ECE 468: Digital Image Processing
Lecture 11
Prof. Sinisa Todorovic
1
Problem with Fourier Transform
•Available frequency content
•But not where that content is in the space domain of 2D signals
•(or in the time domain for 1D signals)
ej2πωx
2
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Digital Image Processing Lecture 11: Scaling Functions and Wavelet Transform - Prof. S. To, Study notes of Electrical and Electronics Engineering
In this lecture of ece 468: digital image processing, prof. Sinisa todorovic discusses the limitations of fourier transform and introduces scaling functions and wavelet transform. The lecture covers the concept of nested function spaces, haar scaling functions, and relations between scaling and wavelet functions. It also explains the haar wavelet functions and wavelet series expansion.
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ECE 468: Digital Image Processing
Lecture 11
Prof. Sinisa Todorovic
1
Problem with Fourier Transform
- Available frequency content
- But not where that content is in the space domain of 2D signals
- (or in the time domain for 1D signals)
e
j 2 πωx
Problem with Fourier Transform
- Available frequency content
- But not where that content is in the space domain of 2D signals
- (or in the time domain for 1D signals)
e
j 2 πωx
just scaling of the generating function w/o translation
2
Scaling Functions
ϕj,k(x) = 2
j/ 2
j
x − k)
Given: ϕ(x)
Example: Haar Scaling Functions
ϕ 0 ,k(x) = √^1 2
ϕ 1 , 2 k(x) + √^1 2 f (x) = 0. 5 ϕ 1 , 0 (x) + ϕ 1 , 1 (x) − 0. 25 ϕ 1 , 4 (x) ϕ^1 ,^2 k+1(x)
Function in V 1
4
Example: Haar Scaling Functions
ϕ 0 ,k(x) = √^1 2
ϕ 1 , 2 k(x) + √^1 2 f (x) = 0. 5 ϕ 1 , 0 (x) + ϕ 1 , 1 (x) − 0. 25 ϕ 1 , 4 (x) ϕ^1 ,^2 k+1(x)
Function in V 1
Nested Function Spaces
5
Any Function
If f (x) ∈ Vj
f (x) =
k
αkϕj,k(x)
f (x) =
k
βkϕj+1,k(x)
f (x) =
k
γkϕj+2,k(x)
Any Function If f (x) ∈ Vj
9
Any Function If f (x) ∈ Vj
f (x) =
∑
k
αkϕj− 1 ,k(x)
︸ ︷︷ ︸ Vj− 1
∑
k
βkψj− 1 ,k(x)
︸ ︷︷ ︸ Vj
Any Function If f (x) ∈ Vj
f (x) =
∑
k
αkϕj− 2 ,k(x) ︸ ︷︷ ︸ Vj− 2
∑
k
βkψj− 2 ,k(x)
︸ ︷︷ ︸ Vj− 1
∑
k
βkψj− 1 ,k(x)
︸ ︷︷ ︸ Vj
f (x) =
∑
k
αkϕj− 1 ,k(x)
︸ ︷︷ ︸ Vj− 1
∑
k
βkψj− 1 ,k(x)
︸ ︷︷ ︸ Vj
9
Any Function If f (x) ∈ Vj
f (x) =
∑
k
αkϕj 0 ,k(x) +
j∑− 1
l=j 0
∑
k
βkψl,k(x)
f (x) =
∑
k
αkϕj− 2 ,k(x) ︸ ︷︷ ︸ Vj− 2
∑
k
βkψj− 2 ,k(x)
︸ ︷︷ ︸ Vj− 1
∑
k
βkψj− 1 ,k(x)
︸ ︷︷ ︸ Vj
f (x) =
∑
k
αkϕj− 1 ,k(x)
︸ ︷︷ ︸ Vj− 1
∑
k
βkψj− 1 ,k(x)
︸ ︷︷ ︸ Vj
Haar Wavelet Functions
f (x) = fa(x) + fd(x)
11
Haar Wavelet Functions
f (x) = fa(x) + fd(x)
Haar Wavelet Functions
Function in V 1
f (x) = fa(x) + fd(x)
11
Haar Wavelet Functions
Function in V 1
f (x) = fa(x) + fd(x)
Example: Haar Wavelet Series Expansion