I am interested in usefull tricks and and banana skins when doing Fourier transformation in python.
Please give introductive code examples to get into the topic, as well as more advise in advanced topics like: frequency filters, continuous FT, high dimensional FT.
One should be carefull with the ordering of frequencies by the numpy.fft library. The fft routine orders first zero, then positive frequencies and then negative ones (Figure 1b, 1c). The function fftshift can be used to establish ordering as expected i.e. from negative to positive values one needs the fftshift routine (Figure 1d, 1e).
Code example:
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable
fig, ((ax1,ax2,ax3),(ax4,ax5,ax6)) = plt.subplots(2,3, figsize=(18.3,12.5))
#Spectrum and FT
spec=np.ndarray(shape=(12,9))
for ii in range(0,spec.shape[0]):
for jj in range(0,spec.shape[1]):
spec[ii,jj]=np.cos(2*np.pi*ii/spec.shape[0])*np.cos(4*np.pi*jj/spec.shape[1])
spec_ft = np.fft.fft2(spec)
spec_ftShifted = np.fft.fftshift(np.fft.fft2(spec))
#Frequencies / labels axes
xFreq=np.around(np.fft.fftfreq(spec.shape[0]),decimals=2)
yFreq=np.around(np.fft.fftfreq(spec.shape[1]),decimals=2)
xFreqShifted=np.around(np.fft.fftshift(np.fft.fftfreq(spec.shape[0])),decimals=2)
yFreqShifted=np.around(np.fft.fftshift(np.fft.fftfreq(spec.shape[1])),decimals=2)
#Plotting
ax=ax1
ax.set_title('1a) Spectrum')
im=ax.imshow(spec.T,origin='lower',cmap='bwr',interpolation='none')
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax2
ax.set_title('1b) FT Spectrum (not shifted): real part')
im=ax.imshow(spec_ft.real.T,origin='lower',cmap='afmhot',interpolation='none')
ax.set_xticks([0,1,int(spec.shape[0]/2),int(spec.shape[0]/2+1),spec.shape[0]-1])
ax.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(spec.shape[0]/2)],xFreq[int(spec.shape[0]/2+1)],xFreq[spec.shape[0]-1]])
ax.set_yticks([0,1,int(spec.shape[1]/2),int(spec.shape[1]/2+1),spec.shape[1]-1])
ax.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(spec.shape[1]/2)],yFreq[int(spec.shape[1]/2+1)],yFreq[spec.shape[1]-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax3
ax.set_title('1c) FT Spectrum (not shifted): imag part')
im=ax.imshow(spec_ft.imag.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,1,int(spec.shape[0]/2),int(spec.shape[0]/2+1),spec.shape[0]-1])
ax.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(spec.shape[0]/2)],xFreq[int(spec.shape[0]/2+1)],xFreq[spec.shape[0]-1]])
ax.set_yticks([0,1,int(spec.shape[1]/2),int(spec.shape[1]/2+1),spec.shape[1]-1])
ax.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(spec.shape[1]/2)],yFreq[int(spec.shape[1]/2+1)],yFreq[spec.shape[1]-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax5
ax.set_title('1d) FT Spectrum (shifted): real part')
im=ax.imshow(spec_ftShifted.real.T,origin='lower',cmap='afmhot',interpolation='none')
ax.set_xticks([0,1,int(spec.shape[0]/2),int(spec.shape[0]/2+1),spec.shape[0]-1])
ax.set_xticklabels([xFreqShifted[0],xFreqShifted[1],xFreqShifted[int(spec.shape[0]/2)],xFreqShifted[int(spec.shape[0]/2+1)],xFreqShifted[spec.shape[0]-1]])
ax.set_yticks([0,1,int(spec.shape[1]/2),int(spec.shape[1]/2+1),spec.shape[1]-1])
ax.set_yticklabels([yFreqShifted[0],yFreqShifted[1],yFreqShifted[int(spec.shape[1]/2)],yFreqShifted[int(spec.shape[1]/2+1)],yFreqShifted[spec.shape[1]-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax6
ax.set_title('1e) FT Spectrum (shifted): imag part')
im=ax.imshow(spec_ftShifted.imag.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,1,int(spec.shape[0]/2),int(spec.shape[0]/2+1),spec.shape[0]-1])
ax.set_xticklabels([xFreqShifted[0],xFreqShifted[1],xFreqShifted[int(spec.shape[0]/2)],xFreqShifted[int(spec.shape[0]/2+1)],xFreqShifted[spec.shape[0]-1]])
ax.set_yticks([0,1,int(spec.shape[1]/2),int(spec.shape[1]/2+1),spec.shape[1]-1])
ax.set_yticklabels([yFreqShifted[0],yFreqShifted[1],yFreqShifted[int(spec.shape[1]/2)],yFreqShifted[int(spec.shape[1]/2+1)],yFreqShifted[spec.shape[1]-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
plt.show()
Output:
The FT for real valued input arrays (rfft) makes use of the fact, that the FT has to be hermitian.
A function F is hermitian, when: F(f)=F(-f)*
Hence only an array of half the size needs to be stored.
Carefull: We will see in 3. that there are some banana skins, when using rfft.
Code example:
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable
fig, ((ax1,ax2,ax3)) = plt.subplots(1,3, figsize=(18.3,12.5))
#Spectrum and FT
spec=np.ndarray(shape=(10,8))
for ii in range(0,spec.shape[0]):
for jj in range(0,spec.shape[1]):
spec[ii,jj]=np.cos(2*np.pi*ii/spec.shape[0])*np.cos(4*np.pi*jj/spec.shape[1])
#spec[ii,jj]=(-1)**(ii+jj)
#spec[:,jj]=np.cos(2*np.pi*jj/spec.shape[1])
#spec[ii,:]=np.cos(2*np.pi*ii/spec.shape[0])
spec_ft=np.fft.fftshift(np.fft.rfft2(spec),axes=0)
#Frequencies / labels axes
xFreq=np.around(np.fft.fftshift(np.fft.fftfreq(spec.shape[0])),decimals=2)
yFreq=np.around(np.fft.rfftfreq(spec.shape[1]),decimals=2)
#Plotting
ax=ax1
ax.set_title('2a) Real spectrum')
im=ax.imshow(spec.T,origin='lower',cmap='bwr',interpolation='none')
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax2
ax.set_title('2b) FT: real part')
im=ax.imshow(spec_ft.real.T,origin='lower',cmap='afmhot',interpolation='none')
ax.set_xticks([0,1,int(len(xFreq)/2),int(len(xFreq)/2+1),len(xFreq)-1])
ax.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(len(xFreq)/2)],xFreq[int(len(xFreq)/2+1)],xFreq[len(xFreq)-1]])
ax.set_yticks([0,1,int(len(yFreq)/2),int(len(yFreq)/2+1),len(yFreq)-1])
ax.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(len(yFreq)/2)],yFreq[int(len(yFreq)/2+1)],yFreq[len(yFreq)-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax3.set_title('2c) FT: imag part')
im3=ax3.imshow(spec_ft.imag.T,origin='lower',cmap='bwr',interpolation='none')
ax3.set_xticks([0,1,int(len(xFreq)/2),int(len(xFreq)/2+1),len(xFreq)-1])
ax3.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(len(xFreq)/2)],xFreq[int(len(xFreq)/2+1)],xFreq[len(xFreq)-1]])
ax3.set_yticks([0,1,int(len(yFreq)/2),int(len(yFreq)/2+1),len(yFreq)-1])
ax3.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(len(yFreq)/2)],yFreq[int(len(yFreq)/2+1)],yFreq[len(yFreq)-1]])
divider0 = make_axes_locatable(ax3)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im3,cax=cax)
plt.show()
Output:
The problem with rfft is, that it not ambiguous. This means, that from the number of pixels n_ft of a FT ndarry, one can not conclude how many pixels the array in real space n_rs has. There are 2 options:
n_rs= (n_ft-1)*2 or n_rs= (n_ft-1)*2+1
Figures 3a and 3d show 2 real valued 2d-arrays. We see that their FT looks exactly the same (3b, 3e). The imaginary parts (3c and 3f) are both almost 0. So from the FT spectrum we can not conclude, what the real space spectrum looks like, if we do not know its dimension / shape. For this reason it might be more universal to use the standard fft instead of rfft.
Code example:
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable
fig, ((ax1,ax2,ax3),(ax4,ax5,ax6)) = plt.subplots(2,3,figsize=(20,10))
#Spectrum and FT
#generating spectrum: this is just used to generate real spectrum option 1/2 and is not plotted
#it is equivalent to: spec_ft1 and spec_ft2
spec_ft=np.zeros(shape=(10,5))
spec_ft[4,2]=20
spec_ft[6,2]=20
#
spec_r1=np.fft.irfft2(np.fft.ifftshift(spec_ft,axes=0),s=(spec_ft.shape[0],(spec_ft.shape[1]-1)*2))
spec_r2=np.fft.irfft2(np.fft.ifftshift(spec_ft,axes=0),s=(spec_ft.shape[0],(spec_ft.shape[1]-1)*2+1))
#
spec_ft1=np.fft.fftshift(np.fft.rfft2(spec_r1),axes=0)
spec_ft2=np.fft.fftshift(np.fft.rfft2(spec_r2),axes=0)
#Frequencies / labels axes
xFreq1=np.around(np.fft.fftshift(np.fft.fftfreq(spec_r1.shape[0])),decimals=2)
yFreq1=np.around(np.fft.rfftfreq(spec_r1.shape[1]),decimals=2)
xFreq2=np.around(np.fft.fftshift(np.fft.fftfreq(spec_r2.shape[0])),decimals=2)
yFreq2=np.around(np.fft.rfftfreq(spec_r2.shape[1]),decimals=2)
#Plotting
ax=ax1
ax.set_title('3a) Spectrum 1')
im=ax.imshow(spec_r1.real.T,origin='lower',cmap='bwr',interpolation='none')
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax2
ax.set_title('3b) FT Spectrum 1: real part')
im=ax.imshow(spec_ft1.real.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,int(len(xFreq1)/2),len(xFreq1)-1])
ax.set_xticklabels([xFreq1[0],xFreq1[int(len(xFreq1)/2)],xFreq1[-1]])
ax.set_yticks([0,len(yFreq1)-1])
ax.set_yticklabels([yFreq1[0],yFreq1[-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax3
ax.set_title('3c) FT Spectrum 1: imag part')
im=ax.imshow(spec_ft1.imag.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,int(len(xFreq1)/2),len(xFreq1)-1])
ax.set_xticklabels([xFreq1[0],xFreq1[int(len(xFreq1)/2)],xFreq1[-1]])
ax.set_yticks([0,len(yFreq1)-1])
ax.set_yticklabels([yFreq1[0],yFreq1[-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax4
ax.set_title('3d) Spectrum 2')
im=ax.imshow(spec_r2.real.T,origin='lower',cmap='bwr',interpolation='none')
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax5
ax.set_title('3e) FT Spectrum 2: real part')
im=ax.imshow(spec_ft2.real.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,int(len(xFreq2)/2),len(xFreq2)-1])
ax.set_xticklabels([xFreq2[0],xFreq2[int(len(xFreq2)/2)],xFreq2[-1]])
ax.set_yticks([0,len(yFreq2)-1])
ax.set_yticklabels([yFreq2[0],yFreq2[-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax6
ax.set_title('3f) FT Spectrum 2: imag part')
im=ax.imshow(spec_ft2.imag.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,int(len(xFreq2)/2),len(xFreq2)-1])
ax.set_xticklabels([xFreq2[0],xFreq2[int(len(xFreq2)/2)],xFreq2[-1]])
ax.set_yticks([0,len(yFreq2)-1])
ax.set_yticklabels([yFreq2[0],yFreq2[-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
plt.show()
Output:
Here is an example how to use FT without including the knowledge of the spectrum beeing real. We see taht the shape of the FT array is the same like the shape of the real space array.
Code example:
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable
fig, ((ax1,ax2,ax3)) = plt.subplots(1,3, figsize=(18.3,12.5))
#Spectrum and FT
spec=np.ndarray(shape=(11,8))
for ii in range(0,spec.shape[0]):
for jj in range(0,spec.shape[1]):
#spec[ii,jj]=(-1)**(ii+jj)
#spec[ii,jj]=np.sin(2*np.pi*ii/spec.shape[0])*np.sin(4*np.pi*jj/spec.shape[1])
spec[ii,jj]=np.cos(2*np.pi*ii/spec.shape[0])*np.cos(4*np.pi*jj/spec.shape[1])
#spec[:,jj]=np.cos(2*np.pi*jj/spec.shape[1])
#spec[ii,:]=np.cos(2*np.pi*ii/spec.shape[0])
spec_ft=np.fft.fftshift(np.fft.fft2(spec))
#Frequencies / labels axes
xFreq=np.around(np.fft.fftshift(np.fft.fftfreq(spec.shape[0])),decimals=2)
yFreq=np.around(np.fft.fftshift(np.fft.fftfreq(spec.shape[1])),decimals=2)
#Plotting
ax=ax1
ax.set_title('4a) Real spectrum')
im=ax.imshow(spec.T,origin='lower',cmap='bwr',interpolation='none')
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax2
ax.set_title('4b) FT: real part')
im=ax.imshow(spec_ft.real.T,origin='lower',cmap='afmhot',interpolation='none')
ax.set_xticks([0,1,int(len(xFreq)/2),int(len(xFreq)/2+1),len(xFreq)-1])
ax.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(len(xFreq)/2)],xFreq[int(len(xFreq)/2+1)],xFreq[len(xFreq)-1]])
ax.set_yticks([0,1,int(len(yFreq)/2),int(len(yFreq)/2+1),len(yFreq)-1])
ax.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(len(yFreq)/2)],yFreq[int(len(yFreq)/2+1)],yFreq[len(yFreq)-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax3
ax.set_title('4c) FT: imag part')
im=ax.imshow(spec_ft.imag.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,1,int(len(xFreq)/2),int(len(xFreq)/2+1),len(xFreq)-1])
ax.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(len(xFreq)/2)],xFreq[int(len(xFreq)/2+1)],xFreq[len(xFreq)-1]])
ax.set_yticks([0,1,int(len(yFreq)/2),int(len(yFreq)/2+1),len(yFreq)-1])
ax.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(len(yFreq)/2)],yFreq[int(len(yFreq)/2+1)],yFreq[len(yFreq)-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
plt.show()
Output:
When we perform the inverse FT, the rsulting real space ndarray will be a complex number. Because of the numeric accuracy the values will have some infinitesimal small imaginary part.
COde example:
import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.axes_grid1 import make_axes_locatable
fig, ((ax1,ax2,ax3)) = plt.subplots(1,3, figsize=(18.3,12.5))
#Spectrum and FT
spec_ft=np.zeros(shape=(10,8))
spec_ft[4,2]=20
spec_ft[4,6]=20
spec_ft[6,2]=20
spec_ft[6,6]=20
spec_ift=np.fft.ifft2(np.fft.ifftshift(spec_ft))
#Frequencies / labels axes
xFreq=np.around(np.fft.fftshift(np.fft.fftfreq(spec_ift.shape[0])),decimals=2)
yFreq=np.around(np.fft.fftshift(np.fft.fftfreq(spec_ift.shape[1])),decimals=2)
#Plotting
ax=ax1
ax.set_title('FT: real part')
im=ax.imshow(spec_ft.real.T,origin='lower',cmap='afmhot',interpolation='none')
ax.set_xticks([0,1,int(len(xFreq)/2),int(len(xFreq)/2+1),len(xFreq)-1])
ax.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(len(xFreq)/2)],xFreq[int(len(xFreq)/2+1)],xFreq[len(xFreq)-1]])
ax.set_yticks([0,1,int(len(yFreq)/2),int(len(yFreq)/2+1),len(yFreq)-1])
ax.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(len(yFreq)/2)],yFreq[int(len(yFreq)/2+1)],yFreq[len(yFreq)-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax2
ax.set_title('FT: imag part')
im=ax.imshow(spec_ft.imag.T,origin='lower',cmap='bwr',interpolation='none')
ax.set_xticks([0,1,int(len(xFreq)/2),int(len(xFreq)/2+1),len(xFreq)-1])
ax.set_xticklabels([xFreq[0],xFreq[1],xFreq[int(len(xFreq)/2)],xFreq[int(len(xFreq)/2+1)],xFreq[len(xFreq)-1]])
ax.set_yticks([0,1,int(len(yFreq)/2),int(len(yFreq)/2+1),len(yFreq)-1])
ax.set_yticklabels([yFreq[0],yFreq[1],yFreq[int(len(yFreq)/2)],yFreq[int(len(yFreq)/2+1)],yFreq[len(yFreq)-1]])
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
#
ax=ax3
ax.set_title('real cosine spectrum')
im=ax.imshow(spec_ift.real.T,origin='lower',cmap='bwr',interpolation='none')
divider0 = make_axes_locatable(ax)
cax=divider0.append_axes("right", size="5%", pad=0.05)
cbar=plt.colorbar(im,cax=cax)
plt.show()
Output:
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