Wavelet Shrinkage - University of South Floridaicons.eng.usf.edu/Pdf and PPt's/David Seebran Wavelet Thresholding Spring 2007.pdfWavelet Shrinkage Wavelet Shrinkage via Thresholding

Wavelet Shrinkage

David Seebran

January 2007

David Seebran () January 2007 1 / 32

Wavelet ShrinkageWavelet Shrinkage via Thresholding

Wavelet thresholding proceeds in a number of distinct steps:

Decompose the noisy data into an orthogonal wavelet basis

Threshold the wavelet coe¢ cients using an estimated discriminatorythreshold to suppress the wavelet coe¢ cients that are smaller than agiven amplitude

Transform the coe¢ cients back into the original time domain via theinverse Wavelet Transform

The non-linear threshold operator can be computed in any orthogonalbasis, and Donoho and Johnstone have proven that its performance isclose to an ideal coe¢ cient selection and attenuation.


Wavelets ShrinkageNoise Modeling

The thresholding operation can be formalized as follows:

Let s(n) be the noise-free signal and x(n) the signal corrupted withgaussian noise, v(n), then

x(n) = s(n) + v(n) n = 1, 2, ...,N

The task of denoising x(n) or, stated another way, removing v(n), issummarized as follows:

Compute the Wavelet Packet Transform of the noisy signal x(n)

Threshold the resulting wavelet coe¢ cients

Compute the Inverse wavelet transform to obtain the denoised signal,x(n)


Wavelets ShrinkageThe Hard Thresholding Operator

The hard thresholding operator T is de�ned as

Th =�wk jwk j � λ0 jwk j < λ

0

0


Wavelets ShrinkageThe Soft Thresholding Operator

The soft thresholding operator

Ts (λ,wk ) =�sgn(wk )(jwk j � λ) jwk j � λ0 jwk j < λ

0

0


Wavelets ShrinkageAdaptive Thresholding Operator

λ is the universal threshold operator and is de�ned as

λ = σp2 logN

and

σ =MAD0.6745

σ is de�ned as the noise level obtained from the �nest scale coe¢ cientsand MAD = median(abs(wk �median(wk )) is the median absolutedeviation of the �rst scale.


Wavelets ShrinkageAdaptive Thresholding

The energies of unvoiced segments in noisy speech, may be comparable tothose of noise. Thus applying a uniform threshold to all waveletcoe¢ cients will not only suppress additional noise but also some speechcomponents, particularly, unvoiced ones. Consequently, the perceptivequality of the denoised speech is degraded.

Need to mitigate this phenomenon. Possible solution:

Compute the energy of the signal

Adapt the discriminative threshold in space and time using the energy


Wavelets ShrinkageAdaptive Thresholding Operator

This can be achieved by:

Modeling the vocal tract as a physical system

The Teager Energy Operator is a physical model that can be used tomodel the energy in speech.


Wavelets ShrinkageThe Teager Energy Operator

The Teager Energy Operator is based on a 2nd order dynamical system.Speci�cally, the motion of a mass m suspended by a spring of forceconstant k.

For dicsrete time signals, the energy operator has a simple form as follows:

E = x2n � xn+1xn�1



An important property of the Teager Energy Operator:

It gives us a good measure of the energy of the oscillating signal whenthe sampling rate of the input speech signal is greater than eighttimes the frequency of oscillation of the signal, i.e. at least twosample points in each quarter cycle of the sinusoidal oscillation.

Thus the above expression forms a simple algorithm to obtain ameasure of the energy in any single component signal.


Wavelets ShrinkageWavelet Packet Decomposition

The Wavelet Packet Transform (WPT) is a generalization of the DiscreteWavelet Transform

At a scale level j , the WPT decomposes the time series signal, x [n], into2j subbands w jk ,m where

w jk ,m = WP [x [n], j ] n = 0, 1, ...,N � 1 (1)

at the scale level j , w jk ,m de�nes the mth coe¢ cient of the k th subband

with m = 1, ..., (N/2j )� 1 and k = 1, ..., 2j

For example, when j = 4, there are k = 2j = 16 subbands.


Wavelets ShrinkageAdaptive Thresholding Algorithm

W11

W10

W 21

W12 W 2

2 W 32 W 4

2

W13

W 23 W8

3

W14 W 2

4 W164

j = 0

j= 1

j = 2

j = 3

j = 4

k = 1 k = 2 j

OriginalTimeseries

Figure: Wavelet packet decomposition for j = 4. This produces 2j = 16 packetsat level 4. Note that, at level 4, W 4

1,m contains the approximation coe¢ cients.



For a wavelet packet decomposition at j = 4 levels, application of theenergy operator to the coe¢ cients at the highest level, w4k ,m , leads to thefollowing

t4k ,m = Ψ�w4k ,m

�k = 1, ..., 16


Wavelets ShrinkageDenoising by Adaptive thresholding operaton

The denoiosing operation proceeds in a number of distinct steps.

For subband k at scale level j , the threshold is

λ4k = σ4k

q2 log(N) k = 1, ..., 16

where σ4k = MAD4k/0.6745 is an estimate of the noise level in the k th

subband at scale j and N is the length of the speech signal.


Wavelets ShrinkageSpace-adaptive Thresholding

Space adaptative thresholding is introduced via the Teager EnergyOperator

t4k ,m = [w4k ,m ]

2 � w4k ,m�1w4k ,m+1

An initial mask is now formed for each subband k at level j = 4 isconstructed by smoothing the TEO coe¢ cients as follows

M4k ,m = t

4k ,m � hk (m)

where hk is a 2nd order IIR low pass �lter.


Wavelets ShrinkageTime adaptive Thresholding

Recall that using a �xed threshold will cause some speech components tobe suppressed, particularly in regions containing unvoiced speech. This willresult in objectionable speech artifacts.

Need to mitigate the introduction of artifacts - di¤erentiate between thepresence of speech or noise and adapting the threshold, λk , for eachsubband.



Speech is determined to be present if there is a signi�cant contrastbetween peaks and valleys of M4

k while its absence will result in a weakercontrast or smoother mask. This is accomplished by de�ning the o¤setparameter, S4k

S4k = abscissa[max(H(M4k ,m))]

where H is precisely the amplitude distribution (or histogram) of M4k . The

mask, M4k , is now preprocessed to suppress the o¤set in order to reduce

the di¤erences between local maxima.

M0kk ,m =

"M4k ,m � S4k

max(M4k ,m � S4k )

#



The time-adapted threshold is now computed for each subband, k.at levelj = 4. This is accomplished by adapting λ4k ,m in the time domain

λ4k ,m = λ4k (1� αM04k ,m)

where λk is the space-dependent threshold, and α = 1 is an adjustmentparameter.


Wavelets ShrinkageSubband Thresholding

Adaptive soft thresholding is applied to the coe¢ cients of each packet, M4k

w4k ,m = Ts (λa,w4k ,m)

where Ts is the soft threshold operator and λa is de�ned as follows

λa =

�λk ,m S4k � 0.5maxM4

k ,mλk otherwise


Wavelets ShrinkageReconstruction

Perform the inverse DWT on the subbands, w4k to reconstruct thedenoised speech signal

s(n) = WP�1�w4k ,m , 4

�Note that the coe¢ cients, w41 (subband) are not thresholded, but are keptintact. Only the detail coe¢ cients packets w4k for k = 2, ..., 2

j arethresholded.


Wavelets ShrinkageSolution of the Re�nement Equation

WP

Framing

TEO

Maskconstruction

Maskprocessing

Time-adaptedThreshold

computation

SoftThresholding

WP- 1j = 4

j = 4

W1, m4

W k ,m4

W16, m4

t k ,m4

M k ,m4

Vk,m4 W1,m

4

Wk,m4

W16,m4

xÝnÞ

x!ÝnÞ

M k ,m' 4


Wavelets ShrinkageDenoising Example

A segment of speech at two di¤erent SNR�s will be denoised. At thehigher SNR, the Berouti spectral subtraction will used for comparison.














Wavelets ShrinkageWhat�s Next - Second Generation Wavelets

Wavelets are basis functions which are localized in both physical space(due to their �nite support).

In contrast, the Fourier transform is based on functions (sines and cosines)that are well localized in frequency but do not provide localization inphysical space due to their global support. Because of this space/scalelocalization, the wavelet transform provides both spatial and scale(frequency) information while the Fourier transform only providesfrequency information.



The need for improvement of wavelets come from a shortcoming that isinherent because of its construction. The main limitation is that the �rstgeneration wavelet works well for in�nite or periodic signals but it is notclear how one should modify it for use in a bounded domain.

In many applications, the domain of interest is not in�nite, and signals arenot periodic. Moreover, even 1-D signals are often not sampled regularly.In higher dimension, domains are often have boundaries, and often themetric is not �at, i.e., we need to analyze functions on manifolds orsurfaces.



The big issue is how to keep the "nice" properties of wavelets, namelytime-frequency localization and fast algorithms, while being able to extendbeyond simple geometries.

The solution is to give up translation and dilation: The construction musttherefore not use any Fourier analysis. Instead, the calculation will beentirely based on a new approach called The Lifting Scheme.



Second generation wavelets are a generalization of �rst generationwavelets that supplies the necessary freedom to deal with complexgeometries, arbitrary boundary conditions, and irregular sampling intervals.Second generation wavelets form a Reisz basis for some function space,with the wavelets being local in both space and frequency and oftenhaving many vanishing polynomial moments, but without the translationand dilation invariance of their �rst generation cousins.

Despite the loss of these two fundamental properties of wavelet bases,second generation wavelets retain many of the useful features of �rstgeneration wavelets, including a fast O(N) transform. The construction ofsecond generation wavelets is based on the lifting scheme


Documents

Wavelet Shrinkage - University of South Floridaicons.eng.usf.edu/Pdf and PPt's/David Seebran Wavelet Thresholding Spring 2007.pdfWavelet Shrinkage Wavelet Shrinkage via Thresholding