import numpy as np
from scipy.interpolate import interp1d
from scipy.constants import golden_ratio
from scipy.fft import fft, ifft, fftfreq, fftshift
from ._warps import warp
from ._radon_transform import sart_projection_update
from .._shared.utils import convert_to_float
from warnings import warn
from functools import partial
__all__ = ['radon', 'order_angles_golden_ratio', 'iradon', 'iradon_sart']
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def radon(image, theta=None, circle=True, *, preserve_range=False):
"""
Calculates the radon transform of an image given specified
projection angles.
Parameters
----------
image : ndarray
Input image. The rotation axis will be located in the pixel with
indices ``(image.shape[0] // 2, image.shape[1] // 2)``.
theta : array, optional
Projection angles (in degrees). If `None`, the value is set to
np.arange(180).
circle : boolean, optional
Assume image is zero outside the inscribed circle, making the
width of each projection (the first dimension of the sinogram)
equal to ``min(image.shape)``.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input
image is converted according to the conventions of `img_as_float`.
Also see https://scikit-image.org/docs/dev/user_guide/data_types.html
Returns
-------
radon_image : ndarray
Radon transform (sinogram). The tomography rotation axis will lie
at the pixel index ``radon_image.shape[0] // 2`` along the 0th
dimension of ``radon_image``.
References
----------
.. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic
Imaging", IEEE Press 1988.
.. [2] B.R. Ramesh, N. Srinivasa, K. Rajgopal, "An Algorithm for Computing
the Discrete Radon Transform With Some Applications", Proceedings of
the Fourth IEEE Region 10 International Conference, TENCON '89, 1989
Notes
-----
Based on code of Justin K. Romberg
(https://www.clear.rice.edu/elec431/projects96/DSP/bpanalysis.html)
"""
if image.ndim != 2:
raise ValueError('The input image must be 2-D')
if theta is None:
theta = np.arange(180)
image = convert_to_float(image, preserve_range)
if circle:
shape_min = min(image.shape)
radius = shape_min // 2
img_shape = np.array(image.shape)
coords = np.array(np.ogrid[: image.shape[0], : image.shape[1]], dtype=object)
dist = ((coords - img_shape // 2) ** 2).sum(0)
outside_reconstruction_circle = dist > radius**2
if np.any(image[outside_reconstruction_circle]):
warn(
'Radon transform: image must be zero outside the '
'reconstruction circle'
)
# Crop image to make it square
slices = tuple(
(
slice(int(np.ceil(excess / 2)), int(np.ceil(excess / 2) + shape_min))
if excess > 0
else slice(None)
)
for excess in (img_shape - shape_min)
)
padded_image = image[slices]
else:
diagonal = np.sqrt(2) * max(image.shape)
pad = [int(np.ceil(diagonal - s)) for s in image.shape]
new_center = [(s + p) // 2 for s, p in zip(image.shape, pad)]
old_center = [s // 2 for s in image.shape]
pad_before = [nc - oc for oc, nc in zip(old_center, new_center)]
pad_width = [(pb, p - pb) for pb, p in zip(pad_before, pad)]
padded_image = np.pad(image, pad_width, mode='constant', constant_values=0)
# padded_image is always square
if padded_image.shape[0] != padded_image.shape[1]:
raise ValueError('padded_image must be a square')
center = padded_image.shape[0] // 2
radon_image = np.zeros((padded_image.shape[0], len(theta)), dtype=image.dtype)
for i, angle in enumerate(np.deg2rad(theta)):
cos_a, sin_a = np.cos(angle), np.sin(angle)
R = np.array(
[
[cos_a, sin_a, -center * (cos_a + sin_a - 1)],
[-sin_a, cos_a, -center * (cos_a - sin_a - 1)],
[0, 0, 1],
]
)
rotated = warp(padded_image, R, clip=False)
radon_image[:, i] = rotated.sum(0)
return radon_image
def _sinogram_circle_to_square(sinogram):
diagonal = int(np.ceil(np.sqrt(2) * sinogram.shape[0]))
pad = diagonal - sinogram.shape[0]
old_center = sinogram.shape[0] // 2
new_center = diagonal // 2
pad_before = new_center - old_center
pad_width = ((pad_before, pad - pad_before), (0, 0))
return np.pad(sinogram, pad_width, mode='constant', constant_values=0)
def _get_fourier_filter(size, filter_name):
"""Construct the Fourier filter.
This computation lessens artifacts and removes a small bias as
explained in [1], Chap 3. Equation 61.
Parameters
----------
size : int
filter size. Must be even.
filter_name : str
Filter used in frequency domain filtering. Filters available:
ramp, shepp-logan, cosine, hamming, hann. Assign None to use
no filter.
Returns
-------
fourier_filter: ndarray
The computed Fourier filter.
References
----------
.. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic
Imaging", IEEE Press 1988.
"""
n = np.concatenate(
(
np.arange(1, size / 2 + 1, 2, dtype=int),
np.arange(size / 2 - 1, 0, -2, dtype=int),
)
)
f = np.zeros(size)
f[0] = 0.25
f[1::2] = -1 / (np.pi * n) ** 2
# Computing the ramp filter from the fourier transform of its
# frequency domain representation lessens artifacts and removes a
# small bias as explained in [1], Chap 3. Equation 61
fourier_filter = 2 * np.real(fft(f)) # ramp filter
if filter_name == "ramp":
pass
elif filter_name == "shepp-logan":
# Start from first element to avoid divide by zero
omega = np.pi * fftfreq(size)[1:]
fourier_filter[1:] *= np.sin(omega) / omega
elif filter_name == "cosine":
freq = np.linspace(0, np.pi, size, endpoint=False)
cosine_filter = fftshift(np.sin(freq))
fourier_filter *= cosine_filter
elif filter_name == "hamming":
fourier_filter *= fftshift(np.hamming(size))
elif filter_name == "hann":
fourier_filter *= fftshift(np.hanning(size))
elif filter_name is None:
fourier_filter[:] = 1
return fourier_filter[:, np.newaxis]
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def iradon(
radon_image,
theta=None,
output_size=None,
filter_name="ramp",
interpolation="linear",
circle=True,
preserve_range=True,
):
"""Inverse radon transform.
Reconstruct an image from the radon transform, using the filtered
back projection algorithm.
Parameters
----------
radon_image : ndarray
Image containing radon transform (sinogram). Each column of
the image corresponds to a projection along a different
angle. The tomography rotation axis should lie at the pixel
index ``radon_image.shape[0] // 2`` along the 0th dimension of
``radon_image``.
theta : array, optional
Reconstruction angles (in degrees). Default: m angles evenly spaced
between 0 and 180 (if the shape of `radon_image` is (N, M)).
output_size : int, optional
Number of rows and columns in the reconstruction.
filter_name : str, optional
Filter used in frequency domain filtering. Ramp filter used by default.
Filters available: ramp, shepp-logan, cosine, hamming, hann.
Assign None to use no filter.
interpolation : str, optional
Interpolation method used in reconstruction. Methods available:
'linear', 'nearest', and 'cubic' ('cubic' is slow).
circle : boolean, optional
Assume the reconstructed image is zero outside the inscribed circle.
Also changes the default output_size to match the behaviour of
``radon`` called with ``circle=True``.
preserve_range : bool, optional
Whether to keep the original range of values. Otherwise, the input
image is converted according to the conventions of `img_as_float`.
Also see https://scikit-image.org/docs/dev/user_guide/data_types.html
Returns
-------
reconstructed : ndarray
Reconstructed image. The rotation axis will be located in the pixel
with indices
``(reconstructed.shape[0] // 2, reconstructed.shape[1] // 2)``.
.. versionchanged:: 0.19
In ``iradon``, ``filter`` argument is deprecated in favor of
``filter_name``.
References
----------
.. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic
Imaging", IEEE Press 1988.
.. [2] B.R. Ramesh, N. Srinivasa, K. Rajgopal, "An Algorithm for Computing
the Discrete Radon Transform With Some Applications", Proceedings of
the Fourth IEEE Region 10 International Conference, TENCON '89, 1989
Notes
-----
It applies the Fourier slice theorem to reconstruct an image by
multiplying the frequency domain of the filter with the FFT of the
projection data. This algorithm is called filtered back projection.
"""
if radon_image.ndim != 2:
raise ValueError('The input image must be 2-D')
if theta is None:
theta = np.linspace(0, 180, radon_image.shape[1], endpoint=False)
angles_count = len(theta)
if angles_count != radon_image.shape[1]:
raise ValueError(
"The given ``theta`` does not match the number of "
"projections in ``radon_image``."
)
interpolation_types = ('linear', 'nearest', 'cubic')
if interpolation not in interpolation_types:
raise ValueError(f"Unknown interpolation: {interpolation}")
filter_types = ('ramp', 'shepp-logan', 'cosine', 'hamming', 'hann', None)
if filter_name not in filter_types:
raise ValueError(f"Unknown filter: {filter_name}")
radon_image = convert_to_float(radon_image, preserve_range)
dtype = radon_image.dtype
img_shape = radon_image.shape[0]
if output_size is None:
# If output size not specified, estimate from input radon image
if circle:
output_size = img_shape
else:
output_size = int(np.floor(np.sqrt((img_shape) ** 2 / 2.0)))
if circle:
radon_image = _sinogram_circle_to_square(radon_image)
img_shape = radon_image.shape[0]
# Resize image to next power of two (but no less than 64) for
# Fourier analysis; speeds up Fourier and lessens artifacts
projection_size_padded = max(64, int(2 ** np.ceil(np.log2(2 * img_shape))))
pad_width = ((0, projection_size_padded - img_shape), (0, 0))
img = np.pad(radon_image, pad_width, mode='constant', constant_values=0)
# Apply filter in Fourier domain
fourier_filter = _get_fourier_filter(projection_size_padded, filter_name)
projection = fft(img, axis=0) * fourier_filter
radon_filtered = np.real(ifft(projection, axis=0)[:img_shape, :])
# Reconstruct image by interpolation
reconstructed = np.zeros((output_size, output_size), dtype=dtype)
radius = output_size // 2
xpr, ypr = np.mgrid[:output_size, :output_size] - radius
x = np.arange(img_shape) - img_shape // 2
for col, angle in zip(radon_filtered.T, np.deg2rad(theta)):
t = ypr * np.cos(angle) - xpr * np.sin(angle)
if interpolation == 'linear':
interpolant = partial(np.interp, xp=x, fp=col, left=0, right=0)
else:
interpolant = interp1d(
x, col, kind=interpolation, bounds_error=False, fill_value=0
)
reconstructed += interpolant(t)
if circle:
out_reconstruction_circle = (xpr**2 + ypr**2) > radius**2
reconstructed[out_reconstruction_circle] = 0.0
return reconstructed * np.pi / (2 * angles_count)
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def order_angles_golden_ratio(theta):
"""Order angles to reduce the amount of correlated information in
subsequent projections.
Parameters
----------
theta : array of floats, shape (M,)
Projection angles in degrees. Duplicate angles are not allowed.
Returns
-------
indices_generator : generator yielding unsigned integers
The returned generator yields indices into ``theta`` such that
``theta[indices]`` gives the approximate golden ratio ordering
of the projections. In total, ``len(theta)`` indices are yielded.
All non-negative integers < ``len(theta)`` are yielded exactly once.
Notes
-----
The method used here is that of the golden ratio introduced
by T. Kohler.
References
----------
.. [1] Kohler, T. "A projection access scheme for iterative
reconstruction based on the golden section." Nuclear Science
Symposium Conference Record, 2004 IEEE. Vol. 6. IEEE, 2004.
.. [2] Winkelmann, Stefanie, et al. "An optimal radial profile order
based on the Golden Ratio for time-resolved MRI."
Medical Imaging, IEEE Transactions on 26.1 (2007): 68-76.
"""
interval = 180
remaining_indices = list(np.argsort(theta)) # indices into theta
# yield an arbitrary angle to start things off
angle = theta[remaining_indices[0]]
yield remaining_indices.pop(0)
# determine subsequent angles using the golden ratio method
angle_increment = interval / golden_ratio**2
while remaining_indices:
remaining_angles = theta[remaining_indices]
angle = (angle + angle_increment) % interval
index_above = np.searchsorted(remaining_angles, angle)
index_below = index_above - 1
index_above %= len(remaining_indices)
diff_below = abs(angle - remaining_angles[index_below])
distance_below = min(diff_below % interval, diff_below % -interval)
diff_above = abs(angle - remaining_angles[index_above])
distance_above = min(diff_above % interval, diff_above % -interval)
if distance_below < distance_above:
yield remaining_indices.pop(index_below)
else:
yield remaining_indices.pop(index_above)
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def iradon_sart(
radon_image,
theta=None,
image=None,
projection_shifts=None,
clip=None,
relaxation=0.15,
dtype=None,
):
"""Inverse radon transform.
Reconstruct an image from the radon transform, using a single iteration of
the Simultaneous Algebraic Reconstruction Technique (SART) algorithm.
Parameters
----------
radon_image : ndarray, shape (M, N)
Image containing radon transform (sinogram). Each column of
the image corresponds to a projection along a different angle. The
tomography rotation axis should lie at the pixel index
``radon_image.shape[0] // 2`` along the 0th dimension of
``radon_image``.
theta : array, shape (N,), optional
Reconstruction angles (in degrees). Default: m angles evenly spaced
between 0 and 180 (if the shape of `radon_image` is (N, M)).
image : ndarray, shape (M, M), optional
Image containing an initial reconstruction estimate. Default is an array of zeros.
projection_shifts : array, shape (N,), optional
Shift the projections contained in ``radon_image`` (the sinogram) by
this many pixels before reconstructing the image. The i'th value
defines the shift of the i'th column of ``radon_image``.
clip : length-2 sequence of floats, optional
Force all values in the reconstructed tomogram to lie in the range
``[clip[0], clip[1]]``
relaxation : float, optional
Relaxation parameter for the update step. A higher value can
improve the convergence rate, but one runs the risk of instabilities.
Values close to or higher than 1 are not recommended.
dtype : dtype, optional
Output data type, must be floating point. By default, if input
data type is not float, input is cast to double, otherwise
dtype is set to input data type.
Returns
-------
reconstructed : ndarray
Reconstructed image. The rotation axis will be located in the pixel
with indices
``(reconstructed.shape[0] // 2, reconstructed.shape[1] // 2)``.
Notes
-----
Algebraic Reconstruction Techniques are based on formulating the tomography
reconstruction problem as a set of linear equations. Along each ray,
the projected value is the sum of all the values of the cross section along
the ray. A typical feature of SART (and a few other variants of algebraic
techniques) is that it samples the cross section at equidistant points
along the ray, using linear interpolation between the pixel values of the
cross section. The resulting set of linear equations are then solved using
a slightly modified Kaczmarz method.
When using SART, a single iteration is usually sufficient to obtain a good
reconstruction. Further iterations will tend to enhance high-frequency
information, but will also often increase the noise.
References
----------
.. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic
Imaging", IEEE Press 1988.
.. [2] AH Andersen, AC Kak, "Simultaneous algebraic reconstruction
technique (SART): a superior implementation of the ART algorithm",
Ultrasonic Imaging 6 pp 81--94 (1984)
.. [3] S Kaczmarz, "Angenäherte auflösung von systemen linearer
gleichungen", Bulletin International de l’Academie Polonaise des
Sciences et des Lettres 35 pp 355--357 (1937)
.. [4] Kohler, T. "A projection access scheme for iterative
reconstruction based on the golden section." Nuclear Science
Symposium Conference Record, 2004 IEEE. Vol. 6. IEEE, 2004.
.. [5] Kaczmarz' method, Wikipedia,
https://en.wikipedia.org/wiki/Kaczmarz_method
"""
if radon_image.ndim != 2:
raise ValueError('radon_image must be two dimensional')
if dtype is None:
if radon_image.dtype.char in 'fd':
dtype = radon_image.dtype
else:
warn(
"Only floating point data type are valid for SART inverse "
"radon transform. Input data is cast to float. To disable "
"this warning, please cast image_radon to float."
)
dtype = np.dtype(float)
elif np.dtype(dtype).char not in 'fd':
raise ValueError(
"Only floating point data type are valid for inverse " "radon transform."
)
dtype = np.dtype(dtype)
radon_image = radon_image.astype(dtype, copy=False)
reconstructed_shape = (radon_image.shape[0], radon_image.shape[0])
if theta is None:
theta = np.linspace(0, 180, radon_image.shape[1], endpoint=False, dtype=dtype)
elif len(theta) != radon_image.shape[1]:
raise ValueError(
f'Shape of theta ({len(theta)}) does not match the '
f'number of projections ({radon_image.shape[1]})'
)
else:
theta = np.asarray(theta, dtype=dtype)
if image is None:
image = np.zeros(reconstructed_shape, dtype=dtype)
elif image.shape != reconstructed_shape:
raise ValueError(
f'Shape of image ({image.shape}) does not match first dimension '
f'of radon_image ({reconstructed_shape})'
)
elif image.dtype != dtype:
warn(f'image dtype does not match output dtype: ' f'image is cast to {dtype}')
image = np.asarray(image, dtype=dtype)
if projection_shifts is None:
projection_shifts = np.zeros((radon_image.shape[1],), dtype=dtype)
elif len(projection_shifts) != radon_image.shape[1]:
raise ValueError(
f'Shape of projection_shifts ({len(projection_shifts)}) does not match the '
f'number of projections ({radon_image.shape[1]})'
)
else:
projection_shifts = np.asarray(projection_shifts, dtype=dtype)
if clip is not None:
if len(clip) != 2:
raise ValueError('clip must be a length-2 sequence')
clip = np.asarray(clip, dtype=dtype)
for angle_index in order_angles_golden_ratio(theta):
image_update = sart_projection_update(
image,
theta[angle_index],
radon_image[:, angle_index],
projection_shifts[angle_index],
)
image += relaxation * image_update
if clip is not None:
image = np.clip(image, clip[0], clip[1])
return image
