1. IMAGE ENHANCEMENT IN THE FREQUENCY DOMAIN

2. Basics of Filtering in the Frequency Domain

3. Steps for Filtering in the Frequency Domain
Given an input image f(x,y) of size MxN, obtain padding parameters P and Q. Typically, P=2M and Q=2N.
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 centre its transform.
Compute the DFT, F(u,v), of the image from step 3.
Generate a real, symmetric filter function, H(u,v), of size PxQ with centre at coordinates (P/2,Q/2). Form the product G(u,v)=H(u,v)F(u,v) using array multiplication.
Obtain the processed image:
gp(x,y)=real[IDFT(G(u,v))]* (-1)x+y
Obtain the final processed result, g(x,y), by extracting the MxN region from the top, left quadrant of gp(x,y)

5. Smoothing Frequency-Domain Filters
The basic model for filtering in the frequency domain
where F(u,v): the Fourier transform of the image to be smoothed
H(u,v): a filter transfer function
Smoothing is fundamentally a lowpass operation in the frequency domain.
There are several standard forms of lowpass filters (LPF).
Ideal lowpass filter
Butterworth lowpass filter
Gaussian lowpass filter

6. IDEAL LOW PASS FILTER
The simplest low pass filter is a filter that “cuts off” all high-frequency components of the Fourier transform that are at a distance greater than a specified distance D0 from the origin of the transform.
The transfer function of an ideal low pass filter
where D(u,v) : the distance from point (u,v) to the center of the frequency rectangle

7. Contd…
Do-cutoff frequency

8. Contd…

10. SPATIAL REPRESENTATION

11. Butterworth Low pass Filters (BLPFs)

12. Butterworth
Low pass
Filters (BLPFs)
n=2
D0=5,15,30,80,and 230

13. SPATIAL REPRESENTATION

14. GAUSSIAN LOWPASS FILTERS (GLPFS)

15. Gaussian Low pass
Filters (GLPFs)
D0=5,15,30,80,and 230

16. Unsharp Masking ,High boost Filtering & High frequency Emphasis Filtering

17. Contd…
High frequency emphasis filtering is used to enhance the X-Ray images.

18. Homomorphic Filtering
f(x , y)=i(x , y)*r(x , y).
i(x,y)-illumination- low frequency
r(x,y)-reflectance-high frequency
g(x,y)=eS(u,v)
S(u , v)=output of IDFT block

19. Contd… The transfer function of the filter is given by
H(u,v)=(ϫH-ϫL)[1-exp(-c[D2(u,v)/Do2])]+ϫL
C-controls the sharpness
ϫL, ϫH controls the slope

20. ϫL<1 and ϫH>1 The parameters are selected like
The Homomorphic filter attenuates the contribution of low frequencies and amplifies the contribution made by high frequencies

21. Selective filtering Selective filtering falls under 3 categories BPF
BRF
Notch filtering

22. Band Reject Filer HBP(u,v)=1-HBR(u,v)
Now the expression for Band pass filter is given by
HBP(u,v)=1-HBR(u,v)

23. BRF BPF

24. THANKYOU