Image Enhancement: Methods. Digital Image Processing. No Explicit definition. Spatial Domain: Frequency Domain:
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1 Image Enhancement: No Explicit definition Methods Spatial Domain: Linear Nonlinear Frequency Domain: Linear Nonlinear 1
2 Spatial Domain Process,, g x y T f x y 2
3 For 1 1 neighborhood: Contrast Enhancement/Stretching/Point Process For w w neighborhood: Filtering/Mask/Kernel/Window/Template Processing s T r 3
4 Intensity Transformation Functions: 4
5 Image Negatives: s L 1 r 5
6 Log Transform: s c log 1r 6
7 Power-Law Transform: s cr 7
8 Gamma Correction: r r r 8
9 Gamma Correction Too Dark Image Original =0.6 =0.4 =0.3 9
10 Gamma Correction Too Bright Image Original =3 =4 =5 10
11 Contrast Stretching Original C. S. THR. 11
12 Gray Level Slicing 12
13 Example Using Fig 3.11 (a) Using Fig 3.11 (b) 13
14 Bit-Plane Slicing: Highlighting effect of a single bit! 14
15 Example: LSB -> MSB (Left to Right and Top to Bottom) 15
16 Image Reconstruction from bit-planes Bits (7,8) Bits (6,7,8) Bits (5,6,7,8) 16
17 Histogram Processing: Enhancement based on statistical Properties: Local Global Histogram Definition:, 0, 1, 0, h r n r L n M N p r k k k k k nk 1 n M N n k 17
18 Histogram Visual Meaning: Dark Light Low Contrast High Contrast 18
19 Histogram Equalization: Continuous Case. Seek for a suitable transform (Except for negative): 19
20 Effect of Point Process on Histogram: s T r P 1 S s Pr r dr r T s ds Special Case (CDF): 0 Uniform Distribution on [0, L -1] r ds dt r s T r L 1 prwdw L 1 pr r dr dr dr 1 1 ps s pr r pr r, 0 s L 1 ds p r L 1 r 20
21 Concept Illustration: 21
22 Discrete Case nk nk prrk, k 0,1,2,, L 1 MN n k k L 1 S T r L 1 p r n, k 0,1,, L 1 k k r j k j0 MN J0 ˆ1 k k 0.5 k S S round S S S min ˆ 2 k k k min L1 0.5 L1Sk S Perfect equalization is NOT possible 22
23 Numerical Example: S 7 p r k k r j j0 S k S k
24 Numerical Examples (Cont.) 24
25 Real Experiment: 25
26 Gray-Level Transfer Function 26
27 Histogram Matching and Modification: Goal: Specify the shape of the histogram:? pr r pz z r s T r L 1 pr wdw z G T r G s z G z L 1 pz tdt 0 Example: Pages:
28 Example (Mars image and its histogram): 28
29 Histogram Matching: Washout Gray 29
30 Histogram Matching: Desired Transform using Initial CDF Transform using Initial Modified CDF 30
31 Local Histogram Enhancement 31
32 Histogram Statistics For Image Enhancement: Use of Global Statistical Measures L1 M N 1 r r m p r f x y m n, n i i i0 MN x1 y1 L1 M N 1 m r p r f x y, i i i0 MN x1 y1 Gross adjustments in overall intensity (m) and contrast (µ 2 ) n 32
33 Histogram Statistics For Image Enhancement: Local mean and local variance: L1 1 m x y r p r f s t S,, Sxy i Sxy i i0 Sxy s, ts L S x, y,,, xy ri ms x y p xy S r xy i f s t ms x y xy i 0 S xy s, ts xy xy : Neighborhood centered on x, y Local information intensity and contrast (edges) xy 1 33
34 A simple enhancement algorithm for SEM image: g x, y,. E. f x, y m x, y k m and k x, y k f x y OW E 4.0, k 0.4, k 0.02, k S 0 G 1 G S 2 G 34
35 Graphical Illustration: Local Mean Local Var E or one 35
36 Enhanced Image Global histogram equalization Use of statistical moments Original Local Histogram Local Statistics 36
37 Fundamentals of Spatial Filtering: a b g x, y w s, t f x s, y t sa tb 37
38 Spatial Correlation () and Convolution (),,,,,,,, Reflection/Rotation in convolution: a w x y f x y w s t f x s y t b sa tb a w x y f x y w s t f x s y t b sa tb
39 Smoothing Spatial Filters: Linear Filters (averaging, lowpass) General Formulation g x, y a b sa tb,, w s t f x s y t a b sa tb w s, t 2 2 x y 2 2 G x, y e 39
40 Square averaging Filter: Denoising/Smoothing Blurring (Equal Value Kernel
41 Blurring Usage: Delete unwanted (small) subjects. Original Smoothed Thresholded 41
42 Order Statistics Filters: Impulsive noise: Mono Level : Salt, Pepper noises Bi Level: Salt-Pepper noises Filter Median Max Min 42
43 Example: Salt-Pepper Noise 3 3 Averaging 3 3 Median 43
44 Sharpening Spatial Filter Highlights Intensity Transitions First and Second order Derivatives 1,,, 1, f x y f x y f x y f x y f f x y f x y x 0.5 1, 1, 2 f f x 1, y 2 f x, y f x 1, y 2 x 44
45 First Order Derivative: Zero in flat region Non-zero at start of step/ramp region Non-zero along ramp 45
46 Second Order Derivative: Zero in flat region Non-zero at start/end of step/ramp region Zero along ramp 46
47 Comparison: 47
48 1 st and 2 nd Order Derivative Comparison: First Derivative: Thicker Edge; Strong Response for step changes; Second Derivative: Strong response for fine details and isolated points; Double response at step changes. 48
49 Laplacian as an isotropic Enhancer: f x f 2 2 Discrete Implementation: f y 1,, 1 1,, 1 4, 2 f f x y f x y f x y f x y f x y isotropic isotropic 49
50 Laplacian Masks Practically use: 50
51 Background Recovering: g x, y 2,, 2,, f x y f x y sign f x y f x y sign isotropic isotropic 51
52 Example: Non-Scaled and scaled Laplacian Sharpened using 90º and 45º degree isotropic Laplacian 52
53 Unsharp Masking and High-Boost Filtering: k 0 mask x y f x y k g x y g x, y f x, y f x, y, f x, y : Blurred image g,,, k=1: Unsharp Masking k>1: High Boost Another mask: mask Laplacian and any highpass filter 53
54 One Dimensional Illustration 54
55 Two Dimensional Example Blurred, 5 5, σ=3 Unsharp mask Unsharp masking, k=1 Highboost, k=4.5 55
56 First Derivative - Gradient: T T f f f Gx G y x y f f f f f x y x y 56
57 Discrete Implementation Roberts Cross Gradient G z z G z z x 9 5 y 8 6 Sobel Gradient G z 2z z z 2z z x G z 2z z z 2z z y
58 Example (Sobel Mask): 58
59 Image Subtraction:,,, g x y f x y f x y Physically static object and background Intensity changes in object but not in background 59
60 Example: Original Discard Some bits Difference Histogram Equalized 60
61 Image Averaging: Consider an additive noise condition: g(x,y)=f(x,y)+η(x,y) Conditions: Noise, η(x,y): Uncorrelated i.i.d Zero Mean Subject, f(x,y): Physical Stationary Repeatable Experiments 61
62 Image Averaging: For i.i.d RV s: N 1 i E E, Var N i1 Var N gi x, y f x, y i x, y E g x, y f x, y N g x, y gi x, y gx, y x, y N i1 N 62
63 Astronomical Application: Repeatable Experiments! Original Noisy N=8 N=16 N=64 N=128 63
64 Difference Histogram: N=8 N=16 N=64 N=128 64
65 Combination: Bone Scan Laplacian Original+Laplacian Soble of Original 65
66 Combination: Smoothed Sobel (Orig. + L.)*S.Sobel Orig.+ (Orig.+L.)*S.Sobel Apply Power-Law 66
67 Fuzzy Image Enhancement: Brief Introduction to Fuzzy set Fuzzy Intensity Transformation Fuzzy Spatial Filtering 67
68 Introduction Logic of real world He is young man is true of false? Fuzzy and Crisp sets: Membership function: 68
69 Fuzzy Set: Definition: A Empty Set: A z 0 Equality: A z B z Subset: A z B z Complement: z 1 A z, z z Z, Z :Set of Elements A z Union (A B, A OR B): z max z, z A C A B Intersection (A B, A AND B): z min z, z C A B 69
70 Graphical Illustration 70
71 Some Membership Functions: 71
72 Using Fuzzy Sets in inferring: Example: Using color to categorize fruits R 1 : If color is green then the fruit is verdant OR R 2 : If color is yellow then the fruit is half-mature OR R 3 : If color is red then the fruit is mature 72
73 We should to do: Fuzzify antecedent and consequent of each rule Evaluate membership function of each rules (R 1, R 2, R 3 ), Rules aggregation, Final crisp value (Defuzzification). 73
74 Fuzzification of antecedents and consequents 74
75 if-then rule fuzzification: Possibility of consequents is less than antecedents Example: if x is A then y is D Our Example: 1 3 z, v min z, v 0 green 0 verd z, v min z, v 2 0 yellow 0 half z, v min z, v 0 red 0 mat 75
76 Rule aggregation R = R 1 OR R 2 OR OR R n 76
77 Rule aggregation in our Example Q = Q 1 OR Q 2 OR Q 3 z, v max z, v, z, v, z, v
78 Defuzzification: Max Membership Center of Gravity. x x : x x, x x x dx, x dx i1 N x N i1 i x x x i i 78
79 Complex antecedent: R: IF x is A AND x is A AND x is A THEN y is B 1 2 n R : IF x is A THEN y is B x min x, x, x A A 1 A 2 A R: IF x is A OR x is A OR x is A THEN y is B 1 2 n R : IF x is A THEN y is B x Max x, x, x A A 1 A 2 A n n 79
80 Fuzzy Intensity Transformation: R 1 : If a pixel is Dark then make it Darker R 2 : If a pixel is Gray then make it Gray R 3 : If a pixel is Bright then make it Brighter 80
81 Results Original Histogram Equalization Fuzzy Enhancement 81
82 Analysis: 82
83 Spatial Filtering Using Fuzzy Sets Goal: Boundary Extraction General Rule: if a pixel belongs to a uniform region, then make it white, else make it black Uniformity: Intensity difference central pixel and its neighbors For a 3 3 maks: d i = z i - z 5 83
84 Filtering Rule: IF d 2 is zero AND d 6 is zero THEN z 5 is white IF d 6 is zero AND d 8 is zero THEN z 5 is white IF d 8 is zero AND d 4 is zero THEN z 5 is white IF d 4 is zero AND d 2 is zero THEN z 5 is white ELSE z 5 is black 84
85 Membership Function 85
86 Rules Graphical Interpretation 86
87 Example: 87
88 MATLAB Command: Image Statistics: means2, std2, corr2, imhist, regionprops Image Intensity Adjustment: imadjust, histeq, adapthisteq, imnoise Linear Filter: imfilter, fspecial, conv2, corr2, Nonlinear filter: medfilt2, ordfilt2, 88
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