Digital Image Processing COSC 6380/4393

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1 Digital Image Processing COSC 6380/4393 Lecture 13 Oct 2 nd, 2018 Pranav Mantini Slides from Dr. Shishir K Shah, and Frank Liu

2 Review f w Zero Padding Initial Position

3 Review f w Full Correlation result

4 Review f w Cropped Correlation result

5 Review f w

6 Review f w Zero Padding Initial Position

7 Review f w Cropped Convolution result

8 8

9 Review: 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. 9

10 Review: Two Smoothing Averaging Filter Masks 10

11 Review: 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) 11

12 Review: Laplace Operator Demo Filters 12

13 Review: 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. 13

14 Review: 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 14

15 Review: 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 15

16 Review: 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 10/3/

17 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)

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

19 The Basic Filtering in the Frequency Domain Modifying the Fourier transform of an image Computing the inverse transform to obtain the processed result g x y 1 (, ) = { H ( u, v) F( u, v)} F( u, v) is the DFT of the input image H ( u, v) is a filter function. 19

20 Diagrams of Convolution All of the following processing is to compute the convolution at a single point (i, j) Consider the two images with image I 1 and its contents shaded at each stage of processing shown: j (0,0) (0,0) i Image I 1 Image I 2 We give them the same coordinate system (superimpose them) (0,0) j i Superimposed 22

21 Diagrams of Convolution The image I 2 is then reversed (reflected), along both axes. This requires that it be defined for negative coordinates, i.e., the periodic extension is used. I 2 reversed (0,0) j i The reversed version of I 2 is then shifted by the amount (i, j) along both j axes: (0,0) i 23 I 2 shifted

22 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 24

23 Review: Periodic Extension of Image 25

24 Diagrams of Convolution The sum extends over 0 m, n N-1 (in blue) so some of the arguments of I 2 (i-m, j-n) fall outside the range 0,..., N-1. What is computed is the summation of the product of [I 1 (m, n) ; 0 m, n N-1] and the periodic extension of [I 2 (i-m, j-n)] as shown: (0,0) j i Overlay of periodic extension of shifted Summation occurs 26 for 0 < I, j < N-1 (inside ) I 2

25 Wraparound Convolution Wraparound convolution is a consequence of the periodic DFT. Wraparound convolution is an artifact of digital processing. Fortunately, wraparound convolution can be used to compute linear convolution. First, let's look at an example of why wraparound convolution is undesirable: 27

26 Linear Convolution by Zero Padding Performing linear convolution by wraparound convolution is a conceptually simple matter. It is accomplished by padding the two image arrays with zero values. Generally, both image arrays must be doubled in size: 0 0 Image I 1 (zero padded) Image I 2 (zero padded) At the edges, no wraparound effect will occur, since the "moving" image will be weighted by zero values only outside the domain. This can be seen by looking at the overlaps when computing the convolution at a point (i, j): 28

27 Linear Convolution by Zero Padding Linear convolution by zero padding Remember, the summations take place only within the blue shaded square (0 i, j, 2N-1). Instead of summing over the periodic extension of the "moving image," zero values are summed with the weighted interior values. 29

28 The Basic Filtering in the Frequency Domain 10/3/

29 Summary: Steps for Filtering in the Frequency Domain 1. Given an input image f(x,y) of size MxN, obtain the padding parameters P and Q. Typically, P = 2M and Q = 2N. 2. Form a padded image, f p (x,y) of size PxQ by appending the necessary number of zeros to f(x,y) 3. Multiply f p (x,y) by (-1) x+y to center its transform 4. Compute the DFT, F(u,v) of the image from step 3 5. Generate a real, symmetric filter function*, H(u,v), of size PxQ with center at coordinates (P/2, Q/2) *generate from a given spatial filter, we pad the spatial filter, multiply the expadded array by (-1) x+y, and compute the DFT of the result to obtain a centered H(u,v). 10/3/

30 Summary: Steps for Filtering in the Frequency Domain 6. Form the product G(u,v) = H(u,v)F(u,v) using array multiplication 7. Obtain the processed image g x y real G u v 1 (, ) (, ) ( 1) x + y p = 8. Obtain the final processed result, g(x,y), by extracting the MxN region from the top, left quadrant of g p (x,y) 10/3/

31 An Example: Steps for Filtering in the Frequency Domain 10/3/

32 Filtering in Frequency Domain Smoothing Sharpening Band Pass and Band Reject

33 Image Smoothing Using Filter Domain Filters 1. Ideal Low Pass Filter 2. Butterworth Low Pass Filter 3. Gaussian Low Pass Filter 35

34 Image Smoothing Using Filter Domain Filters: ILPF Ideal Lowpass Filters (ILPF) 1 if D( u, v) D0 H ( u, v) = 0 if D( u, v) D 0 D is a positive constant and D( u, v) is the distance between a point ( u, v) 0 in the frequency domain and the center of the frequency rectangle D( u, v) = ( u P / 2) + ( v Q / 2) 2 2 1/2 36

35 Image Smoothing Using Filter Domain Filters: ILPF 37

36 ILPF Filtering Example 38

37 ILPF Filtering Example 39

38 The Spatial Representation of ILPF 40

39 Image Smoothing Using Filter Domain Filters: BLPF Butterworth Lowpass Filters (BLPF) of order with cutoff frequency H ( u, v) D 1 = 1 + D( u, v) / D 0 0 2n n and 10/3/

40 10/3/

41 The Spatial Representation of BLPF 43

42 Image Smoothing Using Filter Domain Filters: GLPF Gaussian Lowpass Filters (GLPF) in two dimensions is given H ( u, v) = e D 2 2 ( u, v)/2 By letting = D 0 H ( u, v) = e 2 2 (, )/2 0 D u v D 44

43 Image Smoothing Using Filter Domain Filters: GLPF 45

44 10/3/

45 Image Sharpenning Using Filter Domain Filters 1. Ideal High Pass Filter 2. Butterworth High Pass Filter 3. Gaussian High Pass Filter 47

46 Image Sharpening Using Frequency Domain Filters A highpass filter is obtained from a given lowpass filter using H ( u, v) = 1 H ( u, v) HP LP A 2-D ideal highpass filter (IHPL) is defined as 0 if D( u, v) D H ( u, v) = 1 if D( u, v) D

47 Image Sharpening Using Frequency Domain Filters A 2-D Butterworth highpass filter (BHPL) is defined as H ( u, v) 1 = 1 + / (, ) D D u v 2 0 n A 2-D Gaussian highpass filter (GHPL) is defined as 2 2 (, )/2 0 H ( u, v) = 1 e D u v D 49

48 10/3/

49 The Spatial Representation of Highpass Filters 51

50 Filtering Results by IHPF 52

51 Filtering Results by BHPF 53

52 Filtering Results by GHPF 54

Digital Image Processing COSC 6380/4393

Digital Image Processing COSC 6380/4393 Lecture 11 Oct 3 rd, 2017 Pranav Mantini Slides from Dr. Shishir K Shah, and Frank Liu Review: 2D Discrete Fourier Transform If I is an image of size N then Sin