Real Transforms

This notebook shows how to upsample a signal by performing zero padding in frequency space, using either the FFT, DCT or DST.

By looking at the upsampled signal, the implicit boundary condition of each transform is clear:

• the FFT assumes circulant boundaries, and the upsampled signal wraps around the field of view.
• the DCT-2 assumes reflective boundaries, and the upsampled signal has zero derivatives at the boundaries (Neumann).
• the DST-2 assumes antireflective boundaries, and the upsampled signal has zero values at the boundaries (Dirichlet).

In [15]:

import torch
from torch.fft import fftn, ifftn, fftshift, ifftshift
from bounds import dctn2, dstn2, idctn2, idstn2, pad
import matplotlib.pyplot as plt

import torch from torch.fft import fftn, ifftn, fftshift, ifftshift from bounds import dctn2, dstn2, idctn2, idstn2, pad import matplotlib.pyplot as plt

In [19]:

# Spatial signal
torch.manual_seed(1234)
x = torch.randn([4, 4])

# Fourier coefficients
k = ifftshift(fftn(fftshift(x, dim=[0, 1]), dim=[0, 1]), dim=[0, 1])

# Zero-padded coefficients
kz = pad(k, [14, 14], side='both')

# Back to spatial signal
y = ifftshift(ifftn(fftshift(kz, dim=[0, 1]), dim=[0, 1]), dim=[0, 1]).real

plt.subplot(1, 4, 1)
plt.imshow(x)
plt.axis('off')
plt.subplot(1, 4, 2)
plt.imshow(k.real)
plt.axis('off')
plt.subplot(1, 4, 3)
plt.imshow(kz.real)
plt.axis('off')
plt.subplot(1, 4, 4)
plt.imshow(y)
plt.axis('off')
plt.show()

# Spatial signal torch.manual_seed(1234) x = torch.randn([4, 4]) # Fourier coefficients k = ifftshift(fftn(fftshift(x, dim=[0, 1]), dim=[0, 1]), dim=[0, 1]) # Zero-padded coefficients kz = pad(k, [14, 14], side='both') # Back to spatial signal y = ifftshift(ifftn(fftshift(kz, dim=[0, 1]), dim=[0, 1]), dim=[0, 1]).real plt.subplot(1, 4, 1) plt.imshow(x) plt.axis('off') plt.subplot(1, 4, 2) plt.imshow(k.real) plt.axis('off') plt.subplot(1, 4, 3) plt.imshow(kz.real) plt.axis('off') plt.subplot(1, 4, 4) plt.imshow(y) plt.axis('off') plt.show()

In [20]:

# Spatial signal
torch.manual_seed(1234)
x = torch.randn([4, 4])

# Fourier coefficients
k = dctn2(x, dim=[0, 1])

# Zero-padded coefficients
kz = pad(k, [28, 28], side='post')

# Back to spatial signal
y = idctn2(kz, dim=[0, 1])

plt.subplot(1, 4, 1)
plt.imshow(x)
plt.axis('off')
plt.subplot(1, 4, 2)
plt.imshow(k)
plt.axis('off')
plt.subplot(1, 4, 3)
plt.imshow(kz)
plt.axis('off')
plt.subplot(1, 4, 4)
plt.imshow(y)
plt.axis('off')
plt.show()

# Spatial signal torch.manual_seed(1234) x = torch.randn([4, 4]) # Fourier coefficients k = dctn2(x, dim=[0, 1]) # Zero-padded coefficients kz = pad(k, [28, 28], side='post') # Back to spatial signal y = idctn2(kz, dim=[0, 1]) plt.subplot(1, 4, 1) plt.imshow(x) plt.axis('off') plt.subplot(1, 4, 2) plt.imshow(k) plt.axis('off') plt.subplot(1, 4, 3) plt.imshow(kz) plt.axis('off') plt.subplot(1, 4, 4) plt.imshow(y) plt.axis('off') plt.show()

In [21]:

# Spatial signal
torch.manual_seed(1234)
x = torch.randn([4, 4])

# Fourier coefficients
k = dstn2(x, dim=[0, 1])

# Zero-padded coefficients
kz = pad(k, [28, 28], side='post')

# Back to spatial signal
y = idstn2(kz, dim=[0, 1])

plt.subplot(1, 4, 1)
plt.imshow(x)
plt.axis('off')
plt.subplot(1, 4, 2)
plt.imshow(k)
plt.axis('off')
plt.subplot(1, 4, 3)
plt.imshow(kz)
plt.axis('off')
plt.subplot(1, 4, 4)
plt.imshow(y)
plt.axis('off')
plt.show()

# Spatial signal torch.manual_seed(1234) x = torch.randn([4, 4]) # Fourier coefficients k = dstn2(x, dim=[0, 1]) # Zero-padded coefficients kz = pad(k, [28, 28], side='post') # Back to spatial signal y = idstn2(kz, dim=[0, 1]) plt.subplot(1, 4, 1) plt.imshow(x) plt.axis('off') plt.subplot(1, 4, 2) plt.imshow(k) plt.axis('off') plt.subplot(1, 4, 3) plt.imshow(kz) plt.axis('off') plt.subplot(1, 4, 4) plt.imshow(y) plt.axis('off') plt.show()