Digital Image Processing. Frequency Domain Filtering

Transcription

1 Digital Image Processing Frequency Domain Filtering

2 DFT Matlab demo clear all; close all; a=imread('testpat1.png');b=imdouble(a); figure;imshow(b); Fb = fft(b);fbshift=fftshift(fb); figure;imshow(log(abs(fbshift) ),[]); FMask=zeros(56,56);FMask([96:160],[96:160])=1.0; Fbband=Fbshift.*FMask; figure;imshow(log(abs(fbband) ),[]); Fbband=ifftshift(Fbband);bandrebuilt=ifft(Fbband); figure;imshow(bandrebuilt); FMask=zeros(56,56);FMask([64:19],[64:19])=1.0; Fbband=Fbshift.*FMask; figure;imshow(log(abs(fbband) ),[]); Fbband=ifftshift(Fbband);bandrebuilt=ifft(Fbband); figure;imshow(bandrebuilt); FMask=zeros(56,56);FMask([3:4],[3:4])=1.0; Fbband=Fbshift.*FMask; figure;imshow(log(abs(fbband) ),[]); Fbband=ifftshift(Fbband);bandrebuilt=ifft(Fbband); figure;imshow(bandrebuilt);

3 FT Signal decomposition(coefficient computation) IFT Signal reconstruction

4 IFT IFT

5 The central part of FT, i.e. the low frequency components are responsible for the general gray-level appearance of an image. The high frequency components of FT are responsible for the detail information of an image.

6 Image Frequency Domain (log magnitude) v Detail u General appearance

7 5 % 10 % 0 % 50 %

8 Image Filtering Image filtering techniques: Spatial domain methods Frequency domain methods Spatial (time) domain techniques are techniques that operate directly on pixels. Frequency domain techniques are based on modifying the Fourier transform of an image.

9 Frequency Domain Filtering

10 Frequency Domain Filtering Edges and sharp transitions (e.g., noise) in an image contribute significantly to highfrequency content of FT. Low frequency contents in the FT are responsible to the general appearance of the image over smooth areas. Blurring (smoothing) is achieved by attenuating range of high frequency components of FT.

11 Convolution in Time Domain g(x,y)=h(x,y) f(x,y) M 1 M 1 g( x, y) = h( x', y' ) f( x x', y y' ) x' = 0 y' = 0 f( x, y)* h( x, y) f(x,y) is the input image g(x,y) is the filtered image h(x,y): impulse response

12 Convolution Theorem G(u,v)=F(u,v) H(u,v) g(x,y)= f(x,y) h(x,y) Multiplication in Frequency Domain Convolution in Time Domain Filtering in Frequency Domain with H(u,v) is equivalent to filtering in Spatial Domain with h(x,y).

13 blue line = sum of 3 sinusoids (0, 50, and 80 Hz) + random noise red line = sum of 3 sinusoids without noise blue line = sum of 3 sinusoids after filtering in time domain 1x average [ ] / 5 blue line = sum of 3 sinusoids after filtering in frequency domain cut-off 90 Hz

14 Convolution Property of the Fourier Transform Let functions f ( r, c) and g( r, c) have Fourier Transforms F( u, v) and G( u, v). Then, F { f Moreover, F { f g} = g} = F G. F G. * = convolution = multiplication The The Fourier Transform of of a convolution equals the the product of of the the Fourier Transforms. Similarly, the the Fourier Transform of of a multiplication is is the the convolution of of the the Fourier Transforms

15 Convolution via Fourier Transform Image & Mask Transforms Pixel-wise Product Inverse Transform

16 How to Convolve via FT in Matlab 1. Read the image from a file into a variable, say I.. Read in or create the convolution mask, h. 3. Compute the sum of the mask: s = sum(sum(h)); 4. If s == 0, set s = 1; 5. Create: H = zeros(size(i)); 6. Copy h into the middle of H. 7. Shift H into position: H = ifftshift(h); 8. Take the D FT of I and H: FI=fft(I); FH=fft(H); 9. Pointwise multiply the FTs: FJ=FI.*FH; 10. Compute the inverse FT: J = real(ifft(fj)); 11. Normalize the result: J = J/s; The The mask mask is is usually usually 1-band 1-band For For color color images images you you may may need need to to do do each each step step for for each each band band separately.

17 Coordinate Origin of the FFT Center Center = (floor(r/)+1, (floor(r/)+1, floor(c/)+1) floor(c/)+1) Even Odd Even Odd Image Origin Image Origin Weight Matrix Origin Weight Matrix Origin After FFT shift After FFT shift After IFFT shift After IFFT shift

18 Matlab s fftshift and ifftshift J = fftshift(i): I (1,1) J ( R/ +1, C/ +1) I = ifftshift(j): J ( R/ +1, C/ +1) I (1,1) where x = floor(x) = the largest integer smaller than x.

19 Algorithm Complexity We can compute the DFT directly using the formula An N point DFT would require N floating point multiplications per output point Since there are N output points, the computational complexity of the DFT is N 4 N 4 =4x10 9 for N=56 Bad news! Many hours on a workstation

20 Algorithm Complexity The FFT algorithm was developed in the 60 s for seismic exploration Reduced the DFT complexity to N log N N log N~10 6 for N=56 A few seconds on a workstation F( uv, ) = F( u, v) * original image Fourier log magnitude Fourier phase

21 Examples of Filters Frequency domain Gaussian lowpass filter Gaussian highpass filter Spatial domain

22 Blurring: Averaging / Lowpass Filtering Blurring results from: Pixel averaging in the spatial domain: Each pixel in the output is a weighted average of its neighbors. Is a convolution whose weight matrix sums to 1. Lowpass filtering in the frequency domain: High frequencies are diminished or eliminated Individual frequency components are multiplied by a nonincreasing function of ω such as 1/ω = 1/ (u +v ).

23 Ideal Lowpass Filter 理想低通滤波器 Image Image size: size: 51x51 51x51 FD FD filter filter radius: radius: Multiply by this, or convolve by this Fourier Domain Rep. Rep. Spatial Spatial Representation Central Profile Profile

24 Ideal Lowpass Filter Image Image size: size: 51x51 51x51 FD FD filter filter radius: radius: 8 Multiply by this, or convolve by this Fourier Domain Rep. Rep. Spatial Spatial Representation Central Profile Profile

25 Power Spectrum and Phase of an Image Consider the the image below: Original Image Image Power Power Spectrum Phase Phase

26 Ideal Lowpass Filter Image Image size: size: 51x51 51x51 FD FD filter filter radius: radius: Original Image Image Power Power Spectrum Ideal Ideal LPF LPF in in FD FD

27 Ideal Lowpass Filter Image Image size: size: 51x51 51x51 FD FD filter filter radius: radius: Filtered Image Image Filtered Power Power Spectrum Original Image Image

28 Ideal low-pass filter (ILPF) H( u, v) 1 = 0 D( u, v) D D ( u, v) > D 0 0 1/ Duv (, ) = [( u M/ ) + ( v N/ ) ] (M/,N/): center in frequency domain. D 0 is called the cutoff frequency( 截止频率 ).

29 FT Ideal in frequency domain means non-ideal in spatial domain, vice versa. ringing and blurring

30 Shape of ILPF Frequency domain h(x,y) Spatial domain

31 Fourier transform basis functions Approximating a square wave as the sum of sine waves.

32 F(u,v) H(u,v) f(x,y) h(x,y) The limiting case Frequency domain Spatial Domain H(u,v)=1 h(x,y)= And vice versa

33 Treating the ringing 譬如人染沉疴, 当先用糜粥以饮之, 和药以服之 ; 待其腑脏调和, 形体渐安, 然后用肉食以补之, 猛药以治之 : 则病根尽去, 人得全生也 若不待气脉和缓, 便投以猛药厚味, 欲求安保, 诚为难矣 三国演义 第四十三回

34 Butterworth Lowpass Filters (BLPF) Smooth transfer function, no sharp discontinuity, no clear cutoff frequency. H ( u, v) = n 1+ 1 D( u, v) D0 1

35 Butterworth Lowpass Filters (BLPF) h(x) n=1 n= n=5 n=0 Order=1 Order= Order=5 Order=0 H ( u, v) = n 1+ 1 D( u, v) D0 Spatial representations of BLPF with different orders Notice the ringing of different orders

36 No serious ringing artifacts

37 Gaussian Lowpass Filters (GLPF) Smooth transfer function, smooth impulse response, no ringing H ( u, v) = e D ( u, v) D 0

38 GLPF Frequency domain Gaussian lowpass filter Spatial domain

39 No ringing artifacts

40 Examples of Lowpass Filtering

41 Examples of Lowpass Filtering Low-pass filter H(u,v) Original image and its FT Filtered image and its FT

42 Sharpening: Differencing / Highpass Filtering Sharpening results from adding to the image, a copy of itself that has been: Pixel-differenced in the spatial domain: Each pixel in the output is a difference between itself and a weighted average of its neighbors. Is a convolution whose weight matrix sums to 0. Highpass filtered in the frequency domain: High frequencies are enhanced or amplified. Individual frequency components are multiplied by an increasing function of ω such as αω = α (u +v ), where α is a constant.

43 Ideal Highpass Filter 理想高通滤波器 Image Image size: size: 51x51 51x51 FD FD notch notch radius: radius: Multiply Multiply by by this, this, or or convolve convolve by by this this Fourier Domain Rep. Rep. Spatial Spatial Representation Central Profile Profile

44 Ideal Highpass Filter Image Image size: size: 51x51 51x51 FD FD notch notch radius: radius: Original Image Image Power Power Spectrum Ideal Ideal HPF HPF in in FD FD

45 Ideal Highpass Filter Image Image size: size: 51x51 51x51 FD FD notch notch radius: radius: Filtered Image Image * * Filtered Power Power Spectrum Original Image Image

46 High-pass Filters 高通滤波器 H hp (u,v)=1-h lp (u,v) Ideal: Butterworth: Gaussian: n v D u D v u H 0 ), ( 1 1 ), ( + = ) = 0 1, ( v H u 0 ), ( D v u D > 0 ), ( D v u D 0 )/, ( 1 ), ( D v u D e v u H =

47

48 h(x,y) High-pass Filters

49 Ideal High-pass Filtering ringing artifacts

50 Butterworth High-pass Filtering

51 Gaussian High-pass Filtering

52 Filtered image and its FT Gaussian High-pass Filtering Original image Gaussian filter H(u,v)

53 Laplacian in Frequency Domain ) ( ), ( 1 v u v u H + = ), ( ) ( ] ), ( ), ( [ v u F v u y y x f x y x f + = + I y f x f f + = Laplacian operator Spatial domain Frequency domain

54 The Uncertainty Relation 不确定性关系 space frequency If Δx Δ y is the extent of FT FT the object in space and if Δu Δv is its extent in frequency then, space frequency Δx Δ y Δu Δv 1 16π FT FT A small object in in space has has a large frequency extent and and vice-versa.

55 Ideal Filters Do Not Produce Ideal Results IFT IFT A sharp cutoff in in the the frequency domain causes ringing in in the the spatial domain.

56 Ideal Filters Do Not Produce Ideal Results Ideal LPF LPF Blurring the the image above with with an an ideal lowpass filter distorts the the results with with ringing or or ghosting.

57 Optimal Filter: The Gaussian IFT IFT The The Gaussian filter optimizes the the uncertainty relation. It It provides the the sharpest cutoff with with the the least ringing.

58 One-Dimensional Gaussian g( x) = 1 σ π e ( x μ ) σ

59 Two-Dimensional Gaussian ) ( ) ( 1 ) ( ) ( 1 ) ( ) ( ), ( c r c r r c c r c c r r c r y x y r e e c g r g c r g σ σ μ σ μ σ π σ σ σ μ σ μ π σ σ + = = = ) ( ) ( 1 ), ( σ μ μ π σ + = c r e c r g If μ and σ are different for r & c If μ and σ are different for r & c or if μ and σ are the same for r & c. or if μ and σ are the same for r & c. r c R = 51, C = 51 μ = 57, σ = 64

60 Optimal Filter: The Gaussian Gaussian LPF LPF With a gaussian lowpass filter, the the image above is is blurred without ringing or or ghosting.

61 Gaussian Lowpass Filter 高斯低通滤波器 Image Image size: size: 51x51 51x51 SD SD filter filter sigma sigma = = 8 Multiply Multiply by by this, this, or or convolve convolve by by this this Fourier Domain Rep. Rep. Spatial Spatial Representation Central Profile Profile

62 Gaussian Lowpass Filter Image Image size: size: 51x51 51x51 SD SD filter filter sigma sigma = = Multiply Multiply by by this, this, or or convolve convolve by by this this Fourier Domain Rep. Rep. Spatial Spatial Representation Central Profile Profile

63 Gaussian Lowpass Filter Image Image size: size: 51x51 51x51 SD SD filter filter sigma sigma = = 8 Original Image Image Power Power Spectrum Gaussian LPF LPF in in FD FD

64 Gaussian Lowpass Filter Image Image size: size: 51x51 51x51 SD SD filter filter sigma sigma = = 8 Filtered Image Image Filtered Power Power Spectrum Original Image Image

65 Comparison of Ideal and Gaussian Lowpass Filters Ideal Ideal LPF LPF Original Image Image Gaussian LPF LPF

66 Gaussian Highpass Filter 高斯高通滤波器 Image Image size: size: 51x51 51x51 FD FD notch notch sigma sigma = = 8 Multiply Multiply by by this, this, or or convolve convolve by by this this Fourier Domain Rep. Rep. Spatial Spatial Representation Central Profile Profile

67 Gaussian Highpass Filter Image Image size: size: 51x51 51x51 FD FD notch notch sigma sigma = = 8 Original Image Image Power Power Spectrum Gaussian HPF HPF in in FD FD

68 Gaussian Highpass Filter Image Image size: size: 51x51 51x51 FD FD notch notch sigma sigma = = 8 Filtered Image Image * * Filtered Power Power Spectrum Original Image Image

69 Comparison of Ideal and Gaussian Highpass Filters * * signed signed image; image; 0 mapped mapped to to Ideal Ideal HPF HPF * * Original Image Image Gaussian HPF HPF * *

70 Another Highpass Filter * * signed signed image; image; 0 mapped mapped to to original image image filter filter power power spectrum filtered image image * *

71 Ideal Bandpass Filter 理想带通滤波器 * * signed signed image; image; 0 mapped mapped to to original image image filter filter power power spectrum filtered image image * *

72 Gaussian Bandpass Filter 高斯带通滤波器 Image Image size: size: 51x51 51x51 sigma sigma = = --sigma sigma = = 8 Fourier Domain Rep. Rep. Spatial Spatial Representation Central Profile Profile

73 Gaussian Bandpass Filter Image Image size: size: 51x51 51x51 sigma sigma = = --sigma sigma = = 8 Original Image Image Power Power Spectrum Gaussian BPF BPF in in FD FD

74 Gaussian Bandpass Filter Image Image size: size: 51x51 51x51 sigma sigma = = --sigma sigma = = 8 Filtered Image Image * * Filtered Power Power Spectrum Original Image Image * * signed signed image; image; 0 mapped mapped to to 18 18

75 Comparison of Ideal and Gaussian Bandpass Filters * * signed signed image; image; 0 mapped mapped to to Ideal Ideal BPF BPF * * Original Image Image Gaussian BPF BPF * *

76 DFT Matlab demo clear all;close all; iptsetpref('imshowborder','tight'); bw = imread('text.png'); imshow(bw); a = bw(3:45,88:98); figure, imshow(a); C = real(ifft(fft(bw).* fft(rot90(a,),56,56))); figure, imshow(c,[]) max(c(:)); %cys this call output 68 thresh = 60; % Set a threshold a little less than max. figure, imshow(c > thresh); % highlight pixels over threshold.

77 Homework VII Design your own DFT exploration experimence. It can be either of the following DFT and reconstruction DFT Real/Imagenary part, magnitude/phase Frequency domain filtering DFT based image analysis (Detection/Editing ) Submit your test images, codes, experiment results and documents.

78 End of Lecture