  Prof Duong Anh Duc   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  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 Image     Given a 1-d sequence s[k], k = Fourier transform {…,-1,0,1,2,…,} Fourier transform is periodic with 2 Inverse Fourier transform  How is the Fourier transform of a sequence s[k] related to the Fourier transform of the continuous signal  Continuous-time Fourier transform  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 transform  How is the Fourier transform of a sequence s[m,n] related to the Fourier transform of the continuous signal  Continuous-space 2D Fourier transform f(x,y) |F(u,v)| displayed as image |F(u,v)| displayed in 3-D Image Magnitude Spectrum 10 72    Edges and sharp transitions in grayvalues in an image contribute significantly to high-frequency content of its Fourier transform Regions of relatively uniform grayvalues in an image contribute to lowfrequency content of its Fourier transform Hence, image sharpening in the Frequency domain can be done by attenuating the low-frequency content of its Fourier transform This would be a highpass filter! 73  For simplicity, we will consider only those filters that are real and radially symmetric  An ideal highpass filter with cutoff frequency r0: 74  Note that the origin (0, 0) is at the center and not the corner of the image (recall the “fftshift” operation)  The abrupt transition from to of the transfer function H(u,v) cannot be realized in practice, using electronic components However, it can be simulated on a computer Ideal HPF with r0= 36 75 Original Image Ideal HPF with r0= 18 76 Ideal HPF with r0= 36 Ideal HPF with r0= 26 77  Notice the severe ringing effect in the output images, which is a characteristic of ideal filters It is due to the discontinuity in the filter transfer function 78  A two-dimensional Butterworth highpass filter has transfer function: n: filter order, r0: cutoff frequency 79 80   Frequency response does not have a sharp transition as in the ideal HPF This is more appropriate for image sharpening than the ideal HPF, since this not introduce ringing 81 Original Image HPF with r0= 47 82 HPF with r0= 36 HPF with r0= 81 83  The form of a Gaussian lowpass filter in two-dimensions is given by H u, v    e  D u , v  2 where D u, v   u  v is the distance from the origin in the frequency plane  The parameter s measures the spread or dispersion of the Gaussian curve Larger the value of s, larger the cutoff frequency and more severe the filtering 84 85 86 ... 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... Enhanced Image g(m,n) = h(m,n)*f(m,n) Given Image may be computationally unattractive  We can therefore directly design a transfer function H(u,v) and implement the enhancement in the frequency domain. .. |F(u,v)| displayed as image |F(u,v)| displayed in 3-D Image Magnitude Spectrum 10 Image Magnitude Spectrum 11 Image Magnitude Spectrum 12  As the size of the box increases in spatial domain, the corresponding