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1
Frequency
DomainProcessing
Chapter 4
www.ImageProcessingPlace.com
www.csie.ntnu.edu.tw/~violet/IP93/Chapter04.ppt
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Complex Numbers
http://en.wikipedia.org/wiki/Complex_number
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Polar Coordinates
http://en.wikipedia.org/wiki/Polar_coordinates
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Periodic Function
per iod p
x x+px
A functionfis periodic with period pgreater than zero if
f(x +p) =f(x)
forallvalues ofx in the domain off.
If a functionfis periodic with periodp, then for allx in the domain offand all integers n,
f(x + np) =f(x)
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If a function f is periodic with period p > 0
then thefundamental frequency off is 1/p
p has units of length 1/p has units of cycles/length
Periodic Function
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p is large (large period) 1/p is small (low frequency)
p is small (small period) 1/p is large (high frequency)
p
p
Periodic Function
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Fourier Series & Fourier Transform
Any function thatperiodically repeats itself can
be expressed as the sum of sines and/or cosines of
different frequencies, each multiplied by a
different coefficient (Fourier series).
Even functions that are not periodic (but whose
area under the curve is finite) can be expressed as
the integral of sines and/or cosines multiplied bya weighting function (Fourier transform).
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Fourier Transform & Glass Prism
The prism is a physical device that separates lightinto various color components, each dependingon its wavelength (or frequency) content.
The Fourier transform may be viewed as amathematical prism that separates(characterizes) a function into variouscomponents, also based on frequency content.
This is a powerful concept that lies at the heart oflinear filtering.
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Frequency Domain
The frequency domain
refers to the plane of the
two dimensional discrete
Fourier transform of animage.
The purpose of the
Fourier transform is to
represent a signal as a
linear combination of
sinusoidal signals of
various frequencies.
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Fourier TransformationAnd Its Inverse
The one-dimensional Fourier transform and its inverse
Fourier transform (continuous case)
Inverse Fourier transform:
The two-dimensional Fourier transform and its inverse
Fourier transform (continuous case)
Inverse Fourier transform:
dueuFxf uxj 2)()(
dydxeyxfvuF vyuxj )(2),(),(
1where)()(2
jdxexfuF
uxj
dvduevuFyxf vyuxj )(2),(),(
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The one-dimensional Fourier transform and its inverse
Fourier transform (discrete case)
Inverse Fourier transform:
1,...,2,1,0for)(
1
)(
1
0
/2
MuexfMuF
M
x
Muxj
1,...,2,1,0for)()(1
0
/2
MxeuFxfM
u
Muxj
Discrete Fourier Transformation
One Dimensional
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Since (Eulers formula)
then discrete Fourier transform can be redefined
Frequency (time) domain: the domain (values ofu) over
which the values ofF(u) range; because u determines thefrequency of the components of the transform.
Frequency (time) component: each of theMterms ofF(u).
1,...,2,1,0for Mu
1
0]/2sin/2)[cos(
1
)(
M
xMuxjMuxxfMuF
sincos je j
Discrete Fourier Transformation
One Dimensional
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F(u) can be expressed in polar coordinates:
R(u): the real part ofF(u)
I(u): the imaginary part ofF(u)
Power spectrum:
)()()()( 222
uIuRuFuP
spectrum)phaseorangle(phase)(
)(tan)(
spectrum)or(magnitude)()()(where
)()(
1
22
)(
uR
uIu
uIuRuF
euFuF uj
Discrete Fourier Transformation (DFT)
One Dimensional
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f(0) = 2
f(1) = 3
f(2) = 4
f(3) = 4
The One-Dimensional DFT
Examples
1,...,2,1,0for Mu
1
0
]/2sin/2)[cos(1
)(M
x
MuxjMuxxfM
uF
)()()( 22 uIuRuF
F(0) = 3.25
F(1) = -0.5 +j 0.25
F(2) = 0.25
F(3) = -0.5 -j 0.25
|F(0) | = 3.25
|F(1) | = 0.75
|F(2) | = 0.25
|F(3) | = 0.75
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The One-Dimensional DFT
Examples
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The One-Dimensional DFT
Examples
M= 1024 points, A = 1, K= 8 points
Spectrum is centered at u = 0 (discussed in the following slides)
1,...,2,1,0for Mu
1
0
]/2sin/2)[cos(1
)(M
x
MuxjMuxxfM
uF
)()()( 22 uIuRuF
1
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The One-Dimensional DFT
Examples
Important Features:
The height of the spectrum doubled as the area
under the curve in thex-domain doubled. The number of zeros in the spectrum in the same
interval doubled as the length of the function
doubled. This reciprocal nature of the Fourier transform
pair is most useful in interpreting results of image
processing in the frequency domain.
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The One-Dimensional Fourier TransformSome Examples
The transform of a constant function is a DC value only.
The transform of a delta function is a constant.
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The One-Dimensional Fourier TransformSome Examples
The transform of an infinite train of delta functions spaced by
T is an infinite train of delta functions spaced by 1/T.
The transform of a cosine function is a positive delta at the
appropriate positive and negative frequency.
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The One-Dimensional Fourier TransformSome Examples
The transform of a sin function is a negative complex delta
function at the appropriate positive frequency and a negative
complex delta at the appropriate negative frequency.
The transform of a square pulse is a sinc function.
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The Two Dimensional DFT
The two-dimensional Fourier transform and its inverse
Fourier transform (discrete case) DFT
Inverse Fourier transform:
u, v : the transform or frequency variables
x, y : the spatial or image variables
1,...,2,1,0,1,...,2,1,0for
),(1
),(1
0
1
0
)//(2
NvMu
eyxf
MN
vuFM
x
N
y
NvyMuxj
1,...,2,1,0,1,...,2,1,0for
),(),(
1
0
1
0
)//(2
NyMx
evuFyxf
M
u
N
v
NvyMuxj
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The Two Dimensional DFT
Properties
(shift))2
,2
()1)(,()1(N
vM
uFyxf yx
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We define the Fourier spectrum, phase angle, and power
spectrum as follows:
R(u,v): the real part ofF(u,v)
I(u,v): the imaginary part ofF(u,v)
spectrum)(power),(),(),()(
angle)(phase),(
),(tan),(
spectrum)(),(),(),(
222
1
2
122
vuIvuRvuFu,vP
vuR
vuIvu
vuIvuRvuF
The Two Dimensional DFT
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Magnitude and Phase
Of sinusoidal wave
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Complex Numbers
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A Memory Aid for Evaluating j
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DFT Translation Property
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DFT Rotation Property
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DFT Periodicity
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spectrum)theofcomponentor DC(average
),(1
)0,0(1
0
1
0
M
x
N
y
yxfMN
F
The Two Dimensional DFT
Properties
F(u, v) =F(u+M, v) =F(u, v+N) =F(u+M, v+N)
(infinitely periodic in both u & v directions)
Same is for inverse DFT:
f(x,y) =f(x+M,y) =f(x,y+N) =f(x+M,y+N)
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(a)f(x,y) (b)F(u,y) (c)F(u, v)
The 2D DFTF(u, v) can be obtained by:
1. taking the 1D DFT of every row of imagef(x,y),F(u,y),
2. taking the 1D DFT of every column ofF(u,y)
The Two Dimensional DFT
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shift
The Two Dimensional DFT
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The Two Dimensional DFT
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The Two Dimensional DFT
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Computing & Visualizing
2-D DFT>> f = imread('rectangle.tif');
>> F = fft2(f);
>> Fc = fftshift(F);
>> imshow(abs(Fc), [])>> S2 = log(1+abs(Fc));
>> imshow(S2, [])
Note:[(-1)x+yf(x, y)]
= fftshift(fft2(f))
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The Property of Two-Dimensional DFT
Rotation
DFT
DFT
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The Property of Two-Dimensional DFT
Linear Combination
DFT
DFT
DFT
A
B
0.25 * A
+ 0.75 * B
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The Property of Two-Dimensional DFT
Expansion
DFT
DFT
A
Expanding the original image by a factor of n (n=2), filling
the empty new values with zeros, results in the same DFT.
B
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Two-Dimensional DFT with Different Functions
Sine wave Its DFT
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Two-Dimensional DFT with Different Functions
2D Gaussian
function
Impulses
Its DFT
Its DFT
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Filtering in the Frequency Domain
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1. Since frequency is directly related
to rate of change it can be
associated with frequencies in the
Fourier transform.
2. Slowest varying frequencycomponent (u = v = 0)
corresponds to the average gray
level of an image.
3. As we move further away from
the origin, the higher frequencies
begin to correspond to faster and
faster gray level changes in an
image, which are the edges of
objects or noise etc.
Filtering in the Frequency DomainSome Basic Properties
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Filtering in the Frequency DomainBasics of Filtering in the Frequency Domain
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1) Multiply the input image by (-1)x+y to center the
transform.
2) ComputeF(u,v), the DFT of the image from (1).
3) MultiplyF(u,v) by a filter functionH(u,v).
4) Compute the inverse DFT of the result in (3).
5) Obtain the real part of the result in (4).6) Multiply the result in (5) by (-1)x+y.
Filtering in the Frequency DomainBasics of Filtering in the Frequency Domain
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1. Obtain the padding parameters:
PQ = paddedsize(size(f));
2. Obtain the Fourier transform with padding:
F = fft2(f, PQ(1), PQ(2));
3. Generate a filter function, H, of size PQ using any of the methods
discussed in the later slides. Apply: H = fftshift(H) if necessary.
4. Multiply the transform by the filter:
G = H.*F;5. Obtain the real part of the inverse FFT of G:
g = real(ifft2(G));
6. Crop the top, left rectangle to the original size:
g = g(1:size(f, 1), 1:size(f, 2));
Filtering in the Frequency DomainImplementation Through MATLAB
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Filtering in the spatial domain is described by the following
equation:
g(x,y) = h(x,y)*f(x,y)
wheref(x,y) is the original image, h(x,y) is the filter and * denotes theconvolution operation.
Filtering in the frequency domain is described by the following
equation:
G(u, v) =H(u, v)F(u, v)where
F(u, v) is the Fourier transform of an image
H(u, v) is the frequency domain filter
G(u, v) is the Fourier transform of the filtered image
Filtering in the Spatial and Frequency Domain
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Correspondence between Filtering in
the Spatial and Frequency Domain
Convolution Theorem:
The foundation for linear filtering in both the spatial and
frequency domains is the convolution theorem, which may
be written as:
where
F(u,v) andH(u,v) denote the Fourier transforms off(x,y) and h(x,y). * Indicates the convolution of the two functions.
The discrete convolution of two functionsf(x,y) and h(x,y) of sizeMN
is defined as:
1
0
1
0
),(),(1
),(),(M
m
N
n
nymxhnmf
MN
yxhyxf
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Correspondence between Filtering in
the Spatial and Frequency Domain
The third equivalence relation says that given a filter in the
frequency domain, we can obtain the corresponding filter in the
spatial domain by taking the inverse Fourier transform of the
former. The reverse is also true.
The convolution theorem links filters in both domains (spatial and
frequency). Filtering is often more intuitive in the frequency
domain, since one can easily foresee the results, given a filter.
However, filters in the spatial domain have smaller masks than
those in the frequency domain. That is, one can first specify a
filter in the frequency domain, take its inverse Fourier transform,
and then use the resulting filter in the spatial domain as a guide for
constructing smaller spatial filter masks!
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Spatial domain Frequency domain
Average filter Gaussian lowpass filter
Laplacian Gaussian highpass filter
The convolution theorem establishes a correspondence
between filtering in the spatial and frequency domains as
shown in the following table.
Correspondence between Filtering in
the Spatial and Frequency Domain
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Letf(x,y) and h(x,y) are of sizeA B and C D, resp., then wraparound
error can be avoided by choosing:
P A + C -1
Q B + D -1
Filtering in the Frequency DomainPadding
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Filtering in the Frequency DomainPadding
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Filtering in the Frequency DomainPadding Function
Exercises:
(1) [1, 4.3.1]
Write an M-function:
function PQ = paddedsize(AB, CD, PARAM)
and use it on figure squareBW.tif to visualize the
difference between images with padding and without
padding.
(2) [1, 4.3.3]
Write an M-function for filtering in the frequency
domain.
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Notch Filter: Force the average value of an image to zero.
Multiply all values ofF(u,v) by the filter function:
All this filter would do is setF(0,0) to zero and leave all
frequency components of the Fourier transform untouched.
otherwise.1
)2/,2/(),(if0),(
NMvuvuH
Filtering in the Frequency DomainSome Basic Filters & Their Properties
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Filtering in the Frequency DomainSome Basic Filters & Their Properties
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Filtering in the Frequency DomainSome Basic Filters & Their Properties
Lowpass filter is a filter that attenuates high frequencies
while passing' low frequencies. Low frequencies
correspond to the slowly varying components of an image
(e.g., uniform background).
Highpass filter is a filter that attenuates low frequencies
while passing high frequencies. High frequencies
correspond to the sharply varying components of an image(e.g., edges).
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Filtering in the Frequency DomainSome Basic Filters & Their Properties
Lowpass Filter
Highpass Filter
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Filtering in the Frequency DomainSome Basic Filters & Their Properties
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A 2-D ideal low-pass filter is one whose transfer
function satisfies the relation:
1 ifD(u, v) Do
whereDo is a non-negative quantity and D(u,v) is the distance from
the point (u,v) to the origin of the frequency plane:
Filtering in the Spatial and Frequency DomainIdeal Filters
22
22
),(
Nv
MuvuD
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H(u, v)
D(u, v)0
1
Do
Do
Remove (multiply by zero) all frequencies >Do
Ideal Low Pass Filter
Filtering in the Spatial and Frequency DomainIdeal Filters
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Filtering in the Spatial and Frequency DomainIdeal Filters
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Ideal low pass filter
Spatial domain image Filtered frequency domain imageFrequency domain image
ForwardFFT
Filtered spatial domain image
InverseFFT
Ideal Frequency
Domain Filtering
causes ringing
Filtering in the Spatial and Frequency DomainIdeal Filters
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Filtering in the Spatial and Frequency DomainIdeal Lowpass Filters (ILPF)
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Ideal Lowpass Filters
Digital Image Processing, 2nd ed.www.imageprocessingbook.com
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Spatial domain image Filtered frequency domain imageFrequency domain image
Forward
FFT
Filtered spatial domain image
Inverse
FFT
Ideal Frequency
Domain Filtering
causes ringing
Ideal high pass filter
Filtering in the Spatial and Frequency DomainIdeal Filters
Digital Image Processing, 2nd ed.www.imageprocessingbook.com
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2002 R. C. Gonzalez & R. E. Woods
Butterworth Lowpass Filters (BLPFs)With order n
nDvuDvuH
2
0/),(1
1),(
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Butterworth Lowpass
Filters (BLPFs)
n=2
D0=5,15,30,80,and 230
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Butterworth Lowpass Filters (BLPFs)
Spatial Representation
n=1 n=2 n=5 n=20
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 68
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Gaussian Lowpass Filters (GLPFs)
20
2 2/),(),( DvuDevuH
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 69
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Gaussian Lowpass
Filters (GLPFs)
D0=5,15,30,80,and 230
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 70
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Additional Examples of Lowpass Filtering
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 71
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Additional Examples of Lowpass Filtering
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 72
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2002 R. C. Gonzalez & R. E. Woods
Spatial domain image Filtered frequency domain imageFrequency domain image
Forward
FFT
Filtered spatial domain image
Inverse
FFT
Smoother Gaussian
Filtering in the
Frequency
Domain shows no
ringing
Filtering in the Spatial and Frequency DomainIdeal Filters
Gaussian high pass filter
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 73
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2002 R. C. Gonzalez & R. E. Woods
Sharpening Frequency Domain Filter
),(),( vuHvuH lphp Ideal highpass filter
Butterworth highpass filter
Gaussian highpass filter
),(if1
),(if0),(
0
0
DvuD
DvuDvuH
nvuDDvuH
2
0 ),(/1
1),(
20
2 2/),(1),( DvuDevuH
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 74
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Highpass Filters
Spatial Representations
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Ideal Highpass Filters
),(if1
),(if0),(
0
0
DvuD
DvuDvuH
Digital Image Processing, 2nd ed.www.imageprocessingbook.com 76
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Butterworth Highpass Filters
nvuDDvuH
2
0 ),(/1
1),(
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Gaussian Highpass Filters
20
2 2/),(1),( DvuDevuH