Image Degradation Model (Linear/Additive)
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1 Image Degradation Model (Linear/Additive),,,,,,,, g x y f x y h x y x y G u v F u v H u v N u v 1
2 Source of noise Objects Impurities Image acquisition (digitization) Image transmission Spatial properties of noise Statistical behavior of the gray-level values of pixels Noise parameters, correlation with the image Frequency properties of noise Fourier spectrum White noise (a constant Fourier spectrum) 2
3 Some Noises Distribution: 3
4 Test Pattern Histogram has three spikes (Impulse) For two independent random variables (x, y) z x y p z p x p y Z X Y 4
5 Noisy Images(1): 5
6 Noisy Images (2): 6
7 Periodic 1 Noise: Electronic Devices 1 Disturbance 7
8 Estimation of Noise Parameters Periodic noise Observe the frequency spectrum Random noise with PDFs Case 1: Imaging system is available Capture image of flat environment Case 2: Single noisy image is available Take a strip from constant area Draw the histogram and observe it Estimate PDF parameters or measure the mean and variance 8
9 Noise Estimation: Shape: Histogram of a subimage (Background) 9
10 Noise Estimation: Parameters: N S N Maximum Likelihood me tho d: zi arg max p z, i 1 i i1 Momentum z p z 2 z S i i i 2 z p z i i zis Solve? Power Spectrum Estimation: 2 PDF Parameters N 1 i 2 2 N u, v E x, y x, y N i i1 xy, : Flat (No object) Subimages 10
11 Noise-only spatial filter: Mean Filters:,,,,,, g x y f x y x y G u v F u v N u v A m n mask centered at (x, y): S xy An algebraic operation on the mask pixels! 11
12 Mean Filter: Arithmetic mean: 1 fˆ x, y g s, t Noise Reduction (uncorrelated, zero mean) vs. Blurring Geometric mean: mn ( s, t) S xy Same smoothing with less detail degradation: fˆ x, y g s, t ( s, t) S xy 1 mn 12
13 Mean Filter: Harmonic mean: Salt Reduction (Pepper will increase)/ Good for Gaussian like Contra-harmonic mean: fˆ x, y fˆ x, y Q>0 for pepper and Q<0 for salt noise ( s, t) S xy ( s, t) S xy mn 1 g s, t ( s, t) S xy g s, t g s, t Q1 Q 13
14 Example (1): Additive Noise Mean Arithmetic Geometric Original Noisy More-Less Blurring Arithmetic Geometric 14
15 Example (2): Noisy (Pepper) Noisy (Salt) Impulsive Contra-harmonic Correct use Q=+1.5 Q=
16 Example (3): Impulsive Noise Conrtra-harmonic Wrong use Q=-1.5 Q=
17 Noise-only spatial filter: Mean Filters:,,, g x y f x y x y A m n mask centered at (x, y): S xy An Ordering/Ranking operation on mask members! 17
18 Median: Most Familiar OS filter: fˆ x, y mediang s, t Little Blurring/Single/Double valued noises Max/min: ( s, t) S xy ˆ fˆ x, y Max g s, t, f x, y min g s, t ( s, t) S xy ( s, t) S xy Find Brightest/Darkest Points (Pepper/Salt Reducer) 18
19 Midpoint: OS+Mean Properties (Gaussian, Uniform) 1 fˆ x, y, min, 2 Max g s t g s t ( s, t) S xy ( s, t) S xy 19
20 Alpha Trimmed mean: Delete (d/2) lowest and highest samples d = 0 Mean Filter ˆ 1 f x, y g s, t mn d ( s, t) S d = mn-1 Median Filter Others: Combinational Properties xy r 20
21 Example (1): Noisy (Salt & Pepper) One Pass Impulsive Noise Median Filter passes Less Blurring Two Pass Three Pass 21
22 Example (2): Max and min Filters: 3 3 Max Filter 3 3 min Filter 22
23 Example (3): Noise: Uniform (a) and Impulsive (b) Mask size: 5 5 Filters Arithmetic mean (c) Geometric mean (d) Median (e) Alpha Trimmed (f) 23
24 Adaptive, local noise reduction: 2 If is small, return g(x, y) 2 2 If L,return value close to g(x, y) 2 2 If L,return the arithmetic mean m L 2 Non-stationary noise or poor estimation: 2 L fˆ x, y g x, y g x, y ml x, y xy, 2 max 1 2 L xy, 24
25 Example: N(0,100) Filters: Arithmetic Mean Geometric Mean Local Original Noisy A. Mean G- Mean Local 25
26 Adaptive Median: zmed Median of intensity in Sxy A1 Zmed Zmin, A2 Zmed Zmax zxy Intensity value at ( x, y) if A1 0 and A2< 0 then SMax Maximum allowed size of B1 Z xy Zmin, B2 Z xy Zmax if B1 0 and B2< 0 then Zˆ else ˆ xy Z xy Z xy Zmed else increase Mask size: ( Smax ): Check " A" condition. ( S ˆ max ): Z xy Zm ed z z min max Minimum intensity in S Maximum intensity in S xy xy S xy 26
27 Example: Noise: Salt and Pepper (Pa=Pb=0.25) Noisy Median (7 7) Adaptive Median (Smax=7) 27
28 Band Reject Filter (Periodic Noises) Ideal, Butterworth, and Gaussian IBRF BBRF(1) GBRF 28
29 Band Reject Filter (Periodic Noises) W 1 Du, v D0 2 W W H I u, v 0 D0 D u, v D0 2 2 W 1 Du, v D0 2 1 H B u, v n 2n Du, vw D u, v D0 H G D u, v D 0 u, v 1 exp 2 Du, vw 2 29
30 Example: Noisy Spectrum BBRF(1) DeNoised 30
31 Band Pass Filter (Extract Periodic Noises) Analysis of Spectrum in different bands H u, v 1 H u, v Noise Pattern: bp br 31
32 Notch Reject Filter: INRF GNRF BNRF NPF=1-NRF 32
33 Notch Filter (Single Frequency Removal):, 2 2, 2 2, u, v 2 2 D u v u M u v N v D u v u M u v N v H H H I B G u, v u, v 0 D u, v D and D u, v D 1 O.W., u, v D0 1 D u v D D u v D u, v 1 exp D0 33
34 Example Horizontal Pattern in space Vertical Pattern in Frequency Spectrum Notch in Vertical lines Notch Passed Spectrum Vertical Notch Rejected Spectrum 34
35 Several Single Frequency Noises: Images Spectrum 35
36 Multiple Notch will disturb the image: Extract interference pattern using Notch Pass 1 N( u, v) H NP ( u, v) G( u, v) x, y H NP( u, v) G( u, v) Now Perform Filter Function in Spatial Domain as An Adaptive-Local Point Process: f ˆ x, y g x, y w x, y x, y Optimization Criteria: minimum variance of filtered images min 2ˆ f 36
37 Optimum Notch Filter: 2 fˆ 2 d d 1 x, y ˆ, ˆ f x s y t f x, y 2d 1 sd td 2 fˆ 2 d d ˆ 1 f x, y ˆ 2 fxs, y t 2d 1 sd td d d 1 x, y g x s, y t w x s, y t x s, y t 2d 1 sd td Smoothness of w( x, y):,,, g x y w x y x y w x s, y t w x, y, w x, y x, y w x, y x, y
38 Optimum Notch Filter: d d 1 x, y g x s, y t wx, yx s, y t 2d 1 sd td 2 fˆ 2,,, g x y w x y x y ˆ xy, f g x, y x, y g x, y x, y 0 w x, y w x, y x, y x, y Computed for each pixel using surrounded mask 2 38
39 Result: Former images Spectrum (without fftshift) 39
40 Result: N(u,v) η(x,y) 40
41 Result: Filtered Image 41
42 Linear Degradation: g x, y H f x, y x, y L-System: H f x, y f, H x, y d d h x, y,, H x, y H f x, y f, h x, y dd LSI-System: g x, y f x, y h x, y x, y G u, v F u, v H u, v N u, v 42
43 Degradation Estimation: Image Observation: Look at the image and Experiments: Acquire image using well defined object (pinhole) Modeling: Introduce certain model for certain degradation using physical knowledge. 43
44 Degradation (Using Experimental PSF) Original Object Degraded Object H s u, v G u, v A PSF 44
45 Atmospheric Turbulence: NASA 45
46 Modeling of turbulence in atmospheric images: 2 2, exp H u v k u v 56 46
47 Motion Blurring: Camera Limitation; Object movement. 47
48 Motion Blurring Modeling:,, T g x y f x x t y y t dt 0 T T j 2 ux vy G u, v f x x 0t, y y 0t dt e dxdy 0 j 2 ux t vy t 0 0 G u, v F u, v e dt F u, v H u, v 48
49 Linear one/two dimensional motion blurring: at T j ua x0 t, tmax T H u, v sin uae T ua at bt T x0t, y0t H u, v sin ua vb e T T ua vb j uavb 49
50 Motion Blurring Example (1): a=b=0.1 and T=1 Original (Left) and Notion Blurred (Right) 50
51 Motion Blurring Example (2): 51
52 Uniform Motion Blurring (?): 52
53 Out of Focus Blurring: h x, y circ r 1 circ 2 R 1 r 1 0 r 1 x y 2 2 R 53
54 Performance Metrics in Image Restoration: Definition f(x, y): Clean Image n(x, y): zero mean uncorrelated noise with variance σ 2 g(x, y): Blurred Image, f(x, y)(x, y) f x, y : Restored Image SNR (Signal to Noise Ratio), Image Quality: BSNR (Blurred SNR) SNR 10log BSNR 10log f 2 xy, 10 2 g 2 x, y xy, 10 2 x, y 54
55 Performance Metrics in Image Restoration: ISNR (Improvement in SNR): ISNR 10log,, xy, 10 2 xy, f x y g x y, ˆ, f x y f x y 2 55
56 Inverse Filtering: Without Noise: Hˆ u v ˆ G u, v F u, v H u, v F u, v F u, v Hˆ u, v, Problem of division by zero or NaN! 56
57 Inverse Filtering: With Noise: ˆ ˆ G u, v F u, v H u, v N u, v N u, v F u, v F u, v Hˆ u, v H u, v Hˆ u, v Problem of division by zero! Impossible to recover even if H(.,.) is known!! 57
58 Pseudo Inverse (Constrained) Filtering: Set infinite (large) value to zero; G u, v Hˆ u, v ˆ ˆ F u, v H u, v 0 Hˆ u, v H H THR THR 58
59 Example: Full Band (a) Radius 50 (b) Radius 70 (c) Radius 85 (d) 59
60 Threshold Effect: Ghost 60
61 Phase Problem: 61
62 Wiener Filtering in 2D case: Degradation f(x,y) g(x,y) g(x,y) g x y s x y x y G u v S u v N u v,,,,,, W u v G u v,, ˆ,,,., 2, Fˆ u, v,., E u v F u v F u v F u v W u v G u v E E u v E F WG F WG E FF WGG W WGF FW G 2 2 Restoration E F W G W WGF W G F f(x,y) 62
63 Wiener Filtering in 2D case: W 2 E E u, v P WP W W P WP 2,, 2 FF GG FG GF E E u, v PFG u, v 0 W u, v P u, v P u, v E X :Spectral Estimation XX P u, v E XY : Cross Spectral Estimation XY P u v P u v XY YX GG 63
64 Wiener Filtering in 2D case: Special Cases: Noise Only:,,,,,, g x y f x y x y G u v F u v N u v Uncorrelated "zero mean noise" and "image": FG W u, v * 2 * FF * 2 2 * *, P u v E F F N E F E F E N P, P u v E F G F N E F E N E F E N E F E N P P GG FF NN FF PFF u, v,, P u v P u v NN 64
65 Degradation plus Noise:,,,,,,,, g x y f x y h x y x y G u v F u v H u v N u v Uncorrelated Noise and Image: W FF u, v 2 P H PFF H PNN H 1 H 1 P H P 2 2 H 2 NN H 2 PFF FF 2 H H P P NN FF H P NN SNR 1 65
66 Degradation plus Noise: White Noise W u, v 1 H H 2 H 2 K Select Interavtively 66
67 Wiener Filter is known as: Wiener-Hopf Minimum Mean Square Error Least Square Error Problems with Wiener: P FF P NN 67
68 Phase in Wiener Filter: W H W 2 2 PNN 1 H H PFF P 1 H H 2 H NN PFF W H No Phase compensation! 68
69 Wiener Filter vs. Inverse Filter: 1 H H H 0 W lim W 2 H 2 P P NN NN H H 0 H 0 P FF 69
70 Other Wiener Related Filter: W W S H, 0 1 H 2 PNN H P FF 1 1 exp j 2 P NN H H P FF 70
71 Nonlinear Filter:, 1 ˆ j G u v,,,, F u v G u v e G u v G u v 1 HP like 1 LP like Fˆ u, v j Gu, v log G u, v e G u, v G 0 O.W. min 71
72 Example: Full Inverse Pseudo Inverse Wiener 72
73 Example: Degraded Inverse Wiener Motion Blurring+Noise Noise Decrease 73
74 Algebraic Approach A Matrix Review: Matrix Diagonalization 1 : Each square matrix A (with n linearly independent eigenvectors) can be decomposed into the very special form: A PDP 1 P: Matrix composed of the eigenvectors of. D: Diagonal matrix constructed from the corresponding eigenvalues of A
75 Algebraic Approach: Matrix Representation of Circular Convolution:, Periodicity: In Matrix Form: N 1 y n f n g n f j g n j g i g i N j0 y0 g0 gn 1 gn2 g1 f0 y g g g g f N y 2 g2 g1 g0 g 3 f 2 y g g g g f N 1 N 1 N 2 N 3 0 N 1 C :, A Circulant Matrix! C k j g k j C 75
76 Algebraic Approach: Diagonalization of Circular Convolution: C = WDW WDW, W W W m, n 1 1 e 2 jm n N G G 1 0 D G k 0 0 G N 1 DFT g n K 76
77 Algebraic Approach: Diagonalization of Circular Convolution: f g WDW f WDW f DFT Multiply IDFT Extension to 2D case is similar! 77
78 Algebraic Approach to Noise Reduction: Least Square Approach: Matrix Representation of Degradation Model: g Hf n n g Hf g Hf g Hf fˆ arg min n arg min g Hf f f 1 ˆ T T f H H H g Pseudo Inverse (Left) T 1 T H H H H = I g Hf g Hf g Hf f f T 0 T 2H g Hf 0 T 78
79 Least Square Approach: If H is square and H -1 exists then: This is Inverse Filter! fˆ 1 H g f x, y H u, vg u, v Gu, v 2 Hu, v H u, v * H H H ˆ T T f Hg g 79
80 Constrained Least Square (CLS) Approach: Q is a linear operator (High pass version of image, Laplacian and etc.) Using Lagrange Multiplier: f ˆ arg mi n Qf Const. t o g - Hf n f J f Qf g - Hf n J f f T T 2Q Qf - 2H g - Hf T 1 T T T T 1 T ˆ f H H Q Q H g H H Q Q H g : Lagrange Multiplier which should be be satisfy: g - Hf 2 2 n 80
81 CLS in frequency domain: Diagonalization: 2 1 * T * H = WDHW H H = W DH W T * 1 * * T 2 * H H Q H = WD W H = WD W = WD W Q Q = W D W 1 2 T T T 2 * * * H H Q Qf H g W DH DQ W f WDH W g 1 Multiply by : 2 2 * * * 2 2 * D ˆ H DQ W f DH W g H u, v C u, v F u, v H u, vg u, v W C u, v : Fourier Transform of Q (Linear Operator!) * H u, v, C u, v Fˆ u, v G u, v 2 2 H u v 81
82 CLS in frequency domain: * H uv, 2, C u, v Fˆ u, v G u, v 2 H u v ˆ f H 1 T 1 T T H QQ H g 82
83 Wiener Filter vc. CLS filter: T * H u, v, C u, v T ˆ R E f ff F u, v G 2 2 u, v H u v R E nn n 1 T 1 ˆ T 1 T T Q Q R f Rn f H H Q Q H g 0 Inverse Filter 1 Wiener Filter (General form) 83
84 For Laplacian as a measure of smoothness: P x, y C0 CM 1 CM2 C1 C1 C0 CM 1 C 2 Q C2 C1 C0 C 3 CM 1 CM 2 CM 3 C 0 C j,0, 1,1 Pe j Pe j N Pe j P e j N Pe j, 1,0 84
85 How to estimate γ: Residual Vector: r = g - Hfˆ T It can be shown that r r r r is monotically increasing with. 2 2 We seek for such that r n, is our accuracy 1. Initial Guess on 2. Calculate 3. If criteria r r g - Hfˆ n is not reached : 3.a if r n increase. 3.b if r n decrease. 4. Re-calculate new filter and repeat until criteria is meeted 85
86 How to estimate γ: Computational Tips: 2 2 We need r and n. 2 2,,, ˆ, r, 2 n n M1N1 x0 y0 M1N1 x0 y0 M1N1 2 2 x0 y MN n mn Need only n x, y n x, y x, y and 2 2 n n n 2 M1N1 R u v G u v H u v F u v r x y m n 1 MN 1 MN n n m m x0 y0 86
87 Example: CLS High Noise Mid Noise Low Noise 87
88 Example: Correct Noise Parameter Incorrect Noise Parameter 88
89 Computation Efforts: g Hf n n g Hf T 1 ˆ T f H H H g H is size of MNMN ( )!!!! Gradient Descent Methods of function minimization: x GD min x x x x x x Convergence Condition: 0 : Eignvalue of E ( k 1) k k k xx T 2 max k 89
90 Iterative LS or CLS (Tikhonov-Miller): LS : g Hf n n g Hf Φ f 1 g Hf g Hf Φ 2 f H g Hf 1. f T H g 2 T f f H g Hf H g I H H f f I H H f CLS : T T T T k 1 k k k 0 k J f J f Qf g - Hf n 2 f T T Q Qf - H g - Hf 1. f T H g T T T T T H Q Q 2. f f H g - Hf Q Qf H g I H k k k k k f 90
91 Example (Uniform Noise): Blurred by a 7 7 Disc (BSNR=20dB) Restored Using Generalized Inverse Filter with T=10-3 (ISNR=-32.9dB) 91
92 Example (Gaussian Noise): Blurred by a 5 Gaussian (σ 2 =1) (BSNR=20dB) Restored Using Generalized Inverse Filter with T=10-3 (ISNR=-36.6dB) 92
93 Example (Uniform Noise): Blurred by a 7 7 Disc (BSNR=20dB) Restored Using Generalized Inverse Filter with T=10-1 (ISNR=+0.61 db) 93
94 Example (Gaussian Noise): Blurred by a 5 Gaussian (σ 2 =1) (BSNR=20dB) Restored Using Generalized Inverse Filter with T=10-1 (ISNR=-1.8 db) 94
95 Example (Uniform Noise): Blurred by a 7 7 Disc (BSNR=20dB) Restored Using CLS Filter with ϒ=1 (ISNR=+2.5 db) 95
96 Example (Gaussian Noise): Blurred by a 5 Gaussian (σ 2 =1) (BSNR=20dB) Restored Using CLS Filter with ϒ=1 (ISNR=+1.3 db) 96
97 Example (Uniform Noise): Blurred by a 7 7 Disc (BSNR=20dB) Restored Using CLS Filter with ϒ=10-4 (ISNR=-21.9 db) 97
98 Example (Gaussian Noise): Blurred by a 5 Gaussian (σ 2 =1) (BSNR=20dB) Restored Using CLS Filter with ϒ=10-4 (ISNR=-22.1 db) 98
99 Example: 99
100 Wiener Filter Example (input): Original (Left) and degraded with SNR =7dB (Right) 100
101 Wiener Filter Example (output): Known Noisy Image (Left) and Know Clear Image (Right) 101
102 Iterative Wiener Filter: Wiener in Matrix form (Spatial Filter): g = Hf + n fˆ Wf T 1, T is Hermitian Transfor W R H HR H R R R f n T f f n : Autocorrelation of f : Autocorrelation of n m! 102
103 Iterative Wiener Filter: Conditions: 1. The original image and noise are statistically uncorrelated. 2. The power spectral density (or autocorrelation) of the original image and noise are known. 3. Both original image and the noise are zero mean. 103
104 Iterative Wiener Filter: Iterative Algorithm: 1. R 0 f T i i i T 2. W 1 R H HR H R i i 3. fˆ 1 W 1 g ˆ ˆT i E i i 4. R 1 = f 1 f 1 f = R g 5. Repeat 2,3,4 until convergence. f f n 1 104
105 Example Iterative Wiener Filter): 105
106 Adaptive Wiener Filter: Image are Non Stationary! Need Adaptive WF which is locally optimal. Assume small region which image are stationary Image Model in each region:,,, f x y x y x y : zero-mean white noise with unit variance!, : Constant over each region. f f Noisy Image: f f g x, y f x, y v x, y, v :Constant over each region 106
107 Local Wiener Filter in each region: W a u, v P f,, f xy f x y g x y x y 2 ff f 2 2 ff Pvv f v 2 2 f v fˆ x, y g x, y w x, y f a f ˆ f x, y g x, y f f 2 f 2 2 f v, : Low-pass filtered on noisy image.,,, Zero mean assumption g x, y, : Hi-pass filtered on noisy image. f P 2 wa x y x y ˆ xy, f x, y HP x, y LP x, y 2 f 2 2 f xy, v 107
108 Parameter Estimation: 2 g 2 v Local Noisy Image Variance Variance in a smooth image region or background x y Max x y,,, f g v 108
109 Example (1): 109
110 Example (1): 110
111 Results: 111
112 Results: 112
113 Spatial Transform: x y r(x,y) s(x,y) x y tiepoints f(x,y) g(x,y ) original distorted 113
114 An example of Spatial Transform: x y c c 1 5 x c 2 x c 6 y c 3 y c xy c 7 4 xy c 8 114
115 Gray Level Interpolation: Main Problem: Non-Integer coordination! 115
116 Gray Level Interpolation: Nearest neighbor Bilinear interpolation Use 4 nearest neighbors Cubic convolution interpolation Fit a surface over a large number of neighbors 116
117 Example (1): original Tiepoints after distortion Nearest neighbor restored Bilinear restored 117
118 Example (2): original distorted Difference restored 118
119 Matlab Image Restoration Command: deconvblind: Restore image using blind deconvolution deconvlucy: Restore image using accelerated Richardson-Lucy algorithm deconvreg: Restore image using Regularized filter deconvwnr: Restore image using Wiener filter wiener2: Perform 2-D adaptive noise-removal filtering edgetaper: Taper the discontinuities along the image edges 119
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