mirror of
https://github.com/wassname/scikit-image.git
synced 2026-07-10 21:10:28 +08:00
Jostein Bø Fløystad
2be327815e
Scipy supports all interpolation kinds (nearest, linear) we need, while numpy supports only linear interpolation. The numpy interpolation is substantially faster, so this is used even though it complicates the code slightly.
410 lines
16 KiB
Python
410 lines
16 KiB
Python
# -*- coding: utf-8 -*-
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"""
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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 scipy.interpolate import interp1d
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from ._warps_cy import _warp_fast
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from ._radon_transform import sart_projection_update
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from .. import util
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__all__ = ["radon", "iradon", "iradon_sart"]
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def radon(image, theta=None, circle=False):
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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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circle : boolean, optional
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Assume image is zero outside the inscribed circle, making the
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width of each projection (the first dimension of the sinogram)
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equal to ``min(image.shape)``.
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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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Raises
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------
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ValueError
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If called with ``circle=True`` and ``image != 0`` outside the inscribed
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circle
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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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if circle:
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radius = min(image.shape) // 2
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c0, c1 = np.ogrid[0:image.shape[0], 0:image.shape[1]]
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reconstruction_circle = ((c0 - image.shape[0] // 2)**2
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+ (c1 - image.shape[1] // 2)**2) < radius**2
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if not np.all(reconstruction_circle | (image == 0)):
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raise ValueError('Image must be zero outside the reconstruction'
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' circle')
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slices = []
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for d in (0, 1):
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if image.shape[d] > min(image.shape):
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excess = image.shape[d] - min(image.shape)
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slices.append(slice(int(np.ceil(excess / 2)),
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int(np.ceil(excess / 2)
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+ min(image.shape))))
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else:
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slices.append(slice(None))
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slices = tuple(slices)
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padded_image = image[slices]
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out = np.zeros((min(padded_image.shape), len(theta)))
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dh = padded_image.shape[0] // 2
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dw = padded_image.shape[1] // 2
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else:
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diagonal = np.sqrt(2) * max(image.shape)
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pad = [int(np.ceil(diagonal - s)) for s in image.shape]
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pad_width = [(p // 2, p - p // 2) for p in pad]
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padded_image = util.pad(image, pad_width, mode='constant',
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constant_values=0)
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out = np.zeros((max(padded_image.shape), len(theta)))
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dh = pad[0] // 2 + image.shape[0] // 2
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dw = pad[1] // 2 + image.shape[1] // 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 = _warp_fast(padded_image, build_rotation(theta[i]))
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out[:, i] = rotated.sum(0)
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return out
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def _sinogram_circle_to_square(sinogram):
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diagonal = int(np.ceil(np.sqrt(2) * sinogram.shape[0]))
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pad = diagonal - sinogram.shape[0]
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pad_width = ((pad // 2, pad - pad // 2), (0, 0))
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return util.pad(sinogram, pad_width, mode='constant', constant_values=0)
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def iradon(radon_image, theta=None, output_size=None,
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filter="ramp", interpolation="linear", circle=False):
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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 (N, M)).
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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 are the same as for `scipy.interpolate.interp1d:
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'linear', 'nearest', 'zero', 'slinear', 'quadratic' and 'cubic'.
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circle : boolean, optional
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Assume the reconstructed image is zero outside the inscribed circle.
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Also changes the default output_size to match the behaviour of
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``radon`` called with ``circle=True``.
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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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interpolation_types = ('linear', 'nearest', 'zero', 'slinear',
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'quadratic', 'cubic')
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if not interpolation in interpolation_types:
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raise ValueError("Unknown interpolation: %s" % interpolation)
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if not output_size:
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# If output size not specified, estimate from input radon image
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if circle:
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output_size = radon_image.shape[0]
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else:
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output_size = int(np.floor(np.sqrt((radon_image.shape[0])**2
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/ 2.0)))
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if circle:
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radon_image = _sinogram_circle_to_square(radon_image)
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th = (np.pi / 180.0) * theta
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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 is 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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# Determine the center of the projections (= center of sinogram)
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circle_size = int(np.floor(radon_image.shape[0] / np.sqrt(2)))
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square_size = radon_image.shape[0]
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mid_index = (square_size - circle_size) // 2 + circle_size // 2
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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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if circle:
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radius = (output_size - 1) // 2
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reconstruction_circle = (xpr**2 + ypr**2) < radius**2
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# Reconstruct image by interpolation
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for i in range(len(theta)):
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t = ypr * np.cos(th[i]) - xpr * np.sin(th[i])
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x = np.arange(radon_filtered.shape[0]) - mid_index
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if interpolation == 'linear':
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backprojected = np.interp(t, x, radon_filtered[:, i],
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left=0, right=0)
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else:
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interpolant = interp1d(x, radon_filtered[:, i], kind=interpolation,
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bounds_error=False, fill_value=0)
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backprojected = interpolant(t)
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reconstructed += backprojected
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if circle:
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reconstructed[~reconstruction_circle] = 0.
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return reconstructed * np.pi / (2 * len(th))
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def order_angles_golden_ratio(theta):
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"""
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Order angles to reduce the amount of correlated information
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in subsequent projections.
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Parameters
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----------
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theta : 1D array of floats
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Projection angles in degrees. Duplicate angles are not allowed.
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Returns
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-------
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indices : 1D array of unsigned integers
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Indices into ``theta`` such that ``theta[indices]`` gives the
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approximate golden ratio ordering of the projections.
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Notes
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-----
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The method used here is that of the golden ratio introduced
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by T. Kohler.
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References:
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-Kohler, T. "A projection access scheme for iterative
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reconstruction based on the golden section." Nuclear Science
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Symposium Conference Record, 2004 IEEE. Vol. 6. IEEE, 2004.
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-Winkelmann, Stefanie, et al. "An optimal radial profile order
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based on the Golden Ratio for time-resolved MRI."
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Medical Imaging, IEEE Transactions on 26.1 (2007): 68-76.
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"""
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interval = 180
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def angle_distance(a, b):
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difference = a - b
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return min(abs(difference % interval), abs(difference % -interval))
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remaining = list(np.argsort(theta)) # indices into theta
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# yield an arbitrary angle to start things off
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index = remaining.pop(0)
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angle = theta[index]
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yield index
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# determine subsequent angles using the golden ratio method
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angle_increment = interval * (1 - (np.sqrt(5) - 1) / 2)
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while remaining:
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angle = (angle + angle_increment) % interval
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insert_point = np.searchsorted(theta[remaining], angle)
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index_below = insert_point - 1
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index_above = 0 if insert_point == len(remaining) else insert_point
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distance_below = angle_distance(angle, theta[remaining[index_below]])
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distance_above = angle_distance(angle, theta[remaining[index_above]])
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if distance_below < distance_above:
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yield remaining.pop(index_below)
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else:
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yield remaining.pop(index_above)
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def iradon_sart(radon_image, theta=None, image=None, projection_shifts=None,
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clip=None, relaxation=0.15):
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"""
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Inverse radon transform
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Reconstruct an image from the radon transform, using a single iteration of
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the Simultaneous Algebraic Reconstruction Technique (SART) algorithm.
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Parameters
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----------
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radon_image : 2D array, 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 : 1D array, 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 (N, M)).
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image : 2D array, dtype=float, optional
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Image containing an initial reconstruction estimate. Shape of this
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array should be ``(radon_image.shape[0], radon_image.shape[0])``. The
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default is an array of zeros.
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projection_shifts : 1D array, dtype=float
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Shift the projections contained in ``radon_image`` (the sinogram) by
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this many pixels before reconstructing the image. The i'th value
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defines the shift of the i'th column of ``radon_image``.
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clip : length-2 sequence of floats
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Force all values in the reconstructed tomogram to lie in the range
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``[clip[0], clip[1]]``
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relaxation : float
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Relaxation parameter for the update step. A higher value can
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improve the convergence rate, but one runs the risk of instabilities.
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Values close to or higher than 1 are not recommended.
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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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Algebraic Reconstruction Techniques are based on formulating the tomography
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reconstruction problem as a set of linear equations. Along each ray,
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the projected value is the sum of all the values of the cross section along
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the ray. A typical feature of SART (and a few other variants of algebraic
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techniques) is that it samples the cross section at equidistant points
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along the ray, using linear interpolation between the pixel values of the
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cross section. The resulting set of linear equations are then solved using
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a slightly modified Kaczmarz method.
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When using SART, a single iteration is usually sufficient to obtain a good
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reconstruction. Further iterations will tend to enhance high-frequency
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information, but will also often increase the noise.
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References:
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-AC Kak, M Slaney, "Principles of Computerized Tomographic
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Imaging", IEEE Press 1988.
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-AH Andersen, AC Kak, "Simultaneous algebraic reconstruction technique
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(SART): a superior implementation of the ART algorithm", Ultrasonic
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Imaging 6 pp 81--94 (1984)
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-S Kaczmarz, "Angenäherte auflösung von systemen linearer
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gleichungen", Bulletin International de l’Academie Polonaise des
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Sciences et des Lettres 35 pp 355--357 (1937)
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-Kohler, T. "A projection access scheme for iterative
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reconstruction based on the golden section." Nuclear Science
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Symposium Conference Record, 2004 IEEE. Vol. 6. IEEE, 2004.
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-Kaczmarz' method, Wikipedia,
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http://en.wikipedia.org/wiki/Kaczmarz_method
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"""
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if radon_image.ndim != 2:
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raise ValueError('radon_image must be two dimensional')
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reconstructed_shape = (radon_image.shape[0], radon_image.shape[0])
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if theta is None:
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theta = np.linspace(0, 180, radon_image.shape[1], endpoint=False)
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elif theta.shape != (radon_image.shape[1],):
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raise ValueError('Shape of theta (%s) does not match the '
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'number of projections (%d)'
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% (projection_shifts.shape, radon_image.shape[1]))
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if image is None:
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image = np.zeros(reconstructed_shape, dtype=np.float)
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elif image.shape != reconstructed_shape:
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raise ValueError('Shape of image (%s) does not match first dimension '
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'of radon_image (%s)'
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% (image.shape, reconstructed_shape))
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if projection_shifts is None:
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projection_shifts = np.zeros((radon_image.shape[1],), dtype=np.float)
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elif projection_shifts.shape != (radon_image.shape[1],):
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raise ValueError('Shape of projection_shifts (%s) does not match the '
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'number of projections (%d)'
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% (projection_shifts.shape, radon_image.shape[1]))
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if not clip is None:
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if len(clip) != 2:
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raise ValueError('clip must be a length-2 sequence')
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clip = (float(clip[0]), float(clip[1]))
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relaxation = float(relaxation)
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for angle_index in order_angles_golden_ratio(theta):
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image_update = sart_projection_update(image, theta[angle_index],
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radon_image[:, angle_index],
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projection_shifts[angle_index])
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image += relaxation * image_update
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if not clip is None:
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image = np.clip(image, clip[0], clip[1])
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return image
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