import numpy as np


def integral_image(image):
    r"""Integral image / summed area table.

    The integral image contains the sum of all elements above and to the
    left of it, i.e.:

    .. math::

       S[m, n] = \sum_{i \leq m} \sum_{j \leq n} X[i, j]

    Parameters
    ----------
    image : ndarray
        Input image.

    Returns
    -------
    S : ndarray
        Integral image/summed area table of same shape as input image.

    References
    ----------
    .. [1] F.C. Crow, "Summed-area tables for texture mapping,"
           ACM SIGGRAPH Computer Graphics, vol. 18, 1984, pp. 207-212.

    """
    S = image
    for i in range(image.ndim):
        S = S.cumsum(axis=i)
    return S


def integrate(ii, start, end):
    """Use an integral image to integrate over a given window.

    Parameters
    ----------
    ii : ndarray
        Integral image.
    start : List of tuples, each tuple of length equal to dimension of `ii`
        Coordinates of top left corner of window(s).
        Each tuple in the list contains the starting row, col, ... index
        i.e `[(row_win1, col_win1, ...), (row_win2, col_win2,...), ...]`.
    end : List of tuples, each tuple of length equal to dimension of `ii`
        Coordinates of bottom right corner of window(s).
        Each tuple in the list containing the end row, col, ... index i.e
        `[(row_win1, col_win1, ...), (row_win2, col_win2, ...), ...]`.

    Returns
    -------
    S : scalar or ndarray
        Integral (sum) over the given window(s).


    Examples
    --------
    >>> arr = np.ones((5, 6), dtype=float)
    >>> ii = integral_image(arr)
    >>> integrate(ii, (1, 0), (1, 2))  # sum from (1, 0) to (1, 2)
    array([3.])
    >>> integrate(ii, [(3, 3)], [(4, 5)])  # sum from (3, 3) to (4, 5)
    array([6.])
    >>> # sum from (1, 0) to (1, 2) and from (3, 3) to (4, 5)
    >>> integrate(ii, [(1, 0), (3, 3)], [(1, 2), (4, 5)])
    array([3., 6.])
    """
    start = np.atleast_2d(np.array(start))
    end = np.atleast_2d(np.array(end))
    rows = start.shape[0]

    total_shape = ii.shape
    total_shape = np.tile(total_shape, [rows, 1])

    # convert negative indices into equivalent positive indices
    start_negatives = start < 0
    end_negatives = end < 0
    start = (start + total_shape) * start_negatives + \
             start * ~(start_negatives)
    end = (end + total_shape) * end_negatives + \
           end * ~(end_negatives)

    if np.any((end - start) < 0):
        raise IndexError('end coordinates must be greater or equal to start')

    # bit_perm is the total number of terms in the expression
    # of S. For example, in the case of a 4x4 2D image
    # sum of image from (1,1) to (2,2) is given by
    # S = + ii[2, 2]
    #     - ii[0, 2] - ii[2, 0]
    #     + ii[0, 0]
    # The total terms = 4 = 2 ** 2(dims)

    S = np.zeros(rows)
    bit_perm = 2 ** ii.ndim
    width = len(bin(bit_perm - 1)[2:])

    # Sum of a (hyper)cube, from an integral image is computed using
    # values at the corners of the cube. The corners of cube are
    # selected using binary numbers as described in the following example.
    # In a 3D cube there are 8 corners. The corners are selected using
    # binary numbers 000 to 111. Each number is called a permutation, where
    # perm(000) means, select end corner where none of the coordinates
    # is replaced, i.e ii[end_row, end_col, end_depth]. Similarly, perm(001)
    # means replace last coordinate by start - 1, i.e
    # ii[end_row, end_col, start_depth - 1], and so on.
    # Sign of even permutations is positive, while those of odd is negative.
    # If 'start_coord - 1' is -ve it is labeled bad and not considered in
    # the final sum.

    for i in range(bit_perm):  # for all permutations
        # boolean permutation array eg [True, False] for '10'
        binary = bin(i)[2:].zfill(width)
        bool_mask = [bit == '1' for bit in binary]

        sign = (-1)**sum(bool_mask)  # determine sign of permutation

        bad = [np.any(((start[r] - 1) * bool_mask) < 0)
               for r in range(rows)]  # find out bad start rows

        corner_points = (end * (np.invert(bool_mask))) + \
                         ((start - 1) * bool_mask)  # find corner for each row

        S += [sign * ii[tuple(corner_points[r])] if(not bad[r]) else 0
              for r in range(rows)]  # add only good rows
    return S