"""
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 fftshift, fft, ifft
from ._warps_cy import _warp_fast
from ._radon_transform import sart_projection_update

__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.
    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
    -------
    output : ndarray
        Radon transform (sinogram).

    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')
        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]
        out = np.zeros((min(padded_image.shape), len(theta)))
        dh = padded_image.shape[0] // 2
        dw = padded_image.shape[1] // 2
    else:
        height, width = image.shape
        diagonal = np.sqrt(2) * max(image.shape)
        heightpad = int(np.ceil(diagonal - height))
        widthpad = int(np.ceil(diagonal - width))
        padded_image = np.zeros((int(height + heightpad),
                                int(width + widthpad)))
        y0 = heightpad // 2
        y1 = y0 + height
        x0 = widthpad // 2
        x1 = x0 + width
        padded_image[y0:y1, x0:x1] = image
        out = np.zeros((max(padded_image.shape), len(theta)))
        dh = y0 + height // 2
        dw = x0 + width // 2

    shift0 = np.array([[1, 0, -dw],
                       [0, 1, -dh],
                       [0, 0, 1]])
    shift1 = np.array([[1, 0, dw],
                       [0, 1, dh],
                       [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]))
        out[:, i] = rotated.sum(0)
    return out


def _sinogram_circle_to_square(sinogram):
    size = int(np.ceil(np.sqrt(2) * sinogram.shape[0]))
    sinogram_padded = np.zeros((size, sinogram.shape[1]))
    pad = (size - sinogram.shape[0]) // 2
    sinogram_padded[pad:pad + sinogram.shape[0], :] = sinogram
    return sinogram_padded


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.
    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: nearest, linear.
    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
    -------
    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 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``.")
    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
    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
    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 is 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))
    # Determine the center of the projections (= center of sinogram)
    circle_size = int(np.floor(radon_image.shape[0] / np.sqrt(2)))
    square_size = radon_image.shape[0]
    mid_index = (square_size - circle_size) // 2 + circle_size // 2 + 1

    x = output_size
    y = output_size
    [X, Y] = np.mgrid[0.0:x, 0.0:y]
    xpr = X - int(output_size) // 2
    ypr = Y - int(output_size) // 2
    if circle:
        radius = (output_size - 1) // 2
        reconstruction_circle = (xpr**2 + ypr**2) < radius**2

    # Reconstruct image by interpolation
    if interpolation == "nearest":
        for i in range(len(theta)):
            k = np.round(mid_index + ypr * np.cos(th[i]) - xpr * np.sin(th[i]))
            backprojected = radon_filtered[
                ((((k > 0) & (k < n)) * k) - 1).astype(np.int), i]
            if circle:
                backprojected[~reconstruction_circle] = 0.
            reconstructed += backprojected
    elif interpolation == "linear":
        for i in range(len(theta)):
            t = ypr * np.cos(th[i]) - xpr * np.sin(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)
            backprojected = (t - a) * radon_filtered[b0, i] + \
                            (a - t + 1) * radon_filtered[b1, i]
            if circle:
                backprojected[~reconstruction_circle] = 0.
            reconstructed += backprojected
    else:
        raise ValueError("Unknown interpolation: %s" % interpolation)

    return reconstructed * np.pi / (2 * len(th))


def _sart_order_angles(theta):
    """
    Order angles to reduce the amount of correlated information
    in subsequent projections, i.e. make sure subsequent angles
    are as far away from each other mod 180 degrees as possible.
    Indices into the ``theta`` array are yielded.
    """
    tau = 3.    # time constant for correlations; 0.1 < tau < 100 works well
    used_indices = [0]
    remaining_indices = range(1, len(theta))
    yield 0
    while remaining_indices:
        used = np.array(theta[used_indices])
        used.shape = (-1, 1)
        remaining = np.array(theta[remaining_indices])
        remaining.shape = (1, -1)
        time = (np.arange(used.shape[0]) + 1)[::-1]
        time.shape = (-1, 1)
        difference = used - remaining
        distance = np.minimum(abs(difference % 180), abs(difference % -180))
        cost = np.exp(-distance * time / tau).sum(axis=0).squeeze()
        next_angle_remaining_index = np.argmin(cost)
        next_angle_index = remaining_indices.pop(next_angle_remaining_index)
        used_indices.append(next_angle_index)
        yield next_angle_index


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.
    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
    -------
    output : ndarray
      Reconstructed image.

    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:
        -A. C. Kak, Malcolm Slaney, "Principles of Computerized Tomographic
        Imaging", IEEE Press 1988.
        -AH Andersen, AC Kak, "Simultaneous algebraic reconstruction technique
        (SART): a superior implementation of the ART algorithm", Ultrasonic
        Imaging 6 pp 81--94 (1984)
        -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 _sart_order_angles(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 = clip(image, clip[0], clip[1])
    return image