This document discusses properties of the discrete Fourier transform (DFT) for 2D signals. It outlines properties including periodicity, conjugation, rotation, distributivity, scaling, and convolution/correlation. It also discusses how the fast Fourier transform (FFT) reduces the computational complexity of the DFT from O(N^4) to O(N^2logN) for 2D signals with N samples. The properties of the DFT allow operations in the frequency domain to correspond to operations in the spatial domain, which makes the Fourier transform useful for applications like image processing and computer vision.

1. Properties of Fourier Transform for 2D Signal
Subject: Image Procesing & Computer Vision
Dr. Varun Kumar
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 1 / 13

2. Outlines
1 Properties of DFT
Periodicity and Conjugate
Rotation
Distributivity and Scaling
Convolution and Correlation
2 Fast Fourier transform
3 References
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 2 / 13

3. Properties of Discrete Fourier Transform
3 Periodicity Properties
Let f (x, y) is a 2D image signal and its Fourier transform is F(u, v), where
the number of discrete sample (N) remain same in x and y direction, then
F(u, v) = F(u + N, v) = F(u, v + N) = F(u + N, v + N) (1)
Proof
F(u, v) =
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j 2π
N
(ux+vy)
F(u + N, v) =
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j 2π
N
((u+N)x+vy)
=
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j 2π
N
((ux+vy)
e−j2πx
⇒ x ∈ Integer ⇒ e−j2πx
= 1
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 3 / 13

4. Continued–
4. Conjugate Properties
Let f (x, y) is a 2D image signal and its Fourier transform is F(u, v), where
the number of discrete sample (N) remain same in x and y direction, then
F∗
(u, v) = F(−u, −v) (2)
Proof
Since f (x, y) is a real valued signal. Now
F∗
(u, v) =
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j 2π
N (ux+vy)
∗
=
1
N
N−1
x=0
N−1
y=0
f (x, y)ej 2π
N (ux+vy)
=
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j 2π
N (−ux−vy)
= F(−u, −v)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 4 / 13

5. Continued–
5 Rotation Propety:
Cartesian to polar conversion
1 x = r cos θ and y = r sin θ ⇒ f (x, y) ⇒ f (r, θ)
2 u = w cos φ and v = w sin φ ⇒ F(u, v) ⇒ F(w, φ)
As per the rotation property
f (r, θ + θ0) ⇐⇒ F(w, φ + θ0) (3)
Proof
F(w, φ) =
1
N
rN−1
r=r0
θN
θ1
f (r, θ)e−j 2π
N
(rw(cos θ cos φ+sin θ sin φ=cos(θ−φ)))
F =
1
N
rN−1
r=r0
θN +θ0
θ=θ1+θ0
f (r, θ + θ0)e−j 2π
N
(rw(cos(θ+θ0−φ)))
= F(w, φ + θ0)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 5 / 13

6. Continued–
6 Distributive property
If two 2D signals are f1(x, y) and f2(x, y) and their Fourier transforms are
F1(u, v) and F2(u, v), then
F a1f1(x, y) + a2f2(x, y) = a1F f1(x, y) + a2F f2(x, y)
= a1F1(u, v) + a2F2(u, v)
(4)
Proof
F =
1
N
N−1
x=0
N−1
y=0
a1f1(x, y) + a2f2(x, y) e−j 2π
N
(ux+vy)
=
a1
N
N−1
x=0
N−1
y=0
f1(x, y)ej 2π
N
(ux+vy)
+
a2
N
N−1
x=0
N−1
y=0
f2(x, y)e−j 2π
N
(ux+vy)
= a1F1(u, v) + a2F2(u, v)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 6 / 13

7. Continued–
7 Scaling property
If a 2D signal is f (x, y) and its Fourier transforms is F(u, v). For a new
signal f (ax, by) the Fourier transform is as
f (ax, by) F
−→
1
|ab|
F
u
a
,
v
b
(5)
Proof
F =
1
N
N−1
x=0
N−1
y=0
f (ax, by)e−j 2π
N
(ux+vy)
let ax = z1 and by = z2
=
1
abN
a(N−1)
z1=0
b(N−1)
z2=0
f (z1, z2)ej 2π
N
(u
z1
a
+v
z2
b
)
=
1
ab
F
u
a
,
v
b
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 7 / 13

8. Averaging of 2D signal
Average
Let f (x, y) is a 2D discrete signal and ˆf (x, y) is its average
ˆf (x, y) =
1
N2
N−1
x=0
N−1
y=0
f (x, y)
DC component of Fourier transform
F(0, 0) =
1
N
N−1
x=0
N−1
y=0
f (x, y)
Relation between average of 2D signal and DC component of its Fourier
transform
ˆf (x, y) =
1
N
F(0, 0)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 8 / 13

9. Convolution/Correlation property
Convolution
⇒ g1(x) ∗ g2(x) = G1(u)G2(u) Convolution of 1D space domain signal
⇒ g1(x)g2(x) = G1(u) ∗ G2(u) Multiplication of 1D signal
⇒ g1(x, y) ∗ g2(x, y) = G1(u, v)G2(u, v) Convolution of 2D signal
⇒ g1(x, y)g2(x, y) = G1(u, v) ∗ G2(u, v) Multiplication of 2D signal
Correlation
⇒ g1(x, y) o g2(x, y) = G∗
1 (u, v)G2(u, v) Convolution of 2D signal
⇒ g∗
1 (x, y)g2(x, y) = G1(u, v) o G2(u, v) Multiplication of 2D signal
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 9 / 13

10. Fast Fourier transform FFT
Computational observation
1D signal
F(u) =
1
N
N−1
x=0
f (x)e−j 2π
N
ux
⇒ N2
2D signal
F(u, v) =
1
N
N−1
x=0
N−1
y=0
f (x, y)e−j 2π
N
(ux+vy)
⇒ N4
Note: This massive amount of data processing is not fruitful for sending
image in a band-limited channel.
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 10 / 13

11. Complexity reduction
1D signal
Let N = 2n
and N = 2M, where M = 2n−1
F(u) =
1
N
N−1
x=0
f (x)e−j 2π
N ux
=
1
N
N−1
x=0
f (x)W ux
N ⇒ WN = e−j 2π
N
=
1
2M
2M−1
x=0
f (x)W ux
2M
=
1
2
1
M
M−1
x=0
f (2x)W u2x
2M +
1
M
M−1
x=0
f (2x + 1)W
u(2x+1)
2M
=
1
2
1
M
M−1
x=0
f (2x)W u2x
2M +
1
M
M−1
x=0
f (2x + 1)W
ux)
M W u
2M
=
1
2
Feven(u) + Fodd (u)W u
2M
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 11 / 13

12. Continued–
Conclusion:
In single step processing N2 computation order reduces to
(N2
4 + N2
4 ) = N2
2
In similar fashion even and odd terms can be further reduced.
At the end of computation order reduction, i.e, N2 → N log2(N)
Subject: Image Procesing & Computer Vision Dr. Varun Kumar (IIIT Surat)Lecture 14 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 14 13 / 13