WeekNo.%07 Domain%Transformaon% …Naeem%A.%Mahoto% Time%Domain% Tuesday,%September%8,%2015% •...

Naeem A. Mahoto e-‐mail: [email protected]

Department of So9ware Engineering, Mehran UET

Jamshoro, Sind, Pakistan

Tuesday, September 8, 2015

Week No. 07 Domain TransformaNon (course: Computer Vision)

Naeem A. Mahoto

Transform


•  MathemaNcal transformaNons are applied to signals to obtain a further informaNon from that signal that is not readily available in the raw signal

•  There are number of transformaNons that can be applied, among which the Fourier transforms are probably by far the most popular

Naeem A. Mahoto

Time Domain


•  Time domain is a term used to describe the analysis of mathemaNcal funcNons, or physical signals, with respect to Nme

•  In the Nme domain, the signal or funcNon's value is known at various discrete Nme points; or for all real numbers, for the case of conNnuous Nme

•  An oscilloscope is a tool commonly used to visualize real-‐world signals in the Nme domain

•  The frequency is measured in cycles/second, or with a more common name, in "Hertz"

Naeem A. Mahoto

Time Domain


•  For example the electric power we use in our daily life in the US is 60 Hz (50 Hz elsewhere in the world)

•  This means that if you try to plot the electric current, it will be a sine wave passing through the same point 50 Nmes in 1 second

•  Now, look at the following figures. The first one is a sine wave at 3 Hz, the second one at 10 Hz, and the third one at 50 Hz. Lets, Compare them

Naeem A. Mahoto

Time Domain


Naeem A. Mahoto

Frequency Domain


•  Frequency domain is a term used to describe the analysis of mathemaNcal funcNons or signals with respect to frequency

•  Speaking non-‐technically, a Nme domain graph shows how a signal changes over Nme, whereas a frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies

•  A frequency domain representaNon can also include informaNon on the phase shi9 that must be applied to each sinusoid in order to be able to recombine the frequency components to recover the original Nme signal

Naeem A. Mahoto

Frequency Domain


•  The frequency axis starts from zero, and goes up to infinity. For every frequency, we have an amplitude value

•  For example, if we take the FT of the electric current that we use in our houses, we will have one spike at 50 Hz, and nothing elsewhere, since that signal has only 50 Hz frequency component

•  No other signal, however, has a FT which is this simple. For most pracNcal purposes, signals contain more than one frequency component

•  The following shows the FT of the 50 Hz signal:

Naeem A. Mahoto

Frequency Domain


Naeem A. Mahoto

Time Domain VS Frequency Domain


Naeem A. Mahoto

Fourier Transform


Jean B. Joseph Fourier (1768-1830)

“An arbitrary function, continuous or with discontinuities, defined in a finite interval by an arbitrarily capricious graph can always be expressed as a sum of sinusoids”

J.B.J. Fourier

December, 21, 1807

Naeem A. Mahoto

Fourier Transform


Original Signal Constituent Sinusoids of different frequencies

Naeem A. Mahoto

Fourier Transform


•  The Fourier transform is a certain linear operator that maps funcNons to other funcNons

•  Loosely speaking, the Fourier transform decomposes a funcNon into a conNnuous spectrum of its frequency components, and the inverse transform synthesizes a funcNon from its spectrum of frequency components

•  In mathemaNcal physics, the Fourier transform of a signal x(t) can be thought of as that signal in the "frequency domain"

Naeem A. Mahoto

Fourier Transform


•  For example the following signal:

is a staNonary signal, because it has frequencies of 10, 25, 50, and 100 Hz at any given Nme instant

•  This signal is plobed below:

)1002cos()502cos()252cos()102cos()( tttttx ππππ +++=

Naeem A. Mahoto

Fourier Transform


•  If f(x) is a conNnuous funcNon of a real variable x, then the Fourier Transform of f(x), denoted by , is defined by the equaNon:

•  Given F(u), f(x) can be obtained by using the inverse Fourier Transform:

[ ])(xfℑ

[ ] ∫∞

∞−

−==ℑ dxexfuFxf uxj π2)()()( 1

[ ])()( 1 uFxf −ℑ=

[ ]∫∞

∞−

= duuxjuF π2exp)(

2

Equa6ons (1) and (2) are collec6vely called as Fourier Transform Pair

Naeem A. Mahoto

Fourier Transform


•  As Fourier transform is Complex funcNon, it can be stated as:

Where R(u) and I(u) are, respecNvely, the real and imaginary components of F(u)

•  It is o9en convenient to express equaNon 3 in exponenNal form:

3 )()()( ujIuRuF +=

)()()( ujeuFuF φ= 4

!

F(u) = R2(u) + I2(u) 5

6 and ⎥⎦

⎤⎢⎣

⎡= −

)()(tan)( 1

uRuIuφ

where

The magnitude funcNon is called the Fourier Spectrum of f(x) and φ(u) its phase angle

)(uF

Naeem A. Mahoto

Fourier Transform


•  The square of the spectrum, is o9en referred as ”Power Spectrum” of f(x)

•  Another common term used is “Spectral Density”

2)()( uFuP =

22 )()()( uIuRuP +=

Naeem A. Mahoto

Fourier Transform


•  The variable “u” appearing in the Fourier transform is o9en called as “Frequency Variable”

•  This arises from the fact, that, If we expand the exponenNal term by using Euler’s formula, it is:

•  If we interpret the Integral in EquaNon (1) as a limit-‐summaNon of discrete terms, it is evident that F(u) is composed of an Infinite sum of Sine & Cosine terms, and that each value of “u” determines the frequency of its corresponding sine-‐cosine pair !

e" j 2#ux = cos(2#ux) " j sin(2#ux)

Naeem A. Mahoto

Fourier Transform


Figure (a) Figure (b)

Naeem A. Mahoto

Fourier Transform


•  Consider the funcNon as shown in the Figure (a), Its Fourier transform is obtained from EquaNon (1) as follows:

!

F(u) = f (x)e" j2#uxdx"$

$

%

!

= Ae" j2#uxdx0

X

$

!

="Aj2#u e" j 2#ux[ ]0

X=

"Aj2#u

e" j2#uX "1[ ]

Simplifying: MulNply & Divide by uXje π−

Naeem A. Mahoto

Fourier Transform


!

="Aj2#u (e

" j#uX " e j#uX )e" j#uX

!

=Aj2"u (e

j"uX # e# j"uX )e# j"uX

!

=A"u

e j"uX # e# j"uX

2 je# j"uX

!

ei" # e# i"

2i= sin"

uXjeuXuAuF πππ

−= )sin()(

Since

Therefore

Naeem A. Mahoto

Fourier Transform


•  So we have obtained the Fourier transform, that is Frequency Domain representaNon:

•  As F(u) is a complex term, we can find out the Fourier Spectrum by:

uXjeuXuAuF πππ

−= )sin()(

( )uXuXAX

ππ )sin(

=

A Plot of F(u)

Since, we are only interested in REAL part of the Frequency Spectrum, therefore, |F(u)| will be:

Naeem A. Mahoto

2-‐D Fourier Transform


•  The Fourier transform can be easily extended to a funcNon f(x, y) of two variables

•  If f(x, y) is conNnuous and F (u, v) is integrable, we have that the following Fourier Transform pair:

!

F(u,v) =" f (x,y)[ ]

F(u,v) = f (x,y)e# j2$(ux+vy)#%& dxdy

%&

!

f (x,y) ="#1 F(u,v)[ ]

f (x,y) = F(u,v)e j2$(ux+vy)#%& dudv

%&

7

8

Naeem A. Mahoto



•  As in the one –dimensional case. , the Fourier spectrum, phase, and power spectrum, respecNvely, are:

10

11

!

F(u,v ) = R2 (u,v ) + I 2 (u,v )

⎥⎦

⎤⎢⎣

⎡= −

),(),(tan),( 1

vuRvuIvuφ

),(2),(22

),(),( vuIvuRvuFvuP +==

9

Naeem A. Mahoto



Figure A (a) A two-‐dimensional func6on. (b) It’s a Fourier spectrum (c) the spectrum displayed as an intensity func6on.

Naeem A. Mahoto



•  Consider the funcNon as shown in the Figure A(a) Its Fourier transform is obtained from EquaNon (1) as follows:

1

2

!

F(u,v ) = f (x,y )e" j 2# (ux + vy )

"$% dxdy

$%

!

= e" j 2#ux

dx e" j2#vy

dy0y$

0

X$

Naeem A. Mahoto



3

4

YvyjXuxj

uje

ujeA

0

2

0

2

22 ⎥⎦

⎤⎢⎣

⎡

−⎥⎦

⎤⎢⎣

⎡

−=

−−

ππ

ππ

]1[21]1[

21 22 −

−−

−= −− uYjuXj e

uje

ujA ππ

ππ

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=

−−

uYeuY

uXeuXAXY

uXjuXj

ππ

ππ ππ 22 )sin()sin(

5

As F(u,v) is a complex term, we can find out the Fourier Spectrum (Only REAL Part) by:

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡=uYuY

uXuXAXYvuF

ππ

ππ )sin()sin(),( 6

Naeem A. Mahoto

Short comings in Fourier Transform


•  Fourier Analysis based on overlapping of Sines and Cosines

•  Extend to Infinity and are Non-‐Local (i.e., FT deals images as global and operaNons are performed on whole image unlike SpaNal domain methods where operaNons can be applied locally at subset of images)

•  Poor at ApproximaNng Sharp Spikes and DisconNnuiNes

Naeem A. Mahoto Tuesday, September 8, 2015

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