Question

2) One application of Low-pass filter is noise reduction. However, it has adverse effect on the image edges and may cause Ring effect. To mitigate the aforementioned issues, Butterworth and Gaussian filters have been proposed. Use the image below and apply the following filters. a) Apply Ideal low-pass filter and change its parameters and explain the results on the image edges. b) Apply Butterworth low-pass filter and change its parameters to compare the results with the ideal filter. c) Apply Gaussian low-pass filter and change its parameters to compare the results with the ideal and Butterworth filters. (Hint: use Ipfilter MATLAB command). а a a a a a a a a b Repeat question 2 with the following high-pass filters. a) Ideal filter b) Butterworth filter c) High-frequency-emphasis of Butterworth filter d) Gaussian filter (Hint: use hpfilter MATLAB command).

Answer 1

#ideal Low Pass Filter

# Reading input image : input_image
input_image = imread('[name of input image].[file format]');
% R: no of rows (height of the image)
% C : no of columns (width of the image)
[R, C] = size(input_image);

# Getting Fourier Transform of the input_image
# using MATLAB library function fft2 (2D fast fourier transform)
FT_img = fft2(double(input_image));

#Assign Cut-off Frequency you find comfartable

#change this value accordingly

D0 = 20;

# Designing filter
u = 0:(R-1);
idx = find(u>R/2);
u(idx) = u(idx)-R;
v = 0:(C-1);
idy = find(v>C/2);
v(idy) = v(idy)-N;

# MATLAB library function meshgrid(v, u) returns
# 2D grid which contains the coordinates of vectors
# v and u. Matrix V with each row is a copy
# of v, and matrix U with each column is a copy of u
[V, U] = meshgrid(v, u);

#Calculating Euclidean Distance
D = sqrt(U.^2+V.^2);

#Comparing with the cut-off frequency and
# determining the filtering mask
H = double(D <= D0);

# Convolution between the Fourier Transformed
# image and the mask
G = H.*FT_img;

# Getting the resultant image by Inverse Fourier Transform
# of the convoluted image using MATLAB library function
# ifft2 (2D inverse fast fourier transform)
output_image = real(ifft2(double(G)));

# Displaying Input Image and Output Image
subplot(2, 1, 1), imshow(input_image),
subplot(2, 1, 2), imshow(output_image, [ ]);

Explaination:-

first input and read image file with correct image file format. Then Give small dimension value for image if it is bigger to reduce computation time.Image pixel values will be in matrix of given dimension.Then apply 2D fast fourier transform it will generate a complex array of value which is visualized . The white speckles will indicate high energy in low frequencies.Then set a favorable CUT-off frequency at 20 to50.Then design and apply low pass filter.Apply mesh grid technique and euclidean distance in image pixel matrix is calculate to optimise and reduce the noise on the image. Then converting the resultant information from spectrum back to gray scale image. It could be done by applying inverse shifting and inverse FFT operation.

Image will have low features and details but there will be reduction in image noise.

# Butter worth low pass filter

Similarly to above program we will apply order value

n= 2;

# assigning Cut-off frequency

D0 =20;

%designing filter

u = 0:(R-1);

v = 0:(R-1);

idx = find(u > R/2);

u(idx) = u(idx) - R;

idy = find(v > C/2);

v(idy) = v(idy) - C;

MATLAB library function meshgrid(v, u) returns

# 2D grid which contains the coordinates of vectors

# v and u. Matrix V with each row is a copy of v

# and matrix U with each column is a copy of u

[V, U] = meshgrid(v, u);

# Calculating Euclidean Distance

D = sqrt(U.^2 + V.^2);

#determine filtering mask

H = 1./(1 + (D./D0).^(2*n));

#Inverse Fourieer transform

output_image = real(ifft2(double(G)));

#Displaying Input Image and Output Image

subplot(2, 1, 1), imshow(input_image),

subplot(2, 1, 2), imshow(output_image, [ ]);

##Explaination

It removes high-frequency noise from a digital image and preserves low-frequency components. The transfer function of Butterworth low pass filter of order n is defined to 2

first input and read image file with correct image file format. Then Give small dimension value for image if it is bigger to reduce computation time.Image pixel values will be in matrix of given dimension.Then apply 2D fast fourier transform it will generate a complex array of value which is visualized . The white speckles will indicate high energy in low frequencies.We have to set order of Butterworth filter order to (n=2)Then set a favorable CUT-off frequency at 20 to50.Then design and apply a low pass filter.Apply mesh grid technique and euclidean distance in image pixel matrix.The Filtering mask is applied to reduce noise pixel value  calculate to optimise and reduce the noise on the image. Then converting the resultant information from spectrum back to gray scale image. It could be done by applying inverse shifting and inverse FFT operation.

#Comparsion between Ideal LP filter and LP Butterworth

The image will be more blurry and have less details compared to Ideal low pass filter.The butterworth filter is   It removes high-frequency noise from a digital image more aggressively and preserves low-frequency components.