The document discusses digital image processing and image enhancement in the frequency domain. It provides background on Fourier series and Fourier transforms, explaining that Fourier transforms allow representing even non-periodic functions as integrals of sines and cosines. The Fourier transform converts a signal from the time domain to the frequency domain. Two-dimensional Fourier transforms are used in image processing for applications like image enhancement, restoration, and encoding/decoding. The document also outlines the formulas for one-dimensional and two-dimensional discrete Fourier transforms and their inverses.

1. Digital Image ProcessingDigital Image Processing
Lecture 7:Lecture 7: Image Enhancement inImage Enhancement in
Frequency Domain-IFrequency Domain-I
Naveed Ejaz

2. Introduction

3. Background (Fourier Series)
 Any function that periodically repeats itself can be
expressed as the sum of sines and cosines of
different frequencies each multiplied by a different
coefficient
 This sum is known as Fourier Series
 It does not matter how complicated the function is;
as long as it is periodic and meet some mild
conditions it can be represented by such as a sum
 It was a revolutionary discovery

4. What is the difference between
fourier series and fourier
transform?
The Fourier series is an expression of a pattern (such as
an electrical waveform or signal) in terms of a group of
sine or cosine waves of different frequencies and
amplitude. This is the frequency domain.
The Fourier transform is the process or function used to
convert from time domain (example: voltage samples
over time, as you see on an oscilloscope) to the
frequency domain, which you see on a graphic
equalizer or spectrum analyzer)

6. Background (Fourier Transform)
 Even functions that are not periodic (but whose area under the
curve is finite) can be expressed as the integrals of sines and
cosines multiplied by a weighing function
 This is known as Fourier Transform
 A function expressed in either a Fourier Series or transform can be
reconstructed completely via an inverse process with no loss of
information
 This is one of the important characteristics of these
representations because they allow us to work in the Fourier
Domain and then return to the original domain of the function

7. Fourier Transform
• ‘Fourier Transform’ transforms one function into
another domain , which is called the frequency
domain representation of the original function
• The original function is often a function in the
Time domain
• In image Processing the original function is in the
Spatial Domain
• The term Fourier transform can refer to either the
Frequency domain representation of a function or
to the process/formula that "transforms" one
function into the other.

8. Our Interest in Fourier Transform
• We will be dealing only with functions (images) of
finite duration so we will be interested only in Fourier
Transform

9. Applications of Fourier Transforms
 1-D Fourier transforms are used in Signal Processing
 2-D Fourier transforms are used in Image Processing
 3-D Fourier transforms are used in Computer Vision
 Applications of Fourier transforms in Image processing: –
– Image enhancement,
– Image restoration,
– Image encoding / decoding,
– Image description

10. One Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u) of a single variable, continuous
function f (x) is
 Given F(u) we can obtain f (x) by means of the Inverse
Fourier Transform

11. One Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u) of a single variable, continuous
function f (x) is
 Given F(u) we can obtain f (x) by means of the Inverse
Fourier Transform

12. Discrete Fourier Transforms (DFT)
1-D DFT for M samples is given as
The Inverse Fourier transform in 1-D is given as

13. Discrete Fourier Transforms (DFT)
1-D DFT for M samples is given as
The inverse Fourier transform in 1-D is given as

14. Two Dimensional Fourier Transform
and its Inverse
 The Fourier transform F (u,v) of a two variable, continuous
function f (x,y) is
 Given F(u,v) we can obtain f (x,y) by means of the Inverse
Fourier Transform

15. 2-D DFT

16. Fourier Transform

17. 2-D DFT