fourier_analysis

fourier_analysis

Fourier analysis toolkit with noise introduction, spectral analysis, high/low/pass filters.

Introduction to FFT or Fast Fourier Transforms

Let 1D fft be define as a 1D_FFT(input) where input is a 1D signal. A 1D FFT computes the discrete frequencies in a signal. Effectively it takes a signal and isolates all the possible frequecies. It can be used for manipulation as seen below

For more info see my personal implementation where I solve the math and code from scratch.

A 2D FFT is as follows:: fft_array = []

for row in image: fft_array[row] = 1D_FFT(row)

for col in fft_array: fft_array[col] = = 1D_FFT(col)

Understanding a 2D FFT

### Observing General FFT Trends 
import image_processing as imp 
from image_processing import fourier_analysis,im_processing

###create an image with black lines  
vert_lines = fourier_analysis.gen_line_im(size=(100,100),samp_freq=5,vert=True,balance = True) 
horiz_lines= fourier_analysis.gen_line_im(size=(100,100),samp_freq=5,vert=False,balance = True) 
checker= fourier_analysis.gen_line_im(size=(100,100),samp_freq=5,checker= True) 
rotated = im_processing.rot45(vert_lines)
box= fourier_analysis.gen_square_im(size=(100,100),n=2,val=1)


ims = [vert_lines,horiz_lines,checker,rotated,box]
ims = [fourier_analysis.im_noise(im).poisson() for im in ims]
titles= ['vert_lines','horiz_lines','checker','rotated45','square']

ims.extend([fourier_analysis.magnitude_spectrum(img) for img in ims]) #get FFT

titles.extend([t +' FFT' for t in titles])


imp.im_subplot(ims,shape=[2,5],titles=titles,
	suptitle='Understanding Fast Fourier Transforms (FFT)')

What this script did was add created vertical, horizontal, checkered, and diaganol lined images with a little bit of noise (More detail on noise below. Was done to see clearer FFT's)

General FFT Observations:

• vertical stripes ~ horizontally-oriented frequencies
• diagonal stripes ~ strong (opposite) diagonal components in FFT
• Images with both horizontal and vertical features have both vertical and horizontal components in FFT

• high frequencies are far from the origin, lower are closer

Noise Distribution Class :: im_noise(img)

• Gaussian: add normal distributed noise with set mean and standard deviation | N(mu,sig)

• Salt & Pepper: Replaces random pixels with 0 or 1. Can control amount of noise and ratio of 0/1 (s/p)

• Poisson: Poisson Distribution

• Speckle: img + (img x N(0,1)) where N is gaussian distribution

import image_processing as imp 
from image_processing import fourier_analysis

img = imp.cv_read(file_path='images/img_0.jpg',RGB=True) #Read RGB

noise= fourier_analysis.im_noise(img) #Noise Class 

ims = [img,noise.gaussian(mean=0,sigma=40), noise.salt_pepper(s_vs_p = 0.5, amount = 0.04),noise.poisson(),noise.speckle()] #add noise to ims

titles= ['Orig Image','Gaussian Noise', 'Salt/Pepper Noise', 'Possion Noise','Speckle Noise']

imp.im_subplot(ims,titles=titles,suptitle='Noise') #compare 

##Continued 

ffts= [fourier_analysis.magnitude_spectrum(img) for img in ims] #get fft 2D

titles= [t+' FFT' for t in titles]

imp.im_subplot(ffts,titles=titles,
	suptitle='Noise Fast Fourier Transforms (FFT)')