WAVELETTRANSFORM - vscht.czdsp.vscht.cz/hostalke/upload/WaveletTransform_Lecture.pdf · Wavelets...

Wavelet Transform

Eva Hostalkova

WAVELET TRANSFORM

Eva Hostalkova

Dept of Computing and Control Engineering

Institute of Chemical Technology, Prague

ATHENS Nov 2009

Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 1 / 36

Wavelet Transform

Eva Hostalkova

Table of Contents

1 Introduction to Wavelet Transform2 Fourier Transform

Continuous Fourier TransformShort-Time Fourier Transform (STFT)

3 Wavelet TransformMulti-Resolution Analysis (MRA)Continuous Wavelet Transform (CWT)Wavelet Functions Properties

4 Discrete Wavelet TransformDiscrete and Continuous Wavelet TransformSubband Coding AlgorithmMatrix Interpretation

5 Wavelet Transform ApplicationsDiscontinuity Detection in the ECG SignalImage CompressionImage SegmentationNoise Reduction by Wavelet Shrinkage


Wavelet Transform

Eva Hostalkova

Introduction

Fourier TransformContinuous Fourier Transform

STFT

Wavelet TransformMRA

CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding

Matrix Interpretation

WT ApplicationsDiscontinuity Detection

Image Compression

Image Segmentation

Noise Reduction

References

Introduction

Table of Contents

1 Introduction to Wavelet Transform

2 Fourier TransformContinuous Fourier TransformShort-Time Fourier Transform (STFT)





Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Introduction

Wavelet Transform (WT) History19th cent. - Jean B. Fourier: frequency analysis1909 - Alfred Haar: Haar function (not yet WT)since 1990s - WT related research and applications

Wavelet FunctionsWavelet = a small wave, i.e. an oscillatory function - zerooutside a bounded interval (having compact support)Some real, some complexDesigned as much smooth and symmetric as possibleSeldom analytic expression, mostly parametric (e.g.Morlet, Haar or Daubechies fcns)

10 20 30 40 50 60 70

−0.05

0

0.05

0.1

DAUBECHIES 2 WAVELET

time


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Introduction

WT AnalysisFor non-stationary signals (with time-varying frequencycontent)

WT Applications in Signal ProcessingNoise reductionDetection of trends and discontinuities in higherderivativesCompression (JPEG2000, FBI fingerprints database)Image edge detectionWatermarkingFeatures extraction for image segmentation


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Fourier Transform

Table of Contents







Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Fourier Transform Continuous Fourier Transform

Continuous Fourier Transform (FT)Decomposes signals (real or complex) into the orthogonalbasis of complex exponentials e−j2πft of frequencies fFT for the angular frequency ω=2πf and time t:

X (ω) =

∫ +∞

−∞x(t) e−jωt dt (1)

where X (ω) is the Fourier transform of the function x(t)

Converges for a piece-wise smooth x(t) or finite energy:∫ ∞−∞|x(t)|2dt <∞ (2)

Inverse FT

x(t) =12π

∫ +∞

−∞X (ω) ejωt dω (3)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Fourier Transform Continuous Fourier Transform

FT Analysis of Non-Stationary SignalsDiscrete FT for the discrete freq. index k =0, 1, . . . ,N−1and discrete time n (for the unit sampling frequency)

X (k) =N−1∑n=0

x(n) e−j2πkn/N (4)

0 100 200 300 400 500

−3

−2

−1

0

1

2

3

(a) SIGNAL 1

0 0.1 0.2 0.3 0.4

50

100

150

200

250 (b) MAGNITUDE SPECTRUM 1

0 100 200 300 400 500−1

−0.5

0

0.5

1

time

(c) SIGNAL 2

0 0.1 0.2 0.3 0.4

10

20

30

40

50

60

frequency

(d) MAGNITUDE SPECTRUM 2


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Fourier Transform STFT

Discrete Short-Time Fourier Transform (STFT)Non-stationary signal analysis (localizes frequencycomponents within time intervals)STFT of a discrete signal x(n) for k =0, 1, . . . ,N−1:

X (u, k) =N−1∑n=0

x(n) w(n − u)e−j2πkn/N (5)

where u denotes the window position

Uncertainty (Heisenberg) Principle

∆T ∆f = 1 (6)

∆f = Fs/N . . . frequency resolution (distance betweenadjacent spectral samples)∆T = N Ts . . . time resolution (window length)N . . . number of samples per windowFs = 1/Ts . . . sampling frequency


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Fourier Transform STFT

STFT - Uncertainty Tradeoff

0 100 200 300 400 500−1

−0.5

0

0.5

1

time

(a) SIGNAL 2

510

15

1020

30

2

4

6

time u

(b) STFT: WINDOW L=32

frequency ξ

12

34

2040

6080

100120

10

20

30

time u

(c) STFT: WINDOW L=128

frequency ξ 1

1.5

2

50100

150200

250

10

20

30

time u

(d) STFT: WINDOW L=256

frequency ξ

(Using the Hamming window w - gently dropping to zero)Eva Hostalkova (ICT, Prague) Wavelet Transform Athens 2009 10 / 36

Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Wavelet Transform

Table of Contents







Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Wavelet Transform MRA

Multi-Resolution Analysis (MRA)The wavelet transform deals with the uncertainty principleby MRAAnalyzing signals at different frequency bands withdifferent resolution:

Higher frequencies: good ∆T , poor ∆fLower frequencies: good ∆f , poor ∆T

Convenient for most real-world signals composed of:Long-lasting lower frequencies (approximations)Short-lasting higher frequencies (details - maininformation)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Wavelet Transform CWT

Continuous Wavelet Transform (CWT)FT: correlation of a signal of finite energy with complexexponentials ej2πft of different frequencies fWT: correlation of a signal of finite energy with delatedand shifted versions ψu,s of the mother wavelet ψ:

ψu,s(t) =1√sψ

(t − u

s

)(7)

where u is the shift of ψ along the signal → timeand s is scale (dilation of ψ → the inverse of frequency)1/√

s ensures energy normalization

CWT of the signal x(t):

W ψx (s, u)=〈x , ψu,s〉 =

1√s

∫ +∞

−∞x(t)ψ∗

(t−u

s

)dt (8)

where * denotes the complex conjugate pairand 〈〉 stands for the inner product


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References


CWT Computation Procedure (s =1)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References


CWT Computation Procedure (s =5)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References


CWT Properties

Linearity: implied by the properties of the inner product

Shift invariance:A shift in the signalalong the time axis7→ the equivalentshift of the waveletcoefficients withoutany changesDoes not apply tothe discrete WT(great drawback)

(a) Dual Tree CWT

Input

Wavelets

Level 1

Level 2

Level 3

Level 4

Scaling fn

Level 4

(b) Real DWT

Figure: Comparison of DWT and nearly invariant complex WT(CWT is even better - completely shift invariant)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References


Orthogonality and orthonormality

Not necessary for wavelet analysis (biorthogonality formore freedom in wavelet filters construction)A set of shifted and dilated versions of the mother waveletψu,s , is orthonormal on the interval [a, b] when:

〈ψu,s , ψv ,r 〉 =

∫ b

aψu,s(t)ψ∗v ,r (t) dt =

{1 for u =v , s = r0 otherwise

(9)

A set of ψu,s , is orthogonal on the interval [a, b] when:

〈ψu,s , ψv ,r 〉 =

∫ b

aψu,s(t)ψ∗v ,r (t) dt =

{c for u =v , s = r0 otherwise

(10)where c is a constant, ∗ denotes a complex conjugate


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

Wavelet Transform Wavelets Properties

Wavelet Functions Properties IAdmissibility condition∫ ∞

−∞

|Ψ(ω)|2

|ω|dω < +∞ (11)

where Ψ(ω) is the FT of ψ(t)Required for no loss of information during the analysis northe synthesisThe basis do not have to be orthogonal

Consequences of Admissibility ConditionOscillatory function (zero mean):∫ ∞

−∞ψ(t) dt = 0 (12)

Band-pass spectrum (Ψ(ω) vanishes at ω = 0):

|Ψ(ω)|2| ω=0 = 0 (13)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References


Wavelet Functions Properties II

Time dilation of ψ causes shift and compression of themagnitude frequency spectrum |Ψ|

−50 0 50

−0.5

0

0.5

1DILATION OF THE WAVELET

s = 20

0 0.1 0.2 0.3 0.4 0.5

2

4

6

8

10

norm. frequency

FT OF THE DILATED WAVELET

NEWLAND WAVELET:

ψ(t) =1

j π2 t

(ej π t− ej π2 t)

WAVELET SCALING & DILATION:

ψu,s(t) =1√sψ

(t− u

s

)

−50 0 50

−0.5

0

0.5

1

s = 21

−50 0 50

−0.5

0

0.5

1

time

s = 22

s=20

s=21

s=22


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References


Desired Wavelet Functions Properties

Compact support (non-zero only on a restricted interval) -not necessaryRegularity:

Smoothness of ψ and vanishing of |Ψ(ω)| for large ω(small scales)Vanishing moments concept (see literature)

Symmetry (liner phase)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT

Table of Contents







Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT DWT and CWT

Discrete Wavelet Transform (DWT)CWT provides us with redundant signal representationDWT - derive by critically sampling the CWT (greatreduction in the number of dilations and shifts)

ψu,s(t)=1√sψ

(t−u

s

)⇒ ψj,k(t)=

1√s j0

ψ

(t−k τ0 s j

0

s j0

)(14)

Sampling on the Dyadic Grids0 =2 ⇒ the scale s =2j

τ0 =1 ⇒ the time translation u = k 2j

t denotes time, j , k are integers

ψj,k(t)=1√2jψ

(t−k2j

2j

)(15)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT DWT and CWT

Discrete & Continuous Wavelet Transform

50 100 150 200 250 300 350 400 450 500

2000

2500

3000

3500

(a) ECG SIGNAL

(b) DISCRETE WAVELET TRANSF.: ABSOLUTE COEFFICIENTS

scal

e

50 100 150 200 250 300 350 400 450 500

16

8

4

2

1

20

40

60

(c) CONTINUOUS WAVELET TRANSF.: ABSOLUTE COEFFICIENTS

time

scal

e

50 100 150 200 250 300 350 400 450 500123456789

10111213141516

10

20

30

40

50

60


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT Subband Coding

Wavelet and Scaling Filters

200 400 600

−4

−2

0

2

4

6x 10

−3 (a) Db2 WAVELET FCN

time0 0.1 0.2 0.3 0.4

0

0.2

0.4

0.6

0.8

1

(b) Db2 WAVELET FCN: Freq. Response

Mag

nitu

de

ω/2π

200 400 600

0

2

4

x 10−3 (c) Db2 SCALING FCN

time0 0.1 0.2 0.3 0.4

0

0.2

0.4

0.6

0.8

1

(d) Db2 SCALING FCN: Freq. Response

Mag

nitu

de

ω/2π

The spectrum of the wavelet function corresponds to ahigh-pass filter (or a band-pass filter for higher levels)The spectrum of the scaling function corresponds to alow-pass filter


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT Subband Coding

Subband Coding Algorithm

ld[n]

hd[n]

2

2

ld[n]

hd[n]

2

2

ld[n]

hd[n]

2

2

1D DWT

LEVEL 1 LEVEL 2 LEVEL 3

originalsignal x[n]

detailcoefficients

D1

detailcoefficients

D2

detailcoefficients

D3

approximationcoefficients

A3

CWT versus DWTCorrelationDilating thewavelets

ConvolutionKeeping filters the same anddown-sampling the previous leveloutput by 2


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT Subband Coding

Convolution and Downsampling by 2The taps of the high-pass filter hd and the low-pass ldfilter are derived from the wavelet and the scaling function,resp., of a chosen family (e.g. Daubechies, symlets, etc.)

Convolution and Downsampling by 2Approximation coefficients of the first level

A1[n]=

∞∑k=−∞

ld [k] x [2n − k] (16)

Detail coefficients of the first level

D1[n]=

∞∑k=−∞

hd [k] x [2n − k] (17)

The j-th level Aj [n]=

∞∑k=−∞

ld [k] Aj−1[2n − k] (18)

Dj [n]=

∞∑k=−∞

hd [k] Aj−1[2n − k] (19)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT Subband Coding

Subband Coding & the Frequency Spectrum

Dilated by 2 ⇒ the spectrum is compressed and shiftedFinite number of dilations ⇒ advantageous to use alow-pass filter - derived from the scaling function φ(t)(the counter part of the wavelet filter at each level -creating a filter bank)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT Matrix Interpretation

DWT MatrixThe Haar filters

ld =1/√2 · [1, 1] (20)

hd =1/√2 · [1, −1] (21)

DWT matrix for the Haar filters

W =1√2·

1 1 0 0 . . . 0 0−1 1 0 0 . . . 0 00 0 1 1 . . . 0 00 0 −1 1 . . . 0 0...

......

.... . .

......

0 0 0 0 . . . 1 10 0 0 0 . . . −1 1

(22)

W includes both convolution and down-sampling by 2(the filters are shifted by 2 samples)W is orthonormal ⇒ W ·WT = Iwhere I stands for the identity matrix


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

DWT Matrix Interpretation

DWT of Signal x Using the Haar Filters

1√2

1 1 0 0 . . . 0 0−1 1 0 0 . . . 0 00 0 1 1 . . . 0 00 0 −1 1 . . . 0 0...

......

.... . .

......

0 0 0 0 . . . 1 10 0 0 0 . . . −1 1

·

x(0)x(1)x(2)x(3)...

x(N−1)x(N)

=

A1(0)D1(0)A1(1)D1(1)

...A1( N

2 −1)D1( N

2 −1)

DWT Decomposition and ReconstructionSignal decomposition

W · x = w (23)Signal reconstruction

x = W−1 ·w (24)Signal reconstruction for the orthonormal DWT

x = WT ·w (25)


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

WT Applications

Table of Contents







Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

WT Applications Discontinuity Detection

Discontinuity Detection in the ECG Signal

ld[n]

hd[n]

2

2

ld[n]

hd[n]

2

2

ld[n]

hd[n]

2

2

(a) 1−D DWT

signál x[n]

detaily D

1

detaily D

2

detaily D

3

aproximace A

3

nízkofrekvenènípropust

vysokofrekvenènípropust

0 100 200 300 400 500

(b) EKG SIGNÁL

time

(c) DB4: ABS. DETAILNÍ KOEFICIENTY

time100 200 300 400 500

Nespojitost

Nespojitost


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

WT Applications Image Compression

Image Compression

The Parseval theorem: the energy εx conveyed in thesignal x equals the energy of the coefficients w obtainedthrough an orthonormal transform

εx =N−1∑n=0|xn|2 =

N−1∑k=0|wk |2 (26)

(a) ORIGINAL IMAGE (b) DECOMPOSED IMAGE

0.8%

0.7% 0.1%

94.2% 2%

2% 0.4%

(c) COMPRESSED IMAGE : 94.2%


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

WT Applications Image Segmentation

Magnetic Resonance Image Segmentation

1 Image preprocessing2 Watershed transform3 DWT features extraction4 Features classification using a neural network


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

WT Applications Noise Reduction

Noise Reduction by Wavelet Shrinkage

NoisyImage

NoisyImage

WaveletAnalysisWaveletAnalysisWaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholding procedure type? Threshold level?

Wavelet SynthesisWavelet SynthesisWavelet Synthesis

Image

LoLo2

LoLo1

LoHi1HiLo1

HiHi1LoHi2HiLo2

HiHi2

Wavelet function? Number of levels?Wavelet function? Number of levels?

DenoisedImage

DenoisedImage


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References



NoisyImage

NoisyImage

WaveletAnalysis

WaveletAnalysisWaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients


Wavelet SynthesisWavelet SynthesisWavelet Synthesis

Image

LoLo2

LoLo1

LoHi1HiLo1

HiHi1LoHi2HiLo2

HiHi2


DenoisedImage

DenoisedImage


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References



NoisyImage

NoisyImage

WaveletAnalysis


Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients


Wavelet Synthesis

Wavelet SynthesisWavelet Synthesis

Image

LoLo2

LoLo1

LoHi1HiLo1

HiHi1LoHi2HiLo2

HiHi2


DenoisedImage

DenoisedImage


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References



NoisyImageNoisyImage

WaveletAnalysis


Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients


Wavelet Synthesis


Image

LoLo2

LoLo1

LoHi1HiLo1

HiHi1LoHi2HiLo2

HiHi2


DenoisedImage

DenoisedImage


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References



NoisyImage

NoisyImage

WaveletAnalysis

WaveletAnalysis

WaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients


Wavelet Synthesis

Wavelet Synthesis

Wavelet Synthesis

Image

LoLo2

LoLo1

LoHi1HiLo1

HiHi1LoHi2HiLo2

HiHi2

Wavelet function? Number of levels?


DenoisedImage

DenoisedImage


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References



NoisyImage

NoisyImage


WaveletAnalysis

Image

HiHi1HiLo1

LoHi1LoLo1

HiHi2HiLo2

LoHi2LoLo2

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients

Thresholdingwavelet

coefficients



Wavelet Synthesis

Image

LoLo2

LoLo1

LoHi1HiLo1

HiHi1LoHi2HiLo2

HiHi2



DenoisedImage

DenoisedImage


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References



(a) ORIGINAL (b) WITH ADD. GAUSSIAN NOISE (c) DENOISED


Wavelet Transform

Eva Hostalkova

Introduction


STFT


CWT

Wavelets Properties

DWTDWT and CWT

Subband Coding



Image Compression

Image Segmentation

Noise Reduction

References

References

Further Reading I

S. Mallat.A Wavelet Tour of Signal Processing.Academic Press, 1998.

G. Strang, T. Nguyen. Wavelets and Filter Banks.Wellesley-Cambridge Press, 1996.

M. Barni. Document and Image Compression.Taylor & Francis, 2001.

R. Polikar. The Wavelet Tutorial.http://users.rowan.edu/ polikar/, Iowa State University, 2001.

C. Valens. A Really Friendly Guide to Wavelets.http://pagesperso-orange.fr/polyvalens/, 2003.

www.wavelet.org

The MathWorks, Inc. Matlab Help.www.mathworks.com 2009.


WAVELETTRANSFORM - vscht.czdsp.vscht.cz/hostalke/upload/WaveletTransform_Lecture.pdf · Wavelets...

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