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Transforms
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8
-6
-4
-2
0
2
4
6
8
5*sin (24t)
Amplitude = 5
Frequency = 4 Hz
seconds
A sine wave
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8
-6
-4
-2
0
2
4
6
8
5*sin(24t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256 samples/second
seconds
Sampling duration =1 second
A sine wave signal
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2sin(28t), SR = 8.5 Hz
An undersampled signal
The Nyquist Frequency
• The Nyquist frequency is equal to one-half of the sampling frequency.
• The Nyquist frequency is the highest frequency that can be measured in a signal.
Fourier series
• Periodic functions and signals may be expanded into a series of sine and cosine functions
The Fourier Transform
• A transform takes one function (or signal) and turns it into another function (or signal)
The Fourier Transform
• A transform takes one function (or signal) and turns it into another function (or signal)
• Continuous Fourier Transform:
close your eyes if you don’t like integrals
The Fourier Transform
• A transform takes one function (or signal) and turns it into another function (or signal)
• Continuous Fourier Transform:
dfefHth
dtethfH
ift
ift
2
2
• A transform takes one function (or signal) and turns it into another function (or signal)
• The Discrete Fourier Transform:
The Fourier Transform
1
0
2
1
0
2
1 N
n
Niknnk
N
k
Niknkn
eHN
h
ehH
Fast Fourier Transform
• The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform
• FFT principle first used by Gauss in 18??• FFT algorithm published by Cooley & Tukey in
1965• In 1969, the 2048 point analysis of a seismic trace
took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 20 40 60 80 100 1200
50
100
150
200
250
300
Famous Fourier Transforms
Sine wave
Delta function
Famous Fourier Transforms
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0 50 100 150 200 2500
1
2
3
4
5
6
Gaussian
Gaussian
Famous Fourier Transforms
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
-100 -50 0 50 1000
1
2
3
4
5
6
Sinc function
Square wave
Famous Fourier Transforms
Sinc function
Square wave
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.5
0
0.5
1
1.5
-100 -50 0 50 1000
1
2
3
4
5
6
Famous Fourier Transforms
Exponential
Lorentzian
0 50 100 150 200 2500
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.2
0.4
0.6
0.8
1
FFT of FID
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
f = 8 Hz SR = 256 HzT2 = 0.5 s
2exp2sin
Tt
fttF
FFT of FID
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 20 40 60 80 100 1200
2
4
6
8
10
12
14
f = 8 HzSR = 256 HzT2 = 0.1 s
FFT of FID
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 20 40 60 80 100 1200
50
100
150
200
f = 8 Hz SR = 256 HzT2 = 2 s
Effect of changing sample rate
0 10 20 30 40 50 600
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 10 20 30 40 50 600
5
10
15
20
25
30
35
f = 8 Hz T2 = 0.5 s
Effect of changing sample rate
0 10 20 30 40 50 600
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 10 20 30 40 50 600
5
10
15
20
25
30
35
SR = 256 HzSR = 128 Hz
f = 8 HzT2 = 0.5 s
Effect of changing sample rate
• Lowering the sample rate:– Reduces the Nyquist frequency, which– Reduces the maximum measurable frequency– Does not affect the frequency resolution
Effect of changing sampling duration
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 2 4 6 8 10 12 14 16 18 200
10
20
30
40
50
60
70
f = 8 Hz T2 = .5 s
Effect of changing sampling duration
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 2 4 6 8 10 12 14 16 18 200
10
20
30
40
50
60
70
ST = 2.0 sST = 1.0 s
f = 8 HzT2 = .5 s
Effect of changing sampling duration
• Reducing the sampling duration:– Lowers the frequency resolution– Does not affect the range of frequencies you
can measure
Effect of changing sampling duration
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 2 4 6 8 10 12 14 16 18 200
50
100
150
200
f = 8 Hz T2 = 2.0 s
Effect of changing sampling duration
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-2
-1
0
1
2
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
14
ST = 2.0 sST = 1.0 s
f = 8 Hz T2 = 0.1 s
Measuring multiple frequencies
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
0 20 40 60 80 100 1200
20
40
60
80
100
120
f1 = 80 Hz, T21 = 1 s
f2 = 90 Hz, T22 = .5 s
f3 = 100 Hz, T2
3 = 0.25 s
SR = 256 Hz
Measuring multiple frequencies
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-3
-2
-1
0
1
2
3
0 20 40 60 80 100 1200
20
40
60
80
100
120
f1 = 80 Hz, T21 = 1 s
f2 = 90 Hz, T22 = .5 s
f3 = 200 Hz, T2
3 = 0.25 s
SR = 256 Hz
L: period; u and v are the number of cycles fitting into one horizontal and
vertical period, respectively of f(x,y).
Discrete Fourier Transform
1 12 ( ) /
0 0
1( , ) ( , )
w hj ux vy wh
x y
F u v f x y ewh
1 1
0 0
1 2 ( ) 2 ( )( , ) ( , ) cos sin
w h
x y
ux vy ux vyF u v f x y j
wh wh wh
Discrete Fourier Transform (DFT).
• When applying the procedure to images, we must deal explicitly with the fact that an image is:– Two-dimensional– Sampled– Of finite extent
• These consideration give rise to the The DFT of an NxN image can be written:
Discrete Fourier Transform• For any particular spatial frequency specified by u and v, evaluating
equation 8.5 tell us how much of that particular frequency is present in the image.
• There also exist an inverse Fourier Transform that convert a set of Fourier coefficients into an image.
1
0
/)(21
0
),(1
),(N
x
NvyuxjN
y
evuFN
yxf
PSD• The magnitudes correspond to the amplitudes of the basic images
in our Fourier representation.• The array of magnitudes is termed the amplitude spectrum (or
sometime ‘spectrum’).• The array of phases is termed the phase spectrum.• The power spectrum is simply the square of its amplitude
spectrum:
),(),(),(),( 222vuIvuRvuFvuP
FFT
• The Fast Fourier Transform is one of the most important algorithms ever developed
– Developed by Cooley and Tukey in mid 60s.
– Is a recursive procedure that uses some cool math tricks to combine sub-problem results into the overall solution.
DFT vs FFT
DFT vs FFT
DFT vs FFT
Periodicity assumption• The DFT assumes that an image is part of an infinitely repeated set of
“tiles” in every direction. This is the same effect as “circular indexing”.
Periodicity and Windowing
• Since “tiling” an image causes “fake” discontinuities, the spectrum includes “fake” high-frequency components
Spatial discontinuities
Discrete Cosine Transform
Real-valued
G m n m n g i ki m
N
k n
N
g i k m n G m ni m
N
k n
N
Nm
Nm N
ck
N
i
N
c cn
N
m
N
, ( ) ( ) , cos cos
, ( ) ( ) , cos cos
( ) ( )
2 1
2
2 1
2
2 1
2
2 1
2
01 2
1
0
1
0
1
0
1
0
1
with an inverse
where
and for
DCT in Matrix Form
G CgCc
where the kernel elements are
C mi m
Ni m, cos 2 1
2
Discrete Sine Transform
Most Convenient when N=2 p - 1
G m nN
g i ki m
N
k n
N
g i kN
G m ni m
N
k n
N
k
N
i
N
sn
N
m
N
sin
sin
, , sin sin
, , sin sin
2
1
1 1
1
1 1
1
2
1
1 1
1
1 1
1
0
1
0
1
0
1
0
1
with an inverse
DST in Matrix Form
G TgTc
where the kernel elements are
TN
i k
Ni k, sin2
1
1 1
1
DCT Basis Functions*
(Log Magnitude) DCT Example*
Hartley Transform
• Alternative to Fourier
• Produces N Real Numbers
• Use Cosine Shifted 45o to the Right
cas
cos sin
cos24
Square Hartley Transform
1 1
0 0
1 1
0 0
21, ,
*
with an inverse
2, ,
N N
Hartley Hi k
N N
Hartley Hm n
im knG m n g i k cas
N N N
im kng i k G m n cas
N
Rectangular Hartley Transform
1 1
0 0
1 1
0 0
1 2 2, ,
0.. , 0..
with an inverse
2 2, ,
h w
Hartley Hy x
h w
Hartley Hy x
mx nyG m n g x y cas
wh w h
m h n w
mx nyg m n G x y cas
w h
Hartley in Matrix Form
G TgTHartley
i kTN
ik
N
where the kernel elements are
cas,
1 2
What is an even function?
• the function f is even if the following equation holds for all x in the domain of f:
Hartley Convolution Theorem
• Computational Alternative to Fourier Transform
• If One Function is Even, Convolution in one Domain is Multiplication in Hartley Domain
g x f x h x G F H F Heven odd( ) ( )* ( )
Rectangular Wave Transforms
• Binary Valued {1, -1}
• Fast to Compute
• Examples– Hadamard– Walsh– Slant– Haar
Hadamard Transform
• Consists of elements of +/- 1
• A Normalized N x N Hadamard matrix satisfies the relation H Ht = I
H
HH H
H H
2
2
1
2
1 1
1 1
1
2
Walsh Tx can be constructed as
NN N
N N
Walsh Transform, N=4
*Gonzalez, Wintz
Non-ordered Hadamard Transform H8
H8
1 1
1 1
1 1
1 11 1
1 1
1 1
1 1
1 1
1 1
1 1
1 11 1
1 1
1 1
1 11 1
1 1
1 1
1 11 1
1 1
1 1
1 1
1 1
1 1
1 1
1 11 1
1 1
1 1
1 1
Sequency
• In a Hadamard Transform, the Number of Sign Changes in a Row Divided by Two
• It is Possible to Construct an H matrix with Increasing Sequency per row
Ordered Hadamard Transform
F u vN
F j k
q j k u v g u j g v k
u u
u u u
u u u
u u u
q j k u v
k
N
j
N
i i i ii
N
n
n n
n n
, ,
, , ,
( )
( )
( )
( )
, , ,
11
0
1
0
1
0
1
1
1 2
2 3
1 0
where
and
g
g
g
g
0
1
2
n-1
Ordered Hadamard Transform*
*Gonzalez, Wintz
Haar Transform
• Derived from Haar Matrix• Sampling Process in which Subsequent
Rows Sample the Input Data with Increasing Resolution
• Different Types of Differential Energy Concentrated in Different Regions– Power taken two at a time– Power taken a power of two at a time, etc.
Haar Transform*, H4
H4
1 1 1 1
1 1 1 1
2 2 0 0
0 0 2 2
*Castleman
Karhunen-Loeve Transform
• Variously called the K-L, Hotelling, or Eignevector
• Continuous Form Developed by K-L• Discrete Version Credited to Hotelling• Transforms a Signal into a Set of Uncorrelated
Representational Coefficients• Keep Largest Coefficients for Image Compression
Discrete K-L
F u v F j k A j k u v
A j k u v K j k j k A j k u v
K j k j k
K
k
N
j
N
Fk
N
j
N
F
F
( , ) , , ; ,
, ; , , ; , , ; ,
, ; ,
,
0
1
0
1
0
1
0
1
where the kernel satisfies
u, v
where
is the image covariance function
u, v is a constant for a fixed u, v the eigenvalues of
Singular Value Decomposition
An x matrix A can be expressed as
where
Columns of are Eigenvectors of
Columns of are Eigenvectors of
is the x diagonal matrix of singular values
t
t
N N
N N
t
t
A U V
U AA
V A A
L
U AV
Singular Value Decomposition
• If A is symmetric, then U=V
• Kernel Depends on Image Being Transformed
• Need to Compute AAt and AtA and Find the Eigenvalues
• Small Values can be Ignored to Yield Compression
Transform Domain Filtering
• Similar to Fourier Domain Filtering
• Applicable to Images in which Noise is More Easily Represented in Domain other than Fourier– Vertical and horizontal line detection: Haar
transform produces non-zero entries in first row and/or first column