wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet...
Transcript of wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet...
![Page 1: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/1.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Intro
Wavelet transformfor
dimensionality reduction.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 2: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/2.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Intro
Wavelet transformfor
dimensionality reduction.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 3: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/3.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
A function ψ (t), to be a wavelet must satisfy :
1. The wavelet must be centered at 0 amplitude.∫ ∞
−∞ψ (t) dt = 0
2. The wavelet must have a finite energy. Therefor it is localizedin time (or space). ∫ ∞
−∞|ψ (t)|2 dt <∞
3. Sufficient condition for inverse wavelet transform
c ≡∫ ∞
−∞
|Ψ(w)|2
|w |0 < c <∞
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 4: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/4.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
A function ψ (t), to be a wavelet must satisfy :
1. The wavelet must be centered at 0 amplitude.∫ ∞
−∞ψ (t) dt = 0
2. The wavelet must have a finite energy. Therefor it is localizedin time (or space). ∫ ∞
−∞|ψ (t)|2 dt <∞
3. Sufficient condition for inverse wavelet transform
c ≡∫ ∞
−∞
|Ψ(w)|2
|w |0 < c <∞
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 5: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/5.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
A function ψ (t), to be a wavelet must satisfy :
1. The wavelet must be centered at 0 amplitude.∫ ∞
−∞ψ (t) dt = 0
2. The wavelet must have a finite energy. Therefor it is localizedin time (or space). ∫ ∞
−∞|ψ (t)|2 dt <∞
3. Sufficient condition for inverse wavelet transform
c ≡∫ ∞
−∞
|Ψ(w)|2
|w |0 < c <∞
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 6: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/6.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
A function ψ (t), to be a wavelet must satisfy :
1. The wavelet must be centered at 0 amplitude.∫ ∞
−∞ψ (t) dt = 0
2. The wavelet must have a finite energy. Therefor it is localizedin time (or space). ∫ ∞
−∞|ψ (t)|2 dt <∞
3. Sufficient condition for inverse wavelet transform
c ≡∫ ∞
−∞
|Ψ(w)|2
|w |0 < c <∞
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 7: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/7.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
Popular wavelets which satisfy the previous conditions
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 8: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/8.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
Popular wavelets which satisfy the previous conditions
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 9: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/9.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
Popular wavelets which satisfy the previous conditions
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 10: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/10.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
Popular wavelets which satisfy the previous conditions
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 11: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/11.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
Popular wavelets which satisfy the previous conditions
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 12: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/12.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
we can move and stretch the mother wavelet
ψa,b (t) ≡ 1√|a|ψ
(t − a
b
)a is scale factor while b is a translation factor. 1√
|a|is a
normalization factor to make sure the energy stay the same.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 13: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/13.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
we can move and stretch the mother wavelet
ψa,b (t) ≡ 1√|a|ψ
(t − a
b
)a is scale factor while b is a translation factor. 1√
|a|is a
normalization factor to make sure the energy stay the same.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 14: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/14.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
we can move and stretch the mother wavelet
ψa,b (t) ≡ 1√|a|ψ
(t − a
b
)a is scale factor while b is a translation factor. 1√
|a|is a
normalization factor to make sure the energy stay the same.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 15: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/15.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
we can move and stretch the mother wavelet
ψa,b (t) ≡ 1√|a|ψ
(t − a
b
)a is scale factor while b is a translation factor. 1√
|a|is a
normalization factor to make sure the energy stay the same.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 16: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/16.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
we can move and stretch the mother wavelet
ψa,b (t) ≡ 1√|a|ψ
(t − a
b
)a is scale factor while b is a translation factor. 1√
|a|is a
normalization factor to make sure the energy stay the same.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 17: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/17.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
we can move and stretch the mother wavelet
ψa,b (t) ≡ 1√|a|ψ
(t − a
b
)a is scale factor while b is a translation factor. 1√
|a|is a
normalization factor to make sure the energy stay the same.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 18: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/18.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
we can move and stretch the mother wavelet
ψa,b (t) ≡ 1√|a|ψ
(t − a
b
)a is scale factor while b is a translation factor. 1√
|a|is a
normalization factor to make sure the energy stay the same.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 19: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/19.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
The dot product for continuous function is defined as this :
〈f , g〉 ≡∫ ∞
−∞f (t) g (t)∗ dt
which is similar to the discrete dot product :
u · v ≡n∑
i=1
uivi
The wavelet transform is simply the dot product between thesignal and the wavelet at each translation and each scale.
Wa,b ≡∫ ∞
−∞f (t)ψ∗a,b (t) dt
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 20: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/20.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
The dot product for continuous function is defined as this :
〈f , g〉 ≡∫ ∞
−∞f (t) g (t)∗ dt
which is similar to the discrete dot product :
u · v ≡n∑
i=1
uivi
The wavelet transform is simply the dot product between thesignal and the wavelet at each translation and each scale.
Wa,b ≡∫ ∞
−∞f (t)ψ∗a,b (t) dt
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 21: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/21.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
The dot product for continuous function is defined as this :
〈f , g〉 ≡∫ ∞
−∞f (t) g (t)∗ dt
which is similar to the discrete dot product :
u · v ≡n∑
i=1
uivi
The wavelet transform is simply the dot product between thesignal and the wavelet at each translation and each scale.
Wa,b ≡∫ ∞
−∞f (t)ψ∗a,b (t) dt
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 22: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/22.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 23: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/23.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
What is a wavelet?Scale and translation invariantThe continuous wavelet transformExample using Mexican hat wavelet
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 24: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/24.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I The STFT is the fourier transform computed for every timestep.
I To isolate a particular time step, a window function is used.
I
STFT (t, ω) =
∫ ∞
−∞x (τ) W (τ − t) e−jωτdτ
I The kernel have all the characteristics to be a wavelet
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 25: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/25.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I The STFT is the fourier transform computed for every timestep.
I To isolate a particular time step, a window function is used.
I
STFT (t, ω) =
∫ ∞
−∞x (τ) W (τ − t) e−jωτdτ
I The kernel have all the characteristics to be a wavelet
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 26: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/26.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I The STFT is the fourier transform computed for every timestep.
I To isolate a particular time step, a window function is used.
I
STFT (t, ω) =
∫ ∞
−∞x (τ) W (τ − t) e−jωτdτ
I The kernel have all the characteristics to be a wavelet
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 27: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/27.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I The STFT is the fourier transform computed for every timestep.
I To isolate a particular time step, a window function is used.
I
STFT (t, ω) =
∫ ∞
−∞x (τ) W (τ − t) e−jωτdτ
I The kernel have all the characteristics to be a wavelet
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 28: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/28.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I If we choose a gaussian window for the SFTF, it is exactly theMorlet wavelet.
I The only difference, is when the frequency is changing.I In the case of the wavelet, the widow width is adapting with
the frequency, keeping the number of cycles constant insidethe window.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 29: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/29.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I If we choose a gaussian window for the SFTF, it is exactly theMorlet wavelet.
I The only difference, is when the frequency is changing.
I In the case of the wavelet, the widow width is adapting withthe frequency, keeping the number of cycles constant insidethe window.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 30: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/30.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I If we choose a gaussian window for the SFTF, it is exactly theMorlet wavelet.
I The only difference, is when the frequency is changing.I In the case of the wavelet, the widow width is adapting with
the frequency, keeping the number of cycles constant insidethe window.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 31: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/31.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
Theoretical spectrogram of 2notes repeated 7 times with
200ms of duration.
STFT with gaussian window ofwidth 0.05ms.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 32: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/32.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
Theoretical spectrogram of 2notes repeated 7 times with
200ms of duration.
STFT with gaussian window ofwidth 0.1ms.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 33: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/33.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
Theoretical spectrogram of 2notes repeated 7 times with
200ms of duration.
STFT with gaussian window ofwidth 0.3ms.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 34: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/34.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I Morlet wavelet is similarto the STFT kernel.
I To build a spectrogram,we need a complexwavelet.
I We also need to boost thefrequency resolution.
I
ψ (t) =
(1√πσ2
)e2iπf e−
x2
σ2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 35: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/35.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I Morlet wavelet is similarto the STFT kernel.
I To build a spectrogram,we need a complexwavelet.
I We also need to boost thefrequency resolution.
I
ψ (t) =
(1√πσ2
)e2iπf e−
x2
σ2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 36: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/36.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I Morlet wavelet is similarto the STFT kernel.
I To build a spectrogram,we need a complexwavelet.
I We also need to boost thefrequency resolution.
I
ψ (t) =
(1√πσ2
)e2iπf e−
x2
σ2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 37: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/37.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Short time fourier transform, a reviewScales vs FrequenciesSpectrogram of a sound using STFTWavelet spectrogram
I Morlet wavelet is similarto the STFT kernel.
I To build a spectrogram,we need a complexwavelet.
I We also need to boost thefrequency resolution.
I
ψ (t) =
(1√πσ2
)e2iπf e−
x2
σ2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 38: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/38.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Wavelet are like bandpass filters.
I By computing their fourier transform,we find their frequency response.
I The standard deviation of thefrequency response gives the frequencyresolution.
∆ω =
√√√√∫∞−∞ (ω − ω0)
2 |Ψ(ω)|2 dω∫∞−∞ |Ψ(ω)|2 dω
I similarly, we can find the timeresolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 39: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/39.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Wavelet are like bandpass filters.
I By computing their fourier transform,we find their frequency response.
I The standard deviation of thefrequency response gives the frequencyresolution.
∆ω =
√√√√∫∞−∞ (ω − ω0)
2 |Ψ(ω)|2 dω∫∞−∞ |Ψ(ω)|2 dω
I similarly, we can find the timeresolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 40: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/40.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Wavelet are like bandpass filters.
I By computing their fourier transform,we find their frequency response.
I The standard deviation of thefrequency response gives the frequencyresolution.
∆ω =
√√√√∫∞−∞ (ω − ω0)
2 |Ψ(ω)|2 dω∫∞−∞ |Ψ(ω)|2 dω
I similarly, we can find the timeresolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 41: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/41.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Wavelet are like bandpass filters.
I By computing their fourier transform,we find their frequency response.
I The standard deviation of thefrequency response gives the frequencyresolution.
∆ω =
√√√√∫∞−∞ (ω − ω0)
2 |Ψ(ω)|2 dω∫∞−∞ |Ψ(ω)|2 dω
I similarly, we can find the timeresolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 42: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/42.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Wavelet are like bandpass filters.
I By computing their fourier transform,we find their frequency response.
I The standard deviation of thefrequency response gives the frequencyresolution.
∆ω =
√√√√∫∞−∞ (ω − ω0)
2 |Ψ(ω)|2 dω∫∞−∞ |Ψ(ω)|2 dω
I similarly, we can find the timeresolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 43: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/43.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I But the resolution depends ofthe mother wavelet and thescale.
I Time resolution
∆t → ∆tψ (a) = |a|∆tψ
I Frequency resolution
∆ω → ∆ωψ (a) = ∆ωψ/|a|
I Thus the area of the square isconstant
∆tψ (a) ∆ωψ (a) = ∆tψ∆ωψ = cψ
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 44: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/44.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I But the resolution depends ofthe mother wavelet and thescale.
I Time resolution
∆t → ∆tψ (a) = |a|∆tψ
I Frequency resolution
∆ω → ∆ωψ (a) = ∆ωψ/|a|
I Thus the area of the square isconstant
∆tψ (a) ∆ωψ (a) = ∆tψ∆ωψ = cψ
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 45: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/45.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I But the resolution depends ofthe mother wavelet and thescale.
I Time resolution
∆t → ∆tψ (a) = |a|∆tψ
I Frequency resolution
∆ω → ∆ωψ (a) = ∆ωψ/|a|
I Thus the area of the square isconstant
∆tψ (a) ∆ωψ (a) = ∆tψ∆ωψ = cψ
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 46: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/46.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I But the resolution depends ofthe mother wavelet and thescale.
I Time resolution
∆t → ∆tψ (a) = |a|∆tψ
I Frequency resolution
∆ω → ∆ωψ (a) = ∆ωψ/|a|
I Thus the area of the square isconstant
∆tψ (a) ∆ωψ (a) = ∆tψ∆ωψ = cψ
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 47: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/47.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I But the resolution depends ofthe mother wavelet and thescale.
I Time resolution
∆t → ∆tψ (a) = |a|∆tψ
I Frequency resolution
∆ω → ∆ωψ (a) = ∆ωψ/|a|
I Thus the area of the square isconstant
∆tψ (a) ∆ωψ (a) = ∆tψ∆ωψ = cψ
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 48: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/48.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Thus, the main difference betweenSTFT and CWT is the tiling of theresolution.
I One can tile the time-frequency spacewith any shape of rectangle, even withoverlap.
I The rectangle can have a smaller areawith a different filter, but there is aminimal area.
∆tψ∆ωψ > 1/2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 49: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/49.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Thus, the main difference betweenSTFT and CWT is the tiling of theresolution.
I One can tile the time-frequency spacewith any shape of rectangle, even withoverlap.
I The rectangle can have a smaller areawith a different filter, but there is aminimal area.
∆tψ∆ωψ > 1/2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 50: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/50.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Thus, the main difference betweenSTFT and CWT is the tiling of theresolution.
I One can tile the time-frequency spacewith any shape of rectangle, even withoverlap.
I The rectangle can have a smaller areawith a different filter, but there is aminimal area.
∆tψ∆ωψ > 1/2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 51: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/51.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Time and frequency resolutionMaximum resolutionSTFT resolution VS CWT resolution
I Thus, the main difference betweenSTFT and CWT is the tiling of theresolution.
I One can tile the time-frequency spacewith any shape of rectangle, even withoverlap.
I The rectangle can have a smaller areawith a different filter, but there is aminimal area.
∆tψ∆ωψ > 1/2
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 52: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/52.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I It took about 10 minutes to generate the previous waveletspectrogram (P4 1.8GHz with Matlab wavelet toolbox).
I For each of the m scales, the CWT perform a convolution onthe raw signal of length n.
I The CWT return m · n coefficients in time O (m · n log(n)).
I There is a huge amount of redundancy and for higher scales,we could use a smaller sampling rate.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 53: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/53.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I It took about 10 minutes to generate the previous waveletspectrogram (P4 1.8GHz with Matlab wavelet toolbox).
I For each of the m scales, the CWT perform a convolution onthe raw signal of length n.
I The CWT return m · n coefficients in time O (m · n log(n)).
I There is a huge amount of redundancy and for higher scales,we could use a smaller sampling rate.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 54: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/54.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I It took about 10 minutes to generate the previous waveletspectrogram (P4 1.8GHz with Matlab wavelet toolbox).
I For each of the m scales, the CWT perform a convolution onthe raw signal of length n.
I The CWT return m · n coefficients in time O (m · n log(n)).
I There is a huge amount of redundancy and for higher scales,we could use a smaller sampling rate.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 55: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/55.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I It took about 10 minutes to generate the previous waveletspectrogram (P4 1.8GHz with Matlab wavelet toolbox).
I For each of the m scales, the CWT perform a convolution onthe raw signal of length n.
I The CWT return m · n coefficients in time O (m · n log(n)).
I There is a huge amount of redundancy and for higher scales,we could use a smaller sampling rate.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 56: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/56.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To get a speed up in DWT, instead ofstretching the wavelet to get to abigger scale, we will compress theoriginal signal.
I For that we need a second wavelet,called the scaling function.
I This function is a low-pass filter withfrequency cut half of Nyquist.
I The wavelet is a complementary filterin that it is sensible for the rest of thefrequency.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 57: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/57.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To get a speed up in DWT, instead ofstretching the wavelet to get to abigger scale, we will compress theoriginal signal.
I For that we need a second wavelet,called the scaling function.
I This function is a low-pass filter withfrequency cut half of Nyquist.
I The wavelet is a complementary filterin that it is sensible for the rest of thefrequency.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 58: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/58.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To get a speed up in DWT, instead ofstretching the wavelet to get to abigger scale, we will compress theoriginal signal.
I For that we need a second wavelet,called the scaling function.
I This function is a low-pass filter withfrequency cut half of Nyquist.
I The wavelet is a complementary filterin that it is sensible for the rest of thefrequency.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 59: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/59.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To get a speed up in DWT, instead ofstretching the wavelet to get to abigger scale, we will compress theoriginal signal.
I For that we need a second wavelet,called the scaling function.
I This function is a low-pass filter withfrequency cut half of Nyquist.
I The wavelet is a complementary filterin that it is sensible for the rest of thefrequency.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 60: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/60.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To get a speed up in DWT, instead ofstretching the wavelet to get to abigger scale, we will compress theoriginal signal.
I For that we need a second wavelet,called the scaling function.
I This function is a low-pass filter withfrequency cut half of Nyquist.
I The wavelet is a complementary filterin that it is sensible for the rest of thefrequency.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 61: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/61.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 62: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/62.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 63: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/63.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 64: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/64.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 65: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/65.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 66: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/66.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 67: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/67.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 68: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/68.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the DWT, we start fromthe signal.
I Then we split the signal in tow part.
I Details, using the wavelet.
I Approximation, using the scalingfunction.
I We can start back the decompositionfrom the approximated signal.
I And again...
I All the details is our wavelettransform. But we need to keep thelast approximation for the inversetransform.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 69: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/69.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 70: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/70.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 71: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/71.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 72: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/72.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 73: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/73.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 74: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/74.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 75: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/75.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 76: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/76.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
continuous wavelet transform is very slowscaling functionDWT, a dyadic decompositionThe inverse discrete wavelet transform
I To perform the inverse DWT, we startfrom the details coefficients and thelast approximation.
I Then we combine the lastapproximation with the last details, tofind the second last approximation.
I And we repeat...
I Both inverse and forward take O (n)
I It is thus faster than the fouriertransform.
I But DWT restrict us to an octave offrequency resolution.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 77: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/77.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I We can compute the complete tree orchoose a particular path (waveletpacket).
I We can compute the wavelettransform for n dimensions signals.
I By selecting the highest coefficientson each scales, we can keep only themost important details (imagecompression).
I By making statistics on waveletcoefficients, we can extract the globalstructure and use it as featureextraction for classification algorithms.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 78: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/78.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I We can compute the complete tree orchoose a particular path (waveletpacket).
I We can compute the wavelettransform for n dimensions signals.
I By selecting the highest coefficientson each scales, we can keep only themost important details (imagecompression).
I By making statistics on waveletcoefficients, we can extract the globalstructure and use it as featureextraction for classification algorithms.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 79: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/79.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I We can compute the complete tree orchoose a particular path (waveletpacket).
I We can compute the wavelettransform for n dimensions signals.
I By selecting the highest coefficientson each scales, we can keep only themost important details (imagecompression).
I By making statistics on waveletcoefficients, we can extract the globalstructure and use it as featureextraction for classification algorithms.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 80: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/80.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I We can compute the complete tree orchoose a particular path (waveletpacket).
I We can compute the wavelettransform for n dimensions signals.
I By selecting the highest coefficientson each scales, we can keep only themost important details (imagecompression).
I By making statistics on waveletcoefficients, we can extract the globalstructure and use it as featureextraction for classification algorithms.
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 81: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/81.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I Wavelet statistics are mainly used fortexture classification in imageprocessing.
I But with a little of imagination, wecan see texture everywhere.
I We have used wavelet statistics forgenre classification.
I The texture of a small frame of soundrepresent the timbre.
I The texture of the temporal structurerepresent the rhythm...
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 82: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/82.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I Wavelet statistics are mainly used fortexture classification in imageprocessing.
I But with a little of imagination, wecan see texture everywhere.
I We have used wavelet statistics forgenre classification.
I The texture of a small frame of soundrepresent the timbre.
I The texture of the temporal structurerepresent the rhythm...
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 83: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/83.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I Wavelet statistics are mainly used fortexture classification in imageprocessing.
I But with a little of imagination, wecan see texture everywhere.
I We have used wavelet statistics forgenre classification.
I The texture of a small frame of soundrepresent the timbre.
I The texture of the temporal structurerepresent the rhythm...
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 84: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/84.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I Wavelet statistics are mainly used fortexture classification in imageprocessing.
I But with a little of imagination, wecan see texture everywhere.
I We have used wavelet statistics forgenre classification.
I The texture of a small frame of soundrepresent the timbre.
I The texture of the temporal structurerepresent the rhythm...
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 85: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/85.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
I Wavelet statistics are mainly used fortexture classification in imageprocessing.
I But with a little of imagination, wecan see texture everywhere.
I We have used wavelet statistics forgenre classification.
I The texture of a small frame of soundrepresent the timbre.
I The texture of the temporal structurerepresent the rhythm...
Alexandre Lacoste wavelet transform for dimensionality reduction
![Page 86: wavelet transform for dimensionality reductionlisa/seminaires/22-07-2005.pdf · The wavelet transform is simply the dot product between the signal and the wavelet at each translation](https://reader033.fdocuments.us/reader033/viewer/2022041900/5e5fbaf74d517b052e3c571d/html5/thumbnails/86.jpg)
IntroductionContinuous wavelet transform
CWT vs STFTIncertitude principle
Discrete wavelet transform (DWT)Conclusion
Many other things to talkTexture analysisEnd
that is all !!!questions ?
Alexandre Lacoste wavelet transform for dimensionality reduction