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E Banking is closely associated with computer sciences. In these Lecture Slides, the lecturer has explained the following aspects of Banking : Multiresolution Analysis, Scaling Functions, Wavelets, Complementary Spaces, Nested Spaces, Requirements For Mra, Continuous Time, Box Function, Wavelet Equation, Scaling Function
Typology: Slides
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Continuous time:
φφφφ(t) Box function
t
t t
φφ φφ(2t) Scaling
φφφφ(2t - 1)
Scaling +
Shifting
φφφ φφφ φφφ
φφφ ∑∑∑ φφφ
φφφ
φφφ
φφφ ≤≤≤ <<<
φ φφ
φφφ
∫∫∫ φφφ ∑∑∑ ∫∫∫ φφφ
∑ ∑∑ ∫∫∫ φφφ τττ τττ
φ φφ
For this example:
φ(t) = φ(2t) + φ(2t – 1)
More generally:
N Refinement equation
φ(t) = 2∑ h
0
[k]φ(2t – k)
or
k=
Two-scale difference
equation
φ(t) is called a scaling function
The refinement equation couples the representations
of a continuous-time function at two time scales. The
continuous-time function is determined by a discrete-
time filter, h
0
[n]! For the above (Haar) example:
h
0
[0] = h
0
[1] = ½ (a lowpass filter)
3
Note: (i) Solution to refinement equation may not
always exist. If it does…
(ii) φ(t) has compact support i.e.
φ(t) = 0 outside 0 ≤ t < N
(comes from the FIR filter, h
0
[n])
(iii) φ(t) often has no closed form solution.
(iv) φ(t) is unlikely to be smooth.
Constraint on h
0
[n]:
N
∫ φ(t)dt = 2 ∑ h
0
[k] ∫ φ(2t – k)dt
k=
N
= 2 ∑ h
0
[k] • ½ ∫ φ(τ)dτ
k=
So
N
∑ h
0
[k] = 1 Assumes ∫ φ(t)dt ≠ 0
k=
4
7
Some observations for Haar scaling function and wavelet
t t
φ φφ
φ(t)
φφφφ(t - 1)
∫∫∫∫ φφφφ(t) φφφφ(t – k)dt
1 if k = 0
0 otherwise
= δ δδ
δ[k]
Similarly
∫∫∫∫ w(t) w(t – k)dt δδδδ[k]
Reason: no overlap
8
t
φ φφ
φ(t) w(t)
∫φφφφ(t) w(t)dt =
Reason: +ve and –ve areas cancel each other.
t
∞ ∞∞
∞∞∞
∞∞∞
∞∞∞
9
t
w(t)
t
w(2t)
t
w(2t - 1)
∫ w(t) w(2t)dt = , ∫∫∫∫ w(t) w(2t – 1)dt = 0
Reason: finer scale versions change sign while
coarse scale version remains constant.
Wavelet Bases
Our goal is to use w(t), its scaled versions (dilations)
and their shifts (translates) as building blocks for
continuous-time functions, f(t). Specifically, we are
interested in the class of functions for which we can
define the inner product:
∞
<f(t) , g(t)> = ∫ f(t) g*(t)dt < ∞< ∞
Such functions f(t) must have finite energy:
∞
||f(t)||
2
= ∫f(t)
2
dt << ∞∞
and they are said to belong to the Hilbert space, L
2
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∞ ∞∞
∞ ∞∞
→ → ∞∞
w
jk
(t) form an orthonormal basis for L
2
f(t) = ∑ b
jk
w
jk
(t) ; w
jk
(t) = 2
j/
w(
j
t – k)
j,k
∞
b
jk
∫ f(t) w
jk
(t) dt
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Key ingredients:
0
1
j
j+
2
2
(ℜ) = all functions with finite energy
= {ƒ(t): ∫ ƒ(t)
2
dt < ∞} Hilbert
space
Requirements:
2
(ℜ) and ƒ
j
(t) is the portion of ƒ(t) that lies in
lim
j
, then j →
∞
ƒ
j
(t) = ƒ(t) → ∞
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∞∞∞
∞∞∞
∞∞∞
∞∞∞
→→→ ∞∞∞
Restated as a condition on the subspaces:
∞
j = - ∞
j
2
lim
j → - ∞
|| f
j
(t) || = 0
Restated as a condition on the subspaces:
∞
j
j = - ∞
15
j
such that V
j
j
j+
and V
j
j
= {0} (no overlap)
This is written as
j
j
j+
(Direct sum)
Note: An orthogonal multiresolution will have W
j
orthogonal to V
j
j
j
So orthogonality will ensure that V
j
j
16
∑∑∑ φφφ
So
1
Can we relate this basis for V
1
to the basis for V
0
We know that
0
1
So any function in V
0
can be written as a combination
of the basic functions for V
1
0
, we can write
0
k
This is the Refinement Equation (a.k.a. the Two-
Scale Difference Equation or the Dilation Equation).
19
We also know that
0
1
0
So
0
1
This means that any function in W
0
can also be written
as a combination of the basic functions for V
1
Since w(t) ∈ W
0,
we can write
w(t) = 2∑ h
1
[k] φ(2t – k)
Wavelet
k
Equation
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21
0 0 1 2
2
Coarse
approximation
Level 0 detail
Level 1 detail
Level 2 detail
Finite energy
functions
0
0
1
2
1
Functions:
Images:
Geometry:
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Mesh courtesy of Igor Guskov (Caltech)
