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)

2. 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.

3. Stochastic Resonance
no noise
heavy noise
light noise

4. Image Denoising Where does noise come from? Why do we want to denoise?
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

5. Noisy Image Examples thermal imaging electrical interference
physical interference
ultrasound imaging

6. Noise Removal Techniques
Linear filtering
Nonlinear filtering
Recall
Linear system

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

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)
with probability of p/2
noisy pixels
with probability of p/2
clean pixels
with probability of 1-p
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

9. Numerical Example P=0.1 X Y X Y
Noise level p=0.1 means that approximately 10% of pixels are contaminated by
salt or pepper noise (highlighted by red color)

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’)

11. Impulse Noise Removal Problem
filtering
algorithm
denoised
image ^ X ^
Can we make the denoised image X as close
to the noise-free image X as possible?
Noisy image Y

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 numbers
Example
sorted

13. 1D Median Filtering y(n) W=2T+1 … …
MATLAB command: x=median(y(n-T:n+T));
Note: median operator is nonlinear

14. Numerical Example
T=1:
Boundary
Padding

15. 2D Median Filtering x(m,n) W: (2T+1)-by-(2T+1) window
MATLAB command: x=medfilt2(y,[2*T+1,2*T+1]);

16. Numerical Example ^ Y X
Sorted: [0, 0, 0, 225, 225, 225, 226, 226, 226]

17. Image Example
P=0.1
denoised
image ^
Noisy image Y X
3-by-3 window

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 are
or equivalently where the noisy pixels are
How to detect noisy pixels?
They are black or white dots

19. Median Filtering with Noise Detection
Noisy image Y
Median filtering
x=medfilt2(y,[2*T+1,2*T+1]);
Noise detection
C=(y==0)|(y==255);
Obtain filtering results
xx=c.*x+(1-c).*y;

20. Image Example noisy clean (p=0.2) with w/o noise noise detection

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

22. Additive White Gaussian Noise
Definition
Each pixel in an image is disturbed by a Gaussian random variable
With zero mean and variance 2
X: noise-free image, Y: noisy image
Note: unlike impulse noise situation, every pixel in the image contaminated
by AWGN is noisy

23. Numerical Example 2 =1 X Y 128 128 128 128 128 128 128 128
2 =1 X Y

24. MATLAB Command >Y = IMNOISE(X,’gaussian',m,v)or
>Y = X+m+randn(size(X))*v;
Note:
rand() generates random numbers uniformly distributed over [0,1]
randn() generates random numbers observing Gaussian distribution
N(0,1)

25. Image Denoising filtering denoised ^ algorithm image X
Question: Why not use median filtering?
Hint: the noise type has changed.
Noisy image Y

26. 1D Linear Filtering g(n) h(n) f(n) - Linearity
See review section
Linear convolution
- Linearity
- Time-invariant property

27. Fourier Series forward inverse time-domain convolution
frequency-domain multiplication
Note that the input signal is a discrete sequence
while its FT is a continuous function

28. Filter Examples |H(w)| HP LP w Low-pass (LP) h(n)=[1,1]
|h(w)|=2cos(w/2)
High-pass (LP)
h(n)=[1,-1] w
|h(w)|=2sin(w/2)

29. 2D Linear Filtering g(m,n) h(m,n) f(m,n) 2D convolution
MATLAB function: C = CONV2(A, B)

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
h1 : 1D filter

31. Fourier Series (2D case)
spatial-domain convolution
frequency-domain multiplication
Note that the input signal is discrete
while its FT is a continuous function

32. Filter Examples |h(w1,w2)| Low-pass (LP) h1(n)=[1,1] 1D
|h1(w)|=2cos(w/2)
h(n)=[1,1;1,1] 2D w2
|h(w1,w2)|=4cos(w1/2)cos(w2/2) w1

33. Image DFT Example
Original ray image X
choice 1: Y=fft2(X)

34. Image DFT Example (Con’t)
choice 1: Y=fft2(X)
choice 2: Y=fftshift(fft2(X))
Low-frequency at four corners
Low-frequency at the center
FFTSHIFT Shift zero-frequency component to center of spectrum.

35. Gaussian Filter FT MATLAB code:
>h=fspecial(‘gaussian’, HSIZE,SIGMA);

36. Image Example PSNR=20.2dB PSNR=24.4dB PSNR=22.8dB (=1) (=1.5) noisy
denoised
denoised
PSNR=20.2dB
PSNR=24.4dB
PSNR=22.8dB
(=25)
(=1)
(=1.5)
Matlab functions: imfilter, filter2

37. Gaussian Filter=Heat Diffusion
Linear Heat Flow Equation:
Isotropic diffusion:
A Gaussian filter
with zero mean
and variance of t
scale

38. Basic Idea of Nonlinear Diffusion*y
I(x,y) x
image I
Diffusion should be anisotropic
instead of isotropic
image I viewed as a 3D surface (x,y,I(x,y))

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

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

41. Image Example
spatial
Frequency (note the four pairs of bright dots)

42. Band Rejection Filter w2 w1

43. Image Example
Before filtering
After filtering