Image Enhancement: To improve the quality of images

Transcription

1 Image Enhancement: To improve the quality of images Examples: Noise reduction (to improve SNR or subjective quality) Change contrast, brightness, color etc. Image smoothing Image sharpening Modify image to enhance any desired features

2 Gray-level transformation and spatial domain filtering

3 Gray-level transformation: g( x, y) = T[ f ( x, y)] s = T ( r)

4 Frequently Used Transformations 1. Image negative 2. Logarithm: s = clog( 1+ r) 2. Power law: s = c( r + ε ) γ

5 Negative Transformation S=L-1-r

6 Log Transformation in Frequency Domain S=c log (1+r), c=1

7 Power Law Family s = cr γ

8 Correction of Response of CRT Monitor Response of CRT: 2.5 s = r Gamma correction: 0.4 s = r

9 2002 R. C. Gonzalez & R. E. Woods

10 Power Law Transformation s = cr γ

11 Piecewise Linear Transform

12 Gray-Level Slicing

13 Gray-Level Slicing

14 Bit Plane Representation Bit 7: 0 : gray-level 0 ~ 127 ; 1 : gray-level 128 ~ 255 Bit 6: 0 : 0 ~ 63, 128 ~ 191; 1 : 64 ~ 127, 192 ~ 255 : Bit 0: 0 : levels 2, 4, 254; 1 : 1, 3, 5, 255

15 Bit Plane Representation 2002 R. C. Gonzalez & R. E. Woods

16 Bit plane reconstructed image

17 8 Bit-Plane Images 2002 R. C. Gonzalez & R. E. Woods

18 Histogram of Image Assume T(r) satisfies 2 conditions: (a) T(r) is monotonically increasing function in [0, L-1]. (b) 0 T ( r) L 1 for 0 r L 1.

19 Equalization Algorithm Assume p r (r) : p.d.f. of gray-level; s = T (r) : gray-level transformation. The p.d.f. of transformed image: dr ps s) = pr ( r) ds From Then ds dr ( 1 r= T ( s ) s = T ( r) = ( L 1) pr ( ω) dω. dt ( r) d r = = ( L 1) p ( ) d r = ( L 1) pr ( r) dr dr ω ω 0, r 0 p s ( s) = p r ( r) dr ds = p r 1 ( r) ( L 1) p r ( r) = 1 L 1 Discrete case: nk pr ( rk ) = ( k = 0,1,... L 1) MN s k = T ( r ) k = ( L 1) k j= 0 p r ( r ) j = L 1 MN k j= 0 n j

20 Histogram Equalization

21 Example 3.5

22 Similarly, s 2 =4.55, s 3 =5.67, s 4 =6.23, s 5 =6.65, s 6 =6.86, s 7 =7.00 Next, the new intensity levels are rounded to the nearest integers: s 0 =1, s 1 =3, s 2 =5, s 3 =6, s 4 =6, s 5 =7, s 6 =7, s 7 =7

23 Transforms (c.d.f. of gray-level)

24 Gray-Level Mapping

25 Image & Histogram

26 Histogram Matching Given p.d.f. p r (r), s = T ( r) = ( L 1) pr ( ω) dω For desired p.d.f. p z (z) s = G( z), let = ( L 1) 1 1 Then z = G [ s] = G [ T ( r)]. z 0 p z r 0 ( t) dt Discrete case: To get a mapping from r to z, or from s to z. 1. s k ( k = k k L 1 = T ( rk ) = ( L 1) pr ( rj ) = n MN 0,1,... L 1) j= 0 G( z 2. For a value of q, compute q ( 1) z ( i) ) = j= 0 L such that G( z ) = s ( k = 0,1,... L 1) q k j q where p z ) is the i-th value of the specified histogram. z ( i i= zq = G [ T ( rk )] = G [ sk ], ( q = 0,1,... L 1) p z 2002 R. C. Gonzalez & R. E. Woods That is r k T ( rk ) s k 1 G [ sk ] z q

27 Histogram matching example

28 Example 3.8 First step is to get histogram-equalized values: s 0 =1, s 1 =3, s 2 =5, s 3 =6, s 4 =6, s 5 =7, s 6 =7, s 7 =7 G ( z0 ) = 7 p z ( z j ) = 0 j= Similarly, 1 [ p ( z ) + p ( z )] G( z ) = 7 pz ( z j ) = 7 z 0 z 1 1 = j= 0 G G z2 ) = 0.00, G( z3) = 1.05, G( z ) = ( 4 z5 ) = 4.55, G( z6) = 5.95, G( z ) = ( Rounding to integer, G ( 3 z0) 0, G( z1) 0, G( z2) 0, G( z ) 1 G ( 7 z4) 2, G( z5) 5, G( z6) 6, G( z ) 7

29 Histogram matching

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35 Local histogram equalization Define a window as rectangular (normally square) neighborhood over each pixel. Move centre of the window across the image. At each location, a histogram is calculated in the window and then used for histogram equalization.

36 Enhancement using Histogram Statistics The n-th order moment, ) ( ) ( ) ( 1 0 i L i n i n r p m r r = = µ, ) ( ) ( 1 0 i L i i r p r r E m = = = m and 2 2 ) ( r r σ µ = can be used for local enhancement. It is common practice to estimate mean and variance: = = = ), ( 1 M x N y y x f MN m, = = = ] ), ( [ 1 M x N y r m y x f MN σ Given a small area y x S, centered at (x,y), = = = = ) ( ) ( ), (,,,,, L i i S S i S L i i S i S r p m r r p r m y x y x y x y x y x σ An enhancement algorithm: = otherwise y x f D k D k M k m y x E f y x g G S G G S y x y x ), (, ), ( ), ( 2 1 0,, σ : G M global mean; : G D global standard deviation.

37 Example using local histogram statistics

38 Spatial filtering a b g ( x, y) = w( s, t) f ( x + s, y + t) s= at= b m = 2 a +1, n = 2 b +1.

39 1-D Correlation and convolution

40 2-D Correlation and convolution

41 3x3 filter mask

42 3x3 smoothing filter

43 Image smoothing using smoothing filters of different sizes

44 Image smoothing and thresholding

45 Salt-and pepper noise reduction using 3x3 filter masks

46 Sharpening spatial filters 1 st order derivative: 2 nd order derivative: 1 st order derivative: Produces thicker edges; Generally has a stronger response to a gray-level step; 2 nd order derivative: Stronger response to fine details, thin lines, isolated points; Yields a double response at step edges For most applications, 2 nd order derivative is better.

47 Sharpening spatial filters Laplacian operator, y f x f y x f ), ( + = where ), ( 2 ) 1, ( ) 1, ( 2 2 y x f y x f y x f x f + + = ), ( 2 1), ( 1), ( 2 2 y x f y x f y x f y f + + = 2 (, ) ( 1, ) ( 1, ) (, 1) (, 1) 4 (, ) f x y f x y f x y f x y f xy f xy =

48 1 st and 2 nd derivatives of 1-D image

49 Filter masks to implement the Laplacian

50 Image sharpening using Laplacian

51 Unsharpening and highboost filtering ), ( ), ( ), ( y x f y x f y x g mask = ), ( * ), ( ), ( y x g k y x f y x g mask + =

52 2002 R. C. Gonzalez & R. E. Woods

53 Gradient, = = y f x f y x G G y x f ), ( Or [ ] ), ( y G x G y x f + =, ), ( y G x G y x f + = If using 3x3 mask, ) 2 ( ) 2 ( ) 2 ( ) 2 ( ), ( z z z z z z z z z z z z y x f = Image sharpening using 1 st derivative - Gradient

54 Digital approximation to gradient

55 Edge enhancement using gradient

56 Image enhancement using derivatives

57 Spatial domain image enhancement mixed method

58 Fuzzy set A = { z, µ A( z) z Z} A = {(1,1), (2,1),..., (20,1), (21,0.9), (22,0.8),..., (29,0.1)}

59 Fuzzy set operation Empty set: A fuzzy set is empty iff µ ( z) = 0 for all z Z. A Equality: A=B iff µ ( z) = ( z) for all z Z. A µ B Complement: A is complement of A if µ ( z) = 1 ( z) for all z Z. µ A A Subset: A is a subset of B iff µ ( z) ( z) for all z Z. A µ B Union: U = A B has ( z) max[ µ ( z), ( z)] µ = for all z Z. U A µ B Intersection: I = A B has µ ( z) = min[ µ ( z), ( z)] for all z Z. I A µ B

60 Set operation example

61 Commonly used membership functions

62 Fuzzy set application in color sensation

63 Application of membership function R1: IF the color is green, THEN the fruit is verdant. OR R2: IF the color is yellow, THEN the fruit is half-mature. OR R3: IF the color is red, THEN the fruit is mature.

64 Fuzzy set operation µ 3( z, v) = min{ µ red ( z), µ ( v)} mat

65 Fuzzy set operation Q ( v) = min{ µ ( z ), µ ( z, v)} 3 red Q ( v) = min{ µ ( z ), µ ( z, v)} 2 yellow Q ( v) = min{ µ ( z ), µ ( z, v)} 3 green 0 1 0

66 Membership functions for color representation v= 1 v = Q = Q1 OR Q2 OR Q3. Center of gravity: 0 K v= 1 K vq( v) Q( v)

67 Fuzzy system implementation with more inputs

68 Fuzzy, rule-based contrast enhancement If a pixel is dark, then make it darker. If a pixel is gray, then make it gray. If a pixel is bright, then make it brighter. v 0 = µ dark ( z 0 µ ) v dark d ( z + µ 0 grey ) + µ ( z grey 0 ) v ( z 0 g ) + µ + µ bright bright ( z ( z 0 ) 0 ) v b

69 Contrast enhancement using histogram equalization and fuzzy system

70 Histograms

71 Spatial filtering using fuzzy sets

72 Boundary detection using fuzzy sets: If a pixel belongs to a uniform region, then make it white; Otherwise, make it black.

73 Boundary detection using fuzzy rules

74 Fuzzy spatial filtering for edge detection