import numpy as np from scipy.interpolate import interp1d from scipy.constants import golden_ratio from ._warps import warp from ._radon_transform import sart_projection_update from .._shared.fft import fftmodule from .._shared.utils import deprecate_kwarg, convert_to_float from warnings import warn from functools import partial if fftmodule is np.fft: # fallback from scipy.fft to scipy.fftpack instead of numpy.fft # (fftpack preserves single precision while numpy.fft does not) from scipy.fftpack import fft, ifft else: fft = fftmodule.fft ifft = fftmodule.ifft __all__ = ['radon', 'order_angles_golden_ratio', 'iradon', 'iradon_sart'] def radon(image, theta=None, circle=True, *, preserve_range=False): """ Calculates the radon transform of an image given specified projection angles. Parameters ---------- image : array_like Input image. The rotation axis will be located in the pixel with indices ``(image.shape[0] // 2, image.shape[1] // 2)``. theta : array_like, optional Projection angles (in degrees). If `None`, the value is set to np.arange(180). 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)``. preserve_range : bool, optional Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of `img_as_float`. Also see https://scikit-image.org/docs/dev/user_guide/data_types.html 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``. References ---------- .. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic Imaging", IEEE Press 1988. .. [2] 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 Notes ----- Based on code of Justin K. Romberg (https://www.clear.rice.edu/elec431/projects96/DSP/bpanalysis.html) """ if image.ndim != 2: raise ValueError('The input image must be 2-D') if theta is None: theta = np.arange(180) image = convert_to_float(image, preserve_range) if circle: shape_min = min(image.shape) radius = shape_min // 2 img_shape = np.array(image.shape) coords = np.array(np.ogrid[:image.shape[0], :image.shape[1]], dtype=object) dist = ((coords - img_shape // 2) ** 2).sum(0) outside_reconstruction_circle = dist > radius ** 2 if np.any(image[outside_reconstruction_circle]): warn('Radon transform: image must be zero outside the ' 'reconstruction circle') # Crop image to make it square slices = tuple(slice(int(np.ceil(excess / 2)), int(np.ceil(excess / 2) + shape_min)) if excess > 0 else slice(None) for excess in (img_shape - shape_min)) 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 = np.pad(image, pad_width, mode='constant', constant_values=0) # padded_image is always square if padded_image.shape[0] != padded_image.shape[1]: raise ValueError('padded_image must be a square') center = padded_image.shape[0] // 2 radon_image = np.zeros((padded_image.shape[0], len(theta)), dtype=image.dtype) for i, angle in enumerate(np.deg2rad(theta)): cos_a, sin_a = np.cos(angle), np.sin(angle) R = np.array([[cos_a, sin_a, -center * (cos_a + sin_a - 1)], [-sin_a, cos_a, -center * (cos_a - sin_a - 1)], [0, 0, 1]]) rotated = warp(padded_image, R, clip=False) 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 np.pad(sinogram, pad_width, mode='constant', constant_values=0) def _get_fourier_filter(size, filter_name): """Construct the Fourier filter. This computation lessens artifacts and removes a small bias as explained in [1], Chap 3. Equation 61. Parameters ---------- size: int filter size. Must be even. filter_name: str Filter used in frequency domain filtering. Filters available: ramp, shepp-logan, cosine, hamming, hann. Assign None to use no filter. Returns ------- fourier_filter: ndarray The computed Fourier filter. References ---------- .. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic Imaging", IEEE Press 1988. """ n = np.concatenate((np.arange(1, size / 2 + 1, 2, dtype=int), np.arange(size / 2 - 1, 0, -2, dtype=int))) f = np.zeros(size) f[0] = 0.25 f[1::2] = -1 / (np.pi * n) ** 2 # Computing the ramp filter from the fourier transform of its # frequency domain representation lessens artifacts and removes a # small bias as explained in [1], Chap 3. Equation 61 fourier_filter = 2 * np.real(fft(f)) # ramp filter if filter_name == "ramp": pass elif filter_name == "shepp-logan": # Start from first element to avoid divide by zero omega = np.pi * fftmodule.fftfreq(size)[1:] fourier_filter[1:] *= np.sin(omega) / omega elif filter_name == "cosine": freq = np.linspace(0, np.pi, size, endpoint=False) cosine_filter = fftmodule.fftshift(np.sin(freq)) fourier_filter *= cosine_filter elif filter_name == "hamming": fourier_filter *= fftmodule.fftshift(np.hamming(size)) elif filter_name == "hann": fourier_filter *= fftmodule.fftshift(np.hanning(size)) elif filter_name is None: fourier_filter[:] = 1 return fourier_filter[:, np.newaxis] @deprecate_kwarg(kwarg_mapping={'filter': 'filter_name'}, removed_version="0.19") def iradon(radon_image, theta=None, output_size=None, filter_name="ramp", interpolation="linear", circle=True, preserve_range=True): """Inverse radon transform. Reconstruct an image from the radon transform, using the filtered back projection algorithm. Parameters ---------- radon_image : array 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, 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, optional Number of rows and columns in the reconstruction. filter_name : str, optional 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 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``. preserve_range : bool, optional Whether to keep the original range of values. Otherwise, the input image is converted according to the conventions of `img_as_float`. Also see https://scikit-image.org/docs/dev/user_guide/data_types.html Returns ------- reconstructed : ndarray Reconstructed image. The rotation axis will be located in the pixel with indices ``(reconstructed.shape[0] // 2, reconstructed.shape[1] // 2)``. .. versionchanged :: 0.19 In ``iradon``, ``filter`` argument is deprecated in favor of ``filter_name``. References ---------- .. [1] AC Kak, M Slaney, "Principles of Computerized Tomographic Imaging", IEEE Press 1988. .. [2] 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 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: theta = np.linspace(0, 180, radon_image.shape[1], endpoint=False) angles_count = len(theta) if angles_count != 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 interpolation not in interpolation_types: raise ValueError("Unknown interpolation: %s" % interpolation) filter_types = ('ramp', 'shepp-logan', 'cosine', 'hamming', 'hann', None) if filter_name not in filter_types: raise ValueError("Unknown filter: %s" % filter_name) radon_image = convert_to_float(radon_image, preserve_range) dtype = radon_image.dtype img_shape = radon_image.shape[0] if output_size is None: # If output size not specified, estimate from input radon image if circle: output_size = img_shape else: output_size = int(np.floor(np.sqrt((img_shape) ** 2 / 2.0))) if circle: radon_image = _sinogram_circle_to_square(radon_image) img_shape = radon_image.shape[0] # 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 * img_shape)))) pad_width = ((0, projection_size_padded - img_shape), (0, 0)) img = np.pad(radon_image, pad_width, mode='constant', constant_values=0) # Apply filter in Fourier domain fourier_filter = _get_fourier_filter(projection_size_padded, filter_name) projection = fft(img, axis=0) * fourier_filter radon_filtered = np.real(ifft(projection, axis=0)[:img_shape, :]) # Reconstruct image by interpolation reconstructed = np.zeros((output_size, output_size), dtype=dtype) radius = output_size // 2 xpr, ypr = np.mgrid[:output_size, :output_size] - radius x = np.arange(img_shape) - img_shape // 2 for col, angle in zip(radon_filtered.T, np.deg2rad(theta)): t = ypr * np.cos(angle) - xpr * np.sin(angle) if interpolation == 'linear': interpolant = partial(np.interp, xp=x, fp=col, left=0, right=0) else: interpolant = interp1d(x, col, kind=interpolation, bounds_error=False, fill_value=0) reconstructed += interpolant(t) if circle: out_reconstruction_circle = (xpr ** 2 + ypr ** 2) > radius ** 2 reconstructed[out_reconstruction_circle] = 0. return reconstructed * np.pi / (2 * angles_count) 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 remaining_indices = list(np.argsort(theta)) # indices into theta # yield an arbitrary angle to start things off angle = theta[remaining_indices[0]] yield remaining_indices.pop(0) # determine subsequent angles using the golden ratio method angle_increment = interval / golden_ratio ** 2 while remaining_indices: remaining_angles = theta[remaining_indices] angle = (angle + angle_increment) % interval index_above = np.searchsorted(remaining_angles, angle) index_below = index_above - 1 index_above %= len(remaining_indices) diff_below = abs(angle - remaining_angles[index_below]) distance_below = min(diff_below % interval, diff_below % -interval) diff_above = abs(angle - remaining_angles[index_above]) distance_above = min(diff_above % interval, diff_above % -interval) if distance_below < distance_above: yield remaining_indices.pop(index_below) else: yield remaining_indices.pop(index_above) def iradon_sart(radon_image, theta=None, image=None, projection_shifts=None, clip=None, relaxation=0.15, dtype=None): """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 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, 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, 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, optional 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, optional Force all values in the reconstructed tomogram to lie in the range ``[clip[0], clip[1]]`` relaxation : float, optional 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. dtype : dtype, optional Output data type, must be floating point. By default, if input data type is not float, input is cast to double, otherwise dtype is set to input data type. 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, https://en.wikipedia.org/wiki/Kaczmarz_method """ if radon_image.ndim != 2: raise ValueError('radon_image must be two dimensional') if dtype is None: if radon_image.dtype.char in 'fd': dtype = radon_image.dtype else: warn("Only floating point data type are valid for SART inverse " "radon transform. Input data is cast to float. To disable " "this warning, please cast image_radon to float.") dtype = np.dtype(float) elif np.dtype(dtype).char not in 'fd': raise ValueError("Only floating point data type are valid for inverse " "radon transform.") dtype = np.dtype(dtype) radon_image = radon_image.astype(dtype, copy=False) 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, dtype=dtype) elif len(theta) != radon_image.shape[1]: raise ValueError('Shape of theta (%s) does not match the ' 'number of projections (%d)' % (len(theta), radon_image.shape[1])) else: theta = np.asarray(theta, dtype=dtype) if image is None: image = np.zeros(reconstructed_shape, dtype=dtype) elif image.shape != reconstructed_shape: raise ValueError('Shape of image (%s) does not match first dimension ' 'of radon_image (%s)' % (image.shape, reconstructed_shape)) elif image.dtype != dtype: warn("image dtype does not match output dtype: " "image is cast to {}".format(dtype)) image = np.asarray(image, dtype=dtype) if projection_shifts is None: projection_shifts = np.zeros((radon_image.shape[1],), dtype=dtype) elif len(projection_shifts) != radon_image.shape[1]: raise ValueError('Shape of projection_shifts (%s) does not match the ' 'number of projections (%d)' % (len(projection_shifts), radon_image.shape[1])) else: projection_shifts = np.asarray(projection_shifts, dtype=dtype) if clip is not None: if len(clip) != 2: raise ValueError('clip must be a length-2 sequence') clip = np.asarray(clip, dtype=dtype) 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 clip is not None: image = np.clip(image, clip[0], clip[1]) return image