2-D DFT in images.pdf

Faculty of Electrical Engineering Digital Image Processing, Sec. (3-1-Image Transforms (DFT)) Shahid Beheshti Universit
DIGITAL IMAGE PROCESSING (DIP)
Farah Torkamani Azar
Faculty of Electrical Enginerring(Velenjak)
Shahid Beheshti University
f-torkamani@sbu.ac.ir,
farahtorkamaniazar@gmail.com
Section 3: 1-Image Transforms (DFT)
Fall 1402
Fall 2023
F. Torkamani Azar, https://facultymembers.sbu.ac.ir/torkamaniazar/course/digital-image-processing-fall-2023/ 1

Outlier
In all parts, we will use the definitions in MATLAB.
Students can also use Python software.
Discrete Fourier Transform (DFT)
Discrete Cosine Transform (DCT)
Walsh-Hadamard Transform
Wavelet Transform
Hotelling Transform
Radon Transform

1-D Transform Basis: Basis Vectors
Simple Cartesian Basis Vectors





a0
a1
.
.
.
an−1





= a0





1
0
.
.
.
0





+ a1





0
1
.
.
.
0





+ · · · + an−1





0
0
.
.
.
1





=
n−1
X
i=0
ai.vi,
requires vT
i .vj = δ(i − j)
A vector space V is defined as a subset v0, ..., vn−1 of vectors in V
that are linearly independent and span V. Consequently, every v
vector in V can be uniquely written as
v = a0v0 + a1v1 + ... + an−1vn−1, where a0, ..., an−1 are elements of
the base field.

How do basis vectors vary in different forms and why
is that important?
Each basis vector represents a significant attribute. When the vector
v is composed of different basis vectors, the coefficient ai indicates
how much of the basis vector vi contributes to the vector v.
Each Transform like DFT, DCT, DWT and ... define their basis vectors
with different attributes for different applications.
1-D DFT
X(k) =
N−1
X
n=0
x(n)e−j 2π
N
kn
; x(n) =
N−1
X
k=0
X(k)ej 2π
N
kn
Indeed
ak = X(k) and vk = [ej 2π
N
0k
, ej 2π
N
1k
, ej 2π
N
2k
, · · · , ej 2π
N
k(N−1)
]
is related to frequency ωk = 2π
N k k = 0, 1, ..., N − 1
Therefore, The amount of frequency ωk present in the signal x is
given by the expression X(k).

DFT Basis Vectors
Only real section of vectors
are presented.
K = 0 is related to lowest
frequency and k = 4 is
related to the highest
frequency, and
X(k) = X(N − k)∗. So, it
requires to compute only
N/2 coefficients which are
complex numbers.

2-D Transforms for Images

2-D Transforms for Images: continued
Image Representation
Each image can be represented as the linear combination of different
orthonormal basis images that have the same size as the original
one. These basis images have special properties. To find how much
the original image overlaps with each basis image, it should be
calculated by projection the original image onto basis images by inner
product.
From now on, for simplicity, in the formulas, the size of images is considered
as M × N with the assumption that M = N.

Discrete Fourier Transform: DFT
DFT formulas:
There are N2 basis images in size N × N.

8 × 8 Basis Images
Symmetric Property: F(k1, k2) = F∗(N − k1, N − k2)
because of periodicity: F(k1, k2) = F∗(−k1, −k2)
Is there a problem with DFT?

FFTSHIFT Presentation
We normally use fftshift command in Matlab so that the low frequency
part of the image will be in the center.

Examples:

Importance of DFT-Phase in Image Reconstruction

Properties of Fourier Transform

Filtering: Convolution
Consider: h=Impulse response of filter , kernel, Point Spread
Function (PSF); f=image
2D Convolution: g(m, n) =
P
k,l f(k, l).h(m − k, n − l)
operation extend size g larger than f.
In Matlab using g = conv2(f, h,′ same′) to preserve original image
size.
2D Filtering: g = filter2(h, f) that means:
g(m, n) =
P
k,l f(k, l).h(m + k, n + l)
In case of symmetric filter, ”filter2” and conv2” are the same.

Example:
There are two equivalent ways of carrying out linear spatial filtering
operations:
1. Spatial domain: convolution with a spatial operator;
2. Frequency domain: multiply FT of signal and filter, and compute
inverse FT of product.

Application: find an object in image;
Finding Letter T in the image.

2-D DFT in images.pdf

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