Digital Image Processing. Chapter 4: Image Enhancement in the Frequency Domain

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1 Digital Image Processing Chapter 4: Image Enhancement in the Frequency Domain Image Enhancement in Frequency Domain Objective: To understand the Fourier Transform and frequency domain and how to apply to image enhancement. Fourier Transform. Low pass filters. High pass filters

2 Transform Operation Fourier: a periodic function can be represented by the sum of sines/cosines of different frequencies,multiplied by a different coefficient (Fourier series). Fourier Transform: Definitions

3 2-D Fourier Transform: Properties 1-D Discrete Fourier Transform (DFT) Suppose {f (0), f (1),, f (M - 1) } is a sequence/vector/1-d image of length M. Its M-point DFT is defined as M 1 2 ( ) = π j ux M F u f ( x) e, u = 0,1,2,..., M 1 x= 0 Inverse DFT M 1 1 ( ) = f x F( u) e M u= 0 2π j ux M, x = 0,1,2,..., M 1 Recall: θ e j = cosθ + j sinθ

4 1-D DFT: Example Example: Let f (x) = {1, 1, 2, 3}. (Note that M=4.) 5 ) ( (0) 3 0 *0 4 2 = = = x x j e x f F π j e x f F x x j 4 1 ) ( (1) 3 0 * = = = π 1 ) ( (2) 3 0 *2 4 2 = = = x x j e x f F π j e x f F x x j 4 1 ) ( (3) 3 0 *3 4 2 = = = π Magnitude, Phase and Power Spectrum

5 2-D Discrete Fourier Transform (DFT): Definition Magnitude, Phase and Power Spectrum

6 Displaying the 2-D DFT

7 Example: 2-D DFT of a Rectangle Example: 2-D DFT

8 Basic steps for filtering in the frequency domain Basics of filtering in the frequency domain 1. multiply the input image by (-1) x+y to center the transform to u = M/2 and v = N/2 2. compute F(u,v), the 2-D DFT of the image from (1) 3. multiply F(u,v) by a filter function H(u,v) 4. compute the inverse DFT of the result in (3) 5. obtain the real part of the result in (4) 6. multiply the result in (5) by (-1) x+y to cancel the multiplication of the input image.

9 Low pass filter High pass filter Lowpass Filter (LPF) Edges and sharp transitions in gray values in an image contribute significantly to high-frequency content of its Fourier transform. Regions of relatively uniform gray values in an image contribute to low-frequency content of its Fourier transform. Hence, an image can be smoothed in the Frequency domain by attenuating the high-frequency content of its Fourier transform. This would be a lowpass filter! For simplicity, we will consider only those filters that are real and radially symmetric.

10 Correspondence between filter in spatial and frequency domains Ideal Low Pass Filter Has transfer function Where D(u,v) is the distance from point (u,v) from the origin of the frequency rectangle If the center is at (M/2,N/2)

11 Smoothing Frequency-domain filters: Ideal Lowpass filter (ILPF) Image Power Circles

12 Result of ILPF Notice the severe ringing effect in the blurred images, which is a characteristic of ideal filters. It is due to the discontinuity in the filter transfer function. Example Ideal low pass filter is not practical, because it causes ringing effect. How to avoid this ringing effect?

13 Choice Of Cutoff Frequency in LPF The cutoff frequency D0 of the ideal LPF determines the amount of frequency components passed by the filter. Smaller the value of D0, more the number of image components eliminated by the filter. In general, the value of D0 is chosen such that most components of interest are passed through, while most components not of interest are eliminated. Butterworth Lowpass Filter: BLPF Frequency response does not have a sharp transition more appropriate for image smoothing not introduce ringing. n : filter order D0 : cutoff frequency

14 Example

15 Gaussian Lowpass Filter: GLPF H ( u, v) = e D 2 ( u, v)/ 2D 2 0 Example

16 Example Example

17 Example

18 High Pass Filter Edges and sharp transitions in gray values in an image contribute significantly to high-frequency content of its Fourier transform. Regions of relatively uniform gray values in an image contribute to low-frequency 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 high-pass filter! For simplicity, we will consider only those filters that are real and symmetric. 0 H ( u, v) = 1 Sharpening Frequency Domain Filter: Ideal highpass filter if D(u, v) if D(u, v) D > D Butterworth highpass filter 1 H ( u, v) = 2 1+ [ D D( u, v ] n 0 ) Gaussian highpass filter H ( u, v) = 1 e D ( u, v)/ 2D

19 High Pass Filters (HPF) Butterworth: 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. Gaussian: The parameter D 0 measures the spread of the Gaussian curve. Larger the value of D 0, larger the cutoff frequency. No ringing effect Spatial representation of Ideal, Butterworth and Gaussian highpass filters

20 Example: result of IHPF Example: result of BHPF

21 Example: result of GHPF

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23 Unsharp Masking and High-Boost Filtering in the Frequency Domain

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