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Image Enhancement in the Frequency Domain
Jesus J. Caban
Outline
! Assignment #1
! Paper Presentation & Schedule
! Frequency Domain
! Mathematical Morphology
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Assignment #1 ! Questions?
! How’s OpenCV?
! You have another 48hrs. Keep working!
Paper Presentation & Schedule ! Check the schedule online
for the date you are presenting
! Check the reading list for your paper assignment
! If your name is not there, please let me know
! Let me know about errors
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Presentation: Evaluation Form 1. Delivery
2. Visual Aids & Organization
3. Topic & Papers
Possible change to the syllabus
! Instead of reading and writing a question for each paper, do you want to only read one paper?
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Image Enhancement in the Frequency Domain
Recall: Point Processing ! Based on the intensity of a single pixel only as opposed to a
neighborhood or region
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Moving Window Transform: Example
original 3x3 average
Moving Window Transform: Example
original 3x3 average
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Spatial Filtering: Smoothing
1/16 2/16 1/16
2/16 4/16
2/16
1/16 2/16
1/16
1/9 1/9 1/9
1/9 1/9
1/9
1/9 1/9
1/9
Spatial Filtering: Edge Enhancement
0 1 0
1 -4 1
0 1 0
-1 -2 -1
0 0 0
1 2 1
-1 0 1
-2 0 2
-1 0 1
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Jean Baptiste Joseph Fourier ! Claim (1807): Any periodic function can be rewritten as a weighted sum of sines and cosines of different frequencies
! Published his work in 1822: “La Théorie Analitique de la Chaleur”
! His work was translated into English over 55+ years later (1878): “The Analytic Theory of Heat”
! Researchers didn’t pay too much attention to his work when it was first published
! Fourier Theory then became one of the most important mathematical theories in modern engineering
The General Idea
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• 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 • Resulting collection is called Fourier series
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Frequency
Frequency is the number of occurrences of a repeating event per unit time
t0 t1 t2
Amplitude
The amplitude is the height of the wave.
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Phase shifts
The same wave after a central section underwent a phase shift (e.g. passing through glass)
The phase shift describes how far to the left or right the wave slides.
Frequency Example ! g(t) = sin(2!f t) + (1/3)sin(2! (3f) t)
= +
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Fourier Series - Example
First 60 Fourier approximations
Fourier Transform: Definition
!
! Changes in u and v obtain different basis functions of different frequencies
! Point in the new image F(u,v) represent the contribution of that frequency to the original image
Sum of sine and cosine basis functions
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Fourier Transform: Real & Imaginary Part ! General concept
f(x) F(w) Fourier Transform
• For every w from 0 to inf, • F(!) holds the amplitude A and phase " of the corresponding sine
2D Domain ! Any image can be captured in a single Fourier term that
encodes ! the spatial frequency ! the magnitude (positive or negative) ! the phase
x
y
u
v
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Frequency Domain: Illustration
Frequency Domain: Illustration
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Extension to 2D
(a-jb)
(a+jb) v
u
Frequency Domain: Properties
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Properties of the Fourier Transform ! Rotation: rotated image is equivalent to rotated DFT
! Scaling: a*f(x,y) is equivalent to a*F(u,v)
! F(0,0) = average value of f(x,y)
! Others: symmetry, distributive, etc…
The Discrete Fourier Transform (DFT) 1. Given an image f(x,y) or size MxN
x = 0, 1, 2…M-1 y = 0,1,2…N-1
2. Discrete Fourier Transform of f(x, y) is given by
u = 0, 1, 2…M-1 v = 0, 1, 2…N-1 j= sqrt(-1)
N
M
Origin
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The Inverse DFT ! The key feature of a Fourier Transform is that it’s completely reversible
! The inverse DFT is given by:
x = 0, 1, 2…M-1
y = 0, 1, 2…N-1
The Inverse DFT
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! Concept of convolution
! Filter image by moving the convolution kernel.
Recall: Basics of Spatial Filtering
Convolution Theorem
! Convolution in the spatial domain is equivalent to multiplication in the frequency domain
! Convolution of large kernels to images can be performed by a simple multiplication
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Recall: Smoothing Spatial Filtering 1/9
1/9 1/9
1/9 1/9! 1/9
1/9 1/9
1/9
Origin x
y Image f (x, y)
e = 1/9*106 + 1/9*104 + 1/9*100 + 1/9*108 + 1/9*99 + 1/9*98 + 1/9*95 + 1/9*90 + 1/9*85
= 98.3333
Filter Simple 3*3
Neighbourhood 106
104
99
95
100 108
98
90 85
1/9 1/9
1/9
1/9 1/9
1/9
1/9 1/9
1/9
3*3 Smoothing Filter
104 100 108
99 106 98
95 90 85
Original Image Pixels
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The DFT and Image Processing To filter an image in the frequency domain:
1. Compute F(u,v) the DFT of the image 2. Multiply F(u,v) by a filter function H(u,v) 3. Compute the inverse DFT of the result
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Frequency Domain: Filters
1. A low-pass filter: attenuates high frequencies
2. A high-pass filter: attenuates low frequencies
3. A band-pass filter: attenuates very low and very high frequencies
1. Low-pass Filters
! Attenuates high frequencies and retains low frequencies unchanged. ! high frequencies correspond to sharp intensity changes
! The result in the spatial domain is equivalent to that of a smoothing filter ! fine-scale details and noise in the spatial domain image are
filtered
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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
The transfer function for the ideal low pass filter can be given as:
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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
Ideal Low Pass Filter (cont…)
Original image
Result of filtering with ideal low pass filter of radius 5
Result of filtering with ideal low pass filter of radius 30
Result of filtering with ideal low pass filter of radius 230
Result of filtering with ideal low pass
filter of radius 80
Result of filtering with ideal low pass
filter of radius 15
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Ideal Low Pass Filter (cont…)
Result of filtering with ideal low pass filter of radius 5
Ideal Low Pass Filter (cont…)
Result of filtering with ideal low pass filter of radius 15
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Butterworth Lowpass Filters The transfer function of a Butterworth lowpass filter of order n with cutoff frequency at distance D0 from the origin is defined as:
Butterworth Lowpass Filter (cont…)
Original image
Result of filtering with Butterworth filter of order 2 and cutoff radius 5
Result of filtering with Butterworth filter of order 2 and cutoff radius 30
Result of filtering with Butterworth filter of order 2 and cutoff radius 230
Result of filtering with Butterworth filter of
order 2 and cutoff radius 80
Result of filtering with Butterworth filter of
order 2 and cutoff radius 15
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Gaussian Lowpass Filters The transfer function of a Gaussian lowpass filter is defined as:
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Gaussian Lowpass Filters (cont…)
Original image
Result of filtering with Gaussian filter with cutoff radius 5
Result of filtering with Gaussian filter with cutoff radius 30
Result of filtering with Gaussian filter with cutoff radius 230
Result of filtering with Gaussian filter
with cutoff radius 85
Result of filtering with Gaussian filter
with cutoff radius 15
Lowpass Filters Compared
Result of filtering with ideal low pass
filter of radius 15
Result of filtering with Butterworth filter of order 2 and cutoff radius 15
Result of filtering with Gaussian filter
with cutoff radius 15
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Lowpass Filtering Examples A low pass Gaussian filter is used to connect broken text
Lowpass Filtering Examples
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Lowpass Filtering Examples
Different lowpass Gaussian filters used to remove blemishes in a photograph
1. High-pass Filters
! Attenuates high frequencies and retains low frequencies unchanged. ! high frequencies correspond to sharp intensity changes
! The result in the spatial domain is equivalent to that of a smoothing filter ! fine-scale details and noise in the spatial domain image are
filtered
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High-pass Filters • A highpass filter yields edge enhancement or edge detection
in the spatial domain
• Edges contain mostly high frequencies while other areas of the image are rather constant gray level (i.e. low frequencies) which are suppressed.
Ideal High Pass Filters The ideal high pass filter is given as:
where D0 is the cut off distance as before
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Ideal High Pass Filters (cont…)
Results of ideal high pass filtering with D0
= 15
Results of ideal high pass filtering with D0
= 30
Results of ideal high pass filtering with D0
= 80
Butterworth High Pass Filters The Butterworth high pass filter is given as:
where n is the order and D0 is the cut off distance as before
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Butterworth High Pass Filters (cont…)
Results of Butterworth
high pass filtering of
order 2 with D0 = 15
Results of Butterworth high pass filtering of order 2 with D0 = 80
Results of Butterworth high pass filtering of order 2 with D0 = 30
Gaussian High Pass Filters The Gaussian high pass filter is given as:
where D0 is the cut off distance as before
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Gaussian High Pass Filters (cont…)
Results of Gaussian high pass filtering
with D0 = 15
Results of Gaussian high pass filtering with D0 = 80
Results of Gaussian high pass filtering with D0 = 30
Highpass Filter Comparison
Results of ideal high pass filtering with D0 = 15
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Highpass Filter Comparison
Results of Butterworth high pass filtering of order 2 with D0 = 15
Highpass Filter Comparison
Results of Gaussian high pass filtering with D0 = 15
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Highpass Filter Comparison
Results of ideal high pass filtering with D0
= 15
Results of Gaussian high pass filtering with D0 = 15
Results of Butterworth high pass filtering of order 2 with
D0 = 15
Laplacian In The Frequency Domain
Lapl
acia
n in
the
freq
uenc
y do
mai
n 2-D
image of Laplacian in the
frequency domain
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Frequency Domain Laplacian Example
Original image Laplacian filtered image
Laplacian image scaled
Enhanced image
Some Basic Frequency Domain Filters Low Pass Filter
High Pass Filter
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3. Band-pass Filters • A bandpass filter attenuates very low and very high
frequencies, but retains a middle range band of frequencies.
• Bandpass filtering is used to enhance edges (suppressing low frequencies) while reducing the noise at the same time (attenuating high frequencies).
Band-pass Filters: Example
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Band-pass Filters: Example
Take-home message ! Frequency Domain (Properties)
! Image filtering in Fourier Domain
! Filters: ! Low-pass ! High-pass ! Band-pass
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Basic of filtering: Frequency Domain
! How to filter in the frequency domain: 1. Multiply the input image by (-1)x+y to center the transform 2. Compute F(u,v) (The DFT of the image) 3. Multiply F(u,v) by a filter function H(u,v) 4. Compute the invert DFT of the resulting image 5. Get the real part of the complex image 6. Multiply the resulting image by (-1)x+y
Fast Fourier Transform ! The reason that Fourier based techniques have become so popular is the development of the Fast Fourier Transform (FFT) algorithm ! Allows the Fourier transform to be carried out in a reasonable amount of time ! Reduces the amount of time required to perform a Fourier transform by a factor of 100 – 600 times!
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Summary In this lecture we examined image enhancement in the frequency domain
! The Fourier series & the Fourier transform ! Image Processing in the frequency domain
! Image smoothing ! Image sharpening
! Fast Fourier Transform
Acknowledgements ! Some of the images and diagrams have been taken from the
Gonzalez et al, “Digital Image Processing” book.
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