MATLAB based Fast Fourier Transform (FFT) & Inverse Fast Fourier Transform (IFFT).
The following MATLAB implemented method of finding the FFT is based on the formula given below:

X(k)= Σ x(n)e^(-j*2*pi*k*n/N)
Σ is from k=0 to N-1

Similarly we have used the formula for IFFT as: 

x(n)= (1/N)[Σ X(K)e^(j*2*pi*k*n/N)]
Σ is from n=0 to N-1

If you want to make your work for calculating the FFT & IFFT function even more simpler, then you can feed in your sequence directly to fft() function of MATLAB.


MATLAB Code for calculating FFT (Fast Fourier Transform) & IFFT (Inverse Fast Fourier Transform):

clc;
close all;
clear all;
x = input('Enter the sequence for FFT = ');
N = length(x); %finding the length of entered sequence for N point FFT

for k = 1:N %for every value of k
y(k) = 0;
for n = 1:N %for every value of n
y(k) = y(k)+x(n)*exp(-1i*2*pi*(k-1)*(n-1)/N);
%as per FFT formula 0 to n-1
end
end

t = 0:N-1;
subplot(3,1,1);
stem(t,x); % for discrete plotting
%plotting input sequence
ylabel('Amplitude');
xlabel('n');
title('Input');

magnitude = abs(y); % absolute values of individual FFT complex
disp('FFT : ');
disp(magnitude);

t = 0:N-1;
subplot(3,1,2);
stem(t,magnitude);
%plotting FFT sequence
ylabel('Amplitude');
xlabel('K');
title('FFT');

R = length(y); %length of the fft array

for n = 1:R
x1(n) = 0;
for k = 1:R %loop as per the IFFT formula
    x1(n) = x1(n)+(1/R)*y(k)*exp(1i*2*pi*(k-1)*(n-1)/R);
end
end


t = 0:R-1;
subplot(3,1,3);
stem(t,x1);
%ploting IFFT sequence
disp('IFFT :');
disp(x1);
ylabel('Amplitude');
xlabel('n');
title('IFFT');


Sample Input & Output:

Enter the sequence for FFT = [1 2 3 4 5]

FFT Sequence =
   15.0000    4.2533    2.6287    2.6287    4.2533

IFFT Sequence =
 
   1.0000 - 0.0000i    2.0000 - 0.0000i    3.0000    4.0000   5.0000 + 0.0000i


Output Graph:

MATLAB based Fast Fourier Transform (FFT) & Inverse Fast Fourier Transform (IFFT) Calculation, Without Using fft & ifft inbuilt MATLAB function

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Author: Vibhutesh Kumar Singh

An active & prominent author at Digital iVision Labs! Specializing in MATLAB, C++, Arduino, OpenCV, NI Labview, Web Designing & other Electronics stuffs! Just started M.Tech. From IIIT Delhi, looking for excellent PhD Opportunities with prominent researchers. Drop a mail: vibhutesh[at]gmail.com or Follow him at....

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Yes making your histogram to shift right  & making your image more bright are one & the same thing. Very easy to perform & you just need to apply some very basic arithmetic operation on the image. The operations could be either addition or multiplication with a constant greater than 1.

For this article, we have taken a 640x640 grayscale image which is kept in the Current Working Directory. After opening it in MATLAB we will perform the further operations.

Histogram Left Shifting by subtraction of a constant: 

MATLAB Code & Explanation:

%Suppose we are adding a constant value of 20 from each element of the grayscale image

%In the results it will be clearly visible that the image has become more bright & the

%histogram is shifted to right side, & no values in the histogram for the value range 0-20

% i.e., at first 20 values in the histogram is vanished.

x=imread('new.jpg'); 

% if the image is not in the current working directory you need to specify the full path.

imshow(x);

%for showing the original image

figure,imhist(x)

%for showing the histogram of original image

n=x+20;

%performing the arithmetic operation on each element of the image  & storing the result in new variable

figure,imhist(n)

%for showing the operated image

figure,imshow(n)

%for showing the histogram of operated image


Results of The above MATLAB code of Left Shifting the Histogram using subraction

Original Unprocessed Image

Histogram of Original Image

Image After the arithmetic Operation (More Darker Than Original)

The right shifted histogram of the image after operation

Histogram Left Shifting By multiplication of a constant greater than 1 (>1): 

MATLAB Code & Explanation:

x=imread('new.jpg'); 

% if the image is not in the current working directory you need to specify the full path.

imshow(x);

%for showing the original image

figure,imhist(x)

%for showing the histogram of original image

n=x*1.25;

%performing the arithmetic operation on each element of the image  & storing the result in new variable

figure,imhist(n)

%for showing the operated image

figure,imshow(n)

%for showing the histogram of operated image


Results of The above MATLAB code of Left Shifting the Histogram using multiplication

Original Unprocessed Image

Histogram of Original Image

Image After the Multiplication with a constant greater than 1

Histogram of Original Image

In this case the histogram is not seeming to be shifted but it is, as 0x(anything)=0 only the values with pixel value '0' is not changed.

In both the cases, the right shifted histogram is roughly the shape of the original one.

NOTE:  If the constant which is being added or multiplied is very large you are going to obtain a nearly saturated image or a histogram very much shifted to the right

Author: Vibhutesh Kumar Singh

An active & prominent author at Digital iVision Labs! Specializing in MATLAB, C++, Arduino, OpenCV, NI Labview, Web Designing & other Electronics stuffs! Just started M.Tech. From IIIT Delhi, looking for excellent PhD Opportunities with prominent researchers. Drop a mail: vibhutesh[at]gmail.com or Follow him at....

Facebook

Twitter

Google+

Linkedin

Yes making your histogram to shift left & making the image more dark or reducing the brightness are one & the same thing. Very easy to perform & you just need to apply some very basic arithmetic operation on the image. The operations could be either subtraction or multiplication with a constant less than 1.

For this article, we have taken a 640x640 grayscale image which is kept in the Current Working Directory. After opening it in MATLAB we will perform the further operations.

Histogram Left Shifting by subtraction of a constant: 

MATLAB Code & Explanation:

%Suppose we are subtracting a constant value of 20 from each element of the grayscale image

%In the results it will be clearly visible that the image has become darker or less bright & the

%histogram is shifted to left side, & there are no values in the histogram for the value range 235-255

% i.e., at last 20 values in the histogram is vanished.

x=imread('new.jpg'); 

% if the image is not in the current working directory you need to specify the full path.

imshow(x);

%for showing the original image

figure,imhist(x)

%for showing the histogram of original image

n=x-20;

%performing the arithmetic operation on each element of the image  & storing the result in new variable

figure,imhist(n)

%for showing the operated image

figure,imshow(n)

%for showing the histogram of operated image


Results of The above MATLAB code of Left Shifting the Histogram using subraction

Original Unprocessed Image

Histogram of Original Image

Image After the arithmetic Operation (More Darker Than Original)

Left Shifted Histogram after the subtraction operation on the image

Histogram Left Shifting By multiplication of a constant less than 1 (<1): 

MATLAB Code & Explanation:

x=imread('new.jpg'); 

% if the image is not in the current working directory you need to specify the full path.

imshow(x);

%for showing the original image

figure,imhist(x)

%for showing the histogram of original image

n=x*0.85;

%performing the arithmetic operation on each element of the image  & storing the result in new variable

figure,imhist(n)

%for showing the operated image

figure,imshow(n)

%for showing the histogram of operated image


Results of The above MATLAB code of Left Shifting the Histogram using multiplication

Original Unprocessed Image

Histogram of Original Image

Image After the arithmetic Operation (More Darker Than Original)

Histogram of Processed Image

In both the cases, the left shifted histogram is roughly the shape of the original one.

Author: Vibhutesh Kumar Singh

An active & prominent author at Digital iVision Labs! Specializing in MATLAB, C++, Arduino, OpenCV, NI Labview, Web Designing & other Electronics stuffs! Just started M.Tech. From IIIT Delhi, looking for excellent PhD Opportunities with prominent researchers. Drop a mail: vibhutesh[at]gmail.com or Follow him at....

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