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Chap 4-2. Frequency domain processing Jen-Chang Liu, 2006
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Review of Fourier transform Fourier series: Any function that periodically repeats itself can be expressed as the sum of of sines and/or cosines of different frequencies, each multiplied by a different coefficient Fourier transform: Functions that are not periodic can be expressed as the integral of sines and/or cosines multiplied by a weighting functions Discrete Fourier transform: extends to discrete samples
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1-D Discrete Fourier Transform (DFT) f(x), x=0,1, …,M-1. discrete function F(u), u=0,1, …,M-1. DFT of f(x) Inverse transform (reconstruction): Forward discrete Fourier transform:
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F(u) Complex quantity? Polar coordinate real imaginary m magnitude phase Power spectrum
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Extend to 2-D DFT from 1-D 2-D DFT: 1-D DFT in horizontal then vertical
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Complex Quantities to Real Quantities Useful representation magnitude phase Power spectrum
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Real part 2d DFT basis functions iDFT: 將影像用 合成,其中 (u, v) 代表頻率 DFT
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More DFT basis (real part) (u,v)= (0,2) (0,30) u v (0,63) (1,1) (1,30) (30,30) (1,0)
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Example: reconstruction from DFT coefficients … Zigzag scan
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Example: reconstruction from DFT coefficients http://www.ncnu.edu.tw/~jcliu/course/dip2005/lenaidft.m
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Notes on showing DFT Lena 256x256 F=fft2(I); imshow(abs(F), []) F(1,1)=7761921F(1,127)=334.79+10i imshow(log(abs(F)), [])
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Log transformations s = c log(1+r) Compress the dynamic range of images with large variation in pixel values
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M N M/2 N/2 0 Periodicity and conjugate symmetry property of 2-D DFT
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Outline Frequency domain operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering
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mask coefficients underlying neighborhood X (product) output
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Convolution – 2-D case 2d convolution 旋積 Masking operation
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Convolution theorem f:image Fourier transform F h: filter or mask Fourier transform H Time domain Frequency domain convolution multiplication
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Filtering in the frequency domain fftshift
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Outline Frequency Domain Operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering
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Smoothing frequency-domain filters Design issue G(u,v)=F(u,v) H(u,v) Remove high freq. component (details, noise, …) Ideal low-pass filter Butterworth filter Gaussian filter More smooth in the edge of cut-off frequency
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Ideal low-pass filter Sharp cut-off frequency where D(u,v) is the distance to the center freq.
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Ideal low-pass filter (cont.) Cut-off freq.
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Ideal low-pass filter (con.t) ILPF can not be realized in electronic components, but can be implemented in a computer Decision of cut-off freq.? Measure the percentage of image power within the low freq. Total image power
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ILPF: distribution of image power originalFreq. 99.5 98 96.4 94.6 92
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original =92 D 0 =5 =94.6 D 0 =15 =96.4 D 0 =30 =98 D 0 =80 =99.5 D 0 =230 Ideal low-pass filtering
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Ringing effect
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Effects of ideal low-pass filtering Blurring and ringing ILPF: Freq. F -1 blurring ringing ILPF: spatial
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Effects of ideal low-pass filtering (cont.) spatial impulse ILPF spatial
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Butterworth low-pass filters H=0.5 when D(u,v)=D 0
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Order of butterworth filters n=1n=2n=5n=20 Spatial domain filter of butterworth filters Ringing like Ideal LPF
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Butterworth filters Order = 2 original D 0 =5 D 0 =15D 0 =30 D 0 =80D 0 =230
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Gaussian low-pass filters Variance or cut-off freq. D(u,v)=D 0 H = 0.607
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Gaussian smoothing original D 0 =5 D 0 =15D 0 =30 D 0 =80D 0 =230
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Practical applications: 1 444x508 GLPF, D 0 =80 斷點
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Practical applications: 2 GLPF, D 0 =100 GLPF, D 0 =80 1028x732
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Practical applications: 3 588x600 GLPF, D 0 =30 GLPF, D 0 =10 Scan line
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Outline Frequency Domain Operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering
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Sharpening frequency-domain filters Image details corresponds to high- frequency Sharpening: high-pass filters H hp (u,v)=1-H lp (u,v) Ideal high-pass filters Butterworth high-pass filters Gaussian high-pass filters Difference filters Laplacian filters
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Ideal HPF Butterworth HPF Gaussian HPF
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Spatial-domain HPF ideal Butterworth Gaussian negative
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Ideal high-pass filters D 0 =15D 0 =30D 0 =80 ringing original
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Butterworth high-pass filters n=2,D 0 =15D 0 =30D 0 =80
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Gaussian high-pass filters D 0 =15D 0 =30D 0 =80
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Laplacian frequency-domain filters Spatial-domain Laplacian (2nd derivative) Fourier transform
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Laplacian frequency-domain filters Input f(x,y) Laplacian F(u,v) F F -(u 2 +v 2 )F(u,v) -(u 2 +v 2 ) The Laplacian filter in the frequency domain is H(u,v) = -(u 2 +v 2 )
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0 frequency spatial H(u,v) = -(u 2 +v 2 )
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original Laplacian Scaled Laplacian original+ Laplacian
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Outline Frequency Domain Operations Smoothing Frequency Domain Filters Sharpening Frequency Domain Filters Homomorphic Filtering
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Image Formation Model Illumination source scene reflection eye
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Homomorphic filtering Image formation model f(x,y)=i(x,y) r(x,y) illumination: reflectance: Slow spatial variations vary abruptly, particularly at the junctions of dissimilar objects
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Homomorphic filtering Product term Log of product f(x,y)=i(x,y) r(x,y) => ln f(x,y)=ln i(x,y)+ ln r(x,y) Separation of signal source:
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Homomorphic filtering approach ln i(x,y) ln r(x,y) illumination reflection filtering
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How to identify the illumination and reflection? Illumination -> low frequency Reflection -> high frequency Radius from the origin Example filter: sharpening illumination reflection
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Homomophic filtering: example originalHomomorphic filtering
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