Digital Image Processing Week VI
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Transcript of Digital Image Processing Week VI
Digital Image Processing Week VI
Thurdsak LEAUHATONG
Image RestorationImage Restoration
• Manipulate the image so that it is suitable for a specific application.
• The Concept
•The Image Enhancement vs Restoration
Enhancement
Original Image Enhanced Image
• Restore a degraded image back to the original image.
• The Concept
•The Image Enhancement vs Restoration
Restoration
Degraded Image Original Image
• Model•The Image Degradation
,f x y
Degradation Fiilter
,H u v
,h x y
, ,F u v H u v
, ,f x y h x y
+
,
,
b x y
B u v
DegradedImage
OriginalImage
BlurredImage
AdditiveNoise
,g x y
, , , ,
, , , ,
g x y f x y h x y b x y
G u v F u v H u v B u v
• Model•The Image Degradation
Degraded Image ,g x y
, , , ,
, , ,
, , , ,
i j
g x y f x y h x y b x y
f x i y j h i j b x y
G u v F u v H u v B u v
•The Image Restoration Process
Restoration
Fiilter
DegradedImage
,g x y
,G u v
,w x y
,W u v
DegradedImage
ˆ , , ,f x y g x y w x y
ˆ , , ,F u v G u v W u v
• The Fourier transform of the additive white noise is constant and equal to .
• Frequency Property•Additive White Noise
0, ,2
FT Nb x y B u v
AdditiveWhiteNoise
0 / 2N
• Frequency Property•Additive White Noise
0
2
N
,B u v
0
2
N
,B u v
The ideal white noise contains all frequencies.
The practical white noise is a band-limited signal.
• Autocorrelation function of white noise
Property of Spatial Domain•Additive White Noise
AdditiveWhiteNoise
0
, , ,
, ,
,2
R x y b x y b x y
b x y b x x y y dx dy
Nx y
Property of Spatial Domain•Additive White Noise
0 ,2
Nx y
,R x y
If drops quickly with x and y, then changes quickly with position.
of additive white noise is impulse function, then the additive noise at a pixel is independent from the other pixels.
,R x y ,b x y
,R x y
• The additive white noise cannot be predicted but can be approximately described in statistical way using the probability density function (PDF).
•Statistic Model of Additive White Noise
1
if
0 otherwise
a z bp z b a
22,2 12
b aa b
Uniform
2 2/21
2z a bp z e
b 2,a b Gaussian
for
for
0 otherwise
a
b
P z a
p z P z b
b a
2 22
,a b
a b
aP bP
a P b P
Salt&Pepper
2 2ln /21
20
z a bp z ebz
z
2 2 2/2 2 2, 1a b b a be e e Lognormal
•Statistic Model of Additive White Noise
2 /2
0 z<a
z a bz a e z ap z b
2 4/ 4,
4
ba b
Rayleigh
0
0 0
azae zp z
z
22
1 1,
a a Exponential
1
1 !
0
b baza z
p z eb
z
22
,b b
a a Erlang
(Gamma)
•Statistic Model of Additive White Noise
•Statistic Model of Additive White Noise
Original image
Histogram
Degraded images
),(),(),( yxyxfyxg
•Statistic Model of Additive White Noise
Original image
Histogram
Degraded images
),(),(),( yxyxfyxg
Periodic Noise
Periodic noise looks like dotsIn the frequencydomain
Estimation of Noise
We cannot use the image histogram to estimate noise PDF.
It is better to use the histogram of one areaof an image that has constant intensity to estimate noise PDF.
Periodic Noise Reduction by Freq. Domain Filtering
Band reject filter Restored image
Degraded image DFT
Periodic noisecan be reduced bysetting frequencycomponentscorresponding to noise to zero.
Band Reject Filters
Use to eliminate frequency components in some bands
Periodic noise from theprevious slide that is Filtered out.
Notch Reject Filters A notch reject filter is used to eliminate some frequency components.
Notch Reject Filter:
Degraded image DFTNotch filter
(freq. Domain)
Restored imageNoise
Example: Image Degraded by Periodic Noise Degraded image
DFT(no shift)
Restored imageNoiseDFT of noise
Chapter 5Image Restoration
Chapter 5Image Restoration
Mean Filters
Arithmetic mean filter or moving average filter (from Chapter 3)
xySts
tsgmn
yxf),(
),(1
),(ˆ
Geometric mean filter
mn
Sts xy
tsgyxf
1
),(
),(),(ˆ
mn = size of moving window
Degradation model:
),(),(),(),( yxyxhyxfyxg
To remove this part
Geometric Mean Filter: Example
Original image
Image corrupted by AWGN
Image obtained
using a 3x3geometric mean filter
Image obtained
using a 3x3arithmetic mean filter
AWGN: Additive White Gaussian Noise
Harmonic and Contraharmonic Filters
Harmonic mean filter
xySts tsg
mnyxf
),( ),(1
),(ˆ
Contraharmonic mean filter
xy
xy
Sts
Q
Sts
Q
tsg
tsg
yxf
),(
),(
1
),(
),(
),(ˆ
mn = size of moving window
Works well for salt noisebut fails for pepper noise
Q = the filter order
Positive Q is suitable for eliminating pepper noise.Negative Q is suitable for eliminating salt noise.
Contraharmonic Filters: Example
Image corrupted by pepper noise with prob. = 0.1
Image corrupted
by salt noise with prob. = 0.1
Image obtained
using a 3x3contra-
harmonic mean filter
With Q = 1.5
Image obtained
using a 3x3contra-
harmonic mean filter
With Q=-1.5
Contraharmonic Filters: Incorrect Use Example
Image corrupted by pepper noise with prob. = 0.1
Image corrupted
by salt noise with prob. = 0.1
Image obtained
using a 3x3contra-
harmonic mean filter
With Q=-1.5
Image obtained
using a 3x3contra-
harmonic mean filterWith Q=1.5
Order-Statistic Filters: Revisit
subimage
Original image
Moving window
Statistic parametersMean, Median, Mode, Min, Max, Etc.
Output image
Order-Statistics FiltersMedian filter
),(median),(ˆ),(
tsgyxfxySts
Max filter
),(max),(ˆ),(
tsgyxfxySts
Min filter
),(min),(ˆ),(
tsgyxfxySts
Midpoint filter
),(min),(max2
1),(ˆ
),(),(tsgtsgyxf
xyxy StsSts
Reduce “dark” noise (pepper noise)
Reduce “bright” noise (salt noise)
Median Filter : How it works
A median filter is good for removing impulse, isolated noise
Degraded image
Salt noise
Pepper noise
Movingwindow
Sorted array
Salt noisePepper noise
Median
Filter output
Normally, impulse noise has high magnitude and is isolated. When we sort pixels in the moving window, noise pixels are usually at the ends of the array.
Median Filter : Example
Image corrupted
by salt-and-pepper
noise with pa=pb= 0.1
Images obtained using a 3x3 median filter
1
4
2
3
Max and Min Filters: Example
Image corrupted by pepper noise with prob. = 0.1
Image corrupted
by salt noise with prob. = 0.1
Image obtained
using a 3x3max filter
Image obtained
using a 3x3min filter
Adaptive Filter
-Filter behavior depends on statistical characteristics of local areas inside mxn moving window
- More complex but superior performance compared with “fixed” filters
Statistical characteristics:
General concept:
xySts
L tsgmn
m),(
),(1
Local mean:
Local variance:
xySts
LL mtsgmn ),(
22 )),((1
2
Noise variance:
Adaptive, Local Noise Reduction FilterPurpose: want to preserve edges
1. If sh2 is zero, No noise
the filter should return g(x,y) because g(x,y) = f(x,y)
2. If sL2 is high relative to sh
2, Edges (should be preserved), the filter should return the value close to g(x,y)
3. If sL2 = sh
2, Areas inside objectsthe filter should return the arithmetic mean value mL
LL
myxgyxgyxf ),(),(),(ˆ2
2
Formula:
Concept:
Adaptive Noise Reduction Filter: Example
Image corrupted by additiveGaussian noise with zero mean
and s2=1000
Imageobtained
using a 7x7arithmeticmean filter
Imageobtained
using a 7x7geometricmean filter
Imageobtained
using a 7x7adaptive
noise reduction
filter
Algorithm: Level A: A1= zmedian – zmin
A2= zmedian – zmax
If A1 > 0 and A2 < 0, goto level BElse increase window size
If window size <= Smax repeat level AElse return zxy
Level B: B1= zxy – zmin
B2= zxy – zmax
If B1 > 0 and B2 < 0, return zxy
Else return zmedian
Adaptive Median Filter
zmin = minimum gray level value in Sxy
zmax = maximum gray level value in Sxy
zmedian = median of gray levels in Sxy
zxy = gray level value at pixel (x,y)Smax = maximum allowed size of Sxy
where
Purpose: want to remove impulse noise while preserving edges
Level A: A1= zmedian – zmin
A2= zmedian – zmax
Else Window is not big enough increase window sizeIf window size <= Smax repeat level A
Else return zxy
zmedian is not an impulse
B1= zxy – zmin
B2= zxy – zmax
If B1 > 0 and B2 < 0, zxy is not an impulse return zxy to preserve original details
Else return zmedian to remove impulse
Adaptive Median Filter: How it works
If A1 > 0 and A2 < 0, goto level B
Level B:
Determine whether zmedian
is an impulse or not
Determine whether zxy
is an impulse or not
Adaptive Median Filter: Example
Image corrupted by salt-and-pepper
noise with pa=pb= 0.25
Image obtained using a 7x7median filter
Image obtained using an adaptivemedian filter with
Smax = 7
More small details are preserved