This document summarizes a presentation on implementing and comparing low pass filters in the frequency domain. It introduces low pass filters and their use in smoothing images by reducing high frequencies. It then compares ideal, Butterworth, and Gaussian low pass filters. The document demonstrates implementing each filter type in MATLAB on sample images and analyzing the results. Code examples are provided for applying the different low pass filters using 2D fast Fourier transforms.

1. IMPLEMENTATION AND
COMPARISON OF LOW PASS
FILTERS IN FREQUENCY
DOMAIN
PRESENTERS:
ZARA TARIQ 1573119
SAPNA KUMARI 1573131

2. AGENDA
 Introduction
 Low Pass Filters
 Comparison Between Types of LPF
 Implementation of LPF
 Demonstration of Implementation in MATLAB

3. INTRODUCTION - FILTERS IN FREQENCY DOMAIN
Image filtering in frequency domain can be grouped in three,
depending on the effects:
2. High pass filters (sharpening filters)
3. Notch Filters (band-stop filters)
1. Low pass filters (smoothing filters)

4. LOW PASS FILTERS (LPF)
Why Is It Used?:
 Creates a blurred (or smoothed) image
 Reduces the high frequencies and leave the low frequencies of the
Fourier transformation to relatively unchanged
How Does It Works?
 Low frequency components correspond to slow changes in images
 Used to remove high spatial frequency noise from a digital image
 The low-pass filters usually employ moving window operator which
affects one pixel of the image at a time, changing its value by some
function of a local region (window) of pixels.
 The operator moves over the image to affect all the pixels in the
image.

5. COMPARISON BETWEEN TYPES OF LPF
Ideal LPF:
 Cuts off all components that are greater than distance Do from center
Butterworth LPF:
 The transfer function of a Butterworth low pass filter of order n with cut-off
frequency at distance D0 from the origin
 No clear cut-off between passed & filtered frequencies
Gaussian LPF:
 Does not have sharp discontinuity
 Transfer function is smooth, like Butterworth filter

6. n
DvuD
vuH 2
0 ]/),([1
1
),(


Ideal LPF
Butterworth LPF
Gaussian LPF

7. IMPLEMENTATION OF
ALL TYPES OF LPF
EXAMPLE 1 - NOISY BIRD IMAGE

8. Original Image of noisy miner bathing image

9. Result of Ideal Low Pass Filter

10. Result of Butterworth Low Pass Filter

11. Result of Gaussian Low Pass Filter

12. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Fourier Spectrum of Bird Image

13. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Spectrum of Bird Image

14. IMPLEMENTATION OF
ALL TYPES OF LPF
EXAMPLE 2 - SIBERIAN HUSKY FOX IMAGE

15. Original Image of Siberian Husky Fox image

16. Result of Ideal Low Pass Filter

17. Result of Butterworth Low Pass Filter

18. Result of Gaussian Low Pass Filter

19. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Fourier Spectrum of Siberian Husky Fox Image

20. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Spectrum of Siberian Husky Fox Image

21. IMPLEMENTATION OF
ALL TYPES OF LPF
EXAMPLE 3 - ROSE FLOWER IMAGE

22. Original Image of Rose Flower image

23. Result of Ideal Low Pass Filter

24. Result of Butterworth Low Pass Filter

25. Result of Gaussian Low Pass Filter

26. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Fourier Spectrum of Rose Flower Image

27. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Spectrum of Rose Flower Image

28. IMPLEMENTATION OF
ALL TYPES OF LPF
EXAMPLE 4 - CAT IMAGE

29. Original Image of Wild Cat image

30. Result of Ideal Low Pass Filter

31. Result of Butterworth Low Pass Filter

32. Result of Gaussian Low Pass Filter

33. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Fourier Spectrum of Wild Cat Image

34. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Spectrum of Wild Cat Image

35. IMPLEMENTATION OF
ALL TYPES OF LPF
EXAMPLE 5 - MOVIE POSTER IMAGE

36. Original Image of Movie Poster image

37. Result of Ideal Low Pass Filter

38. Result of Butterworth Low Pass Filter

39. Result of Gaussian Low Pass Filter

40. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Fourier Spectrum of Movie Poster Image

41. Result of Butterworth Low Pass Filter Result of Gaussian Low Pass Filter
Original Image Result of Ideal Low Pass Filter
Spectrum of Movie Poster Image

42. DEMONSTRATION OF
IMPLEMENTATION IN
MATLAB

43. APPENDIX – MATLAB CODE
Low Pass Filter Function
H = lpfilter(type, M, N, D0, n)
[U, V] = dftuv(M, N);
D = sqrt(U.^2 + V.^2);
switch type
case 'ideal'
H = double(D <=D0);
case 'btw'
if nargin == 4
n = 1;
end
H = 1./(1 + (D./D0).^(2*n));
case 'gaussian'
H = exp(-(D.^2)./(2*(D0^2)));
otherwise
error('Unknown filter type.')
end

44. APPENDIX – MATLAB CODE
Ideal Low Pass Filter
fox=imread('fox.jpg');
fox=rgb2gray(fox);
imshow(fox)
PQ = paddedsize(size(fox));
D0 = 0.05*PQ(1);
H = lpfilter('ideal', PQ(1), PQ(2), D0);
F=fft2(double(fox),size(H,1),size(H,2));
LPFS_fox = H.*F;
LPF_fox=real(ifft2(LPFS_fox));
LPF_fox=LPF_fox(1:size(fox,1), 1:size(fox,2));
figure, imshow(LPF_fox, [])
Fc=fftshift(F);
Fcf=fftshift(LPFS_fox);
S1=log(1+abs(Fc));
S2=log(1+abs(Fcf));
figure, imshow(S1,[])
figure, imshow(S2,[])

45. APPENDIX – MATLAB
CODE
Butterworth Low Pass Filter
fox=imread('fox.jpg');
fox=rgb2gray(fox);
imshow(fox)
PQ = paddedsize(size(fox));
D0 = 0.05*PQ(1);
H = lpfilter('btw', PQ(1), PQ(2), D0);
F=fft2(double(fox),size(H,1),size(H,2));
LPFS_fox = H.*F;
LPF_fox=real(ifft2(LPFS_fox));
LPF_fox=LPF_fox(1:size(fox,1), 1:size(fox,2));
figure, imshow(LPF_fox, [])
Fc=fftshift(F);
Fcf=fftshift(LPFS_fox);
S1=log(1+abs(Fc));
S2=log(1+abs(Fcf));
figure, imshow(S1,[])
figure, imshow(S2,[])

46. APPENDIX – MATLAB CODE
Gaussian Low Pass Filter
fox=imread('fox.jpg');
fox=rgb2gray(fox);
imshow(fox)
PQ = paddedsize(size(fox));
D0 = 0.05*PQ(1);
H = lpfilter('gaussian', PQ(1), PQ(2), D0);
F=fft2(double(fox),size(H,1),size(H,2));
LPFS_fox = H.*F;
LPF_fox=real(ifft2(LPFS_fox));
LPF_fox=LPF_fox(1:size(fox,1), 1:size(fox,2));
figure, imshow(LPF_fox, [])
Fc=fftshift(F);
Fcf=fftshift(LPFS_fox);
S1=log(1+abs(Fc));
S2=log(1+abs(Fcf));
figure, imshow(S1,[])
figure, imshow(S2,[])

47. APPENDIX – MATLAB CODE
Meshgrid Frequency Matrices.
function [U, V] = dftuv(M, N)
u = 0:(M-1);
v = 0:(N-1);
idx = find(u > M/2);
u(idx) = u(idx) - M;
idy = find(v > N/2);
v(idy) = v(idy) - N;
[V, U] = meshgrid(v, u);

48. APPENDIX – MATLAB CODE
Padding for Fourier Transform
function PQ = paddedsize(AB, CD, PARAM)
if nargin == 1
PQ = 2*AB;
elseif nargin == 2 & ~ischar(CD)
PQ = AB + CD - 1;
PQ = 2 * ceil(PQ / 2);
elseif nargin == 2
m = max(AB);
P = 2^nextpow2(2*m);
PQ = [P, P];
elseif nargin == 3
m = max([AB CD]);
P = 2^nextpow2(2*m);
PQ = [P, P];
else
error('Wrong number of inputs.')
end

49. THANK YOU!