The Discrete Wavelet Transform

The Story of WaveletsTheory and Engineering Applications

An example on MRA

f(t)

j=0j=-1

j=-2

j=-

j=

j=1 j=2

fj(t)

dftfk

kjj

j

j

)(2

1)(

)1(2

2

Zkj

ktk jjj

,

222

0)(lim

)()(lim

tf

tftf

jj

jj

Let’s suppose we wish to approximate the function f(t) using a simpler function that has constant values over set intervals of 2 j, such that the error ||f(t)-fj(t)|| is minimum for any j. The best approx. would be the average of the function over the interval length. In general, then…

An Example (cont.)

As we go from a value of j to a higher value, we obtain a coarser approximation of the function, hence some detail is lost.

Let’s denote the detail lost at each approximation level with a function g(t):

)()()()( 11 tftftftf jj

)()()( 1 tftftg jjj

Detail functionApproximation function

)()()(

)()()(

010

121

tftftg

tftftg jjj

jj tgtf )()(

That is, the original functioncan be reconstructed simply by adding all the details (!)

)0( f

An example (Cont.)

1. Approximations can be obtained by “averaging” the original function over time the duration of the averaging window determine the resolution of the approximation.

2. Original function can be reconstructed by adding all the details lost in approximations (to the coarsest approximation)

3. Note that the functions at any level can be obtained from a prototype, simply by dilating or compressing the prototype function.

jj tgtf )()( dftf

k

kjj

j

j

)(2

1)(

)1(2

2

Scaling Functions

The prototype function used in the above example was a piecewise linear function. Since we can approximate and reconstruct any function using dilated and translated replicas of this prototype Piecewise linear functions constitute a set of basis functions.

elsewhere

tt

,0

10 ,1)(

(t)

1

1

0

The piecewise linear functions, however,can all be obtained by dilations and translations

of this prototype function. We call this prototype function the scaling function.Among many such prototypes, the one shown here is the simplest, and it is known as the Haar scaling function.

Scaling Functions

dftfkjak

kjj

j

j

)(2

1)(),(

)1(2

2

(t/2)

2

1

0t

(t)

1

1

0 t

(2t)

1/2

1

0t

(2t-3)

3/2

1

0t

2

some j

(2-jt-4)

t0 1.2j 4.2j 5.2j … …

k 0 1 2 3 4 5 6 … …

a(j,0)a(j,1)

a(j,4)

k

jj ktkjatf 2),()(

Approximation coefficients

Scaling Functions

We are now ready to officially define the scaling functions: Orthonormal dyadic discrete wavelets are also associated with scaling functions,

which are used to smooth or obtain approximations of a signal. Scaling functions (t) have a similar form to those of wavelet functions (t):

where is known as the father wavelet.

Scaling functions also have the property Also note that, scaling functions are

orthogonal to their translations, butnot to their dilations

)2(2)( 2/, ktt jjkj )()(0,0 tt

1)(0,0 dtt

0)(0,0 dtt

(Recall that)

Haar scaling function.

Scaling Functions

In general, for any given scaling function, the approximation coefficients can be obtained by the inner product of the signal and the scaling function.

A continuous approximation of the signal at scale j can then be obtained from the discrete sum of these coefficients

Recall and note that as j- xj(t)x(t)

dtttxkja kj )()(),( , Zkj ,Approximation coefficientsat scale j

kkjjj tkjatxtf )(),()()( ,Smoothed, scaling-function-dependent

approximation of x(t) at scale j

Detail Coefficients

So we can reconstruct the signal from its approximation coefficients. But, how about the detail function?

Detail functions, too, can be reconstructed by dilating and translating a prototype function, called the wavelet.

Just like the approximation functions fj(t)=xj(t) can be obtained from scaling functions, the detail functions gj(t) can be obtained from the wavelet functions.

Furthermore, the wavelets corresponding to the scale functions used for approximation, can be obtained from the scaling function:

)()()(

)()()(

010

121

tftftg

tftftg jjj

jj tgtf )()(

)0( f

The Details…

elsewhere ,0

11/2 ,1

1/2t0 ,1

)12()2()( tttt

(2t)

1/2

1

0t

(2t-1)

1/2

1

0t

1

(t)

1/2

1

0 t1

-1

- =

kkjkj

k

jkjj

d

ktdtg

,,

, 2)(

kj

kj

d

dtttxbaW

,

, )()(),(

Zkjktt jjkj , 22)( 2/

,

Putting it All Together

dtttxkja kj )()(),( ,

Approximation coefficientsat scale j

kkjjj tkjatxtf )(),()()( ,

Smoothed, scaling-function-dependentapproximation of x(t) at scale j)2(2)( 2/

, ktt jjkj

)()( ,, tdtgk

kjkjj

kj

kj

d

dtttxbaW

,

, )()(),(

ktt jjkj 22)( 2/

, Detail coefficients

at scale j

Wavelet-function-dependentdetail of x(t) at scale j

Furthermore:

jj tgtf )()( & )()( ,, tdtg

kkjkjj

j kkj tkjdtxtf )(),()()( , Wavelet

Reconstruction

Putting ItAll Together (cont.)

Note that we can also start reconstruction from any level (scale) in between:

j

j

j kkjkjkj tkjdatxtf

0

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

Approximation coefficientsat scale j0

Smoothed, scaling-function-dependentapproximation of x(t) at scale j0

Detail coefficientsat scale j0 and below

Wavelet-function-dependentdetails of x(t) at scales j0 and below