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Image Restoration

Image Restoration. Image restoration vs. image enhancement Enhancement : largely a subjective process Priori knowledge about the degradation is not a must (sometimes no degradation is involved) Procedures are heuristic and take advantage of the psychophysical aspects of human visual system

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Image Restoration

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  1. Image Restoration • Image restoration vs. image enhancement • Enhancement: • largely a subjective process • Priori knowledge about the degradation is not a must (sometimes no degradation is involved) • Procedures are heuristic and take advantage of the psychophysical aspects of human visual system • Restoration: • more an objective process • Images are degraded • Tries to recover the images by using the knowledge about the degradation

  2. 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)

  3. 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)

  4. Stochastic Resonance no noise heavy noise light noise

  5. 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

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

  7. (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

  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

  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

  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)

  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

  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

  15. 2D Median Filtering x(m,n) W: (2T+1)-by-(2T+1) window

  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 ^ X Y 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. Image Example (Con’t) noisy (p=0.2) clean 3-by-3 window 5-by-5 window

  19. 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

  20. 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

  21. 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;

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

  23. Median Filter size =7 x 7 Mean Filter size =7 x 7

  24. The maximum filterselects the largest value within of pixel values, whereas the minimum filterselects the smallest value. Minimum filtering causes the darker regions of an image to swell in size and dominate the lighter regions ( mask size =7 x 7) Minimum filtering ( mask size =3 x 3)

  25. Result from Maximum filtering with mask (7 x 7) Result from Maximum filtering with mask (3 x 3)

  26. Alpha-Trimmed Mean Filter Alpha-Trimmed Mean Filter: We can delete the d/2 lowest and d/2 highest grey levels So gr(s, t) represents the remaining mn – d pixels

  27. 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

  28. 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

  29. 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

  30. 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

  31. 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

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

  33. 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

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

  35. 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.

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

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

  38. Hammer-Nail Analogy salt-pepper/ impulse noise Gaussian filter Gaussian noise median filter periodic noise ???

  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 EE465: Introduction to Digital Image Processing

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

  42. Band Rejection Filter w2 w1

  43. Image Example After filtering Before filtering

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