/
radon_transform.py
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/
radon_transform.py
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"""
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.
"""
import numpy as np
from scipy.fftpack import fftshift, fft, ifft
from ._project import homography
__all__ = ["radon", "iradon"]
def radon(image, theta=None):
"""
Calculates the radon transform of an image given specified
projection angles.
Parameters
----------
image : array_like, dtype=float
Input image.
theta : array_like, dtype=float, optional (default np.arange(180))
Projection angles (in degrees).
Returns
-------
output : ndarray
Radon transform (sinogram).
"""
if image.ndim != 2:
raise ValueError('The input image must be 2-D')
if theta == None:
theta = np.arange(180)
height, width = image.shape
diagonal = np.sqrt(height ** 2 + width ** 2)
heightpad = np.ceil(diagonal - height) + 2
widthpad = np.ceil(diagonal - width) + 2
padded_image = np.zeros((int(height + heightpad),
int(width + widthpad)))
y0, y1 = int(np.ceil(heightpad / 2)), \
int((np.ceil(heightpad / 2) + height))
x0, x1 = int((np.ceil(widthpad / 2))), \
int((np.ceil(widthpad / 2) + width))
padded_image[y0:y1, x0:x1] = image
out = np.zeros((max(padded_image.shape), len(theta)))
h, w = padded_image.shape
shift0 = np.array([[1, 0, -w/2.],
[0, 1, -h/2.],
[0, 0, 1]])
shift1 = np.array([[1, 0, w/2.],
[0, 1, h/2.],
[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 = homography(padded_image,
build_rotation(-theta[i]))
out[:,i] = rotated.sum(0)[::-1]
return out
def iradon(radon_image, theta=None, output_size=None,
filter="ramp", interpolation="linear"):
"""
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.
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 nxm)
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: nearest, linear.
Returns
-------
output : ndarray
Reconstructed image.
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 == None:
m, n = radon_image.shape
theta = np.linspace(0, 180, n, endpoint=False)
th = (np.pi / 180.0) * theta
# if output size not specified, estimate from input radon image
if not output_size:
output_size = 2 * np.floor(radon_image.shape[0] / (2 * np.sqrt(2)))
n = radon_image.shape[0]
img = radon_image.copy()
# resize image to next power of two for fourier analysis
# speeds up fourier and lessens artifacts
order = max(64., 2 ** np.ceil(np.log(2 * n) / np.log(2)))
# zero pad input image
img.resize((order, img.shape[1]))
# construct the fourier filter
freqs = np.zeros((order, 1))
f = fftshift(abs(np.mgrid[-1:1:2 / order])).reshape(-1, 1)
w = 2 * np.pi * f
# start from first element to avoid divide by zero
if filter == "ramp":
pass
elif filter == "shepp-logan":
f[1:] = f[1:] * np.sin(w[1:] / 2) / (w[1:] / 2)
elif filter == "cosine":
f[1:] = f[1:] * np.cos(w[1:] / 2)
elif filter == "hamming":
f[1:] = f[1:] * (0.54 + 0.46 * np.cos(w[1:]))
elif filter == "hann":
f[1:] = f[1:] * (1 + np.cos(w[1:])) / 2
elif filter == None:
f[1:] = 1
else:
raise ValueError("Unknown filter: %s" % filter)
filter_ft = np.tile(f, (1, len(theta)))
# apply filter in fourier domain
projection = fft(img, axis=0) * filter_ft
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))
mid_index = np.ceil(n/2);
x = output_size
y = output_size
[X, Y] = np.mgrid[0.0:x, 0.0:y]
xpr = X - (output_size + 1.0) / 2.0
ypr = Y - (output_size + 1.0) / 2.0
# reconstruct image by interpolation
if interpolation == "nearest":
for i in range(len(theta)):
k = np.round(mid_index + xpr * np.sin(th[i]) - ypr * np.cos(th[i]))
reconstructed += radon_filtered[
((((k > 0) & (k < n)) * k) - 1).astype(np.int), i]
elif interpolation == "linear":
for i in range(len(theta)):
t = xpr*np.sin(th[i]) - ypr*np.cos(th[i])
a = np.floor(t)
b = mid_index + a
b0 = ((((b + 1 > 0) & (b + 1 < n)) * (b + 1)) - 1).astype(np.int)
b1 = ((((b > 0) & (b < n)) * b) - 1).astype(np.int)
reconstructed += (t - a) * radon_filtered[b0, i] + \
(a - t + 1) * radon_filtered[b1, i]
else:
raise ValueError("Unknown interpolation: %s" % interpolation)
return reconstructed * np.pi / (2 * len(th))