Basic Properties of signal, Fourier
Expansion and it’s Applications in
Digital Image processing.
Md. Al Mehedi Hasan
Assistant Professor
Dept. of Computer Science & Engineering
RUET, Rajshahi-6204.
E-mail: [email protected]
Signals
• Signal is defined by its Amplitude, Frequency and
Phase
• Signals can be analog or digital.
• Analog signals can have an infinite number of
values in a range.
• Digital signals can
number of values.
have
only
a
limited
Comparison of analog and digital signals
Periodic Signal
Both analog and digital signals can be of two forms: Periodic and Aperiod.
A signal is a periodic if it completes a pattern within a measurable
time frame, called a period, and repeats that pattern over
identical subsequent period.
Periodic signals (continue)
Periodic signals can be classified as simple or
composite.
simple
composite
Aperiodic Signal
An aperiodic, or nonperiodic, signal has no patterns.
Amplitude
The Amplitude of a signal is the value of the signal at any point on
the wave.
Period and Frequency
Period refers to the amount of time, a signal needs to complete one
cycle. Frequency refers to the number of periods in one second.
Frequency and period are the inverse of each other.
Units of period and frequency
3.10
Phase
The term phase describes the position of the waveform
relative to time zero.
Two signals with the same phase and frequency, but different amplitudes
Two signals with the same amplitude and phase, but different frequencies
3.13
Three sine waves with the same amplitude and frequency, but different phases
3.14
Example
The power we use at home has a frequency of 60 Hz. The period of this sine wave can be
determined as follows:
3.15
Example
The period of a signal is 100 ms. What is its frequency in kilohertz?
Solution
First we change 100 ms to seconds, and then we calculate the frequency from the period
(1 Hz = 10−3 kHz).
3.16
Note
If a signal does not change at all, its
frequency is zero.
If a signal changes instantaneously, its
frequency is infinite.
3.17
Example
A sine wave is offset 1/6 cycle with respect to time 0. What is its phase in degrees and
Solution
We know that 1 complete cycle is 360°. Therefore, 1/6 cycle is
3.18
Sine and Cosine Functions
• Periodic functions
• General form of sine and cosine functions:
Sine and Cosine Functions
Special case: A=1, b=0, α=1
π
Sine and Cosine Functions (cont’d)
• Shifting or translating the sine function by a const b
Cosine is a shifted sine function:
Sine and Cosine Functions (cont’d)
• Changing the amplitude A
Sine and Cosine Functions (cont’d)
• Changing the period T=2π/|α|
e.g., y=cos(αt)
α =4
period 2π/4=π/2
shorter period
higher frequency
(i.e., oscillates faster)
Frequency is defined as f=1/T
Different notation: sin(αt)=sin(2πt/T)=sin(2πft)
One radian is the measure of a central angle that intercepts an arc equal in
length to the radius of the circle. See Figure. Algebraically, this means that
where θ is measured in radians.
In most applications of trigonometry, angles are measured in
of angles is preferred. Radian measure allows us to treat the
trigonometric functions as functions with domains of real numbers,
rather than angles.
Linear speed measures how fast the particle moves,
and angular speed measures how fast the angle
changes.
Frequency and Angular Frequency
Frequency is a metric for expressing the rate of oscillation in a
wave. For planar and longitudinal waves, this often expressed in
oscillations-per-second or Hz. Angular frequency used for
expressing rates of rotation, similar to revolutions-per-second,
and is usually expressed in radians-per-second. It can be thought
of as a wave with a constant amplitude where the amplitude
rotates in a circle in space.
Different Notation of Sine and Cosine
Functions
• Changing the period T=2π/|α|
e.g., y=cos(αt)
α =4
period 2π/4=π/2
shorter period
higher frequency
(i.e., oscillates faster)
Frequency is defined as f=1/T
Different Notation of Sine and Cosine
Functions (continue)
Fundamental Frequency?
Time Domain and Frequency Domain
Time Domain:
The time-domain plot shows changes in signal amplitude with
respect to time. Phase and frequency are not explicitly measure
on a time-domain plot.
Frequency Domain:
The time-domain plot shows changes in signal amplitude with
respect to frequency.
The time-domain and frequency-domain plots of a sine wave
3.36
The time domain and frequency domain of three sine waves
3.37
Frequency Spectrum and Bandwidth
The frequency spectrum of a signal is the combination of all sine
wave signals that make up that signal. The bandwidth of a signal is
the width of the frequency spectrum.
Jean Baptiste Joseph Fourier
Fourier was born in Auxerre,
France in 1768
– Most famous for his work “La
Théorie Analitique de la Chaleur”
published in 1822
– Translated into English in 1878:
“The Analytic Theory of Heat”
Nobody paid much attention when the work was
first published
One of the most important mathematical theories
in modern engineering
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
The Big Idea
=
Any function that periodically repeats itself can be
expressed as a sum of sines and cosines of different
frequencies each multiplied by a different
coefficient – a Fourier series
Fourier analysis
• A single-frequency sine wave is not useful in
some situation
• We need to use a composite signal, a signal
made of many simple sine waves.
• According to Fourier analysis, any composite
signal is a combination of simple sine waves
with different frequencies, amplitudes, and
phases.
3.41
Composite Signals and Periodicity
• If the composite signal is periodic, the
decomposition gives a series of signals with
discrete frequencies.
• If the composite signal is nonperiodic, the
decomposition gives a combination of sine
waves with continuous frequencies.
3.42
Fourier Series of composite
periodic signal
• Every composite periodic signal can be
represented with a series of sine and cosine
functions.
• The functions are integral harmonics of the
fundamental frequency “f” of the composite
signal.
• Using the series we can decompose any
periodic signal into its harmonics.
A composite periodic signal
3.44
Decomposition of a composite periodic signal in the time and
frequency domains
3.45
Nonperiodic signal
The time and frequency domains of a nonperiodic signal
Fourier Series
Fourier Series
Where
Meaning of Coefficients
An equation with many faces
There are several different ways to write the Fourier series.
Examples of Signals and the Fourier Series Representation
Sawtooth Signal
Application
Spatial Frequency in image
When we deal with a one dimensional signal (time series), it is
quite easy to understand what the concept of frequency is.
Frequency is the number of occurrences of a repeating event per
unit time. For example, in the figure below, we have 3 cosine
functions with increasing frequencies cos(t), cos(2t), and cos(3t).
Spatial Frequency in image (con..)
Spatial Frequency in image (con..)
So, we know that a sequence of such numbers gives us the feeling that cos(t) is a low
frequency signal. How we can create an image of these numbers? Let scale the numbers
to the range 0 and 255:
Considering that values are intensity values, we can obtain the following image.
Spatial Frequency in image (con..)
This is our first image with a low frequency component. We have
a smooth transition from white to black and black to white.
However, it is still difficult to say anything since we have not seen
an image with high frequency. If we repeat all the steps for cos(3t)
, we obtain the following image:
where we have sudden jumps to black. You can try the same
experiment for different cosines. By looking at two examples, we
can say that if there are sharp intensity changes in an image,
those regions correspond to high frequency components. On the
other hand, regions with smooth transitions correspond to low
frequency components.
Spatial Frequency in image (con..)
We now have an idea for one dimensional image. It is time to switch to two dimensional
representation of a signal. Let us first define a kind of two dimensional signal: f(x,
y)=cos(kx) cos(ky). For example, the signal for k=1, we have: f(x, y)=cos(x) cos(y).
Spatial Frequency in image (con..)
If you wonder, you can assign different k values (e.g. f(x,y)=cos(x)cos(3y) ) for the base
cosine functions, and plot the result. Our goal is to create an image containing a single
frequency component as much as possible. Let us pick a cosine signal with a low frequency:
cos(t) . Our corresponding two dimensional function will be f(x, y)=cos(x) cos(y). How we
will obtain a two dimensional image from this function? This is the question! We are going
to define a matrix and store the values of f(x,y) for different (x, y) pairs.
Basically, we divide the angle range 0-2∏ into M=512 and N=512 regions for x and y,
respectively.
Spatial Frequency in image (con..)
Here are images for different k values. Values of k represents the level of frequency (from
low to high) for k=0,….,20.
Spatial Frequency in image (con..)
Spatial Frequency in image (con..)
Similar to the 1D case, we can say that if the intensity values in an image changes
dramatically, that image has high frequency components.
Frequency Domain In Images
 Spatial frequency of an image refers to the
rate at which the pixel intensities change
 In picture on right:

High frequences:


Near center
Low frequences:

Corners
The Discrete Fourier Transform (DFT)
2-D case
The Discrete Fourier Transform of f(x, y), for x = 0,
1, 2…M-1 and y = 0,1,2…N-1, denoted by F(u, v),
is given by the equation:
M 1 N 1
F (u, v)   f ( x, y)e
 j 2 ( ux / M vy / N )
x 0 y 0
for u = 0, 1, 2…M-1 and v = 0, 1, 2…N-1.
The Inverse DFT
It is really important to note that the Fourier transform
is completely reversible.
The inverse DFT is given by:
1
f ( x, y) 
MN
M 1 N 1
 F (u, v)e
u 0 v 0
for x = 0, 1, 2…M-1 and y = 0, 1, 2…N-1
j 2 ( ux / M vy / N )
Discrete Fourier transform (2-D)
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
The DFT and Image Processing
How con we connect broken text ?
How can we remove
blemishes in a photograph?
Enhanced image
How can
we get the
enhanced
image from
the original
image?
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Some Basic Frequency Domain Filters
Low Pass Filter
High Pass Filter
Smoothing Frequency Domain Filters
Smoothing is achieved in the frequency domain
by dropping out the high frequency components
The basic model for filtering is:
G(u,v) = H(u,v)F(u,v)
where F(u,v) is the Fourier transform of the
image being filtered and H(u,v) is the filter
transform function
Low pass filters – only pass the low frequencies,
drop the high ones
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Ideal Low Pass Filter
Simply cut off all high frequency components that
are a specified distance D0 from the origin of the
transform
changing the distance changes the behaviour of
the filter
Ideal Low Pass Filter (cont…)
The transfer function for the ideal low pass filter
can be given as:
1 if D(u, v)  D0
H (u, v)  
0 if D(u, v)  D0
where D(u,v) is given as:
D(u, v)  [(u  M / 2)2  (v  N / 2)2 ]1/ 2
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Ideal Low Pass Filter (cont…)
Above we show an image, it’s Fourier spectrum
and a series of ideal low pass filters of radius 5,
15, 30, 80 and 230 superimposed on top of it
Images taken from Gonzalez & Woods, Digital Image Processing (2002)
Ideal Low Pass Filter (cont…)
Original
image
Result of filtering
with ideal low pass
Result of filtering
with ideal low pass
Result of filtering
with ideal low pass
Result of filtering
with ideal low pass
Result of filtering
with ideal low pass