude) spectrum? A: FLAT!

65 CMU SCS 15-826Copyright: C. Faloutsos (2012)65 Arithmetic examples Q: What does this mean?

66 CMU SCS 15-826Copyright: C. Faloutsos (2012)66 Arithmetic examples Q: What does this mean? A: All frequencies are equally important -> –we need n numbers in the frequency domain to represent just one non-zero number in the time domain! – “frequency leak”

67 CMU SCS 15-826Copyright: C. Faloutsos (2012)67 DSP - Detailed outline DFT –what –why –how –Arithmetic examples –properties / observations –DCT –2-d DFT –Fast Fourier Transform (FFT)

68 CMU SCS 15-826Copyright: C. Faloutsos (2012)68 Observations DFT of ‘step’ function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt

69 CMU SCS 15-826Copyright: C. Faloutsos (2012)69 Observations DFT of ‘step’ function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt f = 0

70 CMU SCS 15-826Copyright: C. Faloutsos (2012)70 Observations DFT of ‘step’ function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt f = 0 f=1

71 CMU SCS 15-826Copyright: C. Faloutsos (2012)71 Observations DFT of ‘step’ function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt f = 0 f=1 the more frequencies, the better the approx. ‘ringing’ becomes worse reason: discontinuities; trends

72 CMU SCS 15-826Copyright: C. Faloutsos (2012)72 Observations Ringing for trends: because DFT ‘sub- consciously’ replicates the signal t xtxt

73 CMU SCS 15-826Copyright: C. Faloutsos (2012)73 Observations Ringing for trends: because DFT ‘sub- consciously’ replicates the signal t xtxt

74 CMU SCS 15-826Copyright: C. Faloutsos (2012)74 Observations Ringing for trends: because DFT ‘sub- consciously’ replicates the signal t xtxt

75 CMU SCS 15-826Copyright: C. Faloutsos (2012)75 original

76 CMU SCS 15-826Copyright: C. Faloutsos (2012)76 DC and 1st

77 CMU SCS 15-826Copyright: C. Faloutsos (2012)77 And 2nd DC and 1st

78 CMU SCS 15-826Copyright: C. Faloutsos (2012)78 And 3rd DC and 1st And 2nd

79 CMU SCS 15-826Copyright: C. Faloutsos (2012)79 And 4th DC and 1st And 3rd And 2nd

80 CMU SCS 15-826Copyright: C. Faloutsos (2012)80 Observations Q: DFT of a sinusoid, eg. x t = 3 sin( 2  / 4 t) (t = 0,..., 3) Q: X 0 = ? Q: X 1 = ? Q: X 2 = ? Q: X 3 = ?

81 CMU SCS 15-826Copyright: C. Faloutsos (2012)81 Observations Q: DFT of a sinusoid, eg. x t = 3 sin( 2  / 4 t) (t = 0,..., 3) Q: X 0 = 0 Q: X 1 = -3 j Q: X 2 = 0 Q: X 3 = 3j check ‘symmetry’ check Parseval

82 CMU SCS 15-826Copyright: C. Faloutsos (2012)82 Q: DFT of a sinusoid, eg. x t = 3 sin( 2  / 4 t) (t = 0,..., 3) Q: X 0 = 0 Q: X 1 = -3 j Q: X 2 = 0 Q: X 3 = 3j Observations Does this make sense? f AfAf 0 1 2

83 CMU SCS 15-826Copyright: C. Faloutsos (2012)83 Shifting x in time does NOT change the amplitude spectrum eg., x = { 0 0 0 1} and x’ = { 0 1 0 0 }: same (flat) amplitude spectrum (only the phase spectrum changes) Useful property when we search for patterns that may ‘slide’ Property

84 CMU SCS 15-826Copyright: C. Faloutsos (2012)84 DSP - Detailed outline DFT –what –why –how –Arithmetic examples –properties / observations –DCT –2-d DFT –Fast Fourier Transform (FFT)

85 CMU SCS 15-826Copyright: C. Faloutsos (2012)85 DCT Discrete Cosine Transform motivation#1: DFT gives complex numbers motivation#2: how to avoid the ‘frequency leak’ of DFT on trends? t xtxt Skip

86 CMU SCS 15-826Copyright: C. Faloutsos (2012)86 DCT brilliant solution to both problems: mirror the sequence, do DFT, and drop the redundant entries! t xtxt Skip

87 CMU SCS 15-826Copyright: C. Faloutsos (2012)87 DCT (see Numerical Recipes for exact formulas) Skip

88 CMU SCS 15-826Copyright: C. Faloutsos (2012)88 DCT - properties it gives real numbers as the result it has no problems with trends it is very good when x t and x (t+1) are correlated (thus, is used in JPEG, for image compression)

89 CMU SCS 15-826Copyright: C. Faloutsos (2012)89 DSP - Detailed outline DFT –what –why –how –Arithmetic examples –properties / observations –DCT –2-d DFT –Fast Fourier Transform (FFT)

90 CMU SCS 15-826Copyright: C. Faloutsos (2012)90 2-d DFT Definition:

91 CMU SCS 15-826Copyright: C. Faloutsos (2012)91 2-d DFT Intuition: n1 n2 do 1-d DFT on each row and then 1-d DFT on each column

92 CMU SCS 15-826Copyright: C. Faloutsos (2012)92 2-d DFT Quiz: how do the basis functions look like? for f1= f2 =0 for f1=1, f2=0 for f1=1, f2=1

93 CMU SCS 15-826Copyright: C. Faloutsos (2012)93 2-d DFT Quiz: how do the basis functions look like? for f1= f2 =0 flat for f1=1, f2=0wave on x; flat on y for f1=1, f2=1~ egg-carton

94 CMU SCS 15-826Copyright: C. Faloutsos (2012)94 2-d DFT Quiz: how do the basis functions look like? for f1= f2 =0 flat for f1=1, f2=0wave on x; flat on y for f1=1, f2=1~ egg-carton

95 CMU SCS 15-826Copyright: C. Faloutsos (2012)95 DSP - Detailed outline DFT –what –why –how –Arithmetic examples –properties / observations –DCT –2-d DFT –Fast Fourier Transform (FFT)

96 CMU SCS 15-826Copyright: C. Faloutsos (2012)96 FFT What is the complexity of DFT? Skip

97 CMU SCS 15-826Copyright: C. Faloutsos (2012)97 FFT What is the complexity of DFT? A: Naively, O(n 2 ) Skip

98 CMU SCS 15-826Copyright: C. Faloutsos (2012)98 FFT However, if n is a power of 2 (or a number with many divisors), we can make it O(n log n) Main idea: if we know the DFT of the odd time-ticks, and of the even time-ticks, we can quickly compute the whole DFT Details: in Num. Recipes Skip

99 CMU SCS 15-826Copyright: C. Faloutsos (2012)99 DFT - Conclusions It spots periodicities (with the ‘amplitude spectrum’) can be quickly computed (O( n log n)), thanks to the FFT algorithm. standard tool in signal processing (speech, image etc signals) Freq.

100 CMU SCS 15-826Copyright: C. Faloutsos (2012)100 Detailed outline primary key indexing.. multimedia Digital Signal Processing (DSP) tools –Discrete Fourier Transform (DFT) –Discrete Wavelet Transform (DWT)

101 CMU SCS 15-826Copyright: C. Faloutsos (2012)101 Reminder: Problem: Goal: given a signal (eg., #packets over time) Find: patterns, periodicities, and/or compress year count lynx caught per year (packets per day; virus infections per month)

102 CMU SCS 15-826Copyright: C. Faloutsos (2012)102 Wavelets - DWT DFT is great - but, how about compressing a spike? value time

103 CMU SCS 15-826Copyright: C. Faloutsos (2012)103 Wavelets - DWT DFT is great - but, how about compressing a spike? A: Terrible - all DFT coefficients needed! Freq. Ampl. value time

104 CMU SCS 15-826Copyright: C. Faloutsos (2012)104 Wavelets - DWT DFT is great - but, how about compressing a spike? A: Terrible - all DFT coefficients needed! value time

105 CMU SCS 15-826Copyright: C. Faloutsos (2012)105 Wavelets - DWT Similarly, DFT suffers on short-duration waves (eg., baritone, silence, soprano) time value

106 CMU SCS 15-826Copyright: C. Faloutsos (2012)106 Wavelets - DWT Solution#1: Short window Fourier transform (SWFT) But: how short should be the window? time freq time value

107 CMU SCS 15-826Copyright: C. Faloutsos (2012)107 Wavelets - DWT Answer: multiple window sizes! -> DWT time freq Time domain DFT SWFT DWT

108 CMU SCS 15-826Copyright: C. Faloutsos (2012)108 Haar Wavelets subtract sum of left half from right half repeat recursively for quarters, eight-ths,...

109 CMU SCS 15-826Copyright: C. Faloutsos (2012)109 Wavelets - construction x0 x1 x2 x3 x4 x5 x6 x7

110 CMU SCS 15-826Copyright: C. Faloutsos (2012)110 Wavelets - construction x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... level 1

111 CMU SCS 15-826Copyright: C. Faloutsos (2012)111 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 level 2

112 CMU SCS 15-826Copyright: C. Faloutsos (2012)112 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 etc...

113 CMU SCS 15-826Copyright: C. Faloutsos (2012)113 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 Q: map each coefficient on the time-freq. plane t f

114 CMU SCS 15-826Copyright: C. Faloutsos (2012)114 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 Q: map each coefficient on the time-freq. plane t f

115 CMU SCS 15-826Copyright: C. Faloutsos (2012)115 Haar wavelets - code #!/usr/bin/perl5 # expects a file with numbers # and prints the dwt transform # The number of time-ticks should be a power of 2 # USAGE # haar.pl my @vals=(); my @smooth; # the smooth component of the signal my @diff; # the high-freq. component # collect the values into the array @val while(<>){ @vals = ( @vals, split ); } my $len = scalar(@vals); my $half = int($len/2); while($half >= 1 ){ for(my $i=0; $i< $half; $i++){ $diff [$i] = ($vals[2*$i] - $vals[2*$i + 1] )/ sqrt(2); print "\t", $diff[$i]; $smooth [$i] = ($vals[2*$i] + $vals[2*$i + 1] )/ sqrt(2); } print "\n"; @vals = @smooth; $half = int($half/2); } print "\t", $vals[0], "\n" ; # the final, smooth component

116 CMU SCS 15-826Copyright: C. Faloutsos (2012)116 Wavelets - construction Observation1: ‘+’ can be some weighted addition ‘-’ is the corresponding weighted difference (‘Quadrature mirror filters’) Observation2: unlike DFT/DCT, there are *many* wavelet bases: Haar, Daubechies- 4, Daubechies-6, Coifman, Morlet, Gabor,...

117 CMU SCS 15-826Copyright: C. Faloutsos (2012)117 Wavelets - how do they look like? E.g., Daubechies-4

118 CMU SCS 15-826Copyright: C. Faloutsos (2012)118 Wavelets - how do they look like? E.g., Daubechies-4 ? ?

119 CMU SCS 15-826Copyright: C. Faloutsos (2012)119 Wavelets - how do they look like? E.g., Daubechies-4

120 CMU SCS 15-826Copyright: C. Faloutsos (2012)120 Wavelets - Drill#1: Q: baritone/silence/soprano - DWT? t f time value

121 CMU SCS 15-826Copyright: C. Faloutsos (2012)121 Wavelets - Drill#1: t f time value Q: baritone/silence/soprano - DWT?

122 CMU SCS 15-826Copyright: C. Faloutsos (2012)122 Wavelets - Drill#2: Q: spike - DWT? t f

123 CMU SCS 15-826Copyright: C. Faloutsos (2012)123 Wavelets - Drill#2: t f Q: spike - DWT? 0.00 0.00 0.71 0.00 0.00 0.50 -0.35 0.35

124 CMU SCS 15-826Copyright: C. Faloutsos (2012)124 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f

125 CMU SCS 15-826Copyright: C. Faloutsos (2012)125 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f

126 CMU SCS 15-826Copyright: C. Faloutsos (2012)126 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f

127 CMU SCS 15-826Copyright: C. Faloutsos (2012)127 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f

128 CMU SCS 15-826Copyright: C. Faloutsos (2012)128 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f

129 CMU SCS 15-826Copyright: C. Faloutsos (2012)129 Wavelets - Drill#3: Q: DFT? t f t f DWT DFT

130 CMU SCS 15-826Copyright: C. Faloutsos (2012)130 Wavelets - Drill: Let’s see it live: http://dsp.rice.edu/software/dsp-teaching-tools delta; cosine; cosine2; chirp Haar vs Daubechies-4, -6, etc

131 CMU SCS 15-826Copyright: C. Faloutsos (2012)131 Delta? x(0)=1; x(t)= 0 elsewhere t f

132 CMU SCS 15-826Copyright: C. Faloutsos (2012)132 Delta? x(0)=1; x(t)= 0 elsewhere t f

133 CMU SCS 15-826Copyright: C. Faloutsos (2012)133 2 cosines? x(t) = cos( 2 * pi * 4 * t / 1024 ) + 5 * cos( 2 * pi * 8 * t / 1024 ) t f

134 CMU SCS 15-826Copyright: C. Faloutsos (2012)134 2 cosines? x(t) = cos( 2 * pi * 4 * t / 1024 ) + 5 * cos( 2 * pi * 8 * t / 1024 ) t f Which one is for freq.=4?

135 CMU SCS 15-826Copyright: C. Faloutsos (2012)135 2 cosines? x(t) = cos( 2 * pi * 4 * t / 1024 ) + 5 * cos( 2 * pi * 8 * t / 1024 ) t f Which one is for freq.=4? f~4 f~8

136 CMU SCS 15-826Copyright: C. Faloutsos (2012)136 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f

137 CMU SCS 15-826Copyright: C. Faloutsos (2012)137 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f

138 CMU SCS 15-826Copyright: C. Faloutsos (2012)138 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f SWFT?

139 CMU SCS 15-826Copyright: C. Faloutsos (2012)139 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f SWFT

140 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)140 BGP-lens: Patterns and Anomalies in Internet Routing Updates B. Aditya Prakash et al, SIGKDD 2009 time #updates

141 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)141 freq. Low freq.: omitted t f

142 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)142 freq. ??

143 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)143 ‘Prolonged spike’ t f freq.

144 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)144 freq. 15K msgs, for several hours: 6pm-4am

145 CMU SCS 15-826Copyright: C. Faloutsos (2012)145 Wavelets - Drill Or use ‘R’, ‘octave’ or ‘matlab’ – R: install.packages("wavelets") library("wavelets") X1<-c(1,2,3,4,5,6,7,8) dwt(X1, n.levels=3, filter=“d4”) mra(X1, n.levels=3, filter=“d4”)

146 CMU SCS 15-826Copyright: C. Faloutsos (2012)146 Wavelets - k-dimensions? easily defined for any dimensionality (like DFT, DCT)

147 CMU SCS 15-826Copyright: C. Faloutsos (2012)147 Wavelets - example http://grail.cs.washington.edu/projects/query/ Wavelets achieve *great* compression: 20 100 40016,000 # coefficients

148 CMU SCS 15-826Copyright: C. Faloutsos (2012)148 Wavelets - intuition Edges (horizontal; vertical; diagonal)

149 CMU SCS 15-826Copyright: C. Faloutsos (2012)149 Wavelets - intuition Edges (horizontal; vertical; diagonal) recurse

150 CMU SCS 15-826Copyright: C. Faloutsos (2012)150 Wavelets - intuition Edges (horizontal; vertical; diagonal) http://www331.jpl.nasa.gov/public/wave.ht ml

151 CMU SCS 15-826Copyright: C. Faloutsos (2012)151 Advantages of Wavelets Better compression (better RMSE with same number of coefficients) closely related to the processing of the mammalian eye and ear Good for progressive transmission handle spikes well usually, fast to compute (O(n)!)

152 CMU SCS 15-826Copyright: C. Faloutsos (2012)152 Overall Conclusions DFT, DCT spot periodicities DWT : multi-resolution - matches processing of mammalian ear/eye better All three: powerful tools for compression, pattern detection in real signals All three: included in math packages (matlab, R, mathematica,... )

153 CMU SCS 15-826Copyright: C. Faloutsos (2012)153 Resources Numerical Recipes in C: great description, intuition and code for all three tools xwpl: open source wavelet package from Yale, with excellent GUI.

154 CMU SCS 15-826Copyright: C. Faloutsos (2012)154 Resources (cont’d) (defunct?)http://www.dsptutor.freeuk.com/j sanalyser/FFTSpectrumAnalyser.html : Nice java appletshttp://www.dsptutor.freeuk.com/j sanalyser/FFTSpectrumAnalyser.html http://www.relisoft.com/freeware/freq.html : voice frequency analyzer (needs microphone – MSwindows only)http://www.relisoft.com/freeware/freq.html

155 CMU SCS 15-826Copyright: C. Faloutsos (2012)155 Resources (cont’d) www-dsp.rice.edu/software/EDU/mra.shtml (wavelets and other demos) R (‘install.packages(“wavelets”) )

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