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# -*- coding: utf-8 -*-
"""
radon.py - Radon and inverse radon transforms
Based on code of Justin K. Romberg
(http://www.clear.rice.edu/elec431/projects96/DSP/bpanalysis.html)
J. Gillam and Chris Griffin.
References:
-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.
-A. C. Kak, Malcolm Slaney, "Principles of Computerized Tomographic
Imaging", IEEE Press 1988.
"""
from __future__ import division
import numpy as np
from scipy.fftpack import fft, ifft, fftfreq
from scipy.interpolate import interp1d
from ._warps_cy import _warp_fast
from ._radon_transform import sart_projection_update
from .. import util
from warnings import warn
__all__ = ["radon", "iradon", "iradon_sart"]
def radon(image, theta=None, circle=False):
"""
Calculates the radon transform of an image given specified
projection angles.
Parameters
----------
image : array_like, dtype=float
Input image. The rotation axis will be located in the pixel with
indices ``(image.shape[0] // 2, image.shape[1] // 2)``.
theta : array_like, dtype=float, optional (default np.arange(180))
Projection angles (in degrees).
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)``.
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``.
"""
if image.ndim != 2:
raise ValueError('The input image must be 2-D')
if theta is None:
theta = np.arange(180)
if circle:
radius = min(image.shape) // 2
c0, c1 = np.ogrid[0:image.shape[0], 0:image.shape[1]]
reconstruction_circle = ((c0 - image.shape[0] // 2) ** 2
+ (c1 - image.shape[1] // 2) ** 2)
reconstruction_circle = reconstruction_circle <= radius ** 2
if not np.all(reconstruction_circle | (image == 0)):
warn('Radon transform: image must be zero outside the '
'reconstruction circle')
# Crop image to make it square
slices = []
for d in (0, 1):
if image.shape[d] > min(image.shape):
excess = image.shape[d] - min(image.shape)
slices.append(slice(int(np.ceil(excess / 2)),
int(np.ceil(excess / 2)
+ min(image.shape))))
else:
slices.append(slice(None))
slices = tuple(slices)
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 = util.pad(image, pad_width, mode='constant',
constant_values=0)
# padded_image is always square
assert padded_image.shape[0] == padded_image.shape[1]
radon_image = np.zeros((padded_image.shape[0], len(theta)))
center = padded_image.shape[0] // 2
shift0 = np.array([[1, 0, -center],
[0, 1, -center],
[0, 0, 1]])
shift1 = np.array([[1, 0, center],
[0, 1, center],
[0, 0, 1]])
def build_rotation(theta):
T = np.deg2rad(theta)
R = np.array([[np.cos(T), np.sin(T), 0],
[-np.sin(T), np.cos(T), 0],
[0, 0, 1]])
return shift1.dot(R).dot(shift0)
for i in range(len(theta)):
rotated = _warp_fast(padded_image, build_rotation(theta[i]))
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 util.pad(sinogram, pad_width, mode='constant', constant_values=0)
def iradon(radon_image, theta=None, output_size=None,
filter="ramp", interpolation="linear", circle=False):
"""
Inverse radon transform.
Reconstruct an image from the radon transform, using the filtered
back projection algorithm.
Parameters
----------
radon_image : array_like, dtype=float
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_like, dtype=float, 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
Number of rows and columns in the reconstruction.
filter : str, optional (default ramp)
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 (default 'linear')
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``.
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
-----
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:
m, n = radon_image.shape
theta = np.linspace(0, 180, n, endpoint=False)
else:
theta = np.asarray(theta)
if len(theta) != 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 not interpolation in interpolation_types:
raise ValueError("Unknown interpolation: %s" % interpolation)
if not output_size:
# If output size not specified, estimate from input radon image
if circle:
output_size = radon_image.shape[0]
else:
output_size = int(np.floor(np.sqrt((radon_image.shape[0]) ** 2
/ 2.0)))
if circle:
radon_image = _sinogram_circle_to_square(radon_image)
th = (np.pi / 180.0) * theta
# 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 * radon_image.shape[0]))))
pad_width = ((0, projection_size_padded - radon_image.shape[0]), (0, 0))
img = util.pad(radon_image, pad_width, mode='constant', constant_values=0)
# Construct the Fourier filter
f = fftfreq(projection_size_padded).reshape(-1, 1) # digital frequency
omega = 2 * np.pi * f # angular frequency
fourier_filter = 2 * np.abs(f) # ramp filter
if filter == "ramp":
pass
elif filter == "shepp-logan":
# Start from first element to avoid divide by zero
fourier_filter[1:] = fourier_filter[1:] * np.sin(omega[1:]) / omega[1:]
elif filter == "cosine":
fourier_filter *= np.cos(omega)
elif filter == "hamming":
fourier_filter *= (0.54 + 0.46 * np.cos(omega / 2))
elif filter == "hann":
fourier_filter *= (1 + np.cos(omega / 2)) / 2
elif filter is None:
fourier_filter[:] = 1
else:
raise ValueError("Unknown filter: %s" % filter)
# Apply filter in Fourier domain
projection = fft(img, axis=0) * fourier_filter
radon_filtered = np.real(ifft(projection, axis=0))
# Resize filtered image back to original size
radon_filtered = radon_filtered[:radon_image.shape[0], :]
reconstructed = np.zeros((output_size, output_size))
# Determine the center of the projections (= center of sinogram)
mid_index = radon_image.shape[0] // 2
[X, Y] = np.mgrid[0:output_size, 0:output_size]
xpr = X - int(output_size) // 2
ypr = Y - int(output_size) // 2
# Reconstruct image by interpolation
for i in range(len(theta)):
t = ypr * np.cos(th[i]) - xpr * np.sin(th[i])
x = np.arange(radon_filtered.shape[0]) - mid_index
if interpolation == 'linear':
backprojected = np.interp(t, x, radon_filtered[:, i],
left=0, right=0)
else:
interpolant = interp1d(x, radon_filtered[:, i], kind=interpolation,
bounds_error=False, fill_value=0)
backprojected = interpolant(t)
reconstructed += backprojected
if circle:
radius = output_size // 2
reconstruction_circle = (xpr ** 2 + ypr ** 2) <= radius ** 2
reconstructed[~reconstruction_circle] = 0.
return reconstructed * np.pi / (2 * len(th))
def order_angles_golden_ratio(theta):
"""
Order angles to reduce the amount of correlated information
in subsequent projections.
Parameters
----------
theta : 1D array of floats
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
def angle_distance(a, b):
difference = a - b
return min(abs(difference % interval), abs(difference % -interval))
remaining = list(np.argsort(theta)) # indices into theta
# yield an arbitrary angle to start things off
index = remaining.pop(0)
angle = theta[index]
yield index
# determine subsequent angles using the golden ratio method
angle_increment = interval * (1 - (np.sqrt(5) - 1) / 2)
while remaining:
angle = (angle + angle_increment) % interval
insert_point = np.searchsorted(theta[remaining], angle)
index_below = insert_point - 1
index_above = 0 if insert_point == len(remaining) else insert_point
distance_below = angle_distance(angle, theta[remaining[index_below]])
distance_above = angle_distance(angle, theta[remaining[index_above]])
if distance_below < distance_above:
yield remaining.pop(index_below)
else:
yield remaining.pop(index_above)
def iradon_sart(radon_image, theta=None, image=None, projection_shifts=None,
clip=None, relaxation=0.15):
"""
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 : 2D array, dtype=float
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 : 1D array, dtype=float, optional
Reconstruction angles (in degrees). Default: m angles evenly spaced
between 0 and 180 (if the shape of `radon_image` is (N, M)).
image : 2D array, dtype=float, optional
Image containing an initial reconstruction estimate. Shape of this
array should be ``(radon_image.shape[0], radon_image.shape[0])``. The
default is an array of zeros.
projection_shifts : 1D array, dtype=float
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
Force all values in the reconstructed tomogram to lie in the range
``[clip[0], clip[1]]``
relaxation : float
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.
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,
http://en.wikipedia.org/wiki/Kaczmarz_method
"""
if radon_image.ndim != 2:
raise ValueError('radon_image must be two dimensional')
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)
elif theta.shape != (radon_image.shape[1],):
raise ValueError('Shape of theta (%s) does not match the '
'number of projections (%d)'
% (projection_shifts.shape, radon_image.shape[1]))
if image is None:
image = np.zeros(reconstructed_shape, dtype=np.float)
elif image.shape != reconstructed_shape:
raise ValueError('Shape of image (%s) does not match first dimension '
'of radon_image (%s)'
% (image.shape, reconstructed_shape))
if projection_shifts is None:
projection_shifts = np.zeros((radon_image.shape[1],), dtype=np.float)
elif projection_shifts.shape != (radon_image.shape[1],):
raise ValueError('Shape of projection_shifts (%s) does not match the '
'number of projections (%d)'
% (projection_shifts.shape, radon_image.shape[1]))
if not clip is None:
if len(clip) != 2:
raise ValueError('clip must be a length-2 sequence')
clip = (float(clip[0]), float(clip[1]))
relaxation = float(relaxation)
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 not clip is None:
image = np.clip(image, clip[0], clip[1])
return image
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