Digital Image Processing: Image Enhancement Frequency domain methods - Duong Anh Duc

Digital Image Processing: Image Enhancement Frequency domain methods - Duong Anh Duc provides about Image Enhancement - frequency domain methods; 1-d Fourier transform of a sequence; 2-d Fourier transform of a digital image; Fourier Transform of “Rice” Image; Importance of Phase Information in Images;...
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Digital Image Processing: Image Enhancement Frequency domain methods - Duong Anh Duc Digital Image Processing Image Enhancement Frequency domain methods21/11/15 Duong Anh Duc - Digital Image Processing 1 Image Enhancement: Frequency domain methods The concept of filtering is easier to visualize in the frequency domain. Therefore, enhancement of image f(m,n) can be done in the frequency domain, based on its DFT F(u,v) . This is particularly useful, if the spatial extent of the point-spread sequence h(m,n) is large. In this case, the convolution PSS Enhanced Image g(m,n) = h(m,n)*f(m,n) Given Image may be computationally unattractive.21/11/15 Duong Anh Duc - Digital Image Processing 2 Frequency domain methods We can therefore directly design a transfer function H(u,v) and implement the enhancement in the frequency domain as follows: Transfer Function Enhanced Image G(u,v) = H(u,v)*F(u,v) Given Image21/11/15 Duong Anh Duc - Digital Image Processing 3 1-d Fourier transform of a sequence Given a 1-d sequence s[k], k={…,-1,0,1,2,…,} Fourier transform Fourier transform is periodic with 2 Inverse Fourier transform21/11/15 Duong Anh Duc - Digital Image Processing 4 1-d Fourier transform of a sequence How is the Fourier transform of a sequence s[k] related to the Fourier transform of the continuous signal Continuous-time Fourier transform21/11/15 Duong Anh Duc - Digital Image Processing 5 2-d Fourier transform of a digital image Given a 2-d matrix of image samples s[m,n], m,n Z2 Fourier transform Fourier transform is 2 -periodic both in x and y Inverse Fourier transform21/11/15 Duong Anh Duc - Digital Image Processing 6 2-d Fourier transform of a digital image How is the Fourier transform of a sequence s[m,n] related to the Fourier transform of the continuous signal Continuous-space 2D Fourier transform21/11/15 Duong Anh Duc - Digital Image Processing 7 Fourier Transform Example f(x,y) |F(u,v)| displayed as image21/11/15 Duong Anh Duc - Digital Image Processing 8 Fourier Transform Example |F(u,v)| displayed in 3-D21/11/15 Duong Anh Duc - Digital Image Processing 9Fourier Transform ExampleImage Image Magnitude Spectrum21/11/15 Duong Anh Duc - Digital Image Processing 10Fourier Transform ExampleImage Image Magnitude Spectrum21/11/15 Duong Anh Duc - Digital Image Processing 11Fourier Transform ExampleImage Image Magnitude Spectrum21/11/15 Duong Anh Duc - Digital Image Processing 12Fourier Transform ExampleImage As the size of the box increases in spatial domain, the corresponding “size” in the frequency domain decreases.21/11/15 Duong Anh Duc - Digital Image Processing 13 Fourier Transform of “Rice” Image f(x,y) |F(u,v)|21/11/15 Duong Anh Duc - Digital Image Processing 14 Fourier Transform of “Rice” Image F(u,v)21/11/15 Duong Anh Duc - Digital Image Processing 15 Fourier Transform of “Camera Man” Image g(x,y) |G(u,v)|21/11/15 Duong Anh Duc - Digital Image Processing 16 Fourier Transform of “Camera Man” Image G(u,v)21/11/15 Duong Anh Duc - Digital Image Processing 17 Importance of Phase Information in Images Image formed from magnitude spectrum of Rice and phase spectrum of Camera man21/11/15 Duong Anh Duc - Digital Image Processing 18 Importance of Phase Information in Images Image formed from magnitude spectrum of Camera man and phase spectrum of Rice21/11/15 Duong Anh Duc - Digital Image Processing 19 1-D Discrete Fourier Transform (DFT) For discrete images of finite extent, the analogous Fourier transform is the DFT. We will first study this for the 1-D case, which is easier to visualize. Suppose { f(0), f(1), …, f(N-1)} is a sequence/ vector/1-D image of length N. Its N-point DFT is defined as Inverse DFT (note the normalization):21/11/15 Duong Anh Duc - Digital Image Processing 20