Lecture 13 (Usage of Fourier transform in image processing)

Oct 8, 20200 likes810 views

This document discusses the Fourier transform and its applications in image processing. It begins by explaining the Fourier transform for 1D and 2D continuous and discrete signals. The Fourier transform converts a signal from the time or space domain to the frequency domain. It then covers properties of the Fourier transform such as separability and translation. The document concludes by mentioning references for further reading on image processing and computer vision topics.

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

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

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

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

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

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

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

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

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

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

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

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

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

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The document discusses three scenarios related to property rights and software: 1) Ramesh buying pirated software abroad, 2) a small software company called Bingo having their operating system copied, and 3) a man named Jake improving virus detection software and sharing his modifications. It also defines algorithms, source code, and object code. Property rights and copyright issues regarding software are complex with reasonable arguments on both sides.

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This document discusses Gaussian numerical integration techniques. It describes the Gauss quadrature 2-point and 3-point formulas for numerical integration. The 2-point formula uses two sample points with equal weights of 1 to calculate the integral. The 3-point formula uses three sample points and weights of 5/9, 8/9 and 5/9 to yield more accurate integration over an interval. The document also explains how to apply these formulas when the integral limits differ from [-1,1].

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Introduction to Censorship VARUN KUMAR

Distributed rc ModelVARUN KUMAR

Electrical Wire ModelVARUN KUMAR

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E-democracy or Digital DemocracyVARUN KUMAR

Ethics of Parasitic ComputingVARUN KUMAR

Action Lines of Geneva Plan of ActionVARUN KUMAR

Geneva Plan of ActionVARUN KUMAR

Fair Use in the Electronic AgeVARUN KUMAR

Software as a PropertyVARUN KUMAR

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Patent ProtectionVARUN KUMAR

Property Right and SoftwareVARUN KUMAR

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Lecture 13 (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

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