part i: image transforms digital image processing
TRANSCRIPT
![Page 1: Part I: Image Transforms DIGITAL IMAGE PROCESSING](https://reader031.vdocuments.net/reader031/viewer/2022012402/56649f005503460f94c162db/html5/thumbnails/1.jpg)
Part I: Image Transforms
DIGITAL IMAGE PROCESSING
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}10 ),({ Nxxf
1
0)(),()(
N
xxfxuTug
1-D SIGNAL TRANSFORMGENERAL FORM
10 Nu
fTg
• Scalar form
• Matrix form
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}10 ),({ Nxxf
1
0)(),()(
N
xxfxuTug
1-D SIGNAL TRANSFORM cont.REMEMBER THE 1-D DFT!!!
• General form
10 Nu
1
0
2
)(1
)(N
x
N
xuj
xfeN
ug
• DFT
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1-D INVERSE SIGNAL TRANSFORMGENERAL FORM
• Scalar form
1
0)(),()(
N
uuguxIxf
• Matrix form
gTf 1
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}10 ),({ Nxxf
1-D INVERSE SIGNAL TRANSFORM cont.REMEMBER THE 1-D DFT!!!
• General form
1
0
2
)()(N
x
N
xuj
ugexf
10 Nu
• DFT
1
0)(),()(
N
uuguxIxf
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1-D UNITARY TRANSFORM
fTg
• Matrix form
TTT 1
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SIGNAL RECONSTRUCTION
1
0)(),()(
N
u
TugxuTxfgTf
TTT 1
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IMAGE TRANSFORMS
• Many times, image processing tasks are best performed in a domain other than the spatial domain.
• Key steps:
(1) Transform the image
(2) Carry the task(s) in the transformed domain.
(3) Apply inverse transform to return to the spatial domain.
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2-D (IMAGE) TRANSFORMGENERAL FORM
1
0
1
0),(),,,(),(
N
x
N
yyxfyxvuTvug
1
0
1
0),(),,,(),(
N
u
N
vvugvuyxIyxf
![Page 10: Part I: Image Transforms DIGITAL IMAGE PROCESSING](https://reader031.vdocuments.net/reader031/viewer/2022012402/56649f005503460f94c162db/html5/thumbnails/10.jpg)
2-D IMAGE TRANSFORMSPECIFIC FORMS
• Separable
• Symmetric
),(),(),,,( 21 yvTxuTyxvuT
),(),(),,,( 11 yvTxuTyxvuT
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• Separable and Symmetric
• Separable, Symmetric and Unitary
TTfTg 11
11 TgTfT
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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
yvugyxf
22fg
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ENERGY COMPACTION
Most of the energy of the original data concentrated in only a few of the significant transform coefficients; remaining coefficients are near zero.
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Why is Fourier Transform Useful?
• Easier to remove undesirable frequencies.
• Faster to perform certain operations in the frequency domain than in the spatial domain.
• The transform is independent of the signal.
![Page 15: Part I: Image Transforms DIGITAL IMAGE PROCESSING](https://reader031.vdocuments.net/reader031/viewer/2022012402/56649f005503460f94c162db/html5/thumbnails/15.jpg)
ExampleRemoving undesirable frequencies
remove highfrequencies
reconstructedsignal
frequenciesnoisy signal
zero! to tscoeeficien
ingcorrespond
their set s,frequencie
certain remove To
)(uF
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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
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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
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Visualizing DFT
• Typically, we visualize
• The dynamic range of is typically very large
• Apply stretching:
( is constant)
before scaling after scalingoriginal image
),( vuF),( vuF
]),(1log[),( vuFcvuD c
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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 and Amplitude
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Rewrite as follows:
If we set:
Then:
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
),( vuF
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• How can we compute ?
• How can we compute ?
).( of rows of DFT
),(1
),(),(
1
0
/2
1
0
/2
yxfN
eyxfN
N
eyxfvxG
N
y
Nvyj
N
y
Nvyj
DFT PROPERTIES: SEPARABILITY),( vxG
),( vuF
),( of columns of DFT
),(1
),(1
0
/2
vxGM
evxGM
MvuFM
x
Muxj
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DFT PROPERTIES: SEPARABILITY
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DFT PROPERTIES: PERIODICITY
The DFT and its inverse are periodic with period N
),(),(
),(),(
NvMuFNvuF
vMuFvuF
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DFT PROPERTIES: SYMMETRY
If is real, then),( vuF
odd andimaginary odd and real
even and real even and real
),(),(
),(),(
),(),(),(),( *
vuFyxf
vuFyxf
vuFvuFvuFvuF
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• Translation in spatial domain:
• Translation in frequency domain:
DFT PROPERTIES: TRANSLATION
),(),( 00)//(2 00 vvuuFeyxf NyvMxuj
)//(200
00),(),( NyvMxujevuFyyxxf
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Warning: to show a full period, we need to translate the origin of the transform at
DFT PROPERTIES: TRANSLATION
2/Nu )2/,2/(),( NMvu
)(uF
)2
(N
uF
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DFT PROPERTIES: TRANSLATION
)2/,2/()1)(,(
),(),(
)1(
2/2/
)2/,2/(),(
)(
00)//(2
)()()//(2
00
00
00
nvMuFyxf
vvuuFeyxf
ee
NvMu
NMvuF
yx
NyvMxuj
yxyxjNyvMxuj
Using
case that In
and take
at move To
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DFT PROPERTIES: TRANSLATION
no translation after translation
)2/,2/()1)(,( )( nvMuFyxf yx
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DFT PROPERTIES: ROTATION
by rotates by rotating ),,(),( vuFyxf
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DFT PROPERTIESADDITION-MULTIPLICATION
transform Fourier the is ][ where
)],([)],([)],(),([
)],([)],([)],(),([
yxgyxfyxgyxf
yxgyxfyxgyxf
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DFT PROPERTIES: SCALE
transform Fourier the is ][ where
)],([)],([)],(),([
)],([)],([)],(),([
yxgyxfyxgyxf
yxgyxfyxgyxf
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DFT PROPERTIES: AVERAGE
)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
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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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Magnitude and Phase of DFT
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!)
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Magnitude and Phase of DFT
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Original Image-Fourier AmplitudeKeep Part of the Amplitude Around the Origin and Reconstruct Original
Image (LOW PASS filtering)
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Keep Part of the Amplitude Far from the Origin and Reconstruct Original Image (HIGH PASS filtering)
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Reconstruction from
phase of one image and amplitude of the other
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Reconstruction from
phase of one image and amplitude of the other