Usage of Fourier transform in image processing

1. Usage of Fourier Transform in Image Processing
Subject: Image Procesing & Computer Vision
Dr. Varun Kumar
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 1 / 13

2. Outlines
1 Fourier transform for continuous signal
2 Fourier transform for discrete signal
3 Fast Fourier transform
4 References
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 2 / 13

3. Fourier transform of 1D time varying signal
It is a mathematical operation that convert time domain signal to
frequency domain signal.
Frequency domain signal processing is simpler compare to time
domain.
Mathematical expression:
1 Fourier transform
X(jΩ) =
∞
−∞
x(t)e−jΩt
dt
CTFT
X(ejω
) =
∞
−∞
x(n)e−jωn
DTFT
(1)
2 Inverse Fourier transform
x(t) =
1
2π
∞
−∞
X(jΩ)ejΩt
dΩ
I−CTFT
x(n) =
1
2π
∞
−∞
X(ejω
)ejωn
I−DTFT
(2)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 3 / 13

4. Fourier transform of 1D space varying signal
1 Fourier transform
F(f (x)) = F(u) =
∞
−∞
f (x)e−j2πux
dx (3)
⇒ Here, f (x) must be continuous and integrable.
⇒ F(u) must be integrable.
2 Inverse Fourier transform
F−1
(F(u)) = f (x) =
∞
−∞
F(u)ej2πux
du (4)
⇒ F(u) is a complex variable.
⇒ F(u) = FRe(u) + jFIm(u) = |F(u)|ejφ(u)
⇒ Amplitude spectrum : |F(u)| = FRe(u)2 + FIm(u)2
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 4 / 13

5. Fourier transform of 2D signal
⇒ Phase spectrum : ∠φ(u) = tan−1 FIm(u)
FRe (u)
⇒ Power: P = |F(u)|2 = FRe(u)|2 + |FIm(u)|2
Fourier transform of 2D signal
Image is a 2D space varying signal.
Fourier transform of an image signal
F(u, v) =
∞
−∞
∞
−∞
f (x, y)e−j2π(ux+vy)
dxdy (5)
where f (x, y) is the 2D image signal.
Inverse Fourier transform :
f (x, y) =
∞
−∞
∞
−∞
F(u, v)ej2π(ux+vy)
dudv (6)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 5 / 13

6. Continued–
⇒ F(u, v) = |F(u, v)|ejφ(u,v)
⇒ Amplitude spectrum : |F(u, v)| = FRe(u, v)2 + FIm(u, v)2
⇒ Phase spectrum : φ(u, v) = tan−1 FIm(u,v)
FRe (u,v)
⇒ Power spectrum : P(u, v) = |F(u, v)|2 = FRe(u, v)2 + FIm(u, v)2
Example : Find the Fourier transform of a 2D signal which is as follow
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 6 / 13

7. Continued–
As per the 2D graphical signal f (x, y)
⇒ f (x, y) = A ∀ 0 ≤ x ≤ X and 0 ≤ y ≤ Y
⇒ f (x, y) = 0 ∀ x > X and y > Y
F(u, v) =
X
0
Y
0
Ae−j(ux+vy)
dxdy
= A
X
0
e−j2πux
dx
Y
0
e−j2πvy
dy
= A
1
j2πu
(1 − e−j2πuX
)
1
j2πv
(1 − e−j2πvY
)
(7)
Amplitude spectrum :
|F(u, v)| = AXY
sin(πuX)
πuX
sin(πvY )
πvY
(8)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 7 / 13

8. Graphical representation
2D discrete signal (DTFT) (Finite dimension)
F(u, v) =
∞
−∞
∞
−∞
f (x, y)e−j2π(ux+vy)
(9)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 8 / 13

9. DFT of an image signal
DFT/I-DFT of an 2D signal
DFT
F(u, v) =
1
MN
M−1
x=0
N−1
y=0
f (x, y)e−j2π( ux
M
+vy
N
)
(10)
Here, frequency variable u = 0, 1, ....., M − 1 and v = 0, 1, ...., N − 1
I-DFT
f (x, y) =
1
MN
M−1
u=0
N−1
v=0
F(u, v)ej2π( ux
M
+vy
N
)
(11)
In case of square image, i.e, M = N
F(u, v) =
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j2π( ux
N
+vy
N
)
(12)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 9 / 13

10. Properties of Fourier transform
1 Separability:
In case of DFT
F(u, v) =
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j2π(ux
N
+vy
N
)
⇒
1
N
N−1
x=0
e−j 2πux
N N.
1
N
N−1
y=0
f (x, y)e−j 2πvy
N
⇒
1
N
N−1
x=0
NF(x, v)e−j 2πux
N
(13)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 10 / 13

11. Continued–
In case of I-DFT
f (x, y) =
1
N
N−1
u=0
N−1
v=0
F(u, v)ej2π( ux
N
+vy
N
)
⇒
1
N
N−1
u=0
ej 2πux
N N.
1
N
N−1
v=0
F(u, v)ej 2πvy
N
⇒
1
N
N−1
u=0
Nf (u, y)ej 2πux
N
(14)
2 Translation
f (x, y) ⇒ (x0, y0) ⇒ f (x − x0, y − y0)
F(u, v)|x−x0,y−y0 = F(u, v)|x,y e
−j2π
N
(ux0+vy0)
(15)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 11 / 13

12. Continued–
I-DFT form
F(u − u0, v − v0) ⇒ f (x, y)e
j2π
N
(u0x+v0y)
(16)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 12 / 13

13. References
M. Sonka, V. Hlavac, and R. Boyle, Image processing, analysis, and machine vision.
Cengage Learning, 2014.
D. A. Forsyth and J. Ponce, “A modern approach,” Computer vision: a modern
approach, vol. 17, pp. 21–48, 2003.
L. Shapiro and G. Stockman, “Computer vision prentice hall,” Inc., New Jersey,
2001.
R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using
MATLAB. Pearson Education India, 2004.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 13 13 / 13