mirror of
https://github.com/wassname/scikit-image.git
synced 2026-06-29 03:21:19 +08:00
Stefan van der Walt
209 lines
6.7 KiB
Python
209 lines
6.7 KiB
Python
"""
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radon.py - Radon and inverse radon transforms
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Based on code of Justin K. Romberg
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(http://www.clear.rice.edu/elec431/projects96/DSP/bpanalysis.html)
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J. Gillam and Chris Griffin.
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References:
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-B.R. Ramesh, N. Srinivasa, K. Rajgopal, "An Algorithm for Computing
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the Discrete Radon Transform With Some Applications", Proceedings of
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the Fourth IEEE Region 10 International Conference, TENCON '89, 1989.
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-A. C. Kak, Malcolm Slaney, "Principles of Computerized Tomographic
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Imaging", IEEE Press 1988.
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"""
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from __future__ import division
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import numpy as np
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from scipy.fftpack import fftshift, fft, ifft
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from ._project import homography
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__all__ = ["radon", "iradon"]
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def radon(image, theta=None):
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"""
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Calculates the radon transform of an image given specified
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projection angles.
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Parameters
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----------
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image : array_like, dtype=float
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Input image.
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theta : array_like, dtype=float, optional (default np.arange(180))
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Projection angles (in degrees).
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Returns
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-------
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output : ndarray
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Radon transform (sinogram).
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"""
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if image.ndim != 2:
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raise ValueError('The input image must be 2-D')
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if theta is None:
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theta = np.arange(180)
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height, width = image.shape
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diagonal = np.sqrt(height**2 + width**2)
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heightpad = np.ceil(diagonal - height)
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widthpad = np.ceil(diagonal - width)
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padded_image = np.zeros((int(height + heightpad),
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int(width + widthpad)))
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y0, y1 = int(np.ceil(heightpad / 2)), \
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int((np.ceil(heightpad / 2) + height))
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x0, x1 = int((np.ceil(widthpad / 2))), \
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int((np.ceil(widthpad / 2) + width))
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padded_image[y0:y1, x0:x1] = image
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out = np.zeros((max(padded_image.shape), len(theta)))
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h, w = padded_image.shape
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dh, dw = h // 2, w // 2
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shift0 = np.array([[1, 0, -dw],
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[0, 1, -dh],
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[0, 0, 1]])
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shift1 = np.array([[1, 0, dw],
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[0, 1, dh],
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[0, 0, 1]])
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def build_rotation(theta):
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T = -np.deg2rad(theta)
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R = np.array([[np.cos(T), -np.sin(T), 0],
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[np.sin(T), np.cos(T), 0],
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[0, 0, 1]])
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return shift1.dot(R).dot(shift0)
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for i in range(len(theta)):
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rotated = homography(padded_image,
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build_rotation(-theta[i]))
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out[:, i] = rotated.sum(0)[::-1]
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return out
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def iradon(radon_image, theta=None, output_size=None,
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filter="ramp", interpolation="linear"):
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"""
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Inverse radon transform.
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Reconstruct an image from the radon transform, using the filtered
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back projection algorithm.
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Parameters
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----------
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radon_image : array_like, dtype=float
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Image containing radon transform (sinogram). Each column of
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the image corresponds to a projection along a different angle.
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theta : array_like, dtype=float, optional
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Reconstruction angles (in degrees). Default: m angles evenly spaced
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between 0 and 180 (if the shape of `radon_image` is nxm)
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output_size : int
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Number of rows and columns in the reconstruction.
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filter : str, optional (default ramp)
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Filter used in frequency domain filtering. Ramp filter used by default.
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Filters available: ramp, shepp-logan, cosine, hamming, hann
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Assign None to use no filter.
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interpolation : str, optional (default linear)
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Interpolation method used in reconstruction.
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Methods available: nearest, linear.
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Returns
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-------
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output : ndarray
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Reconstructed image.
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Notes
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-----
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It applies the fourier slice theorem to reconstruct an image by
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multiplying the frequency domain of the filter with the FFT of the
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projection data. This algorithm is called filtered back projection.
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"""
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if radon_image.ndim != 2:
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raise ValueError('The input image must be 2-D')
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if theta is None:
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m, n = radon_image.shape
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theta = np.linspace(0, 180, n, endpoint=False)
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else:
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theta = np.asarray(theta)
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if len(theta) != radon_image.shape[1]:
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raise ValueError("The given ``theta`` does not match the number of "
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"projections in ``radon_image``.")
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th = (np.pi / 180.0) * theta
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# if output size not specified, estimate from input radon image
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if not output_size:
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output_size = int(np.floor(np.sqrt((radon_image.shape[0])**2 / 2.0)))
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n = radon_image.shape[0]
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img = radon_image.copy()
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# resize image to next power of two for fourier analysis
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# speeds up fourier and lessens artifacts
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order = max(64., 2**np.ceil(np.log(2 * n) / np.log(2)))
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# zero pad input image
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img.resize((order, img.shape[1]))
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# construct the fourier filter
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f = fftshift(abs(np.mgrid[-1:1:2 / order])).reshape(-1, 1)
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w = 2 * np.pi * f
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# start from first element to avoid divide by zero
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if filter == "ramp":
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pass
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elif filter == "shepp-logan":
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f[1:] = f[1:] * np.sin(w[1:] / 2) / (w[1:] / 2)
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elif filter == "cosine":
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f[1:] = f[1:] * np.cos(w[1:] / 2)
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elif filter == "hamming":
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f[1:] = f[1:] * (0.54 + 0.46 * np.cos(w[1:]))
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elif filter == "hann":
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f[1:] = f[1:] * (1 + np.cos(w[1:])) / 2
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elif filter == None:
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f[1:] = 1
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else:
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raise ValueError("Unknown filter: %s" % filter)
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filter_ft = np.tile(f, (1, len(theta)))
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# apply filter in fourier domain
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projection = fft(img, axis=0) * filter_ft
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radon_filtered = np.real(ifft(projection, axis=0))
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# resize filtered image back to original size
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radon_filtered = radon_filtered[:radon_image.shape[0], :]
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reconstructed = np.zeros((output_size, output_size))
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mid_index = np.ceil(n / 2.0)
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x = output_size
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y = output_size
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[X, Y] = np.mgrid[0.0:x, 0.0:y]
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xpr = X - int(output_size) // 2
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ypr = Y - int(output_size) // 2
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# reconstruct image by interpolation
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if interpolation == "nearest":
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for i in range(len(theta)):
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k = np.round(mid_index + xpr * np.sin(th[i]) - ypr * np.cos(th[i]))
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reconstructed += radon_filtered[
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((((k > 0) & (k < n)) * k) - 1).astype(np.int), i]
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elif interpolation == "linear":
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for i in range(len(theta)):
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t = xpr * np.sin(th[i]) - ypr * np.cos(th[i])
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a = np.floor(t)
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b = mid_index + a
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b0 = ((((b + 1 > 0) & (b + 1 < n)) * (b + 1)) - 1).astype(np.int)
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b1 = ((((b > 0) & (b < n)) * b) - 1).astype(np.int)
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reconstructed += (t - a) * radon_filtered[b0, i] + \
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(a - t + 1) * radon_filtered[b1, i]
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else:
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raise ValueError("Unknown interpolation: %s" % interpolation)
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return reconstructed * np.pi / (2 * len(th))
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