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Course 18.327 and 1.

Course 18.327 and 1.

Wavelets and Filter Banks

Wavelets and Filter Banks

Mallat

Mallat

pyramid algorithm

pyramid algorithm

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2

Pyramid Algorithm for Computing

Pyramid Algorithm for Computing

Wavelet Coefficients

Wavelet Coefficients

Goal: Given the series expansion for a function Goal: Given the series expansion for a function f

f

jj

(t) in (t) in V

V

jj

f f

jj

(t) = (t) =

aa

j j

[k] [k]

φ φ

j,k j,k

(t) (t)

how do we find the series how do we find the series

f f

j j-

1

(t) = (t) =

a a

j-j

1

[k] [k]

φ φ

j j-

-1,k

1,k

(t) (t)

in in V

V

j j-

1

and the series and the series

g g

jj-

1

(t) = (t) =

bb

j j-

1

[k] [k]w

w

j j-

-1,k

1,k

(t) (t)

in in W

W

j j-

1

such that such that

f f

j j

(t) = (t) = f

f

j j-

1

(t) + (t) + g

g

j-j

1

(t)

(t)

k k

kk

k k

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plus plus

a combination of a combination of

w(t w(t

w(t) w(t)

L

L

L

L

Easy to see because Easy to see because

φ φ

(2t) = (2t) =

[

[

φ φ

(t) + w(t)] (t) + w(t)]

φ φ

(2t (2t

[

[

φ φ

(t) (t)

w(t)] w(t)]

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k k

Suppose that f(t) is a function in LSuppose that f(t) is a function in L

2 2

(R). What are the (R). What are the

coefficients, coefficients, a

a

j j

[k], of the projection of f(t) on to [k], of the projection of f(t) on to V

V

j j

Call the projection Call the projection f

f

j j

(t), (t),

f f

jj

(t) = (t) =

a a

jj

[k] [k]

φ φ

j,k j,k

(t) (t)

a a

j j

[k] must minimize the distance between f(t) and [k] must minimize the distance between f(t) and f

f

j j

(t) (t)

{f(t) {f(t) –

• f

f

jj

(t)} (t)}

22

dt dt = 0

2 {f(t) 2 {f(t) -

a a

jj

[l] [l]

φ φ

j,l j,l

(t)} (t)}

φ φ

j,k j,k

(t) (t)dt

dt = 0

a a

j j

[k] = [k] =

f(t) f(t)

φ φ

j,k j,k

(t) (t)dt

dt

∂ ∂

∂∂

aa

jj

[k][k]

∞ ∞

∞ ∞

∞∞

∞ ∞

ll

f(t) f(t)

f f

j j

(t) (t)

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Multiresolution Multiresolution decomposition equations

decomposition equations

a a

j j-

1

[n] = [n] =

f(t) f(t)

φ φ

j j-

-1,n

1,n

(t) (t) dt

dt

h h

0 0

[

[

l l

]

]

f(t) f(t)

φ φ

j,2n +j,2n +

ll

(t) (t) dt

dt

h h

0 0

[

[

l l

]

] a

a

j j

[2n + [2n +

l l

]

]

So So

a a

j j-

1

[n] = [n] =

h h

00

[k-[k

-2n]

2n]a

a

j j

[k] [k]

Convolution with h Convolution with h

0 0

[

[-

-n] followed by

n] followed by downsampling

downsampling

∞ ∞

∞ ∞

l l

∞ ∞

∞ ∞

ll

k k

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Similarly Similarly

b b

j j-

1

[n]

[n]

f(t) f(t) w

w

j-j

-1,n

1,n

(t) (t) dt

dt

which leads to which leads to

b b

j j-

1

[n] = [n] =

h h

11

[k –[k

• 2n]

2n] a

a

j j

[k] [k]

∞∞

∞ ∞

k k

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φ φ

j j-

-1,k

1,k

(t) (t)

φ φ

j,nj,n

(t) (t) dt

dt =

h h

00

[

[

l l

]

]

φ φ

j,2k+ j,2k+

l l

(t) (t)

φ φ

j,nj,n

(t) (t) dt

dt

h h

00

[

[

l l

]

]

δ δ

[2k + [2k +

l l

• n]

n]

2 h 2 h

00

[n [n –

• 2k]

2k]

l l

∞∞

∞∞

∞∞

∞ ∞

ll

Similarly Similarly

w w

j j-

-1,k

1,k

(t) (t)

φ φ

j,nj,n

(t) dt(t)

dt =

2 h2 h

1 1

[n [n –

–2k]

2k]

Result: Result:

a a

j j

[n] = [n] =

a a

j j-

1

[k]h [k]h

00

[n [n -

• 2k] +

2k] +

b b

j j-

1

[k]h [k]h

11

[n [n –

• 2k]

2k]

∞ ∞

∞∞

k k

k k

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Filter Bank Representation Filter Bank Representation

bb

j j-

1

[n] [n]

a a

jj

[n] [n]

v v

11

[n] [n]

v v

0 0

[n] [n]

2h 2h

00

[n] [n]

2h 2h

1 1

[n] [n]

2h2h

1 1

[n] [n]

2h 2h

00

[n] [n]

time reversal time reversal

h h

0 0

[n] = h [n] = h

0 0

[

[-

-n]

n]

h h

1 1

[n] = h [n] = h

1 1

[

[-

-n]

n]

Verify that filter bank implements MRA equations: Verify that filter bank implements MRA equations:

u u

00

[n] = [n] =

hh

0 0

[n [n -

• k]

k]a

a

jj

[k] [k]

h h

00

[k [k –

• n]

n]a

a

jj

[k] [k]

k k

kk

a a

j j

11

[n] [n]

u u

0 0

[n] [n]

a a

jj

[n] [n]

Analysis Analysis

Synthesis Synthesis

u u

1 1

[n] [n]

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