Image Processing Image Transform and Fourier/Wavelet Transform

108 Pages·2015·10.53 MB·English

Table of Content
Image Processing
Fourier Transform
O. Le Meur
Table of Content
1 Reminder
6 Bi-dimensional Fourier
transformation
2 Image transformation
7 Fast Fourier transform
3 Fourier transformation
8 Discrete unitary transform
4 Time sampling
5 Discrete Fourier Transform
2
Reminder
Image transformation
Fourier transformation
Time sampling
Discrete Fourier Transform
Bi-dimensional Fourier transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Reminder
1 Reminder
6 Bi-dimensional Fourier
transformation
2 Image transformation
7 Fast Fourier transform
3 Fourier transformation
8 Discrete unitary transform
4 Time sampling
5 Discrete Fourier Transform
3
Reminder
Image transformation
Fourier transformation
Time sampling
Discrete Fourier Transform
Bi-dimensional Fourier transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Notations
a continuous signal x(.);
a discrete signal x[.];
a matrix A.
4
Reminder
Image transformation
Fourier transformation
Time sampling
Discrete Fourier Transform
Bi-dimensional Fourier transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Notations
1
f is the frequency (Hz); w = 2πf is the pulsation (radian/s); T = is the period
f
(s);
∫
p +∞ p 2
x ∈ L (R) means that |x(t)| dt < +∞. For instance, the space L (R) is
−∞
∫
+∞ 2
composed of the functions x with a finite energy |x(t)| dt < +∞;
−∞
Euler’s formulas: exp(jθ) = cosθ + jsinθ;
From Wikepedia.
5
Reminder
Image transformation
Fourier transformation
Time sampling
Discrete Fourier Transform
Bi-dimensional Fourier transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Reminder
sinπx
sinc(x) = ;
πx
n th
(−1) = exp (jπn) n root of the unity;
Inner product:
∫
+∞
∗
⟨x, y⟩ = x(t) × y (t)dt
−∞
+∞
∑
∗
⟨x, y⟩ = x(n) × y (n)
n=−∞
6
Reminder
Image transformation
Fourier transformation
Time sampling
Discrete Fourier Transform
Bi-dimensional Fourier transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Reminder
Condition of orthogonality for real basis function:
∫ {
t2
0 n ≠ m
φn(t)φm(t)dt =
t1 λm n = m
where λm is a constant (λm = 1, orthonormal).
Condition of orthogonality for complex basis function:
∫ {
t2
∗ 0 n ≠ m
φn(t)φm(t)dt =
t1 λm n = m
where λm is a constant (λm = 1, orthonormal).
7
Reminder
Image transformation
Presentation
Fourier transformation
3 kinds of transformation
Time sampling
Point to point transformation
Discrete Fourier Transform
Local to point transformation
Bi-dimensional Fourier transformation
Global to point transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Image transformation
1 Reminder
6 Bi-dimensional Fourier
transformation
2 Image transformation
Presentation
7 Fast Fourier transform
3 kinds of transformation
Point to point transformation
Local to point transformation 8 Discrete unitary transform
Global to point transformation
3 Fourier transformation
4 Time sampling
5 Discrete Fourier Transform
8
Reminder
Image transformation
Presentation
Fourier transformation
3 kinds of transformation
Time sampling
Point to point transformation
Discrete Fourier Transform
Local to point transformation
Bi-dimensional Fourier transformation
Global to point transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Image transformation
What is a transformation?
T
im[x, y ] −→ IM[u, v ]
im is the original image;
IM is the transformed image;
x, y (or u, v) represents the spatial coordinates of a pixel.
Goal of a transformation
The goal of a transformation is to get a new representation of the incoming picture.
This new representation can be more convenient for a particular application or can
ease the extraction of particular properties of the picture.
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Reminder
Image transformation
Presentation
Fourier transformation
3 kinds of transformation
Time sampling
Point to point transformation
Discrete Fourier Transform
Local to point transformation
Bi-dimensional Fourier transformation
Global to point transformation
Fast Fourier transform
Discrete unitary transform
Conclusion
Image transformation
There exist 3 types of transformation:
Point to point transformation:
The output value at a specific coordinate
is dependent only on one input value but
not necessarily at the same coordinate;
Local to point transformation:
The output value at a specific coordinate
is dependent on the input values in the
neighborhood of that same coordinate;
Global to point transformation:
The output value at a specific coordinate
is dependent on all the values in the input
image.
Note that the complexity increases with the size of the considered neighborhood...
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