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The concepts of wavelets and filter banks through the mallat pyramid algorithm. It covers the goal of finding wavelet coefficients, the relationship between different functions, and the multiresolution decomposition and reconstruction equations. The document also discusses the importance of filter banks in implementing multiresolution analysis (mra) equations.
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��� φφφ
φ φφ
Goal: Given the series expansion for a function f
j
(t) in V
j
f
j
(t) = � a
j
[k]φ
j,k
(t)
k
how do we find the series
f
j-
(t) = � a
j-
[k]φ
j-1,k
(t)
k
in V
j-
and the series
g
j-
(t) = � b
j-
[k]w
j-1,k
(t)
k
in W
j-
such that
f
j
(t) = f
j-
(t) + g
j-
(t)?
2
3
Example: suppose that φφφφ(t) = box on [0,1]. Then
functions in V
1
can be written either as a combination of
φφφφ(2t)
φφφφ(2t-1)
or as a combination of
φφφφ(t)
φ φφ
φ(t-1)
4
plus a combination of
w(t)
w(t-1)
Easy to see because φφφφ(2t) = ½[φφφφ(t) + w(t)]
φφ φφ(2t –1) = ½[φφφφ(t) - w(t)]
��� φφφ
√√√ ��� ��� φφφ
∞∞∞
∞∞∞
∞∞∞
∞∞∞
Multiresolution decomposition equations
∞
a
j-
[n] = � f(t)φ
j-1,n
(t) dt
∞
= √ 2 � h
0
[] � f(t)φ
j,2n +
(t) dt
- ∞
= √√√√ 2 ���� h
0
[] a
j
[2n + ]
So
a
j-
[n] = √√√√ 2 ���� h
0
[k-2n]a
j
[k]
k
→ Convolution with h
0
[-n] followed by downsampling
7
∞∞∞
∞ ∞∞
Similarly
∞
b
j-
[n] = � f(t) w
j-1,n
(t) dt
which leads to
b
j-
[n] = √√√√ 2 ���� h
1
[k – 2n] a
j
[k]
k
8
φφφ
� �� φφφ ��� φφφ ��� φφφ
��� ��� φφφ φφφ
��� ��� φφφ
∞∞∞
∞ ∞∞
∞ ∞∞
∞∞∞
∞ ∞∞
∞∞∞
∞ ∞∞
∞∞∞
∞∞∞
∞∞∞
Multiresolution reconstruction equation
Start with
f
j
(t) = f
j-
(t) + g
j-
(t)
Multiply by φ
j,n
(t) and integrate
∞
∞ ∞
� f
j
(t) φ
j,n
(t) dt = � f
j-
(t)φ
j,n
(t)dt + � g
j-
(t)φ
j,n
(t) dt
So
∞
a
j
[n] = � a
j-
[k] � φ
j-1,k
(t) φ
j,n
(t) dt +
k
∞
� b
j-
[k] � w
j-1,k
(t) φ
j,n
(t) dt
k
9
10
φφφφ
j-1,k
(t) φφφφ
j,n
(t) dt = √√√√ 2 ���� h
0
[] ���� φφφφ
j,2k+
(t) φφφφ
j,n
(t) dt
= √√√√ 2 ���� h
0
[] δδδδ[2k + - n]
= √√√√2 h
0
[n – 2k]
∞∞∞∞
∞∞∞∞
Similarly
���� w
j-1,k
(t)φφφφ
j,n
(t) dt = √√√√2 h
1
[n –2k]
Result:
a
j
[n] = √√√√ 2 ���� a
j-
[k]h
0
[n - 2k] +
√√√√ 2 ���� b
j-
[k]h
1
[n – 2k]
∞ ∞∞
∞
k
k
