# -*- mode: doctest -*-
"""scikit-fmm is a Python extension module which implements the fast
marching method.

https://github.com/scikit-fmm/scikit-fmm

The fast marching method is used to model the evolution of boundaries
and interfaces in a variety of application areas. More specifically,
the fast marching method is a numerical technique for finding
approximate solutions to boundary value problems of the Eikonal
equation:

F(x) | grad T(x) | = 1.

Typically, such a problem describes the evolution of a closed curve as
a function of time T with speed F(x)>0 in the normal direction at a
point x on the curve. The speed function is specified, and the time at
which the contour crosses a point x is obtained by solving the
equation.

skfmm.distance(phi, dx=1.0, self_test=False, order=2, narrow=0.0,
  periodic=False)

  Return the signed distance from the zero contour of the array phi.

skfmm.travel_time(phi, speed, dx=1.0, self_test=False, order=2,
  narrow=0.0, periodic=False)

  Return the travel from the zero contour of the array phi given the
  scalar velocity field speed.

skfmm.extension_velocities(phi, speed, dx=1.0, self_test=False,
  order=2, ext_mask=None, narrow=0.0, periodic=False)

  Extend the velocities defined at the zero contour of phi, in the
  normal direction, to the rest of the domain. Extend the velocities
  such that grad f_ext dot grad d = 0 where where f_ext is the
  extension velocity and d is the signed distance function.


:Copyright: Copyright 2021 The scikit-fmm team.
:License: BSD-style license. See LICENSE.txt in the source directory.

"""

from __future__ import print_function

__version__ = "2022.02.02"
__docformat__ = 'restructuredtext'

from .pfmm import distance, travel_time, extension_velocities
from .heap import heap

def testing():
    r"""

    These tests are gathered from FiPy_, PyLSMLIB_ and original
    Scikit-fmm_ tests.

    .. _FiPy: http://www.ctcms.nist.gov/fipy/
    .. _PyLSMLIB: https://github.com/ktchu/LSMLIB/tree/master/pylsmlib
    .. _Scikit-fmm: http://packages.python.org/scikit-fmm/
    .. _LSMLIB: http://ktchu.serendipityresearch.org/software/lsmlib/index.html

    **1D Test**

    >>> import numpy as np

    >>> print(np.allclose(distance((-1., -1., -1., -1., 1., 1., 1., 1.), dx=.5),
    ...                   (-1.75, -1.25, -.75, -0.25, 0.25, 0.75, 1.25, 1.75)))
    True

    Small dimensions.

    >>> dx = 1e-10
    >>> print(np.allclose(distance((-1., -1., -1., -1., 1., 1., 1., 1.), dx=dx),
    ...                   np.arange(8) * dx - 3.5 * dx))
    True

    **Bug Fix**

    Test case for a bug in the upwind finite difference scheme for
    negative phi. When computing finite differences we want to
    preferentially use information from the frozen neighbors that
    are closest to the zero contour in each dimension. This means
    that we must compare absolute distances when checking neighbors
    in the negative phi direction.

    The error can result in incorrect values of the updated signed
    distance function for regions close to the minimum contour of
    the level set function, i.e. in the middle of holes.

    To test we use a square matrix for the initial phi field that is
    equal to -1 on the main diagonal and on the three diagonals above
    and below this. The matrix is set to 1 everywhere else. The bug
    results in errors in positions (1,1), (2,2), (3,3), (6,6), (7,7)
    and (8,8) along the main diagonal.

    This error occurs for first- and second-order updates. For
    simplicity, we choose to only test the first-order update.

    >>> phi = np.ones((10, 10))
    >>> i,j = np.indices(phi.shape)
    >>> phi[i==j-3] = -1
    >>> phi[i==j-2] = -1
    >>> phi[i==j-1] = -1
    >>> phi[i==j] = -1
    >>> phi[i==j+1] = -1
    >>> phi[i==j+2] = -1
    >>> phi[i==j+3] = -1
    >>> phi = distance(phi, order=1)
    >>> print(np.allclose(phi[1, 1], -2.70464, atol=1e-4))
    True
    >>> print(np.allclose(phi[2, 2], -2.50873, atol=1e-4))
    True
    >>> print(np.allclose(phi[3, 3], -2.47487, atol=1e-4))
    True
    >>> print(np.allclose(phi[6, 6], -2.47487, atol=1e-4))
    True
    >>> print(np.allclose(phi[7, 7], -2.50873, atol=1e-4))
    True
    >>> print(np.allclose(phi[8, 8], -2.70464, atol=1e-4))
    True

    **Bug Fix**

    A 2D test case to test trial values for a pathological case.

    >>> dx = 1.
    >>> dy = 2.
    >>> vbl = -dx * dy / np.sqrt(dx**2 + dy**2) / 2.
    >>> vbr = dx / 2
    >>> vml = dy / 2.
    >>> crossProd = dx * dy
    >>> dsq = dx**2 + dy**2
    >>> top = vbr * dx**2 + vml * dy**2
    >>> sqrt = crossProd**2 *(dsq - (vbr - vml)**2)
    >>> sqrt = np.sqrt(max(sqrt, 0))
    >>> vmr = (top + sqrt) / dsq
    >>> print(np.allclose(distance(((-1., 1., -1.), (1., 1., 1.)), dx=(dx, dy), order=1),
    ...                   ((vbl, vml, vbl), (vbr, vmr, vbr))))
    True

    **Test Extension Field Calculation**

    >>> tmp = 1 / np.sqrt(2)
    >>> phi = np.array([[-1., 1.], [1., 1.]])
    >>> phi, ext =  extension_velocities(phi,
    ...                                    [[-1, .5], [2., -1.]],
    ...                                    ext_mask=phi < 0,
    ...                                    dx=1., order=1)
    >>> print(np.allclose(phi, ((-tmp / 2, 0.5), (0.5, 0.5 + tmp))))
    True
    >>> print(np.allclose(ext, [[1.25, .5], [2., 1.25]]))
    True

    >>> phi = np.array(((-1., 1., 1.), (1., 1., 1.), (1., 1., 1.)))
    >>> phi, ext = extension_velocities(phi,
    ...                                   ((-1., 2., -1.),
    ...                                    (.5, -1., -1.),
    ...                                    (-1., -1., -1.)),
    ...                                   ext_mask=phi < 0,
    ...                                   order=1)
    >>> v1 = 0.5 + tmp
    >>> v2 = 1.5
    >>> tmp1 = (v1 + v2) / 2 + np.sqrt(2. - (v1 - v2)**2) / 2
    >>> tmp2 = tmp1 + 1 / np.sqrt(2)
    >>> print(np.allclose(phi, ((-tmp / 2, 0.5, 1.5),
    ...                         (0.5, 0.5 + tmp, tmp1),
    ...                         (1.5, tmp1, tmp2))))
    True
    >>> print(np.allclose(ext, ((1.25, 2., 2.),
    ...                         (.5, 1.25, 1.5456),
    ...                         (.5, 0.9544, 1.25)),
    ...                   rtol = 1e-4))
    True

    **Bug Fix**

    Test case for a bug that occurs when initializing the distance
    variable at the interface. Currently it is assumed that adjacent
    cells that are opposite sign neighbors have perpendicular normal
    vectors. In fact the two closest cells could have opposite
    normals.

    >>> print(np.allclose(distance((-1., 1., -1.)), (-0.5, 0.5, -0.5)))
    True

    Testing second order. This example failed with Scikit-fmm_.

    >>> phi = ((-1., -1., 1., 1.),
    ...        (-1., -1., 1., 1.),
    ...        (1., 1., 1., 1.),
    ...        (1., 1., 1., 1.))
    >>> answer = ((-1.30473785, -0.5, 0.5, 1.49923009),
    ...           (-0.5, -0.35355339, 0.5, 1.45118446),
    ...           (0.5, 0.5, 0.97140452, 1.76215286),
    ...           (1.49923009, 1.45118446, 1.76215286, 2.33721352))
    >>> print(np.allclose(distance(phi),
    ...                   answer,
    ...                   rtol=1e-9))
    True

    **A test for a bug in both LSMLIB and Scikit-fmm**

    The following test gives different results depending on whether
    LSMLIB_ or Scikit-fmm_ is used. This issue occurs when calculating
    second order accurate distance functions. When a value becomes
    "known" after previously being a "trial" value it updates its
    neighbors' values. In a second order scheme the neighbors one step
    away also need to be updated (if the cell between the new "known"
    cell and the cell required for second order accuracy also happens
    to be "known"), but are not updated in either package.  By luck
    (due to trial values having the same value), the values calculated
    in Scikit-fmm_ for the following example are correct although an
    example that didn't work for Scikit-fmm_ could also be
    constructed.

    >>> phi = distance([[-1, -1, -1, -1],
    ...                                [ 1,  1, -1, -1],
    ...                                [ 1,  1, -1, -1],
    ...                                [ 1,  1, -1, -1]], order=2)
    >>> phi = distance(phi, order=2)

    The following values come form Scikit-fmm_.

    >>> answer = [[-0.5,        -0.58578644, -1.08578644, -1.85136395],
    ...           [ 0.5,         0.29289322, -0.58578644, -1.54389939],
    ...           [ 1.30473785,  0.5,        -0.5,        -1.5       ],
    ...           [ 1.49547948,  0.5,        -0.5,        -1.5       ]]

    The 3rd and 7th element are different for LSMLIB_. This is because
    the 15th element is not "known" when the "trial" value for the 7th
    element is calculated. Scikit-fmm_ calculates the values in a
    slightly different order so gets a seemingly better answer, but
    this is just chance.

    >>> print(np.allclose(phi, answer, rtol=1e-9))
    True

    The following tests for the same issue but is a better test case
    guaranteed to fail.

    >>> phi = np.array([[-1,  1,  1,  1,  1, -1],
    ...                 [-1, -1, -1, -1, -1, -1],
    ...                 [-1, -1, -1, -1, -1, -1]])

    >>> phi = distance(phi)
    >>> print(phi[2, 2] == phi[2, 3])
    True

    >>> phi = distance(phi)
    >>> print(phi[2, 2] == phi[2, 3])
    True

    **Circle Example**

    Solve the level set equation in two dimensions for a circle.

    The 2D level set equation can be written,

    .. math::

        |\nabla \phi| = 1

    and the boundary condition for a circle is given by, :math:`\phi = 0` at
    :math:`(x - L / 2)^2 + (y - L / 2)^2 = (L / 4)^2`.

    The solution to this problem will be demonstrated in the following
    script. Firstly, setup the parameters.

    >>> def mesh(nx=1, ny=1, dx=1., dy=1.):
    ...     y, x = np.mgrid[0:nx,0:ny]
    ...     x = x * dx + dx / 2
    ...     y = y * dy + dy / 2
    ...     return x, y

    >>> dx = 1.
    >>> N = 11
    >>> L = N * dx
    >>> x, y = mesh(nx=N, ny=N, dx=dx, dy=dx)
    >>> phi = -np.ones(N * N, 'd')
    >>> phi[(x.flatten() - L / 2.)**2 + (y.flatten() - L / 2.)**2 < (L / 4.)**2] = 1.
    >>> phi = np.reshape(phi, (N, N))
    >>> phi = distance(phi, dx=dx, order=1).flatten()

    >>> dX = dx / 2.
    >>> m1 = dX * dX / np.sqrt(dX**2 + dX**2)
    >>> def evalCell(phix, phiy, dx):
    ...     aa = dx**2 + dx**2
    ...     bb = -2 * ( phix * dx**2 + phiy * dx**2)
    ...     cc = dx**2 * phix**2 + dx**2 * phiy**2 - dx**2 * dx**2
    ...     sqr = np.sqrt(bb**2 - 4. * aa * cc)
    ...     return ((-bb - sqr) / 2. / aa,  (-bb + sqr) / 2. / aa)
    >>> v1 = evalCell(-dX, -m1, dx)[0]
    >>> v2 = evalCell(-m1, -dX, dx)[0]
    >>> v3 = evalCell(m1,  m1,  dx)[1]
    >>> v4 = evalCell(v3, dX, dx)[1]
    >>> v5 = evalCell(dX, v3, dx)[1]
    >>> MASK = -1000.
    >>> trialValues = np.array((
    ...     MASK,  MASK, MASK, MASK, MASK, MASK, MASK, MASK, MASK, MASK, MASK,
    ...     MASK,  MASK, MASK, MASK,-3*dX,-3*dX,-3*dX, MASK, MASK, MASK, MASK,
    ...     MASK,  MASK, MASK,   v1,  -dX,  -dX,  -dX,   v1, MASK, MASK, MASK,
    ...     MASK,  MASK,   v2,  -m1,   m1,   dX,   m1,  -m1,   v2, MASK, MASK,
    ...     MASK, -dX*3,  -dX,   m1,   v3,   v4,   v3,   m1,  -dX,-dX*3, MASK,
    ...     MASK, -dX*3,  -dX,   dX,   v5, MASK,   v5,   dX,  -dX,-dX*3, MASK,
    ...     MASK, -dX*3,  -dX,   m1,   v3,   v4,   v3,   m1,  -dX,-dX*3, MASK,
    ...     MASK,  MASK,   v2,  -m1,   m1,   dX,   m1,  -m1,   v2, MASK, MASK,
    ...     MASK,  MASK, MASK,   v1,  -dX,  -dX,  -dX,   v1, MASK, MASK, MASK,
    ...     MASK,  MASK, MASK, MASK,-3*dX,-3*dX,-3*dX, MASK, MASK, MASK, MASK,
    ...     MASK,  MASK, MASK, MASK, MASK, MASK, MASK, MASK, MASK, MASK, MASK), 'd')

    >>> phi[trialValues == MASK] = MASK
    >>> print(np.allclose(phi, trialValues))
    True

    **Square Example**

    Here we solve the level set equation in two dimensions for a square. The equation is
    given by:

    .. math::

       |\nabla \phi| &= 1 \\
       \phi &= 0 \qquad \text{at} \qquad \begin{cases}
           x = \left( L / 3, 2 L / 3 \right)
           & \text{for $L / 3 \le y \le 2 L / 3$} \\
           y = \left( L / 3, 2 L / 3 \right)
           & \text{for $L / 3 \le x \le 2 L / 3$}
       \end{cases}

    >>> dx = 0.5
    >>> dy = 2.
    >>> nx = 5
    >>> ny = 5
    >>> Lx = nx * dx
    >>> Ly = ny * dy

    >>> x, y = mesh(nx=nx, ny=ny, dx=dx, dy=dy)
    >>> x = x.flatten()
    >>> y = y.flatten()
    >>> phi = -np.ones(nx * ny, 'd')
    >>> phi[((Lx / 3. < x) & (x < 2. * Lx / 3.)) & ((Ly / 3. < y) & (y < 2. * Ly / 3))] = 1.
    >>> phi = np.reshape(phi, (nx, ny))
    >>> phi = distance(phi, dx=(dy, dx), order=1).flatten()

    >>> def evalCell(phix, phiy, dx, dy):
    ...     aa = dy**2 + dx**2
    ...     bb = -2 * ( phix * dy**2 + phiy * dx**2)
    ...     cc = dy**2 * phix**2 + dx**2 * phiy**2 - dx**2 * dy**2
    ...     sqr = np.sqrt(bb**2 - 4. * aa * cc)
    ...     return ((-bb - sqr) / 2. / aa,  (-bb + sqr) / 2. / aa)
    >>> val = evalCell(-dy / 2., -dx / 2., dx, dy)[0]
    >>> v1 = evalCell(val, -3. * dx / 2., dx, dy)[0]
    >>> v2 = evalCell(-3. * dy / 2., val, dx, dy)[0]
    >>> v3 = evalCell(v2, v1, dx, dy)[0]
    >>> v4 = dx * dy / np.sqrt(dx**2 + dy**2) / 2
    >>> arr = np.array((
    ...     v3           , v2      , -3. * dy / 2.   , v2      , v3,
    ...     v1           , val     , -dy / 2.        , val     , v1           ,
    ...     -3. * dx / 2., -dx / 2., v4              , -dx / 2., -3. * dx / 2.,
    ...     v1           , val     , -dy / 2.        , val     , v1           ,
    ...     v3           , v2      , -3. * dy / 2.   , v2      , v3           ))
    >>> print(np.allclose(arr, phi))
    True

    **Assertion Errors**

    >>> distance([[-1, 1],[1, 1]], dx=(1, 2, 3))
    Traceback (most recent call last):
      ...
    ValueError: dx must be of length len(phi.shape)
    >>> extension_velocities([[-1, 1],[1, 1]], speed=[1, 1])
    Traceback (most recent call last):
      ...
    ValueError: phi and speed must have the same shape

    **Test for 1D equality between `distance` and `travel_time`**

    >>> phi = np.arange(-5, 5) + 0.499
    >>> d = distance(phi)
    >>> t = travel_time(phi, speed=np.ones_like(phi))
    >>> np.testing.assert_allclose(t, np.abs(d))

    **Tests taken from FiPy**

    >>> phi = np.array(((-1, -1, 1, 1),
    ...                 (-1, -1, 1, 1),
    ...                 (1, 1, 1, 1),
    ...                 (1, 1, 1, 1)))
    >>> o1 = distance(phi, order=1, self_test=True)
    >>> dw_o1 =   [[-1.20710678, -0.5,         0.5,         1.5],
    ...            [-0.5,        -0.35355339,  0.5,         1.5],
    ...            [ 0.5,         0.5,         1.20710678,  2.04532893],
    ...            [ 1.5,         1.5,         2.04532893,  2.75243571]]
    >>> np.testing.assert_allclose(o1, dw_o1)

    >>> phi = np.array(((-1, -1, 1, 1),
    ...                 (-1, -1, 1, 1),
    ...                 (1, 1, 1, 1),
    ...                 (1, 1, 1, 1)))
    >>> o1 = travel_time(phi, np.ones_like(phi), order=1, self_test=True)
    >>> dw_o1 =   [[-1.20710678, -0.5,         0.5,         1.5],
    ...            [-0.5,        -0.35355339,  0.5,         1.5],
    ...            [ 0.5,         0.5,         1.20710678,  2.04532893],
    ...            [ 1.5,         1.5,         2.04532893,  2.75243571]]
    >>> np.testing.assert_allclose(o1, np.abs(dw_o1))

    >>> phi = np.array(((-1, -1, 1, 1),
    ...                (-1, -1, 1, 1),
    ...                (1, 1, 1, 1),
    ...                (1, 1, 1, 1)))
    >>> o2 = distance(phi, self_test=True)
    >>> dw_o2 = [[-1.30473785,  -0.5,          0.5,         1.49923009],
    ...          [-0.5,        -0.35355339,    0.5,         1.45118446],
    ...          [ 0.5,         0.5,         0.97140452,  1.76215286],
    ...          [ 1.49923009,  1.45118446,  1.76215286,  2.33721352]]

    >>> np.testing.assert_allclose(o2, dw_o2)
    >>> phi = np.array(((-1, -1, 1, 1),
    ...                (-1, -1, 1, 1),
    ...                (1, 1, 1, 1),
    ...                (1, 1, 1, 1)))
    >>> o2 = travel_time(phi, np.ones_like(phi), self_test=True)
    >>> dw_o2 = [[-1.30473785,  -0.5,          0.5,         1.49923009],
    ...          [-0.5,        -0.35355339,    0.5,         1.45118446],
    ...          [ 0.5,         0.5,         0.97140452,  1.76215286],
    ...          [ 1.49923009,  1.45118446,  1.76215286,  2.33721352]]
    >>> np.testing.assert_allclose(o2, np.abs(dw_o2))

    >>> distance([-1,1], order=0)
    Traceback (most recent call last):
      ...
    ValueError: order must be 1 or 2

    >>> distance([-1,1], order=3)
    Traceback (most recent call last):
      ...
    ValueError: order must be 1 or 2

    **Extension velocity tests**

    Test 1d extension constant.

    >>> phi =   [-1,-1,-1,1,1,1]
    >>> speed = [1,1,1,1,1,1]
    >>> d, f_ext  = extension_velocities(phi, speed, self_test=True)
    >>> np.testing.assert_allclose(speed, f_ext)

    Test the 1D extension block.

    >>> phi =   np.ones(10)
    >>> phi[0] =- 1
    >>> speed = np.ones(10)
    >>> speed[0:3] = 5
    >>> d, f_ext  = extension_velocities(phi, speed, self_test=True)
    >>> np.testing.assert_allclose(f_ext, 5)

    Test that a uniform speed value is preserved.

    >>> N     = 50
    >>> X, Y  = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> r     = 0.25
    >>> dx    = 2.0 / (N - 1)
    >>> phi   = (X) ** 2 + (Y) ** 2 - r ** 2
    >>> speed = np.ones_like(phi)
    >>> d, f_ext = extension_velocities(phi, speed, dx, self_test=True)
    >>> np.testing.assert_allclose(f_ext, 1.0)

    Constant value march-out test

    >>> speed[abs(Y)<0.3] = 10.0
    >>> d, f_ext = extension_velocities(phi, speed, dx, self_test=True)
    >>> np.testing.assert_allclose(f_ext, 10.0)

    Test distance from extension

    >>> speed = np.ones_like(phi)
    >>> d, f_ext = extension_velocities(phi, speed, dx, self_test=True)
    >>> d2 = distance(phi, dx, self_test=True)
    >>> np.testing.assert_allclose(d, d2)

    Test for extension velocity bug

    >>> N     = 150
    >>> X, Y  = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> r     = 0.5
    >>> dx    = 2.0 / (N - 1)
    >>> phi   = (X) ** 2 + (Y) ** 2 - r ** 2
    >>> speed = np.ones_like(phi)
    >>> speed[X>0.25] = 3.0
    >>> d2, f_ext = extension_velocities(phi, speed, dx)

    >>> assert (f_ext <= 3.0000001).all()
    >>> assert (f_ext >= 1).all()

    >>> np.testing.assert_almost_equal(f_ext[137, 95], 1, 3)
    >>> np.testing.assert_almost_equal(f_ext[103, 78], 1, 2)
    >>> np.testing.assert_almost_equal(f_ext[72, 100], 3, 3)
    >>> np.testing.assert_almost_equal(f_ext[72, 86], 3, 3)
    >>> np.testing.assert_almost_equal(f_ext[110, 121], 3, 3)

    Simple two point tests

    >>> np.testing.assert_array_equal(distance([-1, 1]),
    ...                                        [-0.5, 0.5])
    >>> np.testing.assert_allclose(distance([-1, -1, -1, 1, 1, 1]),
    ...                                     [-2.5, -1.5, -0.5, 0.5, 1.5, 2.5])
    >>> np.testing.assert_allclose(distance([1, 1, 1, -1, -1, -1]),
    ...                                     [2.5, 1.5, 0.5, -0.5, -1.5, -2.5])

    Three point test case

    >>> np.testing.assert_array_equal(distance([-1, 0, 1]),           [-1, 0, 1])
    >>> np.testing.assert_array_equal(distance([-1, 0, 1], dx=[2]),   [-2, 0, 2])
    >>> np.testing.assert_array_equal(distance([-1, 0, 1], dx=2),     [-2, 0, 2])
    >>> np.testing.assert_array_equal(distance([-1, 0, 1], dx=2.0),   [-2, 0, 2])
    >>> np.testing.assert_array_equal(travel_time([1, 0, -1], [1, 1, 1]),
    ...                                           [1, 0, 1])
    >>> np.testing.assert_array_equal(travel_time([-1, 0, 1], [1, 1, 1]),
    ...                                           [1, 0, 1])
    >>> np.testing.assert_array_equal(travel_time([1, 0, -1], [1, 1, 1], dx=2),
    ...                                           [2, 0, 2])
    >>> np.testing.assert_array_equal(travel_time([1, 0, -1], [1, 1, 1], dx=[2]),
    ...                                           [2, 0, 2])
    >>> np.testing.assert_array_equal(travel_time([1, 0, -1], [1, 1, 1], dx=2.0),
    ...                                           [2, 0, 2])

    Travel time tests 1

    >>> np.testing.assert_allclose(travel_time([0, 1, 1, 1, 1], [2, 2, 2, 2, 2]),
    ...                                        [0, 0.5, 1.0, 1.5, 2.0])
    >>> np.testing.assert_array_equal(travel_time([1, 0, -1], [2, 2, 2]),
    ...                                           [0.5, 0, 0.5])

    Travel time tests 2

    >>> phi   = [1, 1, 1, -1, -1, -1]
    >>> t     = travel_time(phi, np.ones_like(phi))
    >>> exact = [2.5, 1.5, 0.5, 0.5, 1.5, 2.5]
    >>> np.testing.assert_allclose(t, exact)

    Travel time tests 3

    >>> phi   = [-1, -1, -1, 1, 1, 1]
    >>> t     = travel_time(phi, np.ones_like(phi))
    >>> exact = [2.5, 1.5, 0.5, 0.5, 1.5, 2.5]
    >>> np.testing.assert_allclose(t, exact)

    Corner case

    >>> np.testing.assert_array_equal(distance([0, 0]), [0, 0])
    >>> np.testing.assert_array_equal(travel_time([0, 0], [1, 1]), [0, 0])

    Test zero

    >>> distance([1, 0, 1, 1], 0)
    Traceback (most recent call last):
      ...
    ValueError: dx must be greater than zero

    Test dx shape

    >>> distance([0, 0, 1, 0, 0], [0, 0, 1, 0, 0])
    Traceback (most recent call last):
      ...
    ValueError: dx must be of length len(phi.shape)

    Test for small speeds

    Test catching speeds which are too small. Speeds less than the
    machine epsilon are masked off to avoid an overflow.

    >>> t = travel_time([-1, -1, 0, 1, 1], [1, 1, 1, 1, 0])
    >>> assert isinstance(t, np.ma.MaskedArray)
    >>> np.testing.assert_array_equal(t.data[:-1], [2, 1, 0, 1])
    >>> np.testing.assert_array_equal(t.mask, [False, False, False, False, True])

    >>> t2 = travel_time([-1, -1, 0, 1, 1], [1, 1, 1, 1, 1e-300])
    >>> np.testing.assert_array_equal(t, t2)

    Mask test

    Test that when the mask cuts off the solution, the cut off points
    are also masked.

    >>> ma    = np.ma.MaskedArray([1, 1, 1, 0], [False, True, False, False])
    >>> d     = distance(ma)
    >>> exact = np.ma.MaskedArray([0, 0, 1, 0], [True, True, False, False])
    >>> np.testing.assert_array_equal(d.mask, exact.mask)
    >>> np.testing.assert_array_equal(d, exact)

    Circular level set

    >>> N     = 50
    >>> X, Y  = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> r     = 0.5
    >>> dx    = 2.0 / (N - 1)
    >>> phi   = (X) ** 2 + (Y) ** 2 - r ** 2
    >>> d     = distance(phi, dx, self_test=True)
    >>> exact = np.sqrt(X ** 2 + Y ** 2) - r
    >>> np.testing.assert_allclose(d, exact, atol=dx)

    Planar level set

    >>> N         = 50
    >>> X, Y      = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> dx        = 2.0 / (N - 1)
    >>> phi       = np.ones_like(X)
    >>> phi[0, :] = -1
    >>> d         = distance(phi, dx, self_test=True)
    >>> exact     = Y + 1 - dx / 2.0
    >>> np.testing.assert_allclose(d, exact)

    Masked input

    >>> N         = 50
    >>> X, Y      = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> dx        = 2.0 / (N - 1)
    >>> phi       = np.ones_like(X)
    >>> phi[0, 0] = -1
    >>> mask      = np.logical_and(abs(X) < 0.25, abs(Y) < 0.25)
    >>> mphi      = np.ma.MaskedArray(phi.copy(), mask)
    >>> d0        = distance(phi, dx, self_test=True)
    >>> d         = distance(mphi, dx, self_test=True)
    >>> d0[mask]  = 0
    >>> d[mask]   = 0
    >>> shadow    = d0 - d
    >>> bsh       = abs(shadow) > 0.001
    >>> diff      = (bsh).sum()

    >>> assert diff > 635 and diff < 645

    Test Eikonal solution

    >>> N     = 50
    >>> X, Y  = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> r     = 0.5
    >>> dx    = 2.0 / (N - 1)
    >>> phi   = (X) ** 2 + (Y) ** 2 - r ** 2
    >>> speed = np.ones_like(phi) * 2
    >>> t     = travel_time(phi, speed, dx)
    >>> exact = 0.5 * np.abs(np.sqrt(X ** 2 + Y ** 2) - 0.5)

    >>> np.testing.assert_allclose(t, exact, atol=dx)

    Test 1d

    >>> N          = 100
    >>> X          = np.linspace(-1.0, 1.0, N)
    >>> dx         = 2.0 / (N - 1)
    >>> phi        = np.zeros_like(X)
    >>> phi[X < 0] = -1
    >>> phi[X > 0] = 1
    >>> d          = distance(phi, dx, self_test=True)

    >>> np.testing.assert_allclose(d, X)

    Test 3d

    >>> N            = 15
    >>> X            = np.linspace(-1, 1, N)
    >>> Y            = np.linspace(-1, 1, N)
    >>> Z            = np.linspace(-1, 1, N)
    >>> phi          = np.ones((N, N, N))
    >>> phi[0, 0, 0] = -1.0
    >>> dx           = 2.0 / (N - 1)
    >>> d            = distance(phi, dx, self_test=True)

    >>> exact        = np.sqrt((X + 1) ** 2 +
    ...                        (Y + 1)[:, np.newaxis] ** 2 +
    ...                        (Z + 1)[:, np.newaxis, np.newaxis] ** 2)

    >>> np.testing.assert_allclose(d, exact, atol=dx)

    Test default dx

    >>> N     = 50
    >>> X, Y  = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> r     = 0.5
    >>> phi   = (X) ** 2 + (Y) ** 2 - r ** 2
    >>> speed = np.ones_like(phi) * 2
    >>> out = travel_time(phi, speed, self_test=True)

    Test non-square grid and dx different in different directions

    >>> N      = 50
    >>> NX, NY = N, 5 * N
    >>> X, Y   = np.meshgrid(np.linspace(-1, 1, NY), np.linspace(-1, 1, NX))
    >>> r      = 0.5
    >>> phi    = X ** 2 + Y ** 2 - r ** 2
    >>> dx     = [2.0 / (NX - 1), 2.0 / (NY - 1)]
    >>> d      = distance(phi, dx, self_test=True)
    >>> exact  = np.sqrt(X ** 2 + Y ** 2) - r

    >>> np.testing.assert_allclose(d, exact, atol=1.3*max(dx))


    No zero level set test

    >>> distance([1, 1], self_test=False)
    Traceback (most recent call last):
      ...
    ValueError: the array phi contains no zero contour (no zero level set)

    Shape mismatch test

    >>> travel_time([-1, 1], [2])
    Traceback (most recent call last):
      ...
    ValueError: phi and speed must have the same shape

    Speed wrong type test

    >>> travel_time([0, 0, 1, 1], 2)
    Traceback (most recent call last):
      ...
    ValueError: speed must be a 1D to 12-D array of doubles

    dx mismatch test

    >>> travel_time([-1, 1], [2, 2], [2, 2, 2, 2])
    Traceback (most recent call last):
      ...
    ValueError: dx must be of length len(phi.shape)

    Test c error handling

    >>> distance([-1, 1], self_test=44)
    Traceback (most recent call last):
      ...
    ValueError: self_test must be 0 or 1

    Check array type test

    >>> distance(np.array(["a", "b"]))
    Traceback (most recent call last):
      ...
    ValueError: phi must be a 1 to 12-D array of doubles

    >>> from skfmm import heap
    >>> h = heap(10,True)
    >>> h.push(0,0.2)
    0
    >>> h.push(1,0.3)
    1
    >>> h.push(2,0.1)
    2
    >>> h.update(1, 0.01)
    >>> h.pop()
    (1, 0.01)
    >>> h.pop()
    (2, 0.1)
    >>> h.pop()
    (0, 0.2)
    >>> h.empty()
    True
    >>> h.pop()
    Traceback (most recent call last):
      ...
    RuntimeError: heap pop error: empty heap
    <BLANKLINE>

    Test narrow optional argument.

    >>> phi = np.array([-1,-1,-1,1,1,1])
    >>> d = distance(phi, narrow=1.0)
    >>> d.data[2:-2]
    array([-0.5,  0.5])
    >>> assert (d.mask == [ True,  True, False, False,  True,  True]).all()
    >>> N     = 50
    >>> X, Y  = np.meshgrid(np.linspace(-1, 1, N), np.linspace(-1, 1, N))
    >>> r     = 0.5
    >>> dx    = 2.0 / (N - 1)
    >>> phi   = (X) ** 2 + (Y) ** 2 - r ** 2
    >>> exact = np.sqrt(X ** 2 + Y ** 2) - r

    >>> d = distance(phi, dx)
    >>> bandwidth=5*dx
    >>> d2 = distance(phi, dx, narrow=bandwidth)
    >>> np.testing.assert_allclose(d, exact, atol=dx)
    >>> # make sure we get a normal array if there are no points outside the narrow band
    >>> np.testing.assert_allclose(distance(phi, dx, narrow=1000*dx), exact, atol=dx)

    >>> assert  (d2<bandwidth).all()
    >>> assert type(d2) is np.ma.masked_array

    >>> speed = np.ones_like(phi)
    >>> speed[X>0] = 1.8
    >>> t = travel_time(phi, speed, dx)
    >>> t2 = travel_time(phi, speed, dx, narrow=bandwidth)
    >>> assert type(t2) is np.ma.masked_array
    >>> assert (t2<bandwidth).all()

    >>> speed = np.ones_like(phi)
    >>> speed = X + Y
    >>> ed, ev = extension_velocities(phi, speed, dx)
    >>> ed2, ev2 = extension_velocities(phi, speed, dx, narrow=bandwidth)
    >>> assert type(ed2) is np.ma.masked_array
    >>> assert type(ev2) is np.ma.masked_array
    >>> assert (ed2<bandwidth).all()
    >>> assert ed2.shape == ed.shape == ev.shape == ev2.shape
    >>> distance(phi, dx, narrow=-1)
    Traceback (most recent call last):
      ...
    ValueError: parameter "narrow" must be greater than or equal to zero.
    >>> # make sure the narrow options works with an existing mask.
    >>> mask = abs(X)<0.25
    >>> d3 = distance(np.ma.masked_array(phi, mask), dx, narrow=bandwidth)
    >>> assert (d3.mask[mask]==True).all()
    >>> assert d3.mask.sum() > mask.sum()

    Testing periodic argument

    >>> X, Y = np.meshgrid(np.linspace(-2,2,200), np.linspace(-2,2,200))
    >>> phi = -1*np.ones_like(X); phi[X**2+(Y-0.9)**2<0.5] = 1.0
    >>> speed = np.ones_like(X); speed[(X-0.9)**2+Y**2<1.0] = 2.0
    >>> np.allclose(distance(phi),distance(phi,periodic=False)) and np.allclose(distance(phi),distance(phi,periodic=(0,0)))
    True
    >>> np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=False)) and np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=(0,0)))
    True
    >>> phi = -1*np.ones_like(X); phi[X**2+Y**2<0.5] = 1.0
    >>> speed = np.ones_like(X); speed[X**2+Y**2<1.0] = 2.0
    >>> np.allclose(distance(phi),distance(phi,periodic=True)) and np.allclose(distance(phi),distance(phi,periodic=(1,1)))
    True
    >>> np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=True)) and np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=(1,1)))
    True
    >>> np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=True)) and np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=[1,1]))
    True
    >>> np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=True)) and np.allclose(travel_time(phi,speed),travel_time(phi,speed,periodic=[True,True]))
    True
    >>> phi = -1*np.ones_like(X); phi[X**2+(Y-0.9)**2<0.5] = 1.0
    >>> speed = np.ones_like(X); speed[(X-0.9)**2+Y**2<1.0] = 2.0
    >>> np.allclose(distance(np.roll(phi,137,axis=0),periodic=True),np.roll(distance(phi,periodic=True),137,axis=0))
    True
    >>> np.allclose(travel_time(np.roll(phi,-77,axis=1),np.roll(speed,-77,axis=1),periodic=True),np.roll(travel_time(phi,speed,periodic=True),-77,axis=1))
    True

    >>> phi=[1,-1,1,1,1,1]
    >>> speed=[4,1,2,2,2,2]
    >>> np.allclose(extension_velocities(phi,speed)[1],(2.5,2.5,1.5,1.5,1.5,1.5))
    True
    >>> np.allclose(extension_velocities(phi,speed,periodic=True)[1],(2.5,2.5,1.5,1.5,1.5,2.5))
    True


    This is issue #18,

    >>> phi = np.array([[ 1,  1, -1, -1, -1, -1,  1],
    ...                 [ 3,  1, -1, -1, -1, -1,  1],
    ...                 [ 3,  3,  1,  1,  1,  1,  3],
    ...                 [ 3,  3,  3,  3,  3,  3,  3],
    ...                 [ 3,  3,  3,  3,  3,  3,  3]])
    >>> speed = np.array([[ 1.   ,  1.   ,  0.057,  0.128,  0.037,  0.039,  1.   ],
    ...                   [ 1.   ,  1.   ,  0.199,  0.408,  0.997,  0.688,  1.   ],
    ...                   [ 1.   ,  1.   ,  1.   ,  1.   ,  1.   ,  1.   ,  1.   ],
    ...                   [ 1.   ,  1.   ,  1.   ,  1.   ,  1.   ,  1.   ,  1.   ],
    ...                   [ 1.   ,  1.   ,  1.   ,  1.   ,  1.   ,  1.   ,  1.   ]])
    >>> assert not travel_time(phi,speed)[0][3] == 0

    >>> phi = np.array([[1, -1, -1],
    ...                 [0,  0, -1],
    ...                 [0,  0,  1]])
    >>> value = 0.01
    >>> speed = np.array([[ 1, value, 0.1],
    ...                   [ 0, 0,     1. ],
    ...                  [ 0, 0,     1. ]])
    >>> phi = np.ma.MaskedArray(phi, phi==0)
    >>> tt = travel_time(phi,speed)
    >>> tt.fill_value=0
    >>> np.testing.assert_allclose(tt.data, ((0.5, 50, 7.5),
    ...                                      (0,    0, 0.5),
    ...                                      (0,    0, 0.5)))

    This is from Pull Request #57:

    >>> a = np.array([[[600,  399,  641], [607,  605,  796], [602,  641,  797], [602,  658,  814]], [[272,   -1,  398], [208,  282,  539], [209,  285,  513], [208,  298,  519]], [[-19,   51,  307], [-191,   2,  403], [-171,   5,  325], [-172,   3,  318]]])

    >>> d = distance(a)
    >>> assert not (d==0).any()

    A 2D version of this bug was also discovered

    >>> a = np.array([[399,  -1, 51,   10.0], [605, 282,  2,   -3], [641, 285,  5, -100], [658, 298,  3,   -3]])
    >>> d = distance(a)
    >>> assert not (d==0).any()

    """

def test(verbose=None):
    r"""
    Run all the doctests available.
    """
    import doctest
    import skfmm
    fail0, test0 = doctest.testmod(skfmm,      verbose=verbose)
    fail1, test1 = doctest.testfile("heap.py", verbose=verbose)

    print ("Summary: {} tests run {} failures".format(test0+test1,
                                                      fail0+fail1))
    return fail0+fail1
