def test_convex_hull(self):
# Simple check that the convex hull seems to works
points = np.array([(0, 0), (0, 1), (1, 1), (1, 0)], dtype=np.double)
tri = qhull.Delaunay(points)
# +---+
# |\ 0|
# | \ |
# |1 \|
# +---+
assert_equal(tri.convex_hull, [[3, 2], [1, 2], [1, 0], [3, 0]])
def test_regression_2359(self):
# Check regression --- for certain point sets, gradient
# estimation could end up in an infinite loop
points = np.load(data_file('estimate_gradients_hang.npy'))
values = np.random.rand(points.shape[0])
tri = qhull.Delaunay(points)
# This should not hang
with warnings.catch_warnings():
warnings.simplefilter('ignore',
category=interpnd.GradientEstimationWarning)
interpnd.estimate_gradients_2d_global(tri, values, maxiter=1)
def test_tripoints_input_rescale(self):
# Test at single points
x = np.array([(0,0), (-5,-5), (-5,5), (5, 5), (2.5, 3)],
dtype=np.double)
y = np.arange(x.shape[0], dtype=np.double)
y = y - 3j*y
tri = qhull.Delaunay(x)
yi = interpnd.LinearNDInterpolator(tri.points, y)(x)
yi_rescale = interpnd.LinearNDInterpolator(tri.points, y,
rescale=True)(x)
assert_almost_equal(yi, yi_rescale)
def test_regression_2359(self):
# Check regression --- for certain point sets, gradient
# estimation could end up in an infinite loop
points = np.load(data_file('estimate_gradients_hang.npy'))
values = np.random.rand(points.shape[0])
tri = qhull.Delaunay(points)
# This should not hang
with suppress_warnings() as sup:
sup.filter(interpnd.GradientEstimationWarning,
"Gradient estimation did not converge")
interpnd.estimate_gradients_2d_global(tri, values, maxiter=1)
def test_tri_input_rescale(self):
# Test at single points
x = np.array([(0, 0), (-5, -5), (-5, 5), (5, 5), (2.5, 3)],
dtype=np.double)
y = np.arange(x.shape[0], dtype=np.double)
y = y - 3j * y
tri = qhull.Delaunay(x)
match = ("Rescaling is not supported when passing a "
"Delaunay triangulation as ``points``.")
with pytest.raises(ValueError, match=match):
interpnd.CloughTocher2DInterpolator(tri, y, rescale=True)(x)
def interp_weights(xyz, uvw):
"""The initial part of griddata (using linear interpolation) -
Creates a Delaunay mesh triangulation of the source points,
each point in the mesh is transformed to the new mesh
using an interpolation of each point inside a triangle is done using barycentric coordinates"""
tri = qhull.Delaunay(xyz)
simplex = tri.find_simplex(uvw)
vertices = np.take(tri.simplices, simplex, axis=0)
temp = np.take(tri.transform, simplex, axis=0)
delta = uvw - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
return vertices, np.hstack((bary, 1 - bary.sum(axis=1, keepdims=True)))
def check(name, chunksize):
points = DATASETS[name]
ndim = points.shape[1]
opts = None
nmin = ndim + 2
if name == 'some-points':
# since Qz is not allowed, use QJ
opts = 'QJ Pp'
elif name == 'pathological-1':
# include enough points so that we get different x-coordinates
nmin = 12
obj = qhull.Delaunay(points[:nmin], incremental=True,
qhull_options=opts)
for j in xrange(nmin, len(points), chunksize):
obj.add_points(points[j:j+chunksize])
obj2 = qhull.Delaunay(points)
obj3 = qhull.Delaunay(points[:nmin], incremental=True,
qhull_options=opts)
obj3.add_points(points[nmin:], restart=True)
# Check that the incremental mode agrees with upfront mode
if name.startswith('pathological'):
# XXX: These produce valid but different triangulations.
# They look OK when plotted, but how to check them?
assert_array_equal(np.unique(obj.simplices.ravel()),
np.arange(points.shape[0]))
assert_array_equal(np.unique(obj2.simplices.ravel()),
np.arange(points.shape[0]))
else:
assert_unordered_tuple_list_equal(obj.simplices, obj2.simplices,
tpl=sorted_tuple)
assert_unordered_tuple_list_equal(obj2.simplices, obj3.simplices,
tpl=sorted_tuple)
def check(name):
points = DATASETS[name]
tri = qhull.Delaunay(points)
hull = qhull.ConvexHull(points)
assert_hulls_equal(points, tri.convex_hull, hull.simplices)
# Check that the hull extremes are as expected
if points.shape[1] == 2:
assert_equal(np.unique(hull.simplices), np.sort(hull.vertices))
else:
assert_equal(np.unique(hull.simplices), hull.vertices)
def interp_weights(xy, uv,d=2):
'''calculate triangulation and coordinates transformation using qhull.Delaunay
1) Triangulate the irregular grid coordinates xy;
2) For each point in the new grid uv, search which simplex does it lay
3) Calculate barycentric coordinates with respect to the vertices of enclosing simplex
'''
tri = qhull.Delaunay(xy)
simplex = tri.find_simplex(uv)
vertices = np.take(tri.simplices, simplex, axis=0)
temp = np.take(tri.transform, simplex, axis=0)
delta = uv - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
return vertices, np.hstack((bary, 1 - bary.sum(axis=1, keepdims=True)))
def interp_weights_nd(xyz, uvw, d=2):
# Check for right shape:
if xyz.shape[1] != d:
xyz = xyz.T
if uvw.shape[1] != d:
uvw = uvw.T
tri = qhull.Delaunay(xyz)
simplex = tri.find_simplex(uvw)
vertices = np.take(tri.simplices, simplex, axis=0)
temp = np.take(tri.transform, simplex, axis=0)
delta = uvw - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
return vertices, np.hstack((bary, 1 - bary.sum(axis=1, keepdims=True)))
def test_coplanar(self):
# Check that the coplanar point output option indeed works
points = np.random.rand(10, 2)
points = np.r_[points, points] # duplicate input data
tri = qhull.Delaunay(points)
assert_(len(np.unique(tri.simplices.ravel())) == len(points) // 2)
assert_(len(tri.coplanar) == len(points) // 2)
assert_(len(np.unique(tri.coplanar[:, 2])) == len(points) // 2)
assert_(np.all(tri.vertex_to_simplex >= 0))
def computeV(self,values):
"""
Compute rotation matrix _V, and triangulation self.tri
:param values: Nested array with the data values
"""
if not self._V is None:
return
Morig= [self.flattenArray(pt[0]) for pt in values]
aM = np.array(Morig)
MT = aM.T.tolist()
self.delta_x = np.array([[ sum(x)/len(Morig) for x in MT ]])
M = []
for Mx in Morig:
m=(np.array([Mx]) - self.delta_x).tolist()[0]
M.append(m)
try:
## we dont need thousands of points for SVD
n = int(math.ceil(len(M)/2000.))
Vt=svd(M[::n])[2]
except Exception as e:
raise SModelSError("exception caught when performing singular value decomposition: %s, %s" %(type(e), e))
V=Vt.T
self._V= V ## self.round ( V )
Mp=[]
## the dimensionality of the whole mass space, disrespecting equal branches
## assumption
self.full_dimensionality = len(Morig[0])
self.dimensionality=0
for m in M:
mp=np.dot(m,V)
Mp.append ( mp )
nz=self.countNonZeros(mp)
if nz>self.dimensionality:
self.dimensionality=nz
MpCut=[]
for i in Mp:
MpCut.append(i[:self.dimensionality].tolist() )
if self.dimensionality > 1:
self.tri = qhull.Delaunay(MpCut)
else:
self.tri = Delaunay1D(MpCut)
def test_more_barycentric_transforms(self):
# Triangulate some "nasty" grids
eps = np.finfo(float).eps
npoints = {2: 70, 3: 11, 4: 5, 5: 3}
for ndim in xrange(2, 6):
# Generate an uniform grid in n-d unit cube
x = np.linspace(0, 1, npoints[ndim])
grid = np.c_[list(
map(np.ravel, np.broadcast_arrays(*np.ix_(*([x] * ndim)))))].T
err_msg = "ndim=%d" % ndim
# Check using regular grid
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True)
# Check with eps-perturbations
np.random.seed(1234)
m = (np.random.rand(grid.shape[0]) < 0.2)
grid[m, :] += 2 * eps * (np.random.rand(*grid[m, :].shape) - 0.5)
tri = qhull.Delaunay(grid)
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True,
unit_cube_tol=2 * eps)
# Check with duplicated data
tri = qhull.Delaunay(np.r_[grid, grid])
self._check_barycentric_transforms(tri,
err_msg=err_msg,
unit_cube=True,
unit_cube_tol=2 * eps)
def _get_subset_nodes(self, grid_x, grid_y, target_x, target_y):
"""
Retuns grid nodes that are necessary for intepolating onto target_x,y
"""
orig_shape = grid_x.shape
grid_xy = np.array((grid_x.ravel(), grid_y.ravel())).T
target_xy = np.array((target_x.ravel(), target_y.ravel())).T
tri = qhull.Delaunay(grid_xy)
simplex = tri.find_simplex(target_xy)
vertices = np.take(tri.simplices, simplex, axis=0)
nodes = np.unique(vertices.ravel())
nodes_x, nodes_y = np.unravel_index(nodes, orig_shape)
return nodes, nodes_x, nodes_y
def test_nd_simplex(self):
# simple smoke test: triangulate a n-dimensional simplex
for nd in xrange(2, 8):
points = np.zeros((nd+1, nd))
for j in xrange(nd):
points[j,j] = 1.0
points[-1,:] = 1.0
tri = qhull.Delaunay(points)
tri.vertices.sort()
assert_equal(tri.vertices, np.arange(nd+1, dtype=np.int)[None,:])
assert_equal(tri.neighbors, -1 + np.zeros((nd+1), dtype=np.int)[None,:])
def _interp_weights(xyz, uvw, d=None):
"""
:param xyz: flattened coords of current grid
:param uvw: flattened coords of target grid
:param d: number of dimensions of new grid
:return: triangulisation lookup table, point weights
"""
tri = qhull.Delaunay(xyz)
simplex = tri.find_simplex(uvw)
vertices = np.take(tri.simplices, simplex, axis=0)
temp = np.take(tri.transform, simplex, axis=0)
delta = uvw - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
return vertices, np.hstack((bary, 1 - bary.sum(axis=1, keepdims=True)))
def interp_weights(xy, uv, d=2): # pragma: no cover
tri = qhull.Delaunay(xy)
simplex = tri.find_simplex(uv)
vertices = np.take(tri.simplices, simplex, axis=0)
temp = np.take(tri.transform, simplex, axis=0)
delta = uv - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
return vertices, np.hstack((bary, 1 - bary.sum(axis=1, keepdims=True)))
def check(name):
chunks, opts = INCREMENTAL_DATASETS[name]
points = np.concatenate(chunks, axis=0)
obj = qhull.Delaunay(chunks[0],
incremental=True,
qhull_options=opts)
for chunk in chunks[1:]:
obj.add_points(chunk)
obj2 = qhull.Delaunay(points)
obj3 = qhull.Delaunay(chunks[0],
incremental=True,
qhull_options=opts)
if len(chunks) > 1:
obj3.add_points(np.concatenate(chunks[1:], axis=0),
restart=True)
# Check that the incremental mode agrees with upfront mode
if name.startswith('pathological'):
# XXX: These produce valid but different triangulations.
# They look OK when plotted, but how to check them?
assert_array_equal(np.unique(obj.simplices.ravel()),
np.arange(points.shape[0]))
assert_array_equal(np.unique(obj2.simplices.ravel()),
np.arange(points.shape[0]))
else:
assert_unordered_tuple_list_equal(obj.simplices,
obj2.simplices,
tpl=sorted_tuple)
assert_unordered_tuple_list_equal(obj2.simplices,
obj3.simplices,
tpl=sorted_tuple)
def test_hull_consistency_tri(self, name):
# Check that a convex hull returned by qhull in ndim
# and the hull constructed from ndim delaunay agree
points = DATASETS[name]
tri = qhull.Delaunay(points)
hull = qhull.ConvexHull(points)
assert_hulls_equal(points, tri.convex_hull, hull.simplices)
# Check that the hull extremes are as expected
if points.shape[1] == 2:
assert_equal(np.unique(hull.simplices), np.sort(hull.vertices))
else:
assert_equal(np.unique(hull.simplices), hull.vertices)
def test_triangle(self):
points = np.array([(0,0), (0,1), (1,0)], dtype=np.double)
tri = qhull.Delaunay(points)
# 1
# +
# |\
# | \
# |0 \
# +---+
# 0 2
self._check_ridges(tri, 0, [(0, 1), (0, 2)])
self._check_ridges(tri, 1, [(1, 0), (1, 2)])
self._check_ridges(tri, 2, [(2, 0), (2, 1)])
def __init__(self, grid_xyz, target_xyz, fill_mode=None, fill_value=np.nan, normalize=False):
"""
:arg grid_xyz: Array of source grid coordinates, shape (npoints, 2) or (npoints, 3)
:arg target_xyz: Array of target grid coordinates, shape (n, 2) or (n, 3)
"""
self.fill_value = np.nan
self.fill_mode = fill_mode
self.normalize = normalize
if self.normalize:
def get_norm_params(x, scale=None):
min = x.min()
max = x.max()
if scale is None:
scale = max - min
a = 1./scale
b = -min*a
return a, b
ax, bx = get_norm_params(target_xyz[:, 0])
ay, by = get_norm_params(target_xyz[:, 1])
az, bz = get_norm_params(target_xyz[:, 2])
self.norm_a = np.array([ax, ay, az])
self.norm_b = np.array([bx, by, bz])
ngrid_xyz = self.norm_a*grid_xyz + self.norm_b
ntarget_xyz = self.norm_a*target_xyz + self.norm_b
else:
ngrid_xyz = grid_xyz
ntarget_xyz = target_xyz
d = ngrid_xyz.shape[1]
tri = qhull.Delaunay(ngrid_xyz)
# NOTE this becomes expensive in 3D for npoints > 10k
simplex = tri.find_simplex(ntarget_xyz)
vertices = np.take(tri.simplices, simplex, axis=0)
temp = np.take(tri.transform, simplex, axis=0)
delta = ntarget_xyz - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
self.vtx = vertices
self.wts = np.hstack((bary, 1 - bary.sum(axis=1, keepdims=True)))
self.outside = np.nonzero(np.any(self.wts < 0, axis=1))[0]
self.fill_nearest = self.fill_mode == 'nearest' and len(self.outside) > 0
if self.fill_nearest:
# find nearest neighbor in the data set
from scipy.spatial import cKDTree
dist, ix = cKDTree(ngrid_xyz).query(ntarget_xyz[self.outside])
self.outside_to_nearest = ix
def test_tri_input_rescale(self):
# Test at single points
x = np.array([(0, 0), (-5, -5), (-5, 5), (5, 5), (2.5, 3)],
dtype=np.double)
y = np.arange(x.shape[0], dtype=np.double)
y = y - 3j * y
tri = qhull.Delaunay(x)
try:
interpnd.CloughTocher2DInterpolator(tri, y, rescale=True)(x)
except ValueError as a:
if str(a) != ("Rescaling is not supported when passing a "
"Delaunay triangulation as ``points``."):
raise
except:
raise
def test_rectangle(self):
points = np.array([(0,0), (0,1), (1,1), (1,0)], dtype=np.double)
tri = qhull.Delaunay(points)
# 1 2
# +---+
# |\ 0|
# | \ |
# |1 \|
# +---+
# 0 3
self._check_ridges(tri, 0, [(0, 1), (0, 3)])
self._check_ridges(tri, 1, [(1, 0), (1, 3), (1, 2)])
self._check_ridges(tri, 2, [(2, 1), (2, 3)])
self._check_ridges(tri, 3, [(3, 0), (3, 1), (3, 2)])
def test_degenerate_barycentric_transforms(self):
# The triangulation should not produce invalid barycentric
# transforms that stump the simplex finding
data = np.load(os.path.join(os.path.dirname(__file__), 'data',
'degenerate_pointset.npz'))
points = data['c']
data.close()
tri = qhull.Delaunay(points)
# Check that there are not too many invalid simplices
bad_count = np.isnan(tri.transform[:,0,0]).sum()
assert_(bad_count < 20, bad_count)
# Check the transforms
self._check_barycentric_transforms(tri)
def get_interp_weights(points, xi, fill_value=np.nan):
'''
Compute vertices and weights for 3D linear interpolation.
Arguments
---------
points : ndarray
Coordinates of known points, shape = (N, 3).
xi : ndarray
Coordinates of points to be interpolated,
shape = (M, 3).
Keyword Arguments
-----------------
fill_value : scalar
Value used in output `weights` array for invalid
points (outside convex hull).
Returns
-------
vertices : ndarray
Shape = (M, 4).
weights : ndarray
Shape (M, 4).
Notes
-----
Adapted from https://stackoverflow.com/questions/20915502/ .
'''
# Only 3d interpolation for now.
d = 3
if points.shape[1] != d or xi.shape[1] != d:
raise ValueError('Input shape not 3d.')
T = qhull.Delaunay(points)
simplex = T.find_simplex(xi) # Same shape as xi. Points outside get -1.
vertices = np.take(T.simplices, simplex, axis=0)
temp = np.take(T.transform, simplex, axis=0)
delta = xi - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
weights = np.hstack((bary, 1 - bary.sum(axis=1, keepdims=True)))
weights[simplex == -1] *= fill_value
return vertices, weights
def test_find_simplex(self):
# Simple check that simplex finding works
points = np.array([(0, 0), (0, 1), (1, 1), (1, 0)], dtype=np.double)
tri = qhull.Delaunay(points)
# +---+
# |\ 0|
# | \ |
# |1 \|
# +---+
assert_equal(tri.vertices, [[1, 3, 2], [3, 1, 0]])
for p in [(0.25, 0.25, 1), (0.75, 0.75, 0), (0.3, 0.2, 1)]:
i = tri.find_simplex(p[:2])
assert_equal(i, p[2], err_msg='%r' % (p, ))
j = qhull.tsearch(tri, p[:2])
assert_equal(i, j)
def _get_subset_nodes(grid_x, grid_y, target_x, target_y):
"""
Retuns grid nodes that are necessary for intepolating onto target_x,y
"""
orig_shape = grid_x.shape
grid_xy = np.array((grid_x.ravel(), grid_y.ravel())).T
target_xy = np.array((target_x.ravel(), target_y.ravel())).T
tri = qhull.Delaunay(grid_xy)
simplex = tri.find_simplex(target_xy)
vertices = np.take(tri.simplices, simplex, axis=0)
nodes = np.unique(vertices.ravel())
nodes_x, nodes_y = np.unravel_index(nodes, orig_shape)
# x and y bounds for reading a subset of the netcdf data
ind_x = slice(nodes_x.min(), nodes_x.max() + 1)
ind_y = slice(nodes_y.min(), nodes_y.max() + 1)
return nodes, ind_x, ind_y
def generate_triangulation(self):
xc = self.x.ravel()
yc = self.y.ravel()
zc = self.z.ravel()
# Remove any NaN values as the triangulation can't handle this
nans = np.isnan(zc)
xc = xc[~nans]
yc = yc[~nans]
self.no_nan_values = zc[~nans]
# Normalize the coordinates. This improves the triangulation results
# in cases where the data ranges on both axes are very different
# in magnitude
xmin, xmax, ymin, ymax, _, _ = self.get_limits()
xc = (xc - xmin) / (xmax - xmin)
yc = (yc - ymin) / (ymax - ymin)
self.tri = qhull.Delaunay(np.column_stack((xc, yc)))
def interp_weights(xy, uv, d=2):
"""Compute the nearest vertices & weights (for reuse)"""
tri = qhull.Delaunay(xy)
simplex = tri.find_simplex(uv)
if not np.all(simplex >= 0):
if not self.params.mesh.periodic_bc:
log.error("Some points missing in interpolation "
"of velocity obs to function space.")
else:
log.warning("Some points missing in interpolation "
"of velocity obs to function space.")
vertices = np.take(tri.simplices, simplex, axis=0)
temp = np.take(tri.transform, simplex, axis=0)
delta = uv - temp[:, d]
bary = np.einsum('njk,nk->nj', temp[:, :d, :], delta)
return vertices, np.hstack(
(bary, 1 - bary.sum(axis=1, keepdims=True)))
def triangulate(xy):
"""
triangulate(xy)
Compute the D-D Delaunay triangulation of a grid.
Parameters
----------
xy : 2-D ndarray of floats with shape (n, D), or length D tuple of 1-D ndarrays with shape (n,).
Data point coordinates.
Returns
-------
tri : Delaunay object
2D Delaunay triangulation of `xy`.
"""
tri = qhull.Delaunay(xy)
return tri
