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Limitation of Imaging Technology

Limitation of 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).

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Limitation of Imaging Technology

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  1. Limitation of 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) EE465: Introduction to Digital Image Processing

  2. What is Noise? • Wiki definition: 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 (see stochastic resonance example in the next slide) EE465: Introduction to Digital Image Processing

  3. Stochastic Resonance no noise heavy noise light noise EE465: Introduction to Digital Image Processing

  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 EE465: Introduction to Digital Image Processing

  5. Noisy Image Examples thermal imaging electrical interference physical interference ultrasound imaging EE465: Introduction to Digital Image Processing

  6. (Ad-hoc) Noise Modeling • Simplified assumptions • Noise is independent of signal • Noise types • Independent of spatial location • Impulse noise • Additive white Gaussian noise • Spatially dependent • Periodic noise EE465: Introduction to Digital Image Processing

  7. Noise Removal Techniques Linear filtering Nonlinear filtering Recall Linear system EE465: Introduction to Digital Image Processing

  8. 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 EE465: Introduction to Digital Image Processing

  9. 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 EE465: Introduction to Digital Image Processing

  10. Numerical Example P=0.1 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 128 255 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 X Y Noise level p=0.1 means that approximately 10% of pixels are contaminated by salt or pepper noise (highlighted by red color) EE465: Introduction to Digital Image Processing

  11. 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; If X is uint8, MATLAB would do the normalization automatically  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’) EE465: Introduction to Digital Image Processing

  12. 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 EE465: Introduction to Digital Image Processing

  13. 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 EE465: Introduction to Digital Image Processing

  14. 1D Median Filtering y(n) … … W=2T+1 MATLAB command: x=median(y(n-T:n+T)); Note: median operator is nonlinear EE465: Introduction to Digital Image Processing

  15. Numerical Example T=1: Boundary Padding EE465: Introduction to Digital Image Processing

  16. 2D Median Filtering x(m,n) W: (2T+1)-by-(2T+1) window MATLAB command: x=medfilt2(y,[2*T+1,2*T+1]); EE465: Introduction to Digital Image Processing

  17. 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 ^ X Y Sorted: [0, 0, 0, 225, 225, 225, 226, 226, 226] EE465: Introduction to Digital Image Processing

  18. Image Example P=0.1 denoised image ^ Noisy image Y X 3-by-3 window EE465: Introduction to Digital Image Processing

  19. Image Example (Con’t) noisy (p=0.2) clean 3-by-3 window 5-by-5 window EE465: Introduction to Digital Image Processing

  20. Reflections • What is good about median operation? • Since we know impulse noise appears as black (minimum) or white (maximum) dots, taking median effectively suppresses the noise • What is bad about median operation? • It affects clean pixels as well • Noticeable edge blurring after median filtering EE465: Introduction to Digital Image Processing

  21. 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 EE465: Introduction to Digital Image Processing

  22. 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; EE465: Introduction to Digital Image Processing

  23. Image Example noisy (p=0.2) clean with noise detection w/o noise detection EE465: Introduction to Digital Image Processing

  24. 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 EE465: Introduction to Digital Image Processing

  25. 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 EE465: Introduction to Digital Image Processing

  26. Numerical Example 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 2=1 X Y EE465: Introduction to Digital Image Processing

  27. MATLAB Command >Y = IMNOISE(X,’gaussian',m,v) or >Y = X+m+randn(size(X))*v; rand() generates random numbers uniformly distributed over [0,1] Note: randn() generates random numbers observing Gaussian distribution N(0,1) EE465: Introduction to Digital Image Processing

  28. Image Denoising filtering algorithm denoised image ^ X Question: Why not use median filtering? Hint: the noise type has changed. Noisy image Y EE465: Introduction to Digital Image Processing

  29. 1D Linear Filtering g(n) h(n) f(n) See review section Linear convolution - Linearity - Time-invariant property EE465: Introduction to Digital Image Processing

  30. 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 EE465: Introduction to Digital Image Processing

  31. Filter Examples |H(w)| Low-pass (LP) h(n)=[1,1] HP LP |h(w)|=2cos(w/2) High-pass (LP) h(n)=[1,-1] w |h(w)|=2sin(w/2) EE465: Introduction to Digital Image Processing

  32. 1D Discrete Fourier Transform forward transform inverse transform • Properties - periodic Proof: - conjugate symmetric Proof: EE465: Introduction to Digital Image Processing

  33. Matrix Representation of 1D DFT Im Re DFT: EE465: Introduction to Digital Image Processing

  34. Fast Fourier Transform (FFT)* Invented by Tukey and Cooley in 1965 Basic idea: divide-and-conquer Reduce the complexity of N-point DFT from O(N2) to O(Nlog2N) N-point DFT N/2-point DFT N/2-point DFT EE465: Introduction to Digital Image Processing

  35. Filtering in the Frequency Domain H(k) g(n) F(k) G(k) h(n) f(n) DFT convolution in the time domain is equivalent to multiplication in the frequency domain EE465: Introduction to Digital Image Processing

  36. 2D Linear Filtering g(m,n) h(m,n) f(m,n) 2D convolution MATLAB function: C = CONV2(A, B) EE465: Introduction to Digital Image Processing

  37. 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 EE465: Introduction to Digital Image Processing

  38. Numerical Example h1(m)=[1,1], h1(n)=[1,-1] 1D filter MATLAB command: >h1=[1,1];h2=[1,-1]; >conv2(h1,h2) >conv2(h2,h1) EE465: Introduction to Digital Image Processing

  39. Fourier Series (2D case) spatial-domain convolution frequency-domain multiplication Note that the input signal is discrete while its FT is a continuous function EE465: Introduction to Digital Image Processing

  40. 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 EE465: Introduction to Digital Image Processing

  41. Image DFT Example choice 1: Y=fft2(X) Original ray image X EE465: Introduction to Digital Image Processing

  42. 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. EE465: Introduction to Digital Image Processing

  43. Gaussian Filter FT MATLAB code: >h=fspecial(‘gaussian’, HSIZE,SIGMA); EE465: Introduction to Digital Image Processing

  44. Image Example denoised noisy denoised PSNR=20.2dB PSNR=24.4dB PSNR=22.8dB (=1) (=1.5) (=25) Matlab functions: imfilter, filter2 EE465: Introduction to Digital Image Processing

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

  46. 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))

  47. Experimental Results noisy linear diffusion nonlinear diffusion PSNR=20.2dB PSNR=24.4dB PSNR=27.5dB (Gaussian filtering) (TV filtering) (=25) EE465: Introduction to Digital Image Processing

  48. Hammer-Nail Analogy salt-pepper/ impulse noise Gaussian filter Gaussian noise median filter periodic noise ??? EE465: Introduction to Digital Image Processing

  49. 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 EE465: Introduction to Digital Image Processing

  50. 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 EE465: Introduction to Digital Image Processing

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