CoE4TN4 Image Processing Chapter 4 Filtering in the Frequency Domain
Fourier Transform Sections 4.1 to 4.5 will be done on the board 2
2D Fourier Transform 3
2D Sampling and Aliasing 4
2D Sampling and Aliasing Perfect reconstruction of a bank-limited image from a set of its samples requires 2-D convolution in the spatial domain with a sinc function. To reduce aliasing it is a good idea to blur an image before shrinking or sampling 5
2D Sampling and Aliasing 6
2D Sampling and Aliasing In images with strong edge content, the effects of aliasing are called jaggies 7
2D Sampling and Aliasing Moire patterns happens in sampling scenes with periodic or nearly perodic components Scanning of media prints such as newspaper 8
2D Sampling and Aliasing Newspapers and other printed materials use halftone dots Halftone: black dots or ellipses whose sizes are used to simulate gray tones 9
2D Sampling and Aliasing 10
2D DFT 11
2D DFT 12
Fourier Transform Fourier Transform (FT) of a 2-D signal: 13
Fourier Transform Properties of Fourier Transform (FT) of a 2-D real, signal: 14
Fourier Transform It is common to multiply input image by (-1) x+y prior to computing the Fourier Transform. This shift the center of the FT to (M/2,N/2) 15
Fourier Transform 16
2D DFT 17
2D DFT 18
Discrete Fourier transform When working with discrete Fourier transform, we have to keep in mind the periodicity of functions involved. This periodicity is a byproduct of the way in which discrete FT (DFT) is defined Using DFT allows us to perform convolution in the frequency domain but the functions are treated as periodic. 19
2D DFT 20
2D DFT 21
2D DFT 22
Frequency domain filtering 1. Multiply the input image by (-1) x+y to center the transform Compute the DFT of the image from step 1 2. Multiply the result by the transfer function of the filter (centered) 3. Take the inverse transform. 4. Multiply the result by (-1) x+y 23
H(u,v)is 0 at the center of the transform and 1 elsewhere 25
Low frequencies: slowly varying components in an image High frequencies: caused by sharp transitions in intensity such as edges and noise
Steps for filtering in frequency domain For input image f(x,y) of size MxN obtain the padding parameters P and Q Form a padded image fp(x,y) of size PxQ by appending the necessary number of zeros to f(x,y) Multiply fp(x,y) by (-1) x+y to center the Fourier transform Compute DFT of f, F(u,v) Generate a real, symmetric filter H(u,v) of size PxQ. Form product G(u,v)=H(u,v)F(u,v) Obtain the processed image: g p (x, y) ={real(f 1 (G(u, v)))}( Obtain the final processed result by extracting the MxN region from the top left quadrant of gp(x,y) 1) x+y 27
G(u,v)=F(u,v)H(u,v) Frequency domain filtering Based on convolution theorem: g(x,y)=h(x,y)*f(x,y) f(x,y) is the input image g(x,y) is the processed image h(x,y): impulse response or point spread function Based on the form of H(u,v), the output image exhibits some features of f(x,y) 28
Convolution h(x,y) f(x,y) Frequency domain filtering Implementing a mask w(x,y) f(x,y) 1) Flip 2) Shift, multiply, add 29
Frequency domain filtering Convolution with a filter and implementing a mask are very similar The only difference is the flipping operation If the impulse response of the filter is symmetric about the origin the two operations are the same. Instead of filtering in the frequency domain, we can approximate the impulse response of the filter by a mask, and use the mask in the spatial domain. 30
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Frequency domain filtering Edges and sharp transitions (e.g., noise) in the gray levels of an image contribute significantly to high-frequency content of FT. Low frequency in the Fourier transform of an image are responsible for the general gray-level appearance of the image over smooth areas. Blurring (smoothing) is achieved by attenuating range of high frequency components of FT. We consider 3 types of lowpass filters: ideal, Butterworth and Gaussian 32
Ideal low-pass filter Ideal: all the frequencies inside a circle of radius D 0 are passed and all the frequencies outside this circle are completely removed. H H u u v M/2-D 0 M/2+D 0 33
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Ideal low-pass filter Effects of ideal low-pass filtering: blurring and ringing 36
Ideal low-pass filter The radii of the concentric rings in h(x,y) are proportional to 1/D 0 D0 1/D0 Less blurring, Less ringing (amplitude of rings drop) D0 1/D0 More blurring, More ringing (amplitude of rings increase) 37
Butterworth Lowpass Filters Smooth transfer function, no sharp discontinuity, no clear cutoff frequency. 38
Butterworth Lowpass Filters 39
Butterworth Lowpass Filters 40
Gaussian Lowpass Filters Smooth transfer function, smooth impulse response, no ringing 41
Gaussian Lowpass Filters 42
Applications of Lowpass Filters 43
Applications of Lowpass Filters 44
High-pass filtering Image sharpening can be achieved by a high-pass filtering process. H hp (u,v)=1-h lp (u,v) Ideal: Butterworth: Gaussian: 45
High-pass filtering 46
High-pass filtering 47
High-pass filtering 48
High-pass filtering 49
High-pass filtering 50
Laplacian in Frequency Domain 51
Laplacian in Frequency Domain 52
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Unsharp Masking, High-boost Filtering Unsharp masking: f hp (x,y)=f(x,y)-f lp (x,y) H hp (u,v)=1-h lp (u,v) High-boost filtering: f hb (x,y)=af(x,y)-f lp (x,y) f hb (x,y)=(a-1)f(x,y)+f hp (x,y) H hb (u,v)=(a-1)+h hp (u,v) 54
Homomorphic filtering In some images, the quality of the image has reduced because of non-uniform illumination Homomorphic filtering can be used to perform illumination correction We can view an image f(x,y) as a product of two components: This equation cannot be used directly in order to operate separately on the frequency components of illumination and reflectance 55
Homomorphic filtering 56
Homomorphic filtering The key to the approach is that separation of the illumination and reflectance components is achieved. The homomorphic filter can then operate on these components separately Illumination component of an image generally has slow variations, while the reflectance component vary abruptly By removing the low frequencies (highpass filtering) the effects of illumination can be removed 57
Homomorphic filtering 58
Selective Filtering Bandreject filters remove or attenuate a band of frequencies 59
Selective Filtering 60
Selective Filtering 61
A bandpass filter performs the opposite of a bandreject filter. H bp (u,v)=1-h br (u,v) Bandpass and band reject filters A notch filter rejects (or passes) frequencies in predefined neighborhood about a center frequency Due to symmetry of the FT, notch filters must appear in symmetric pairs about the origin in order to obtain meaningful results 62
Periodic noise reduction 63
Periodic noise reduction 64
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