noise in imaging technology
DESCRIPTION
Noise in Imaging Technology. Two plagues in image acquisition Noise interference Blur (motion, out-of-focus, hazy weather) Difficult to obtain high-quality images as imaging goes Beyond visible spectrum Micro-scale (microscopic imaging) Macro-scale (astronomical imaging). What is Noise?. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/1.jpg)
1
Noise in Imaging Technology
Two plagues in image acquisition Noise interference Blur (motion, out-of-focus, hazy
weather) Difficult to obtain high-quality
images as imaging goes Beyond visible spectrum Micro-scale (microscopic imaging) Macro-scale (astronomical imaging)
![Page 2: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/2.jpg)
What is Noise? noise means any unwanted signal One person’s signal is another
one’s noise Noise is not always random and
randomness is an artificial term Noise is not always bad.
2
![Page 3: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/3.jpg)
Stochastic Resonance
3
no noise heavy noiselight noise
![Page 4: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/4.jpg)
4
Image Denoising Where does noise come from?
Sensor (e.g., thermal or electrical interference)
Environmental conditions (rain, snow etc.)
Why do we want to denoise? Visually unpleasant Bad for compression Bad for analysis
![Page 5: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/5.jpg)
5
electrical interference
Noisy Image Examples
thermal imaging
ultrasound imaging
physical interference
![Page 6: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/6.jpg)
6
Noise Removal Techniques
Linear filtering Nonlinear filtering
Recall
Linear system
![Page 7: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/7.jpg)
7
Image Denoising
Introduction Impulse noise removal
Median filtering Additive white Gaussian noise
removal 2D convolution and DFT
Periodic noise removal Band-rejection and Notch filter
![Page 8: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/8.jpg)
8
Impulse Noise (salt-pepper Noise)
Definition
Each pixel in an image has the probability of p/2 (0<p<1) being contaminated by either a white dot (salt) or a black dot (pepper)
WjHi
jiX
jiY
1,1
),(
0
255
),(
with probability of p/2
with probability of p/2
with probability of 1-p
noisy pixels
clean pixels
X: noise-free image, Y: noisy image
Note: in some applications, noisy pixels are not simply black or white, which makes the impulse noise removal problem more difficult
![Page 9: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/9.jpg)
9
Numerical Example
P=0.1
128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128
X
128 128 255 0 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 0 128 128 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 0 128 128 128 128 255 128 128 128 128 128 128 128 128 128 128 128 128 128 255 128 128 128 128 128 128 128 255 128 128
Y
Noise level p=0.1 means that approximately 10% of pixels are contaminated bysalt or pepper noise (highlighted by red color)
![Page 10: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/10.jpg)
10
MATLAB Command
>Y = IMNOISE(X,'salt & pepper',p)
Notes:
The intensity of input images is assumed to be normalized to [0,1].If X is double, you need to do normalization first, i.e., X=X/255;
The default value of p is 0.05 (i.e., 5 percent of pixels are contaminated)
imnoise function can produce other types of noise as well (you need to change the noise type ‘salt & pepper’)
![Page 11: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/11.jpg)
11
Impulse Noise Removal Problem
Noisy image Y
filteringalgorithm
Can we make the denoised image X as closeto the noise-free image X as possible?
^
X̂denoised
image
![Page 12: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/12.jpg)
12
Median Operator
Given a sequence of numbers {y1,…,yN} Mean: average of N numbers Min: minimum of N numbers Max: maximum of N numbers Median: half-way of N numbersExample ]56,55,54,255,52,0,50[y
54)( ymedian
]255,56,55,54,52,50,0[y
sorted
![Page 13: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/13.jpg)
13
y(n)
W=2T+1
)](),...,(),...,([)(ˆ TnynyTnymediannx
1D Median Filtering
… …
Note: median operator is nonlinear
MATLAB command: x=median(y(n-T:n+T));
![Page 14: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/14.jpg)
14
Numerical Example
]56,55,54,255,52,0,50[yT=1:
],56,55,54,255,52,0,50,[ 5650y
]55,55,54,54,52,50,50[ˆ x
BoundaryPadding
![Page 15: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/15.jpg)
15
x(m,n)
W: (2T+1)-by-(2T+1) window
2D Median Filtering
)],(),...,,(),...,,(
),...,,(),...,,([),(ˆ
TnTmyTnTmynmy
TnTmyTnTmymediannmx
MATLAB command: x=medfilt2(y,[2*T+1,2*T+1]);
![Page 16: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/16.jpg)
16
Numerical Example
225 225 225 226 226 226 226 226 225 225 255 226 226 226 225 226 226 226 225 226 0 226 226 255 255 226 225 0 226 226 226 226 225 255 0 225 226 226 226 255 255 225 224 226 226 0 225 226 226 225 225 226 255 226 226 228 226 226 225 226 226 226 226 226
0 225 225 226 226 226 226 226 225 225 226 226 226 226 226 226 225 226 226 226 226 226 226 226 226 226 225 225 226 226 226 226 225 225 225 225 226 226 226 226 225 225 225 226 226 226 226 226 225 225 225 226 226 226 226 226 226 226 226 226 226 226 226 226
Y X̂
Sorted: [0, 0, 0, 225, 225, 225, 226, 226, 226]
![Page 17: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/17.jpg)
17
Image Example
P=0.1
Noisy image Y X̂denoised
image3-by-3 window
![Page 18: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/18.jpg)
18
Idea of Improving Median Filtering
Can we get rid of impulse noise without affecting clean pixels? Yes, if we know where the clean pixels
areor equivalently where the noisy pixels
are How to detect noisy pixels?
They are black or white dots
![Page 19: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/19.jpg)
19
Median Filtering with Noise Detection
Noisy image Y
x=medfilt2(y,[2*T+1,2*T+1]);
Median filtering
Noise detection
C=(y==0)|(y==255);
xx=c.*x+(1-c).*y;
Obtain filtering results
![Page 20: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/20.jpg)
20
Image Example
cleannoisy(p=0.2)
w/onoisedetection
withnoisedetection
![Page 21: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/21.jpg)
21
Image Denoising
Introduction Impulse noise removal
Median filtering Additive white Gaussian noise
removal 2D convolution and DFT
Periodic noise removal Band-rejection and Notch filter
![Page 22: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/22.jpg)
22
Additive White Gaussian Noise
Definition
Each pixel in an image is disturbed by a Gaussian random variableWith zero mean and variance 2
WjHiNjiN
jiNjiXjiY
1,1),,0(~),(
),,(),(),(2
X: noise-free image, Y: noisy image
Note: unlike impulse noise situation, every pixel in the image contaminatedby AWGN is noisy
![Page 23: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/23.jpg)
23
Numerical Example
2 =1
X Y
128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128
128 128 129 127 129 126 126 128 126 128 128 129 129 128 128 127 128 128 128 129 129 127 127 128 128 129 127 126 129 129 129 128 127 127 128 127 129 127 129 128 129 130 127 129 127 129 130 128 129 128 129 128 128 128 129 129 128 128 130 129 128 127 127 126
![Page 24: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/24.jpg)
24
MATLAB Command
>Y = IMNOISE(X,’gaussian',m,v)
>Y = X+m+randn(size(X))*v;
or
Note:rand() generates random numbers uniformly distributed over [0,1]
randn() generates random numbers observing Gaussian distributionN(0,1)
![Page 25: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/25.jpg)
25
Image Denoising
Noisy image Y
filteringalgorithm
Question: Why not use median filtering?Hint: the noise type has changed.
X̂denoised
image
![Page 26: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/26.jpg)
26
f(n) h(n) g(n)
)()()()()()()( nhnfnfnhknfkhngk
- Linearity
- Time-invariant property
)()()()( 22112211 nganganfanfa
)()( 00 nngnnf
Linear convolution
1D Linear Filtering
See review section
![Page 27: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/27.jpg)
27
jwnenfwF )()(
dwewFnf jwn)(
2
1)(
forward
inverse
Note that the input signal is a discrete sequence while its FT is a continuous function
)()( nhnf time-domain convolution
)()( wHwF
frequency-domain multiplication
Fourier Series
![Page 28: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/28.jpg)
28
Filter Examples
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
LP HP
w
|H(w)|Low-pass (LP)
h(n)=[1,1]
|h(w)|=2cos(w/2)
High-pass (LP)
h(n)=[1,-1]
|h(w)|=2sin(w/2)
![Page 29: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/29.jpg)
29
2D Linear Filtering
f(m,n) h(m,n) g(m,n)
),(),(),(),(),(,
nmfnmhlnkmflkhnmglk
2D convolution
MATLAB function: C = CONV2(A, B)
![Page 30: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/30.jpg)
30
2D Filtering=Two Sequential 1D Filtering
Just as we have observed with 2D transform, 2D (separable) filtering can be viewed as two sequential 1D filtering operations: one along row direction and the other along column direction
The order of filtering does not matter
)()()()(),( 1111 mhnhnhmhnmh h1 : 1D filter
![Page 31: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/31.jpg)
31
Fourier Series (2D case)
m n
nwmwjenmfwwF )(21
21),(),(
Note that the input signal is discrete while its FT is a continuous function
),(),( nmhnmf spatial-domain convolution
),(),( 2121 wwHwwF
frequency-domain multiplication
![Page 32: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/32.jpg)
32
Filter Examples
Low-pass (LP)
h1(n)=[1,1]
|h1(w)|=2cos(w/2)
1D
h(n)=[1,1;1,1]
|h(w1,w2)|=4cos(w1/2)cos(w2/2)
2D
w1
w2
|h(w1,w2)|
![Page 33: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/33.jpg)
33
Image DFT Example
Original ray image X choice 1: Y=fft2(X)
![Page 34: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/34.jpg)
34
Image DFT Example (Con’t)
choice 1: Y=fft2(X) choice 2: Y=fftshift(fft2(X))Low-frequency at the centerLow-frequency at four corners
FFTSHIFT Shift zero-frequency component to center of spectrum.
![Page 35: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/35.jpg)
35
Gaussian Filter
)2
exp(),(2
22
21
21 ww
wwH
)2
exp(),(2
22
nm
nmh
FT
>h=fspecial(‘gaussian’, HSIZE,SIGMA);MATLAB code:
![Page 36: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/36.jpg)
36
(=1)PSNR=24.4dB
Image Example
PSNR=20.2dB
noisy
(=25)
denoised denoised
(=1.5)PSNR=22.8dB
Matlab functions: imfilter, filter2
![Page 37: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/37.jpg)
37
Gaussian Filter=Heat Diffusion
2
2
2
2 ),,(),,(),,(
),,(
y
tyxI
x
tyxItyxI
t
tyxI
Linear Heat Flow Equation:
)()0,,(),,( tGyxItyxI
scale A Gaussian filterwith zero mean and variance of t
Isotropic diffusion:
![Page 38: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/38.jpg)
38
Basic Idea of Nonlinear Diffusion*
x
y I(x,y)
image I
image I viewed as a 3D surface (x,y,I(x,y))
Diffusion should be anisotropicinstead of isotropic
![Page 39: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/39.jpg)
39
Image Denoising
Introduction Impulse noise removal
Median filtering Additive white Gaussian noise
removal 2D convolution and DFT
Periodic noise removal Band-rejection and Notch filter
![Page 40: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/40.jpg)
40
Periodic Noise Source: electrical or electromechanical
interference during image acquistion Characteristics
Spatially dependent Periodic – easy to observe in frequency
domain Processing method
Suppressing noise component in frequency domain
![Page 41: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/41.jpg)
41
Image Example
spatial
Frequency (note the four pairs of bright dots)
![Page 42: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/42.jpg)
42
Band Rejection Filter
otherwise
WDww
WDwwH
122
0),(22
21
21
w1
w2
![Page 43: Noise in Imaging Technology](https://reader036.vdocument.in/reader036/viewer/2022062304/56813dd0550346895da79956/html5/thumbnails/43.jpg)
43
Image Example
Before filtering After filtering