MATHEMATICAL MODEL OF IMAGE DEGRADATION. = s

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2 MATHEMATICAL MODEL OF IMAGE DEGRADATION H s u v G s u v F s u v ^ F u v G u v H s u v

3 Gaussian Kernel Source: C. Rasmussen

4 Gaussian filters pixel 5 pixels 0 pixels 30 pixels

5 Gaussian filter Removes high-frequency components from the image lo-pass filter Convolution ith self is another Gaussian * Convolving to times ith Gaussian kernel of idth convolving once ith kernel of idth Source: K. Grauman

6 Sharpening revisited What does blurring take aay? original Let s add it back: smoothed 5x5 detail + α original detail sharpened Source: S. Lazebnik

7 Sharpen filter image blurred image unit impulse identity scaled impulse Gaussian Laplacian of Gaussian

8 Sharpen filter unfiltered filtered

9 Convolution in the real orld Camera shake * Source: Fergus et al. Removing Camera Shake from a Single Photograph SIGGRAPH 006 Bokeh: Blur in out-of-focus regions of an image. Source:

10 Image Sharpening Idea: compute intensity differences in local image regions. Useful for emphasizing transitions in intensity e.g. in edge detection. st derivative of Gaussian

11 Sharpening Source: D. Loe

12 Filtering as matrix multiplication What kind of filter is this?

13 Multiplying ro and column vectors?

14 Filtering as matrix multiplication

15 A model of the image degradation/restoration process gxyfxy*hxy+ηxy GuvFuvHuv+Nuv

16 Histogram is an estimate of PDF Measure the mean and variance S z i i i z p z μ S z i i i z p z μ σ Gaussian: μ σ Uniform: a b

17 Additive noise only gxyfxy+ηxy GuvFuv+Nuv

18 Estimation by image observation Take a indo in the image Simple structure Strong signal content Estimate the original image in the indo H s u v G Fˆ s s u v u v knon estimate

19 Inverse filtering With the estimated degradation function Huv > GuvFuvHuv+Nuv ˆ G u v F u v F u v + H u v N u v H u v Unknon noise Estimate of original image Problem: 0 or small values Sol: limit the frequency around the origin

20 Atmospheric Turbulence Blur Obtain restoration as: F u v H u v G u v Minimize:

21 Modeling Blurring Process Linear degradation model xmn hmn + m n ymn h m n blurring filter m n ~ N0 σ additive hite Gaussian noise

22 The Curse of Noise xmn zmn hmn + ymn m n ~ N0 σ Blurring SNR BSNR 0 log 0 σ σ z

23 Blind vs. Nonblind Deblurring Blind deblurring deconvolution: blurring kernel hmn is unknon Nonblind deconvolution: blurring kernel hmn is knon In this course e only cover the nonblind case the easier case 3

24 Image Deblurring Introduction Inverse filtering Suffer from noise amplification Wiener filtering Tradeoff beteen image recovery and noise suppression Iterative deblurring* Landeber algorithm 4

25 5 Inverse Filter hmn blurring filter h I mn xmn ymn inverse filter k l I I combi n m n m l k h l n k m h n m h n m h n m h δ H H I To compensate the blurring e require h combi mn xmn ^

26 6 Inverse Filtering Con t hmn + xmn ymn n m h I mn inverse filter xmn ^ Spatial: ˆ n m h n m n m h n m x n m h n m y n m x I I + Frequency: ˆ H W X H W H X H Y X I + + amplified noise

27 7 Pseudo-inverse Filter Basic idea: To handle zeros in H e treat them separately hen performing the inverse filtering > δ δ 0 H H H H

28 Image Deblurring Introduction Inverse filtering Suffer from noise amplification Wiener filtering Tradeoff beteen image recovery and noise suppression Iterative deblurring* Landeber algorithm 8

Norbert Wiener 894-964 The renoned MIT professor Norbert Wiener as famed for his absent-mindedness. While crossing the MIT campus one day he as stopped by a student ith a mathematical problem.

29 Norbert Wiener The renoned MIT professor Norbert Wiener as famed for his absent-mindedness. While crossing the MIT campus one day he as stopped by a student ith a mathematical problem. The perplexing question ansered Norbert folloed ith one of his on: "In hich direction as I alking hen you stopped me?" he asked prompting an anser from the curious student. "Ah" Wiener declared "then I've had my lunch Anecdote of Norbert Wiener 9

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31 Wiener Filtering Also called Minimum Mean Square Error MMSE or Least-Square LS filtering H * H mmse H + K constant Example choice of K: σ K σ z K0 inverse filtering noise energy signal energy 3

32 3 Constrained Least Square Filtering * C H H H mmse γ + Similar to Wiener but a different ay of balancing the tradeoff beteen Example choice of C: n m C Laplacian operator γ0 inverse filtering

33 Image Deblurring Introduction Inverse filtering Suffer from noise amplification Wiener filtering Tradeoff beteen image recovery and noise suppression Iterative deblurring* Landeber algorithm 33

34 Method of Successive Substitution A poerful technique for finding the roots of any function fx Basic idea Rerite fx0 into an equivalent equation xgx x is called fixed point of gx Successive substitution: x i+ gx i Under certain condition the iteration ill converge to the desired solution 34

35 35 Numerical Example 3 + x x x f x x To roots: x g x x x x x f i i x x successive substitution:

36 Numerical Example Con t Note that iteration quickly converges to x 36

37 Landeber Iteration Linear blurring Y H X We ant to find the root of f X Y HX f X 0 X X + βf X X + β Y HX g X β relaxation parameter controls convergence property Successive substitution: X 0 0 X X + Y HX n+ n β n 37

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Computer Vision & Digital Image Processing

Computer Vision & Digital Image Processing Image Restoration and Reconstruction I Dr. D. J. Jackson Lecture 11-1 Image restoration Restoration is an objective process that attempts to recover an image