• Filter implementation of the Haar wavelet

• Multiresolution approximation in general

• Filter implementation of DWT

• Applications - Compression

The Story of WaveletsTheory and Engineering Applications

DWT Using Filtering

a(j-1,2k)

a(j-1, 2k+1)

f(t)

fj-1(t)

fj(t)

a(j,k)

][2

1

12,12,12

1

12,12,1

,

kjkj

kj

aa

kjakjaa

Note that at the next finer level, intervals are half as long, so you need 2k to get the same interval.

h’[n]aj-1,k yj-1,k

]1[][2

1][' nnnh

1,1,1,1 2

1 njnjkj aay kjkj ya 2,1,

h’[n] 2aj-1,k

aj,kaj,2k

Approx. coefficients at any level j can be obtained by filtering coef. at level j-1 (next finer level) by h’[n] and downsampling by 2

Filter Implementation of Haar Wavelet

We showed that aj,k can be obtained from aj-1,k through filtering by using a filter

]1[][2

1][' nnnh

followed by a downsampling operation (drop every other sample). Similarly, d j,k can also be obtained from aj-1,k using the filter g’[n] followed by down sampling by 2…

]1[][2

1][' nnng

This is called decomposition in the wavelet jargon.

g’[n] 2aj-1,k

dj,kdj,2k

Detail coefficients at any level j can be obtained by filtering approximation coefficients at level j-1 (next finer level) by g’[n] and downsampling by 2

Decomposition Filters

If we take the FT of h’[n] and g’[n]…

)2/cos()('

)2/cos()('2/

2/

j

j

jejG

ejH

LPF HPF

Decomposition / Reconstruction Filters

We can obtain the coarser level coefficients aj,k or dj,k by filtering aj-1,k with h’[n] or g’[n], respectively, followed by downsampling by 2.

Would any LPF and HPF work? No! There are certain requirements that the filters need to satisfy. In fact, the filters are obtained from scaling and wavelet functions using dilation (two-scale) equations (coming soon…)

Can we go the other way? Can we obtain aj-1,k from aj,k and dj,k from a set of filters. YES!. This process is called reconstruction.Upsample a(k,n) and d(k,n) by 2 (insert zeros

between every sample) and use filters h[n]=[n]+ [n-1] and g[n]= [n]- [n-1]. Add the filter outputs !

The Discrete Wavelet Transform

g`[n]

h`[n]

2

2 g`[n]

h`[n]

2

2

2

2

g[n]

h[n]

+

2

2

g[n]

h[n]

+

aj,k

dj+1,k

Decomposition Reconstruction

We have only shown the above implementation for the Haar Wavelet, however, as we willsee later, this implementation – subband coding – is applicable in general.

dj+2,k

aj+2,k

aj+1,kaj+1,k

aj,k

We see that app. and detail coefficients can be obtained through filtering operations, but where do scaling and wavelet functions appear in the subband coding DWT implementation?

Clearly, these functions are somehow hidden in the filter coefficients, but how?

To find out, we need to know little bit more about these scaling and wavelet functions

j

j

j kkjkjkj tkjdatxtf

0

00)(),()()( ,,,

Let’s suppose that the function f(t) is sampled at N points to give the sequence f[n], and further suppose that kth resolution is the highest resolution (we will compute approximations at k+1, k+2, …etc. Then:

Multiplying and integrating

MRA on Discrete Functions

1

0,1,1

1

0,1,1

1

0,,

)()(

)(][

N

kkjkj

N

kkjkj

N

kkjkj

tdta

tanf

1

2

dttdtt kjkj )( )( ,1,1 1 2

N

kkjkjkjkj

N

kkjkjkjnj

dtttad

dtttaa

0),1(,,,1

0),1(,,1

)()(

)()(

h[k]

g[k]

1)()(

0)()(

11

11

dttt

dttt

kk

kk

From MRA to Filters

This substitution gives us level j+1 approximation and detail

coefficients in terms of level j coefficients :

we can put the above expressions in convolution (filter) form as

k

kjkjk

kjkj kgadkhaa ][ and ][ ,),1(,),1(

m

mjkjm

mjkj mkgadmkhaa ]2[ and ]2[~

,),1(

~

,),1(

H~

aj,k

G~

2

21-level of DWT decomposition

h[n]=h[-n], and g[n]=g[-n]~ ~ So where do these filters really come from…?

aj+1,k

dj+1,k

Dilation / Two-scale Equations

Two scale (dilation) equations for the scaling and wavelet functions determine the filters associated with these functions. In particular:

The coefficients c(n) can be obtained as

Recall that

In some books, h[k]= c(k)/√2. Then the two-scale equation becomes

k

ktkct )2()()( or more generally

k

jj ktkct )2()()2( )1(

)2(),(2)( kttkc

)2(][~

2)( ktkhtk

dxxgxfgf )()(,

2/][][~

,][~

)()(][ ),1(,),1(, kckhkhdtttkh kjkjkjkj

Similarly, the two-scale equation for the wavelet function:

Then:

Dilation / Two-scale Equations

k

ktkbt )2()()(

k

jj ktkbt )2()()2( )1(

)2(][2)(~

ktkgtk

2/][][~,][~)()(][ ),1(,),1(, kbkgkgdtttkg kjkjkjkj

In some books, g[k]= b(k)/√2. Then the two-scale equation becomes~

Two-Scale Equations

These two equations determine the coefficients of all 4 filters:

h[n]: Reconstruction, lowpass filter

g[n]: Reconstruction, highpass filter

h[n]: Decomposition, lowpass filter

g[n]: Decomposition, highpass filter The following observations can therefore be made

)2(][2)(~

ntnhtn

)2(][2)(~

ntngtn

~

~

highpass isG 1)()( ,1)( 0,G(0)

lowpass is H 1)()( ,0)(,1)0(

][2

1)G(j 0][][

][2

1)H(j 2][][

22

22

~

~

jGjGG

jHjHHH

engngng

enhnhnh

n

nj

n

nj Note : |H(jw)|= |H(jw)|~

Quadrature Mirror Filters

It can be shown that

that is, h[] and g[] filters are related to each other:

in fact, that is, h[] and g[] are mirrors of each other, with every other coefficient negated. Such filters are called quadrature mirror filters. For example, Daubechies wavelets with 4 vanishing moments…..

1)()(1)()(

2~2~22 jGjHjGjH

][)1(]1[ ngnLh n

DB-4 Wavelets

h = -0.0106 0.0329 0.0308 -0.1870 -0.0280 0.6309 0.7148 0.2304

g = -0.2304 0.7148 -0.6309 -0.0280 0.1870 0.0308 -0.0329 -0.0106

h = 0.2304 0.7148 0.6309 -0.0280 -0.1870 0.0308 0.0329 -0.0106

g = -0.0106 -0.0329 0.0308 0.1870 -0.0280 -0.6309 0.7148 -0.2304

~

~

][][ ],[][

][)1(]1[~~

ngngnhnh

ngnLh n

L: filter length (8, in this case)

Matlab command “wfilters()” Use freqz() to see its freq. response

DWT implementation:Subband Coding

G

H

2

2 G

H

2

2

2

2

G

H

+

2

2

G

H

+

x[n]x[n]

Decomposition Reconstruction

~

~ ~

~

n

high kngnxky ]2[][][~

n

low knhnxky ]2[][][~

k

high kngky ]2[][

k

high kngky ]2[][

DWT Decomposition

x[n] Length: 512B: 0 ~

g[n] h[n]

g[n] h[n]

g[n] h[n]

2

d1: Level 1 DWTCoeff.

Length: 256B: 0 ~ /2 Hz

Length: 256B: /2 ~ Hz

Length: 128B: 0 ~ /4 HzLength: 128

B: /4 ~ /2 Hz

d2: Level 2 DWTCoeff.

d3: Level 3 DWTCoeff.

…….

Length: 64B: 0 ~ /8 HzLength: 64

B: /8 ~ /4 Hz

2

2 2

22

|H(jw)|

w/2-/2

|G(jw)|

w- /2-/2

Applications

Detect discontinuities

Applications

Detect hiddendiscontinuities

Applications

Simpledenoising

Compression

DWT is commonly used for compression, since most DWT are very small, can be zeroed-out!

Compression

Compression

Compression - ECG

ECG - Compression

ECG- Compression