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filters for operations on regular grids (in fourier space)

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Martin Diehl 2019-11-26 10:25:39 +01:00
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import numpy as np
def curl(size,field):
"""Calculate curl of a vector or tensor field in Fourier space."""
shapeFFT = np.array(np.shape(field))[0:3]
grid = np.array(np.shape(field)[2::-1])
N = grid.prod() # field size
n = np.array(np.shape(field)[3:]).prod() # data size
field_fourier = np.fft.rfftn(field,axes=(0,1,2),s=shapeFFT)
curl_fourier = np.empty(field_fourier.shape,'c16')
k_sk = np.where(np.arange(grid[2])>grid[2]//2,np.arange(grid[2])-grid[2],np.arange(grid[2]))/size[0]
if grid[2]%2 == 0: k_sk[grid[2]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
k_sj = np.where(np.arange(grid[1])>grid[1]//2,np.arange(grid[1])-grid[1],np.arange(grid[1]))/size[1]
if grid[1]%2 == 0: k_sj[grid[1]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
k_si = np.arange(grid[0]//2+1)/size[2]
kk, kj, ki = np.meshgrid(k_sk,k_sj,k_si,indexing = 'ij')
k_s = np.concatenate((ki[:,:,:,None],kj[:,:,:,None],kk[:,:,:,None]),axis = 3).astype('c16')
e = np.zeros((3, 3, 3))
e[0, 1, 2] = e[1, 2, 0] = e[2, 0, 1] = +1.0 # Levi-Civita symbol
e[0, 2, 1] = e[2, 1, 0] = e[1, 0, 2] = -1.0
curl_fourier = np.einsum('slm,ijkl,ijkm, ->ijks', e,k_s,field_fourier)*2.0j*np.pi if n == 3 else# vector, 3 -> 3
np.einsum('slm,ijkl,ijknm,->ijksn',e,k_s,field_fourier)*2.0j*np.pi # tensor, 3x3 -> 3x3
return np.fft.irfftn(curl_fourier,axes=(0,1,2),s=shapeFFT).reshape([N,n])
def divergence(size,field):
"""Calculate divergence of a vector or tensor field in Fourier space."""
shapeFFT = np.array(np.shape(field))[0:3]
grid = np.array(np.shape(field)[2::-1])
N = grid.prod() # field size
n = np.array(np.shape(field)[3:]).prod() # data size
field_fourier = np.fft.rfftn(field,axes=(0,1,2),s=shapeFFT)
div_fourier = np.empty(field_fourier.shape[0:len(np.shape(field))-1],'c16')
k_sk = np.where(np.arange(grid[2])>grid[2]//2,np.arange(grid[2])-grid[2],np.arange(grid[2]))/size[0]
if grid[2]%2 == 0: k_sk[grid[2]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
k_sj = np.where(np.arange(grid[1])>grid[1]//2,np.arange(grid[1])-grid[1],np.arange(grid[1]))/size[1]
if grid[1]%2 == 0: k_sj[grid[1]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
k_si = np.arange(grid[0]//2+1)/size[2]
kk, kj, ki = np.meshgrid(k_sk,k_sj,k_si,indexing = 'ij')
k_s = np.concatenate((ki[:,:,:,None],kj[:,:,:,None],kk[:,:,:,None]),axis = 3).astype('c16')
div_fourier = np.einsum('ijkl,ijkl ->ijk', k_s,field_fourier)*2.0j*np.pi if n == 3 else # vector, 3 -> 1
np.einsum('ijkm,ijklm->ijkl',k_s,field_fourier)*2.0j*np.pi # tensor, 3x3 -> 3
return np.fft.irfftn(div_fourier,axes=(0,1,2),s=shapeFFT).reshape([N,n//3])
def gradient(size,field):
"""Calculate gradient of a vector or scalar field in Fourier space."""
shapeFFT = np.array(np.shape(field))[0:3]
grid = np.array(np.shape(field)[2::-1])
N = grid.prod() # field size
n = np.array(np.shape(field)[3:]).prod() # data size
field_fourier = np.fft.rfftn(field,axes=(0,1,2),s=shapeFFT)
grad_fourier = np.empty(field_fourier.shape+(3,),'c16')
k_sk = np.where(np.arange(grid[2])>grid[2]//2,np.arange(grid[2])-grid[2],np.arange(grid[2]))/size[0]
if grid[2]%2 == 0: k_sk[grid[2]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
k_sj = np.where(np.arange(grid[1])>grid[1]//2,np.arange(grid[1])-grid[1],np.arange(grid[1]))/size[1]
if grid[1]%2 == 0: k_sj[grid[1]//2] = 0 # Nyquist freq=0 for even grid (Johnson, MIT, 2011)
k_si = np.arange(grid[0]//2+1)/size[2]
kk, kj, ki = np.meshgrid(k_sk,k_sj,k_si,indexing = 'ij')
k_s = np.concatenate((ki[:,:,:,None],kj[:,:,:,None],kk[:,:,:,None]),axis = 3).astype('c16')
grad_fourier = np.einsum('ijkl,ijkm->ijkm', field_fourier,k_s)*2.0j*np.pi if n == 1 else # scalar, 1 -> 3
np.einsum('ijkl,ijkm->ijklm',field_fourier,k_s)*2.0j*np.pi # vector, 3 -> 3x3
return np.fft.irfftn(grad_fourier,axes=(0,1,2),s=shapeFFT).reshape([N,3*n])
#--------------------------------------------------------------------------------------------------
def displacementFluctFFT(F,size):
"""Calculate displacement field from deformation gradient field."""
integrator = 0.5j * size / np.pi
kk, kj, ki = np.meshgrid(np.where(np.arange(grid[2])>grid[2]//2,np.arange(grid[2])-grid[2],np.arange(grid[2])),
np.where(np.arange(grid[1])>grid[1]//2,np.arange(grid[1])-grid[1],np.arange(grid[1])),
np.arange(grid[0]//2+1),
indexing = 'ij')
k_s = np.concatenate((ki[:,:,:,None],kj[:,:,:,None],kk[:,:,:,None]),axis = 3)
k_sSquared = np.einsum('...l,...l',k_s,k_s)
k_sSquared[0,0,0] = 1.0 # ignore global average frequency
#--------------------------------------------------------------------------------------------------
# integration in Fourier space
displacement_fourier = -np.einsum('ijkml,ijkl,l->ijkm',
np.fft.rfftn(F,axes=(0,1,2)),
k_s,
integrator,
) / k_sSquared[...,np.newaxis]
#--------------------------------------------------------------------------------------------------
# backtransformation to real space
return np.fft.irfftn(displacement_fourier,grid[::-1],axes=(0,1,2))