Digital Image Procesing - Imperial College Londontania/teaching/DIP 2014/DFT.pdf · Digital Image...
Transcript of Digital Image Procesing - Imperial College Londontania/teaching/DIP 2014/DFT.pdf · Digital Image...
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Digital Image Procesing
Unitary Transforms
Discrete Fourier Trasform (DFT) in Image Processing
DR TANIA STATHAKI READER (ASSOCIATE PROFFESOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON
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}10 ),({ Nxxf
1
0
)(),()(N
x
xfxuTug
1-D Signal Transforms
10 Nu
fTg
Scalar form
Matrix form
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}10 ),({ Nxxf
1
0
)(),()(N
x
xfxuTug
1-D Signal Transforms-Remember the 1-D DFT
General form
DFT
10 Nu
1
0
2
)(1
)(N
x
N
xuj
xfeN
ug
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1-D Inverse Signal Transforms-General Form
Scalar form
Matrix form
1
0
)(),()(N
u
uguxIxf
gTf 1
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}10 ),({ Nxxf
1-D Inverse Signal Transforms-Remember the 1D DFT
General form
Inverse DFT
1
0
2
)()(N
x
N
xuj
ugexf
10 Nu
1
0
)(),()(N
u
uguxIxf
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1-D Unitary Transforms
fTg
Matrix form
T
TT
1
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Signal Reconstruction
1
0
)(),()(N
u
T
ugxuTxfgTf
T
TT
1
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Why do we use Image Transforms?
Often, image processing tasks are best performed in a
domain other than the spatial domain.
Key steps:
• Transform the image
• Carry the task(s) of interest in the transformed domain.
• Apply inverse transform to return to the spatial domain.
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2-D (Image) Transforms-General Form
1
0
1
0
),(),,,(),(N
x
N
y
yxfyxvuTvug
1
0
1
0
),(),,,(),(N
u
N
v
vugvuyxIyxf
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2-D (Image) Transforms-Special Cases
Separable
Symmetric
),(),(),,,( 21 yvTxuTyxvuT
),(),(),,,( 11 yvTxuTyxvuT
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Separable and Symmetric
Separable, Symmetric and Unitary
TTfTg 11
11 TgTf
T
2-D (Image) Transforms-Special Cases (cont.)
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Energy Preservation
1-D
2-D
1
0
1
0
21
0
1
0
2),(),(
M
u
N
v
M
x
N
y
vugyxf
22fg
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Energy Compaction
• Most of the energy of the original data is
concentrated in only a few transform
coefficients, which are placed close to the
origin; remaining coefficients have small
values.
• This property facilitate compression of the
original image.
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Let’s talk about DFT in images: Why is it useful?
• It is easier for removing undesirable
frequencies.
• It is faster to perform certain operations in
the frequency domain than in the spatial
domain.
• The transform is independent of the signal.
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Example: Removing undesirable frequencies
Result after removing high frequencies
frequencies noisy signal Example of
removing a high
frequency using
the transform
domain
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How do frequencies show up in an image?
• Low frequencies correspond to slowly
varying information (e.g., continuous
surface).
• High frequencies correspond to quickly
varying information (e.g., edges)
Original Image Low-passed version
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2-D Discrete Fourier Transform
1
0
1
0
)//(2),(),(M
x
N
y
NvyMuxjeyxfvuF
1
0
1
0
)//(2),(1
),(M
u
N
v
NvyMuxjevuFMN
yxf
• It is separable, symmetric and unitary
• It results in a sequence of two 1-D DFT
operations (prove this)
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Visualizing DFT
• The dynamic range of F(u,v) is typically very large
• Small values are not distinguishable
• We apply a logarithmic transformation to enhance
small values.
original image before transformation after transformation
constant a is ],),(1log[),( cvuFcvuD
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Amplitude and Log of the Amplitude
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Amplitude and Log of the Amplitude
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Original Image and Log of the Amplitude
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1
0
/2
1
0
/2
1
0
1
0
/2/2
),(),(
),(),(
),(),(
M
x
Muxj
N
y
Nvyj
M
x
N
y
NvyjMuxj
vxGevuF
eyxfvxG
eyxfevuF
DFT properties: Separability
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).( of rows of DFT
),(1
),(),(
1
0
/2
1
0
/2
yxfN
eyxfN
N
eyxfvxG
N
y
Nvyj
N
y
Nvyj
),( of columns of DFT
),(1
),(1
0
/2
vxGM
evxGM
MvuFM
x
Muxj
DFT properties: Separability
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DFT properties: Separability
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The DFT and its inverse are periodic.
),(),(),(),( NvMuFNvuFvMuFvuF
DFT properties: Separability
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odd andimaginary ),( odd and real ),(
even and real ),( even and real ),(
),(),(
integersany , with ),,(),( *
vuFyxf
vuFyxf
vuFvuF
qpqNvpMuFvuF
DFT properties: Conjugate Symmetry
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Translation in spatial domain:
Translation in frequency domain:
),(),( 00)//(2 00 vvuuFeyxf
NyvMxuj
)//(200
00),(),(NyvMxuj
evuFyyxxf
DFT properties: Translation
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Warning: to show a full period, we need to translate
the origin of the transform at )2/,2/(),( NMvu
)(uF
)2
(N
uF
DFT properties: Translation
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)2/,2/()1)(,(
),(),(
Using
)1(
caseIn that
2/ and 2/ replace
)2/,2/(at ),( move To
)(
00)//(2
)()()//(2
00
00
00
nvMuFyxf
vvuuFeyxf
ee
NvMu
NMvuF
yx
NyvMxuj
yxyxjNyvMxuj
DFT properties: Translation
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without translation after translation
)2/,2/()1)(,( )( nvMuFyxf yx
DFT properties: Translation
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by rotates by rotating ),,(),( vuFyxf
DFT properties: Rotation
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transform Fourier the is ][ where
)],([)],([)],(),([
)],([)],([)],(),([
yxgyxfyxgyxf
yxgyxfyxgyxf
DFT properties: Addition and Multiplication
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)0,0(1
),(
),()0,0(
),(1
),(
1
0
1
0
1
0
1
0
FMN
yxf
yxfF
yxfMN
yxf
M
x
N
y
M
x
N
y
:image the of value Average
DFT properties: Average value of the signal
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Original Image Fourier Amplitude Fourier Phase
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Magnitude and Phase of DFT
What is more important?
Hint: use inverse DFT to reconstruct the image using magnitude or phase only information
magnitude phase
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Reconstructed image using
magnitude only
(i.e., magnitude determines the
contribution of each component!)
Reconstructed image using
phase only
(i.e., phase determines
which components are present!)
Magnitude and Phase of DFT
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Magnitude and Phase of DFT
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Low pass filtering using DFT
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High pass filtering using DFT
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Experiment: Verify the importance of phase in images
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Reconstruction from
phase of one image and amplitude of the other
phase of cameraman phase of grasshopper
amplitude of grasshopper amplitude of cameraman
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Experiment: Verify the importance of phase in images
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Reconstruction from
phase of one image and amplitude of the other
phase of buffalo phase of rocks
amplitude of rocks amplitude of buffalo
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DFT of a single edge
• Consider DFT of image with single edge.
• For display, DC component is shifted to the centre.
• Log of magnitude of Fourier Transform is displayed
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DFT of a box
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DFT of rotated box
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DFT computation: Extended image
• DFT computation assumes image repeated
horizontally and vertically.
• Large intensity transition at edges = vertical and
horizontal line in middle of spectrum after shifting.
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Windowing
• Can multiply image by windowing function before
DFT to reduce sharp transitions between borders of
repeated images.
• Ideally, causes image to drop off towards ends,
reducing transitions