IMAGE PROCESSING

" Two roads diverged in a wood, and I,
I took the one less traveled by,
And that has made all the difference "-Robert Frost

UPSAMPLING IN MATLAB

Upsampling is the process of inserting zeros in between the signal value in order to increase the size of the matrix.  We will discuss about upsampling in both spatial and time domain.
1.
1.1  Upsampling using MATLAB built-in function
1.2  Upsampling in 1D
1.3  Upsampling in 2D or image matrix
2.
2.1  Upsampling a 1D signal
2.2  Upsampling a image matrix

UPSAMPLING IN SPATIAL DOMAIN:

Given: 1-D array of size 1xN
Upsample by factor of 2:

Output: 1-D array of size 1x(2*N)
Where N represents length of the array, n represents the index starting from 0,1,2…,N
%UPSAMPLING USING MATLAB BUILT-IN FUNCTION 'UPSAMPLE'

A = 1:25;
B = upsample(A,2);

EXPLANATION:

The above MATLAB function will insert zeros in between the samples.

To upsample an array by ratio 2, update the output array as follows:
1.      If index(n) of the output array is divisible by 2(ratio), then update the output array with the value of the input array with index n/2
2.      Otherwise, insert zero

STEPS TO PERFORM:
1.      Consider an array A of size 1xM
2.      Obtain the upsample Ratio N
3.      Pre-allocate an array B of size 1x(M*N)
4.      If the index is divisible by N then update the array B with value of A else zero

MATLAB CODE:

%UPSAMPLING IN SPATIAL DOMAIN

A = 1:10;

%UPSAMPLING WITH RATIO 2
B = zeros([1 size(A,2)*2]);
B(1:2:end)=A;

%UPSAMPLING WITH RATIO N
N=3;
B=zeros([1 size(A,2)*N]);
B(1:N:end)=A;

EXPLANATION:
Let A = [1 2 3 4 5]
Let us upsample try to upsample A with ratio 2
Pre-allocate output matrix B = [0 0 0 0 0 0 0 0 0 0]
Substitute the value of the matrix A in indices divisible by the ratio i.e. 2 of matrix B
B(1:2:end) = A
Now B(1,3,5,7,9) =[1 2 3 4 5]
Thus B = [1 0 2 0 3 0 4 0 5 0]

NOTE:
Definition of upsampling is usually given assuming the index starts from zero. But in case of MATLAB, the index starts with one that is why the odd positions are considered instead of even.

STEPS TO PERFORM TO UPSAMPLE A 2D MATRIX:

MATLAB CODE:

%IMAGE UPSAMPLING

M = 2;
N = 3;

B = zeros([size(A,1)*M size(A,2)*N]);
B(1:M:end,1:N:end) = A;

figure,imagesc(B);colormap(gray);

sz = M*N;
H = fspecial('average',[sz sz]);
C = conv2(B,H,'same');

figure,imagesc(C);colormap(gray);

EXPLANATION:

The above image is the pixel representation of the zero inserted image. In each row, two zeros are inserted between the pixels and in the each column; single zero is inserted between the pixels. In spatial domain, inserting zeros will be quite visible so it’s advisable to perform any low pass filtering or approximation like spatial averaging or applying Gaussian.
In the above example, spatial averaging is done. In order to understand how spatial averaging is done using convolution check the following link:

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UPSAMPLING IN FREQUENCY DOMAIN

1.1  Upsampling using MATLAB built-in function
1.2  Upsampling in 1D
1.3  Upsampling in 2D or image matrix
2.1  Upsampling a 1D signal
2.2  Upsampling a image matrix

In Frequency domain, upsampling means nothing but the padding of zeros at the end of high frequency components on both sides of the signal.

STEPS TO PERFORM:

2.      Obtain the ratio to upsample
3.      Perform Fast Fourier Transform
4.      Shift the Low frequency components to the centre and High frequency components outside.
5.      Add zeros on both the sides of the image
6.      Shift the High frequency components to the centre and Low frequency components to the exterior (Inverse of fftshift)
7.      Perform Inverse Fast Fourier Transform
8.      Display the Upsampled Image

MATLAB CODE:

%UPSAMPLING IN FREQUENCY DOMAIN
%1D UPSAMPLING

FS = 100;
t  = 0:(1/FS):1;

A = 10*sin(2*pi*5*t);
figure,plot(t,A);

%FOURIER DOMAIN

FT = fft(A);
fq =linspace(-1/FS,1/FS,101);
figure,plot(fq,abs(FT));title('Before FFTSHIFT');

FT_s = fftshift(FT);
figure,plot(fq,abs(FT_s));title('After FFTSHIFT');

fq =linspace(-1/FS,1/FS,201);

%INVERSE FOURIER TRANSFORM

%UPSAMPLED SIGNAL
t1 = linspace(0,1,201);
figure,plot(t1,(IFT*2),'r',t,A,'g');

EXPLANATION:

Amplitude of the input original signal is 10 and the frequency is 5.
Similarly, the amplitude of the upsampled signal is 10 and the frequency is 5. The number of samples used to plot the signal is increased in the later case.

IMAGE UPSAMPLING IN FOURIER DOMAIN

MATLAB CODE:

%RATIO
RatioM = 3;
RatioN = 2;

%UPSAMPLING OVER EACH ROW

mnrow = round(size(A,2)*(RatioM-1)/2);
% 1D FFT ON EACH ROW
row_fft = fft(A,[],2);

%PAD WITH ZEROS ON BOTH SIDES OF EACH ROW

 Logarthmic scale
%UPSAMPLING OVER EACH COLUMN
mncol = round(size(A,1)*(RatioN-1)/2);

% 1D FFT ON EACH COLUMN

%PAD WITH ZEROS ON BOTH SIDES OF EACH COLUMN

 Logarthmic scale
%PERFORM 1D IFFT ON EACH COLUMN

%PERFORM 1D IFFT ON EACH ROW
ifft_col_row = ifft(ifftshift(ifft_col,2),[],2);

%DISPLAY THE IMAGE AFTER UPSAMPLING
res = abs(ifft_col_row);
res = uint8(res*(numel(res)/numel(A)));

figure,imagesc(res);

SIMPLE VERSION :

%RATIO
RatioM = 3;
RatioN = 2;

mnrow = round(size(A,2)*(RatioM-1)/2);
mncol = round(size(A,1)*(RatioN-1)/2);

%FFT ON 2D MATRIX
FT = fftshift(fft2(A));

%INVERSE FOURIER TRANSFORM

Img = uint8(abs(IFT)*(numel(IFT)/numel(A)));
figure,imagesc(Img);

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Fast Fourier Transform on 2 Dimensional matrix using MATLAB

Fast Fourier transformation on a 2D matrix can be performed using the MATLAB built in function ‘fft2()’.

Fourier transform is one of the various mathematical transformations known which is used to transform signals from time domain to frequency domain.
The main advantage of this transformation is it makes life easier for many problems when we deal a signal in frequency domain rather than time domain.

Example:
%FOURIER TRANSFORM ON A MATRIX
A  = zeros(5);
A ( : ) = 1:25;
display(A);
F_FFT  = fft2(A);
display(F_FFT);
%INVERSE FOURIER TRANSFORM

I_FFT = ifft2(F_FFT);
display(abs(I_FFT));

NOTE on ABSOLUTE VALUE:
When we use FFT2() or FFT(),the result we obtain in the frequency domain is of complex data type.
i.e It contains both the real  as well as the imaginary part.
Let   A=10+5i
A is a complex number as it contains both real and imaginary part.In this particular case ‘10’ is the real part and ‘5’ is the imaginary part.
abs(A) = 11.1803 is the absolute (also called modulus in few books or notations) value of A which is nothing but the magnitude. It can be arrived by using the below mentioned formula:
abs(A) = sqrt(real part^2+imaginary part^2).
= sqrt(10^2+5^2)
= sqrt(125)
= 11.1803 (approx)
Let’s try to understand how the Fourier transform on 2 dimensional data works with a simple example.
This method will be helpful to understand the up sampling and down sampling in both spatial and frequency domain.

1.       Consider a matrix A

2.       Perform 1 D Fast Fourier transform(FFT) on each row

1D FFT on first row (Note that the absolute value is only displayed and not the actual imaginary number):

Similarly, perform  1D FFT on each row:

NOTE: The figure represents the 1 D FFT on each row and the result is the absolute value of the complex data obtained using FFT.
3.       Perform 1 D Fast Fourier transform on each column.
On the matrix obtained from the previous step, compute 1D FFT column wise.

4.       Display the results obtained.

Flow Chart for Fast Fourier Transform on 2D :

INVERSE FOURIER TRANSFORM:

1.       Perform Inverse Fourier Transform on each column

2.       Perform IFFT on each row

3.       Display the original data

MATLAB CODE:
A=[110 20 140 0 220;
60 34 23 198 20;
15 12 126 230 15;
140 28 10 28 10;
11 12 19 85 100];

FFT_row = zeros(size(A));
FFT_col = zeros(size(A));

%Perform FFT on each row
for i=1:size(A,1)
FFT_row(i,:) = fft(A(i,:));
end

display(FFT_row);
%display(abs(FFT_row));

%Perform FFT on each column

for i=1:size(A,2)
FFT_col(:,i) = fft(FFT_row(:,i));
end

display(FFT_col);
%display(abs(FFT_col));

%INVERSE FOURIER TRANSFORM

IFFT_row = zeros(size(A));
IFFT_col = zeros(size(A));

%Perform Inverse Fourier Transform on each column
for i=1:size(A,2)
IFFT_col(:,i) = ifft(FFT_col(:,i));
end

%Perform IFFT on each row

for i=1:size(A,2)
IFFT_row(:,i) = ifft(IFFT_col(:,i));
end

display(abs(A))

ALTERNATE METHOD FOR INVERSE FOURIER TRANSFORM:
Instead of using ifft2() or ifft(), we can also use the following method to obtain the original data from the Fast Fourier transformed result :
1.       Obtain the conjugate of the Forward FFT
2.       Perform Forward fast Fourier transform
3.       Obtain the conjugate of the result from step 2.
4.       Divide it by the number of elements present in the matrix
5.       Obtain the original matrix

MATLAB CODE:
Conj_F = conj(F_FFT);
Conj_FFT = fft2(Conj_F);
IFFT_conj = conj(Conj_FFT)/numel(Conj_FFT)
display(abs(IFFT_conj));

Reference: Digital Image Processing  by Rafael C.Gonzalez, fourth Chapter.

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