def make_sinogram(mat, angles, pad_rate=0.5, pad_mode="edge"):
"""
Create a sinogram (series of 1D projections) from a 2D image based on the
Fourier slice theorem (Ref. [1]).
Parameters
----------
mat : array_like
Square array.
angles : array_like
1D array. List of angles (in radian) for projecting.
pad_rate : float
To apply padding before the FFT. The padding width equals to
(pad_rate * image_width).
pad_mode : str
Padding method. Full list can be found at numpy.pad documentation.
References
----------
.. [1] https://doi.org/10.1071/PH560198
"""
(nrow0, ncol0) = mat.shape
if nrow0 != ncol0:
raise ValueError("Width and height of the image are not the same")
if np.max(np.abs(angles)) > np.pi:
print("!!! Warning !!!\nMaking sure that the angles are converted to "
"Radian and in the range of [0; Pi]")
pad = int(pad_rate * nrow0)
mat_pad = np.pad(mat, pad, mode=pad_mode)
if mat_pad.shape[0] % 2 == 0:
mat_pad = np.pad(mat_pad, ((0, 1), (0, 1)), mode="edge")
(nrow, ncol) = mat_pad.shape
xcenter = (ncol - 1.0) * 0.5
ycenter = (nrow - 1.0) * 0.5
r_max = np.floor(max(xcenter, ycenter))
r_list = np.linspace(-r_max, r_max, ncol)
theta_list = -np.asarray(angles)
r_mat, theta_mat = np.meshgrid(r_list, theta_list)
x_mat = np.float32(
np.clip(xcenter + r_mat * np.cos(theta_mat), 0, ncol - 1))
y_mat = np.float32(
np.clip(ycenter + r_mat * np.sin(theta_mat), 0, nrow - 1))
mat_fft = fft.fftshift(fft.fft2(fft.ifftshift(mat_pad)))
mat_real = np.real(mat_fft)
mat_imag = np.imag(mat_fft)
sino_real = util.mapping(mat_real, x_mat, y_mat, order=5, mode="reflect")
sino_imag = util.mapping(mat_imag, x_mat, y_mat, order=5, mode="reflect")
sinogram = np.real(
fft.fftshift(fft.ifft(fft.ifftshift(sino_real + 1j * sino_imag,
axes=1)),
axes=1))
return sinogram[:, pad:ncol0 + pad]
def remove_ring_based_wavelet_fft(mat,
level=5,
size=1,
wavelet_name="db9",
sort=False):
"""
Remove ring artifacts in a reconstructed image by combining the polar
transform and the wavelet-fft-based method (Ref. [1]).
Parameters
----------
mat : array_like
Square array. Reconstructed image
level : int
Wavelet decomposition level.
size : int
Damping parameter. Larger is stronger.
wavelet_name : str
Name of a wavelet. Search pywavelets API for a full list.
sort : bool, optional
Apply sorting (Ref. [2]) if True.
Returns
-------
array_like
Ring-removed image.
References
----------
.. [1] https://doi.org/10.1364/OE.17.008567
.. [2] https://doi.org/10.1364/OE.26.028396
"""
(nrow, ncol) = mat.shape
if nrow != ncol:
raise ValueError(
"Width and height of the reconstructed image are not the same")
mask = util.make_circle_mask(ncol, 1.0)
(x_mat, y_mat) = util.rectangular_from_polar(ncol, ncol, ncol, ncol)
(r_mat, theta_mat) = util.polar_from_rectangular(ncol, ncol, ncol, ncol)
polar_mat = util.mapping(mat, x_mat, y_mat)
polar_mat = remo.remove_stripe_based_wavelet_fft(polar_mat,
level,
size,
wavelet_name,
sort=sort)
mat_rec = util.mapping(polar_mat, r_mat, theta_mat)
return mat_rec * mask
def remove_ring_based_fft(mat, u=20, n=8, v=1, sort=False):
"""
Remove ring artifacts in the reconstructed image by combining the polar
transform and the fft-based method.
Parameters
----------
mat : array_like
Square array. Reconstructed image
u : int
Cutoff frequency.
n : int
Filter order.
v : int
Number of rows (* 2) to be applied the filter.
sort : bool, optional
Apply sorting (Ref. [2]) if True.
Returns
-------
array_like
Ring-removed image.
References
----------
.. [1] https://doi.org/10.1063/1.1149043
.. [2] https://doi.org/10.1364/OE.26.028396
"""
(nrow, ncol) = mat.shape
if nrow != ncol:
raise ValueError(
"Width and height of the reconstructed image are not the same")
mask = util.make_circle_mask(ncol, 1.0)
(x_mat, y_mat) = util.rectangular_from_polar(ncol, ncol, ncol, ncol)
(r_mat, theta_mat) = util.polar_from_rectangular(ncol, ncol, ncol, ncol)
polar_mat = util.mapping(mat, x_mat, y_mat)
polar_mat = remo.remove_stripe_based_fft(polar_mat, u, n, v, sort=sort)
mat_rec = util.mapping(polar_mat, r_mat, theta_mat)
return mat_rec * mask
def generate_tilted_sinogram_chunk(data, start_index, stop_index, angle,
**option):
"""
Generate a chunk of tilted sinograms of a 3D tomographic dataset or a
hdf/nxs object.
Parameters
----------
data : array_like or hdf object
3D array.
start_index : int
Starting index of sinograms.
stop_index : int
Stopping index of sinograms.
angle : float
Tilted angle in degree.
option : list or tuple of int
To extract subset data along axis 0 from a hdf object. E.g option =
(start, stop, step)
Returns
-------
array_like
3D array. Chunk of tilted sinograms.
"""
if data.ndim != 3:
raise ValueError("Input must be a 3D data !!!")
(depth, height, width) = data.shape
if len(option) != 0:
opt = option[list(option.keys())[0]]
start, stop, step = opt
else:
start, stop, step = 0, depth, 1
list_idx = range(start, stop, step)
x_cen = (width - 1.0) / 2
y_cen = (height - 1.0) / 2
angle = angle * np.pi / 180.0
x_list = np.arange(0, width) - x_cen
y_list = np.arange(start_index, stop_index + 1) - y_cen
x_mat, y_mat = np.meshgrid(x_list, y_list)
x_mat1 = x_mat * np.cos(angle) - y_mat * np.sin(angle)
y_mat1 = x_mat * np.sin(angle) + y_mat * np.cos(angle)
x_mat1 = np.clip(x_cen + x_mat1, 0, width - 1)
y_mat1 = np.clip(y_cen + y_mat1, 0, height - 1)
y_min = np.int16(np.floor(np.amin(y_mat1)))
y_max = np.int16(np.ceil(np.amax(y_mat1))) + 1
y_mat1 = y_mat1 - y_min
sino_chunk = np.asarray([
util.mapping(data[i, y_min:y_max, :], x_mat1, y_mat1) for i in list_idx
])
return sino_chunk
def generate_tilted_profile_chunk(mat, start_index, stop_index, angle):
"""
Generate a chunk of tilted horizontal intensity-profiles of an image.
Parameters
----------
mat : array_like
2D array.
start_index : int
Starting index of lines.
stop_index : int
Stopping index of lines.
angle : float
Tilted angle in degree.
Returns
-------
array_like
2D array.
"""
if mat.ndim != 2:
raise ValueError("Input must be a 2D data !!!")
(height, width) = mat.shape
x_cen = (width - 1.0) / 2
y_cen = (height - 1.0) / 2
angle = angle * np.pi / 180.0
x_list = np.arange(0, width) - x_cen
y_list = np.arange(start_index, stop_index + 1) - y_cen
x_mat, y_mat = np.meshgrid(x_list, y_list)
x_mat1 = x_mat * np.cos(angle) - y_mat * np.sin(angle)
y_mat1 = x_mat * np.sin(angle) + y_mat * np.cos(angle)
x_mat1 = np.clip(x_cen + x_mat1, 0, width - 1)
y_mat1 = np.clip(y_cen + y_mat1, 0, height - 1)
y_min = np.int16(np.floor(np.amin(y_mat1)))
y_max = np.int16(np.ceil(np.amax(y_mat1))) + 1
y_mat1 = y_mat1 - y_min
profile_chunk = util.mapping(mat[y_min:y_max, :], x_mat1, y_mat1)
return profile_chunk
def unwarp_sinogram_chunk(data, start_index, stop_index, xcenter, ycenter,
list_fact, **option):
"""
Unwarp chunk of sinograms [:, start_index: stop_index, :]
of a 3D tomographic dataset or a hdf/nxs object.
Parameters
----------
data : array_like or hdf object
3D array.
start_index : int
Starting index of sinograms.
stop_index : int
Stopping index of sinograms.
xcenter : float
Center of distortion in x-direction.
ycenter : float
Center of distortion in y-direction.
list_fact : list of float
Polynomial coefficients of the backward model.
option : list or tuple of int
To extract subset data along axis 0 from a hdf object. E.g option =
[start, stop, step]
Returns
-------
array_like
3D array. Distortion corrected.
"""
if data.ndim != 3:
raise ValueError("Input must be a 3D data !!!")
(depth, height, width) = data.shape
if len(option) != 0:
opt = option[list(option.keys())[0]]
start, stop, step = opt
else:
start, stop, step = 0, depth, 1
list_idx = range(start, stop, step)
if stop_index == -1:
stop_index = height
xu_list = np.arange(0, width) - xcenter
yu1 = start_index - ycenter
ru_list = np.sqrt(xu_list**2 + yu1**2)
flist = np.sum(np.asarray(
[factor * ru_list**i for i, factor in enumerate(list_fact)]),
axis=0)
yd_list1 = np.clip(ycenter + flist * yu1, 0, height - 1)
yu2 = stop_index - ycenter
ru_list = np.sqrt(xu_list**2 + yu2**2)
flist = np.sum(np.asarray(
[factor * ru_list**i for i, factor in enumerate(list_fact)]),
axis=0)
yd_list2 = np.clip(ycenter + flist * yu2, 0, height - 1)
yd_min = np.int16(np.floor(np.amin(yd_list1)))
yd_max = np.int16(np.ceil(np.amax(yd_list2)))
yu_list = np.arange(start_index, stop_index) - ycenter
xu_mat, yu_mat = np.meshgrid(xu_list, yu_list)
ru_mat = np.sqrt(xu_mat**2 + yu_mat**2)
fact_mat = np.sum(np.asarray(
[factor * ru_mat**i for i, factor in enumerate(list_fact)]),
axis=0)
xd_mat = np.float32(np.clip(xcenter + fact_mat * xu_mat, 0, width - 1))
yd_mat = np.float32(np.clip(ycenter + fact_mat * yu_mat, 0,
height - 1)) - yd_min
sino_chunk = np.asarray([
util.mapping(data[i, yd_min:yd_max, :], xd_mat, yd_mat)
for i in list_idx
])
return sino_chunk
def dfi_reconstruction(sinogram,
center,
angles=None,
ratio=1.0,
filter_name="hann",
pad_rate=0.25,
pad_mode="edge",
apply_log=True):
"""
Apply the DFI (direct Fourier inversion) reconstruction method to a
sinogram image (Ref. [1]_). The method is a practical and direct
implementation of the Fourier slice theorem (Ref. [2]_).
Parameters
----------
sinogram : array_like
2D array. Sinogram image.
center : float
Center of rotation.
angles : array_like
1D array. List of angles (in radian) corresponding to the sinogram.
ratio : float
To apply a circle mask to the reconstructed image.
filter_name : {None, "hann", "bartlett", "blackman", "hamming", "nuttall",\\
"parzen", "triang"}
Apply a smoothing filter.
pad_rate : float
To apply padding before the FFT. The padding width equals to
(pad_rate * image_width).
pad_mode : str
Padding method. Full list can be found at numpy.pad documentation.
apply_log : bool
Apply the logarithm function to the sinogram before reconstruction.
Returns
-------
array_like
Square array. Reconstructed image.
References
----------
.. [1] https://doi.org/10.1364/OE.418448
.. [2] https://doi.org/10.1071/PH560198
"""
if apply_log is True:
sinogram = -np.log(sinogram)
(nrow, ncol) = sinogram.shape
if ncol % 2 == 0:
sinogram = np.pad(sinogram, ((0, 0), (0, 1)), mode="edge")
ncol1 = sinogram.shape[1]
xshift = (ncol1 - 1) / 2.0 - center
sinogram = shift(sinogram, (0, xshift), mode='nearest')
if angles is not None:
t_ang = np.sum(np.abs(np.diff(angles * 180.0 / np.pi)))
if abs(t_ang - 360) < 10:
nrow = nrow // 2 + 1
sinogram = (sinogram[:nrow] + np.fliplr(sinogram[-nrow:])) / 2
step = np.mean(np.abs(np.diff(angles)))
b_ang = angles[0] - (angles[0] // (2 * np.pi)) * (2 * np.pi)
sino_360 = np.vstack((sinogram[:nrow - 1], np.fliplr(sinogram)))
sinogram = shift(sino_360, (b_ang / step, 0), mode='wrap')[:nrow]
if angles[-1] < angles[0]:
sinogram = np.flipud(np.fliplr(sinogram))
num_pad = int(pad_rate * ncol1)
sinogram = np.pad(sinogram, ((0, 0), (num_pad, num_pad)), mode=pad_mode)
ncol2 = sinogram.shape[1]
mask = util.make_circle_mask(ncol2, 1.0)
(r_mat, theta_mat) = generate_mapping_coordinate(ncol2, nrow, ncol2, ncol2)
sino_fft = fft.fftshift(fft.fft(fft.ifftshift(sinogram, axes=1)), axes=1)
if filter_name is not None:
window = make_smoothing_window(filter_name, ncol2)
sino_fft = sino_fft * np.tile(window, (nrow, 1))
mat_real = np.real(sino_fft)
mat_imag = np.imag(sino_fft)
reg_real = util.mapping(
mat_real, r_mat, theta_mat, order=5, mode="reflect") * mask
reg_imag = util.mapping(
mat_imag, r_mat, theta_mat, order=5, mode="reflect") * mask
recon = np.real(
fft.fftshift(fft.ifft2(
fft.ifftshift(reg_real + 1j * reg_imag))))[num_pad:ncol + num_pad,
num_pad:ncol + num_pad]
if ratio is not None:
if ratio == 0.0:
ratio = min(center, ncol - center) / (0.5 * ncol)
mask = util.make_circle_mask(ncol, ratio)
recon = recon * mask
return recon