python中griddata的外插值_griddata二维插值[通俗易懂]

“””Simple N-D interpolation

.. versionadded:: 0.9″””

#

#Copyright (C) Pauli Virtanen, 2010.#

#Distributed under the same BSD license as Scipy.#

#

#Note: this file should be run through the Mako template engine before#feeding it to Cython.#

#Run “generate_qhull.py“ to regenerate the “qhull.c“ file#cimport cythonfromlibc.float cimport DBL_EPSILONfromlibc.math cimport fabs, sqrtimportnumpy as npimportscipy.spatial.qhull as qhull

cimport scipy.spatial.qhull as qhullimportwarnings#——————————————————————————#Numpy etc.#——————————————————————————

cdef externfrom “numpy/ndarrayobject.h”:

cdef enum:

NPY_MAXDIMS

ctypedef fused double_or_complex:

double

double complex#——————————————————————————#Interpolator base class#——————————————————————————

classNDInterpolatorBase(object):”””Common routines for interpolators.

.. versionadded:: 0.9″””

def __init__(self, points, values, fill_value=np.nan, ndim=None,

rescale=False, need_contiguous=True, need_values=True):”””Check shape of points and values arrays, and reshape values to

(npoints, nvalues). Ensure the `points` and values arrays are

C-contiguous, and of correct type.”””

ifisinstance(points, qhull.Delaunay):#Precomputed triangulation was passed in

ifrescale:raise ValueError(“Rescaling is not supported when passing”

“a Delaunay triangulation as “points“.”)

self.tri=points

points=points.pointselse:

self.tri=None

points=_ndim_coords_from_arrays(points)

values=np.asarray(values)

_check_init_shape(points, values, ndim=ndim)ifneed_contiguous:

points= np.ascontiguousarray(points, dtype=np.double)ifneed_values:

self.values_shape= values.shape[1:]if values.ndim == 1:

self.values=values[:,None]elif values.ndim == 2:

self.values=valueselse:

self.values=values.reshape(values.shape[0],

np.prod(values.shape[1:]))#Complex or real?

self.is_complex =np.issubdtype(self.values.dtype, np.complexfloating)ifself.is_complex:ifneed_contiguous:

self.values=np.ascontiguousarray(self.values,

dtype=np.complex128)

self.fill_value=complex(fill_value)else:ifneed_contiguous:

self.values= np.ascontiguousarray(self.values, dtype=np.double)

self.fill_value=float(fill_value)if notrescale:

self.scale=None

self.points=pointselse:#scale to unit cube centered at 0

self.offset = np.mean(points, axis=0)

self.points= points -self.offset

self.scale= self.points.ptp(axis=0)

self.scale[~(self.scale > 0)] = 1.0 #avoid division by 0

self.points /=self.scaledef_check_call_shape(self, xi):

xi=np.asanyarray(xi)if xi.shape[-1] != self.points.shape[1]:raise ValueError(“number of dimensions in xi does not match x”)returnxidef_scale_x(self, xi):if self.scale isNone:returnxielse:return (xi – self.offset) /self.scaledef __call__(self, *args):”””interpolator(xi)

Evaluate interpolator at given points.

Parameters

———-

x1, x2, … xn: array-like of float

Points where to interpolate data at.

x1, x2, … xn can be array-like of float with broadcastable shape.

or x1 can be array-like of float with shape “(…, ndim)“”””xi= _ndim_coords_from_arrays(args, ndim=self.points.shape[1])

xi=self._check_call_shape(xi)

shape=xi.shape

xi= xi.reshape(-1, shape[-1])

xi= np.ascontiguousarray(xi, dtype=np.double)

xi=self._scale_x(xi)ifself.is_complex:

r=self._evaluate_complex(xi)else:

r=self._evaluate_double(xi)return np.asarray(r).reshape(shape[:-1] +self.values_shape)

cpdef _ndim_coords_from_arrays(points, ndim=None):”””Convert a tuple of coordinate arrays to a (…, ndim)-shaped array.”””cdef ssize_t j, nif isinstance(points, tuple) and len(points) == 1:#handle argument tuple

points =points[0]ifisinstance(points, tuple):

p= np.broadcast_arrays(*points)

n=len(p)for j in range(1, n):if p[j].shape !=p[0].shape:raise ValueError(“coordinate arrays do not have the same shape”)

points= np.empty(p[0].shape + (len(points),), dtype=float)for j, item inenumerate(p):

points[…,j]=itemelse:

points=np.asanyarray(points)if points.ndim == 1:if ndim isNone:

points= points.reshape(-1, 1)else:

points= points.reshape(-1, ndim)returnpoints

cdef _check_init_shape(points, values, ndim=None):”””Check shape of points and values arrays”””

if values.shape[0] !=points.shape[0]:raise ValueError(“different number of values and points”)if points.ndim != 2:raise ValueError(“invalid shape for input data points”)if points.shape[1] < 2:raise ValueError(“input data must be at least 2-D”)if ndim is not None and points.shape[1] !=ndim:raise ValueError(“this mode of interpolation available only for”

“%d-D data” %ndim)#——————————————————————————#Linear interpolation in N-D#——————————————————————————

classLinearNDInterpolator(NDInterpolatorBase):”””LinearNDInterpolator(points, values, fill_value=np.nan, rescale=False)

Piecewise linear interpolant in N dimensions.

.. versionadded:: 0.9

Methods

——-

__call__

Parameters

———-

points : ndarray of floats, shape (npoints, ndims); or Delaunay

Data point coordinates, or a precomputed Delaunay triangulation.

values : ndarray of float or complex, shape (npoints, …)

Data values.

fill_value : float, optional

Value used to fill in for requested points outside of the

convex hull of the input points. If not provided, then

the default is “nan“.

rescale : bool, optional

Rescale points to unit cube before performing interpolation.

This is useful if some of the input dimensions have

incommensurable units and differ by many orders of magnitude.

Notes

—–

The interpolant is constructed by triangulating the input data

with Qhull [1]_, and on each triangle performing linear

barycentric interpolation.

Examples

——–

We can interpolate values on a 2D plane:

>>> from scipy.interpolate import LinearNDInterpolator

>>> import matplotlib.pyplot as plt

>>> np.random.seed(0)

>>> x = np.random.random(10) – 0.5

>>> y = np.random.random(10) – 0.5

>>> z = np.hypot(x, y)

>>> X = np.linspace(min(x), max(x))

>>> Y = np.linspace(min(y), max(y))

>>> X, Y = np.meshgrid(X, Y) # 2D grid for interpolation

>>> interp = LinearNDInterpolator(list(zip(x, y)), z)

>>> Z = interp(X, Y)

>>> plt.pcolormesh(X, Y, Z, shading=’auto’)

>>> plt.plot(x, y, “ok”, label=”input point”)

>>> plt.legend()

>>> plt.colorbar()

>>> plt.axis(“equal”)

>>> plt.show()

See also

——–

griddata :

Interpolate unstructured D-D data.

NearestNDInterpolator :

Nearest-neighbor interpolation in N dimensions.

CloughTocher2DInterpolator :

Piecewise cubic, C1 smooth, curvature-minimizing interpolant in 2D.

References

———-

.. [1] http://www.qhull.org/”””

def __init__(self, points, values, fill_value=np.nan, rescale=False):

NDInterpolatorBase.__init__(self, points, values, fill_value=fill_value,

rescale=rescale)if self.tri isNone:

self.tri=qhull.Delaunay(self.points)def_evaluate_double(self, xi):return self._do_evaluate(xi, 1.0)def_evaluate_complex(self, xi):return self._do_evaluate(xi, 1.0j)

@cython.boundscheck(False)

@cython.wraparound(False)def _do_evaluate(self, const double[:,::1] xi, double_or_complex dummy):

cdef const double_or_complex[:,::1] values =self.values

cdef double_or_complex[:,::1] out

cdef const double[:,::1] points =self.points

cdef const int[:,::1] simplices =self.tri.simplices

cdef double c[NPY_MAXDIMS]

cdef double_or_complex fill_value

cdef int i, j, k, m, ndim, isimplex, inside, start, nvalues

cdef qhull.DelaunayInfo_t info

cdef double eps, eps_broad

ndim= xi.shape[1]

start=0

fill_value=self.fill_value

qhull._get_delaunay_info(&info, self.tri, 1, 0, 0)

out= np.empty((xi.shape[0], self.values.shape[1]),

dtype=self.values.dtype)

nvalues= out.shape[1]

eps= 100 *DBL_EPSILON

eps_broad=sqrt(DBL_EPSILON)

with nogil:for i inrange(xi.shape[0]):#1) Find the simplex

isimplex= qhull._find_simplex(&info, c,&xi[0,0] + i*ndim,&start, eps, eps_broad)#2) Linear barycentric interpolation

if isimplex == -1:#don’t extrapolate

for k inrange(nvalues):

out[i,k]=fill_valuecontinue

for k inrange(nvalues):

out[i,k]=0for j in range(ndim+1):for k inrange(nvalues):

m=simplices[isimplex,j]

out[i,k]= out[i,k] + c[j] *values[m,k]returnout#——————————————————————————#Gradient estimation in 2D#——————————————————————————

classGradientEstimationWarning(Warning):pass@cython.cdivision(True)

cdef int _estimate_gradients_2d_global(qhull.DelaunayInfo_t*d, double *data,

int maxiter, double tol,

double*y) nogil:”””Estimate gradients of a function at the vertices of a 2d triangulation.

Parameters

———-

info : input

Triangulation in 2D

data : input

Function values at the vertices

maxiter : input

Maximum number of Gauss-Seidel iterations

tol : input

Absolute / relative stop tolerance

y : output, shape (npoints, 2)

Derivatives [F_x, F_y] at the vertices

Returns

——-

num_iterations

Number of iterations if converged, 0 if maxiter reached

without convergence

Notes

—–

This routine uses a re-implementation of the global approximate

curvature minimization algorithm described in [Nielson83] and [Renka84].

References

———-

.. [Nielson83] G. Nielson,

”A method for interpolating scattered data based upon a minimum norm

network”.

Math. Comp., 40, 253 (1983).

.. [Renka84] R. J. Renka and A. K. Cline.

”A Triangle-based C1 interpolation method.”,

Rocky Mountain J. Math., 14, 223 (1984).”””cdef double Q[2*2]

cdef double s[2]

cdef double r[2]

cdef int ipoint, iiter, k, ipoint2, jpoint2

cdef double f1, f2, df2, ex, ey, L, L3, det, err, change#initialize

for ipoint in range(2*d.npoints):

y[ipoint]=0# #Main point:

# #Z = sum_T sum_{E in T} int_E |W”|^2 = min!

# #where W” is the second derivative of the Clough-Tocher

#interpolant to the direction of the edge E in triangle T.

# #The minimization is done iteratively: for each vertex V,

#the sum

# #Z_V = sum_{E connected to V} int_E |W”|^2

# #is minimized separately, using existing values at other V.

# #Since the interpolant can be written as

# #W(x) = f(x) + w(x)^T y

# #where y = [ F_x(V); F_y(V) ], it is clear that the solution to

#the local problem is is given as a solution of the 2×2 matrix

#equation.

# #Here, we use the Clough-Tocher interpolant, which restricted to

#a single edge is

# #w(x) = (1 – x)**3 * f1

#+ x*(1 – x)**2 * (df1 + 3*f1)

#+ x**2*(1 – x) * (df2 + 3*f2)

#+ x**3 * f2

# #where f1, f2 are values at the vertices, and df1 and df2 are

#derivatives along the edge (away from the vertices).

# #As a consequence, one finds

# #L^3 int_{E} |W”|^2 = y^T A y + 2 B y + C

# #with

# #A = [4, -2; -2, 4]

#B = [6*(f1 – f2), 6*(f2 – f1)]

#y = [df1, df2]

#L = length of edge E

# #and C is not needed for minimization. Since df1 = dF1.E, df2 = -dF2.E,

#with dF1 = [F_x(V_1), F_y(V_1)], and the edge vector E = V2 – V1,

#we have

# #Z_V = dF1^T Q dF1 + 2 s.dF1 + const.

# #which is minimized by

# #dF1 = -Q^{-1} s

# #where

# #Q = sum_E [A_11 E E^T]/L_E^3 = 4 sum_E [E E^T]/L_E^3

#s = sum_E [ B_1 + A_21 df2] E /L_E^3

#= sum_E [ 6*(f1 – f2) + 2*(E.dF2)] E / L_E^3

#

#Gauss-Seidel

for iiter inrange(maxiter):

err=0for ipoint inrange(d.npoints):for k in range(2*2):

Q[k]=0for k in range(2):

s[k]=0#walk over neighbours of given point

for jpoint2 inrange(d.vertex_neighbors_indptr[ipoint],

d.vertex_neighbors_indptr[ipoint+1]):

ipoint2=d.vertex_neighbors_indices[jpoint2]#edge

ex = d.points[2*ipoint2 + 0] – d.points[2*ipoint +0]

ey= d.points[2*ipoint2 + 1] – d.points[2*ipoint + 1]

L= sqrt(ex**2 + ey**2)

L3= L*L*L#data at vertices

f1 =data[ipoint]

f2=data[ipoint2]#scaled gradient projections on the edge

df2 = -ex*y[2*ipoint2 + 0] – ey*y[2*ipoint2 + 1]#edge sum

Q[0] += 4*ex*ex /L3

Q[1] += 4*ex*ey /L3

Q[3] += 4*ey*ey /L3

s[0]+= (6*(f1 – f2) – 2*df2) * ex /L3

s[1] += (6*(f1 – f2) – 2*df2) * ey /L3

Q[2] = Q[1]#solve

det= Q[0]*Q[3] – Q[1]*Q[2]

r[0]= ( Q[3]*s[0] – Q[1]*s[1])/det

r[1] = (-Q[2]*s[0] + Q[0]*s[1])/det

change= max(fabs(y[2*ipoint + 0] +r[0]),

fabs(y[2*ipoint + 1] + r[1]))

y[2*ipoint + 0] = -r[0]

y[2*ipoint + 1] = -r[1]#relative/absolute error

change /= max(1.0, max(fabs(r[0]), fabs(r[1])))

err=max(err, change)if err

#Didn’t converge before maxiter

return0

@cython.boundscheck(False)

@cython.wraparound(False)

cpdef estimate_gradients_2d_global(tri, y, int maxiter=400, double tol=1e-6):

cdef const double[:,::1] data

cdef double[:,:,::1] grad

cdef qhull.DelaunayInfo_t info

cdef int k, ret, nvalues

y=np.asanyarray(y)if y.shape[0] !=tri.npoints:raise ValueError(“‘y’ has a wrong number of items”)ifnp.issubdtype(y.dtype, np.complexfloating):

rg= estimate_gradients_2d_global(tri, y.real, maxiter=maxiter, tol=tol)

ig= estimate_gradients_2d_global(tri, y.imag, maxiter=maxiter, tol=tol)

r= np.zeros(rg.shape, dtype=complex)

r.real=rg

r.imag=igreturnr

y_shape=y.shapeif y.ndim == 1:

y=y[:,None]

y= y.reshape(tri.npoints, -1).T

y= np.ascontiguousarray(y, dtype=np.double)

yi= np.empty((y.shape[0], y.shape[1], 2))

data=y

grad=yi

qhull._get_delaunay_info(&info, tri, 0, 0, 1)

nvalues=data.shape[0]for k inrange(nvalues):

with nogil:

ret=_estimate_gradients_2d_global(&info,&data[k,0],

maxiter,

tol,&grad[k,0,0])if ret ==0:

warnings.warn(“Gradient estimation did not converge,”

“the results may be inaccurate”,

GradientEstimationWarning)return yi.transpose(1, 0, 2).reshape(y_shape + (2,))#——————————————————————————#Cubic interpolation in 2D#——————————————————————————

@cython.cdivision(True)

cdef double_or_complex _clough_tocher_2d_single(qhull.DelaunayInfo_t*d,

int isimplex,

double*b,

double_or_complex*f,

double_or_complex*df) nogil:”””Evaluate Clough-Tocher interpolant on a 2D triangle.

Parameters

———-

d :

Delaunay info

isimplex : int

Triangle to evaluate on

b : shape (3,)

Barycentric coordinates of the point on the triangle

f : shape (3,)

Function values at vertices

df : shape (3, 2)

Gradient values at vertices

Returns

——-

w :

Value of the interpolant at the given point

References

———-

.. [CT] See, for example,

P. Alfeld,

”A trivariate Clough-Tocher scheme for tetrahedral data”.

Computer Aided Geometric Design, 1, 169 (1984);

G. Farin,

”Triangular Bernstein-Bezier patches”.

Computer Aided Geometric Design, 3, 83 (1986).”””cdef double_or_complex \

c3000, c0300, c0030, c0003, \

c2100, c2010, c2001, c0210, c0201, c0021, \

c1200, c1020, c1002, c0120, c0102, c0012, \

c1101, c1011, c0111

cdef double_or_complex \

f1, f2, f3, df12, df13, df21, df23, df31, df32

cdef double g[3]

cdef double \

e12x, e12y, e23x, e23y, e31x, e31y, \

e14x, e14y, e24x, e24y, e34x, e34y

cdef double_or_complex w

cdef double minval

cdef double b1, b2, b3, b4

cdef int k, itri

cdef double c[3]

cdef double y[2]#XXX: optimize + refactor this!

e12x= (+ d.points[0 + 2*d.simplices[3*isimplex + 1]]- d.points[0 + 2*d.simplices[3*isimplex +0]])

e12y= (+ d.points[1 + 2*d.simplices[3*isimplex + 1]]- d.points[1 + 2*d.simplices[3*isimplex +0]])

e23x= (+ d.points[0 + 2*d.simplices[3*isimplex + 2]]- d.points[0 + 2*d.simplices[3*isimplex + 1]])

e23y= (+ d.points[1 + 2*d.simplices[3*isimplex + 2]]- d.points[1 + 2*d.simplices[3*isimplex + 1]])

e31x= (+ d.points[0 + 2*d.simplices[3*isimplex +0]]- d.points[0 + 2*d.simplices[3*isimplex + 2]])

e31y= (+ d.points[1 + 2*d.simplices[3*isimplex +0]]- d.points[1 + 2*d.simplices[3*isimplex + 2]])

e14x= (e12x – e31x)/3e14y= (e12y – e31y)/3e24x= (-e12x + e23x)/3e24y= (-e12y + e23y)/3e34x= (e31x – e23x)/3e34y= (e31y – e23y)/3f1=f[0]

f2= f[1]

f3= f[2]

df12= +(df[2*0+0]*e12x + df[2*0+1]*e12y)

df21= -(df[2*1+0]*e12x + df[2*1+1]*e12y)

df23= +(df[2*1+0]*e23x + df[2*1+1]*e23y)

df32= -(df[2*2+0]*e23x + df[2*2+1]*e23y)

df31= +(df[2*2+0]*e31x + df[2*2+1]*e31y)

df13= -(df[2*0+0]*e31x + df[2*0+1]*e31y)

c3000=f1

c2100= (df12 + 3*c3000)/3c2010= (df13 + 3*c3000)/3c0300=f2

c1200= (df21 + 3*c0300)/3c0210= (df23 + 3*c0300)/3c0030=f3

c1020= (df31 + 3*c0030)/3c0120= (df32 + 3*c0030)/3c2001= (c2100 + c2010 + c3000)/3c0201= (c1200 + c0300 + c0210)/3c0021= (c1020 + c0120 + c0030)/3

# #Now, we need to impose the condition that the gradient of the spline

#to some direction `w` is a linear function along the edge.

# #As long as two neighbouring triangles agree on the choice of the

#direction `w`, this ensures global C1 differentiability.

#Otherwise, the choice of the direction is arbitrary (except that

#it should not point along the edge, of course).

# #In [CT]_, it is suggested to pick `w` as the normal of the edge.

#This choice is given by the formulas

# #w_12 = E_24 + g[0] * E_23

#w_23 = E_34 + g[1] * E_31

#w_31 = E_14 + g[2] * E_12

# #g[0] = -(e24x*e23x + e24y*e23y) / (e23x**2 + e23y**2)

#g[1] = -(e34x*e31x + e34y*e31y) / (e31x**2 + e31y**2)

#g[2] = -(e14x*e12x + e14y*e12y) / (e12x**2 + e12y**2)

# #However, this choice gives an interpolant that is *not*

#invariant under affine transforms. This has some bad

#consequences: for a very narrow triangle, the spline can

#develops huge oscillations. For instance, with the input data

# #[(0, 0), (0, 1), (eps, eps)], eps = 0.01

#F = [0, 0, 1]

#dF = [(0,0), (0,0), (0,0)]

# #one observes that as eps -> 0, the absolute maximum value of the

#interpolant approaches infinity.

# #So below, we aim to pick affine invariant `g[k]`.

#We choose

# #w = V_4′ – V_4

# #where V_4 is the centroid of the current triangle, and V_4′ the

#centroid of the neighbour. Since this quantity transforms similarly

#as the gradient under affine transforms, the resulting interpolant

#is affine-invariant. Moreover, two neighbouring triangles clearly

#always agree on the choice of `w` (sign is unimportant), and so

#this choice also makes the interpolant C1.

# #The drawback here is a performance penalty, since we need to

#peek into neighbouring triangles.

#

for k in range(3):

itri= d.neighbors[3*isimplex +k]if itri == -1:#No neighbour.

#Compute derivative to the centroid direction (e_12 + e_13)/2.

g[k] = -1./2

continue

#Centroid of the neighbour, in our local barycentric coordinates

y[0]= (+ d.points[0 + 2*d.simplices[3*itri +0]]+ d.points[0 + 2*d.simplices[3*itri + 1]]+ d.points[0 + 2*d.simplices[3*itri + 2]]) / 3y[1] = (+ d.points[1 + 2*d.simplices[3*itri +0]]+ d.points[1 + 2*d.simplices[3*itri + 1]]+ d.points[1 + 2*d.simplices[3*itri + 2]]) / 3qhull._barycentric_coordinates(2, d.transform + isimplex*2*3, y, c)#Rewrite V_4′-V_4 = const*[(V_4-V_2) + g_i*(V_3 – V_2)]

#Now, observe that the results can be written *in terms of

#barycentric coordinates*. Barycentric coordinates stay

#invariant under affine transformations, so we can directly

#conclude that the choice below is affine-invariant.

if k ==0:

g[k]= (2*c[2] + c[1] – 1) / (2 – 3*c[2] – 3*c[1])elif k == 1:

g[k]= (2*c[0] + c[2] – 1) / (2 – 3*c[0] – 3*c[2])elif k == 2:

g[k]= (2*c[1] + c[0] – 1) / (2 – 3*c[1] – 3*c[0])

c0111= (g[0]*(-c0300 + 3*c0210 – 3*c0120 +c0030)+ (-c0300 + 2*c0210 – c0120 + c0021 + c0201))/2c1011= (g[1]*(-c0030 + 3*c1020 – 3*c2010 +c3000)+ (-c0030 + 2*c1020 – c2010 + c2001 + c0021))/2c1101= (g[2]*(-c3000 + 3*c2100 – 3*c1200 +c0300)+ (-c3000 + 2*c2100 – c1200 + c2001 + c0201))/2c1002= (c1101 + c1011 + c2001)/3c0102= (c1101 + c0111 + c0201)/3c0012= (c1011 + c0111 + c0021)/3c0003= (c1002 + c0102 + c0012)/3

#extended barycentric coordinates

minval =b[0]for k in range(3):if b[k]

minval=b[k]

b1= b[0] -minval

b2= b[1] -minval

b3= b[2] -minval

b4= 3*minval#evaluate the polynomial — the stupid and ugly way to do it,

#one of the 4 coordinates is in fact zero

w = (b1**3*c3000 + 3*b1**2*b2*c2100 + 3*b1**2*b3*c2010 +

3*b1**2*b4*c2001 + 3*b1*b2**2*c1200 +

6*b1*b2*b4*c1101 + 3*b1*b3**2*c1020 + 6*b1*b3*b4*c1011 +

3*b1*b4**2*c1002 + b2**3*c0300 + 3*b2**2*b3*c0210 +

3*b2**2*b4*c0201 + 3*b2*b3**2*c0120 + 6*b2*b3*b4*c0111 +

3*b2*b4**2*c0102 + b3**3*c0030 + 3*b3**2*b4*c0021 +

3*b3*b4**2*c0012 + b4**3*c0003)returnwclassCloughTocher2DInterpolator(NDInterpolatorBase):”””CloughTocher2DInterpolator(points, values, tol=1e-6)

Piecewise cubic, C1 smooth, curvature-minimizing interpolant in 2D.

.. versionadded:: 0.9

Methods

——-

__call__

Parameters

———-

points : ndarray of floats, shape (npoints, ndims); or Delaunay

Data point coordinates, or a precomputed Delaunay triangulation.

values : ndarray of float or complex, shape (npoints, …)

Data values.

fill_value : float, optional

Value used to fill in for requested points outside of the

convex hull of the input points. If not provided, then

the default is “nan“.

tol : float, optional

Absolute/relative tolerance for gradient estimation.

maxiter : int, optional

Maximum number of iterations in gradient estimation.

rescale : bool, optional

Rescale points to unit cube before performing interpolation.

This is useful if some of the input dimensions have

incommensurable units and differ by many orders of magnitude.

Notes

—–

The interpolant is constructed by triangulating the input data

with Qhull [1]_, and constructing a piecewise cubic

interpolating Bezier polynomial on each triangle, using a

Clough-Tocher scheme [CT]_. The interpolant is guaranteed to be

continuously differentiable.

The gradients of the interpolant are chosen so that the curvature

of the interpolating surface is approximatively minimized. The

gradients necessary for this are estimated using the global

algorithm described in [Nielson83]_ and [Renka84]_.

Examples

——–

We can interpolate values on a 2D plane:

>>> from scipy.interpolate import CloughTocher2DInterpolator

>>> import matplotlib.pyplot as plt

>>> np.random.seed(0)

>>> x = np.random.random(10) – 0.5

>>> y = np.random.random(10) – 0.5

>>> z = np.hypot(x, y)

>>> X = np.linspace(min(x), max(x))

>>> Y = np.linspace(min(y), max(y))

>>> X, Y = np.meshgrid(X, Y) # 2D grid for interpolation

>>> interp = CloughTocher2DInterpolator(list(zip(x, y)), z)

>>> Z = interp(X, Y)

>>> plt.pcolormesh(X, Y, Z, shading=’auto’)

>>> plt.plot(x, y, “ok”, label=”input point”)

>>> plt.legend()

>>> plt.colorbar()

>>> plt.axis(“equal”)

>>> plt.show()

See also

——–

griddata :

Interpolate unstructured D-D data.

LinearNDInterpolator :

Piecewise linear interpolant in N dimensions.

NearestNDInterpolator :

Nearest-neighbor interpolation in N dimensions.

References

———-

.. [1] http://www.qhull.org/

.. [CT] See, for example,

P. Alfeld,

”A trivariate Clough-Tocher scheme for tetrahedral data”.

Computer Aided Geometric Design, 1, 169 (1984);

G. Farin,

”Triangular Bernstein-Bezier patches”.

Computer Aided Geometric Design, 3, 83 (1986).

.. [Nielson83] G. Nielson,

”A method for interpolating scattered data based upon a minimum norm

network”.

Math. Comp., 40, 253 (1983).

.. [Renka84] R. J. Renka and A. K. Cline.

”A Triangle-based C1 interpolation method.”,

Rocky Mountain J. Math., 14, 223 (1984).”””

def __init__(self, points, values, fill_value=np.nan,

tol=1e-6, maxiter=400, rescale=False):

NDInterpolatorBase.__init__(self, points, values, ndim=2,

fill_value=fill_value, rescale=rescale)if self.tri isNone:

self.tri=qhull.Delaunay(self.points)

self.grad=estimate_gradients_2d_global(self.tri, self.values,

tol=tol, maxiter=maxiter)def_evaluate_double(self, xi):return self._do_evaluate(xi, 1.0)def_evaluate_complex(self, xi):return self._do_evaluate(xi, 1.0j)

@cython.boundscheck(False)

@cython.wraparound(False)def _do_evaluate(self, const double[:,::1] xi, double_or_complex dummy):

cdef const double_or_complex[:,::1] values =self.values

cdef const double_or_complex[:,:,:] grad=self.grad

cdef double_or_complex[:,::1] out

cdef const double[:,::1] points =self.points

cdef const int[:,::1] simplices =self.tri.simplices

cdef double c[NPY_MAXDIMS]

cdef double_or_complex f[NPY_MAXDIMS+1]

cdef double_or_complex df[2*NPY_MAXDIMS+2]

cdef double_or_complex w

cdef double_or_complex fill_value

cdef int i, j, k, m, ndim, isimplex, inside, start, nvalues

cdef qhull.DelaunayInfo_t info

cdef double eps, eps_broad

ndim= xi.shape[1]

start=0

fill_value=self.fill_value

qhull._get_delaunay_info(&info, self.tri, 1, 1, 0)

out= np.zeros((xi.shape[0], self.values.shape[1]),

dtype=self.values.dtype)

nvalues= out.shape[1]

eps= 100 *DBL_EPSILON

eps_broad=sqrt(eps)

with nogil:for i inrange(xi.shape[0]):#1) Find the simplex

isimplex= qhull._find_simplex(&info, c,&xi[i,0],&start, eps, eps_broad)#2) Clough-Tocher interpolation

if isimplex == -1:#outside triangulation

for k inrange(nvalues):

out[i,k]=fill_valuecontinue

for k inrange(nvalues):for j in range(ndim+1):

f[j]=values[simplices[isimplex,j],k]

df[2*j] =grad[simplices[isimplex,j],k,0]

df[2*j+1] = grad[simplices[isimplex,j],k,1]

w= _clough_tocher_2d_single(&info, isimplex, c, f, df)

out[i,k]=wreturn out