Using Fourier Transform to increase the spatial resolution

I have showed how to use bilinear interpolation of pixel values to resample the pixel size of an image. Here I will show how to resample the pixel size using the sampling theory in Fourier Transforms.

import pyfits
import matplotlib.pyplot as plt
import numpy as np
from numpy.fft import fft2, fftshift, ifft2, ifftshift

Load the target (suppose for the sake of simplicity that it's a quadratic image).

path = '/home/fatima/kahil_data/project_4/grey_simulation_100/analysis_mu=1/'
data = pyfits.getdata(path+'synth_continuum_profile.fits')

fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(1,1,1) 
ax.imshow(data,cmap='gray')

<matplotlib.image.AxesImage at 0x7feb88d82e50>

This is a synthesized brightness image of a quiet sun target, so it has a pefect spatial resolution. Let us degrade it a bit to mimic actual data

from scipy.ndimage.filters import gaussian_filter as gf

def Sigma(FWHM_arcsec, pixel_size):
    
    FWHM_pixels = FWHM_arcsec /pixel_size
    Sigma = FWHM_pixels /(2*np.sqrt(2*np.log(2)))
    return Sigma

data_deg = gf(data,sigma=Sigma(0.23,0.1),mode='nearest')

fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(1,1,1) 
ax.imshow(data_deg,cmap='gray')

<matplotlib.image.AxesImage at 0x7feb73840390>

The idea is simple, we know that in the Fourier transform image, pixels away from the center correspond to high spatial frequncey (small-scale features), so when we enlarge the FT of an image, we are adding higher spatial frequency components and therefore enhacing the resolution.

w_in, h_in = data.shape
w = h = 2*w_in
## FFTing the image
a = np.fft.fft2(data_deg)
a = np.fft.fftshift(a)
## embeding in a larger array
b = np.zeros((h,w),dtype=complex)
b[(h-h_in)/2:(h+h_in)/2,(w-w_in)/2:(w+w_in)/2] = a 
## iFFT
b = np.fft.ifftshift(b)
b = np.fft.ifft2(b)
data_resamp = np.abs(b)

fig = plt.figure(figsize=(10,10))
ax = fig.add_subplot(1,1,1) 
ax.imshow(data_resamp,cmap='gray')

<matplotlib.image.AxesImage at 0x7feb79139590>

This is a simple mathematical proof on why we have finer sampling after this procedure:

Let's suppose that the sampling of the image (of dimensions $N_1 \times N_1$) was $\Delta s$. After FFT the sampling becomes according to the sampling theory $$\frac{1}{N_1\times \Delta s}$$. After embeding the image in a larger array of dimension $N_2$ and iFFTing, the sampling becomes: $$\Delta s \times \frac{N_1}{N_2}$$ which is smaller than $\Delta s$ since $N_1/N_2 < 1$