# -*- 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 __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``. Raises ------ ValueError If called with ``circle=True`` and ``image != 0`` outside the inscribed circle """ 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) <= radius**2 if not np.all(reconstruction_circle | (image == 0)): raise ValueError('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