Digital Image Processing COSC 6380/4393

Transcription

1 Digital Image Processing COSC 6380/4393 Lecture 11 Oct 3 rd, 2017 Pranav Mantini Slides from Dr. Shishir K Shah, and Frank Liu

2 Review: 2D Discrete Fourier Transform If I is an image of size N then Sin image Cos image Let ሚI be the DFT of the I N 1 N 1 ሚI u, v = I i, j cos[ 2π (ui + vj)] N 1 i=0 j=0 N 1 N 1 N 1 N 1 i=0 I i, j sin[ 2π (ui + vj)] N ሚI u, v = I i, j (cos[ 2π (ui + vj)] N 1sin[2π (ui + vj)]) N i=0 j=0 N 1 N 1 ሚI u, v = I i, j e 12π N (ui+vj) i=0 j=0 0 F u = f x e 1ux

3 Review: 2D Inverse Discrete Fourier Transform Let ሚI be the DFT of the I N 1 N 1 I i, j = u=0 u=0 ሚI u, v e 12π N (ui+vj) f x = F u e iux

4 Review: Properties of DFT Matrix We can understand the DFT matrix better by studying some of its properties. Any image I of interest to us is composed of real integers. However, the DFT of I is generally complex. It can be written in the form

5 Review: Symmetry of DFT ሚI N u, N v = ሚI u, v The DFT of an image I is conjugate symmetric: The magnitude DFT of an image I is symmetric:

6 Review: Symmetry of DFT Depiction of the symmetry of the DFT (magnitude). The highest frequencies are represented near (u, v) = (N/2, N/2). 6

7 Review: Periodicity of DFT We have defined the DFT matrix as finite in extent (N x N): However, if the arguments are allowed to take values outside the range 0 u, v N-1, we find that the DFT is periodic in both the u- and v-directions, with period N: For any integers m, n This is called the periodic extension of the DFT. It is defined for all integer frequencies u, v. 7

8 Review: Periodic Extension of DFT 8

9 Review: Periodic Extension of Image The IDFT equation implies the periodic extension of the image I as well (with period N), simply by letting the arguments (i, j) take any integer value. Note that for any integers n, m In a sense, the DFT implies that the image I is already periodic. This will be extremely important when we consider convolution 9

10 Review: Periodic Extension of Image 10

11 Review: Displaying the DFT Usually, the DFT is displayed with its center coordinate (u, v) = (0, 0) at the center of the image. This way, the lower frequency information (which usually dominates an image) is clustered together near the origin at the center of the display. This can be accomplished in practice by taking the DFT of the alternating image (for display purposes only!) Observe that [(-1) i+j I(i,j) ; 0 i, j N-1] A simple shift of the DFT by half its length in both directions. 11

12 Review: Centered DFT 12

13 Review: Granularity Large values near the DFT origin correspond to large smooth regions in the image or a strong background. Since images are positive (implying an additive offset), any image has a large peak at (u, v) = (0, 0). Masking the DFT Suppose that we define several zero-one images: Masking the DFT with these will produce IDFT images with only low, mid-, or high frequencies remaining (slides). Of course, if we add up the results, we get the original image back. 13

14 Review: Directionality If the DFT is brighter along a specific orientation, then the image must contains highly oriented components in that direction. Masking the DFT Suppose that we define several oriented zero-one images: Masking the DFT with these will produce IDFT images with only highly oriented frequencies remaining (slides). Again, if we add up the results, we get the original image back. 14

15 Review: Periodic Noise removal

16 Spatial Correlation Operator The correlation of a filter w( x, y) of size m n with an image f ( x, y), denoted as w( x, y) f ( x, y) a w( x, y) f ( x, y) w( s, t) f ( x s, y t) b s a t b 10/6/

17 f w

18 f w

19 f w Zero Padding Initial Position

20 f w Zero Padding

21 f w Zero Padding Position after one shift 0 0

22 f w Zero Padding Position after one shift 0 0

23 f w Zero Padding Position after four shift

24 f w Zero Padding Final Position

25 f w Full Correlation result

26 f w Cropped Correlation result

27 Spatial Convolution Operator The convolution of a filter w( x, y) of size m n with an image f ( x, y), denoted as w( x, y) f ( x, y) a w( x, y) f ( x, y) w( s, t) f ( x s, y t) b s a t b 10/6/

28 f w

29 f w w rotated by

30 f w Zero Padding Initial Position

31 f w Zero Padding

32 f w Zero Padding Position after four shift

33 f w Full Convolution result

34 f w Cropped Convolution result

35 Spatial Filtering A spatial filter consists of (a) a neighborhood, and (b) a predefined operation Linear spatial filtering of an image of size MXN with a filter of size mxn is given by the expression a b g( x, y) w( s, t) f ( x s, y t) s a t b 35

36 Spatial Filtering 36

37 Spatial Correlation Operator 37

38 38

39 Linear Systems And Linear Image Filtering A process that accepts a signal or image I as input and transforms it by an act of linear convolution is a type of linear system Example 39

40 Goals of Linear Image Filtering Process sampled, quantized images to transform them into - images of better quality (by some criteria) - images with certain features enhanced - images with certain features de-emphasized or eradicated 40

41 Some Specific Goals smoothing - remove noise from bit errors, transmission, etc deblurring - increase sharpness of blurred images sharpening - emphasize significant features, such as edges combinations of these 41

42 Smoothing Spatial Filters Smoothing filters are used for blurring and for noise reduction Blurring is used in removal of small details and bridging of small gaps in lines or curves Smoothing spatial filters include linear filters and nonlinear filters. 42

43 Spatial Smoothing Linear Filters The general implementation for filtering an M N image with a weighted averaging filter of size m n is given g( x, y) a b s a t b w( s, t) f ( x s, y t) w( s, t) where m 2a 1, n 2b 1. a b s a t b 43

44 Two Smoothing Averaging Filter Masks 44

45 Two Smoothing Averaging Filter Masks Demo Filters 45

46 46

47 Example: Gross Representation of Objects Blur an image for getting a gross representation. Small objects get blended with background 47

48 Example: Gross Representation of Objects Blur an image for getting a gross representation. Small objects get blended with background Demo stars 48

49 Laplace Operator: Foundation The first-order derivative of a one-dimensional function f(x) is the difference f f ( x 1) f ( x) x The second-order derivative of f(x) as the difference 2 f f ( x 1) f ( x 1) 2 f ( x) 2 x 50

50 10/6/

51 Laplace Operator The second-order isotropic derivative operator is the Laplacian for a function (image) f(x,y) f x f y f f f ( x 1, y) f ( x 1, y) 2 f ( x, y) 2 x 2 f f ( x, y 1) f ( x, y 1) 2 f ( x, y) 2 y 2 f f x y f x y f x y f x y ( 1, ) ( 1, ) (, 1) (, 1) - 4 f ( x, y) 52

52 Laplace Operator Demo Filters 53

53 Sharpening Spatial Filters: Laplace Operator Image sharpening in the way of using the Laplacian: g x y f x y c f x y where, f ( x, y) is input image, g( x, y) is sharpenend images, c 2 (, ) (, ) (, ) 2-1 if f ( x, y) corresponding to Fig. 3.37(a) or (b) and c 1 if either of the other two filters is used. 54

54 55

55 Unsharp Masking and Highboost Filtering Unsharp masking Sharpen images consists of subtracting an unsharp (smoothed) version of an image from the original image e.g., printing and publishing industry Steps 1. Blur the original image 2. Subtract the blurred image from the original 3. Add the mask to the original 56

56 Unsharp Masking and Highboost Filtering Let f ( x, y) denote the blurred image, unsharp masking is g ( x, y) f ( x, y) f ( x, y) mask Then add a weighted portion of the mask back to the original g( x, y) f ( x, y) k * g ( x, y) k 0 mask when k 1, the process is referred to as highboost filtering. 57

57 Unsharp Masking: Demo 58

58 Unsharp Masking and Highboost Filtering: Example Demo Sharpen 59

59 Convolution Theorem Let f be and image and h a filtering window Lets us consider the convolution f(t) h(t)

60 Convolution Theorem Let f and h be two function Lets us consider the convolution f t h t = f τ h t τ τ= From the definition of convolution a w( x, y) f ( x, y) w( s, t) f ( x s, y t) b s a t b

61 Convolution Theorem Let f and h be two function Lets us consider the convolution f t h t = f τ h t τ F[f t h t ] τ= = [ f τ h t τ ]e 12πμt t= τ= Computing the Fourier transform of the convolution F u = f x e 1ux

62 Convolution Theorem Let f and h be two function Lets us consider the convolution f t h t = f τ h t τ τ= F[f t h t ] = [ f τ h t τ ]e 12πμt t= τ= = f τ [ h t τ e 12πμt ] τ= t=

63 Convolution Theorem Let f and h be two function Lets us consider the convolution f t h t = f τ h t τ τ= F[f t h t ] = [ f τ h t τ ]e 12πμt t= τ= = f τ [ h t τ e 12πμt ] τ= t= = f τ [ h t τ e 12πμ(t τ) ] e 12πμτ τ= t=

64 Convolution Theorem Let f and h be two function Lets us consider the convolution f t h t = F[f t h t ] = = = τ= τ= = τ= f τ [ τ= [ t= τ= f τ [ t= t= f τ h t τ f τ h t τ ]e 12πμt h t τ e 12πμt ] h t τ e 12πμ(t τ) ] e 12πμτ f τ [H(μ)] e 12πμ(τ) = H(μ) = H μ F(μ) τ= f τ e 12πμ(τ)

65 Convolution Theorem Fourier transform pairs f t h t H μ F(μ) f t h t H μ F(μ)