__all__ = ['RegularGridInterpolator', 'interpn'] import itertools import cupy as cp from cupyx.scipy.interpolate._bspline2 import make_interp_spline from cupyx.scipy.interpolate._cubic import PchipInterpolator def _ndim_coords_from_arrays(points, ndim=None): """ Convert a tuple of coordinate arrays to a (..., ndim)-shaped array. """ if isinstance(points, tuple) and len(points) == 1: # handle argument tuple points = points[0] if isinstance(points, tuple): p = cp.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 = cp.empty(p[0].shape + (len(points),), dtype=float) for j, item in enumerate(p): points[..., j] = item else: points = cp.asanyarray(points) if points.ndim == 1: if ndim is None: points = points.reshape(-1, 1) else: points = points.reshape(-1, ndim) return points def _check_points(points): descending_dimensions = [] grid = [] for i, p in enumerate(points): # early make points float # see https://github.com/scipy/scipy/pull/17230 p = cp.asarray(p, dtype=float) if not cp.all(p[1:] > p[:-1]): if cp.all(p[1:] < p[:-1]): # input is descending, so make it ascending descending_dimensions.append(i) p = cp.flip(p) p = cp.ascontiguousarray(p) else: raise ValueError( "The points in dimension %d must be strictly " "ascending or descending" % i) grid.append(p) return tuple(grid), tuple(descending_dimensions) def _check_dimensionality(points, values): if len(points) > values.ndim: raise ValueError("There are %d point arrays, but values has %d " "dimensions" % (len(points), values.ndim)) for i, p in enumerate(points): if not cp.asarray(p).ndim == 1: raise ValueError("The points in dimension %d must be " "1-dimensional" % i) if not values.shape[i] == len(p): raise ValueError("There are %d points and %d values in " "dimension %d" % (len(p), values.shape[i], i)) class RegularGridInterpolator: """ Interpolation on a regular or rectilinear grid in arbitrary dimensions. The data must be defined on a rectilinear grid; that is, a rectangular grid with even or uneven spacing. Linear and nearest-neighbor interpolations are supported. After setting up the interpolator object, the interpolation method may be chosen at each evaluation. Parameters ---------- points : tuple of ndarray of float, with shapes (m1, ), ..., (mn, ) The points defining the regular grid in n dimensions. The points in each dimension (i.e. every elements of the points tuple) must be strictly ascending or descending. values : ndarray, shape (m1, ..., mn, ...) The data on the regular grid in n dimensions. Complex data can be acceptable. method : str, optional The method of interpolation to perform. Supported are "linear", "nearest", "slinear", "cubic", "quintic" and "pchip". This parameter will become the default for the object's ``__call__`` method. Default is "linear". bounds_error : bool, optional If True, when interpolated values are requested outside of the domain of the input data, a ValueError is raised. If False, then `fill_value` is used. Default is True. fill_value : float or None, optional The value to use for points outside of the interpolation domain. If None, values outside the domain are extrapolated. Default is ``cp.nan``. Notes ----- Contrary to scipy's `LinearNDInterpolator` and `NearestNDInterpolator`, this class avoids expensive triangulation of the input data by taking advantage of the regular grid structure. In other words, this class assumes that the data is defined on a *rectilinear* grid. If the input data is such that dimensions have incommensurate units and differ by many orders of magnitude, the interpolant may have numerical artifacts. Consider rescaling the data before interpolating. Examples -------- **Evaluate a function on the points of a 3-D grid** As a first example, we evaluate a simple example function on the points of a 3-D grid: >>> from cupyx.scipy.interpolate import RegularGridInterpolator >>> import cupy as cp >>> def f(x, y, z): ... return 2 * x**3 + 3 * y**2 - z >>> x = cp.linspace(1, 4, 11) >>> y = cp.linspace(4, 7, 22) >>> z = cp.linspace(7, 9, 33) >>> xg, yg ,zg = cp.meshgrid(x, y, z, indexing='ij', sparse=True) >>> data = f(xg, yg, zg) ``data`` is now a 3-D array with ``data[i, j, k] = f(x[i], y[j], z[k])``. Next, define an interpolating function from this data: >>> interp = RegularGridInterpolator((x, y, z), data) Evaluate the interpolating function at the two points ``(x,y,z) = (2.1, 6.2, 8.3)`` and ``(3.3, 5.2, 7.1)``: >>> pts = cp.array([[2.1, 6.2, 8.3], ... [3.3, 5.2, 7.1]]) >>> interp(pts) array([ 125.80469388, 146.30069388]) which is indeed a close approximation to >>> f(2.1, 6.2, 8.3), f(3.3, 5.2, 7.1) (125.54200000000002, 145.894) **Interpolate and extrapolate a 2D dataset** As a second example, we interpolate and extrapolate a 2D data set: >>> x, y = cp.array([-2, 0, 4]), cp.array([-2, 0, 2, 5]) >>> def ff(x, y): ... return x**2 + y**2 >>> xg, yg = cp.meshgrid(x, y, indexing='ij') >>> data = ff(xg, yg) >>> interp = RegularGridInterpolator((x, y), data, ... bounds_error=False, fill_value=None) >>> import matplotlib.pyplot as plt >>> fig = plt.figure() >>> ax = fig.add_subplot(projection='3d') >>> ax.scatter(xg.ravel().get(), yg.ravel().get(), data.ravel().get(), ... s=60, c='k', label='data') Evaluate and plot the interpolator on a finer grid >>> xx = cp.linspace(-4, 9, 31) >>> yy = cp.linspace(-4, 9, 31) >>> X, Y = cp.meshgrid(xx, yy, indexing='ij') >>> # interpolator >>> ax.plot_wireframe(X.get(), Y.get(), interp((X, Y)).get(), rstride=3, cstride=3, alpha=0.4, color='m', label='linear interp') >>> # ground truth >>> ax.plot_wireframe(X.get(), Y.get(), ff(X, Y).get(), rstride=3, cstride=3, ... alpha=0.4, label='ground truth') >>> plt.legend() >>> plt.show() See Also -------- interpn : a convenience function which wraps `RegularGridInterpolator` scipy.ndimage.map_coordinates : interpolation on grids with equal spacing (suitable for e.g., N-D image resampling) References ---------- [1] Python package *regulargrid* by Johannes Buchner, see https://pypi.python.org/pypi/regulargrid/ [2] Wikipedia, "Trilinear interpolation", https://en.wikipedia.org/wiki/Trilinear_interpolation [3] Weiser, Alan, and Sergio E. Zarantonello. "A note on piecewise linear and multilinear table interpolation in many dimensions." MATH. COMPUT. 50.181 (1988): 189-196. https://www.ams.org/journals/mcom/1988-50-181/S0025-5718-1988-0917826-0/S0025-5718-1988-0917826-0.pdf """ # this class is based on code originally programmed by Johannes Buchner, # see https://github.com/JohannesBuchner/regulargrid _SPLINE_DEGREE_MAP = {"slinear": 1, "cubic": 3, "quintic": 5, 'pchip': 3} _SPLINE_METHODS = list(_SPLINE_DEGREE_MAP.keys()) _ALL_METHODS = ["linear", "nearest"] + _SPLINE_METHODS def __init__(self, points, values, method="linear", bounds_error=True, fill_value=cp.nan): if method not in self._ALL_METHODS: raise ValueError("Method '%s' is not defined" % method) elif method in self._SPLINE_METHODS: self._validate_grid_dimensions(points, method) self.method = method self.bounds_error = bounds_error self.grid, self._descending_dimensions = _check_points(points) self.values = self._check_values(values) self._check_dimensionality(self.grid, self.values) self.fill_value = self._check_fill_value(self.values, fill_value) if self._descending_dimensions: self.values = cp.flip(values, axis=self._descending_dimensions) def _check_dimensionality(self, grid, values): _check_dimensionality(grid, values) def _validate_grid_dimensions(self, points, method): k = self._SPLINE_DEGREE_MAP[method] for i, point in enumerate(points): ndim = len(cp.atleast_1d(point)) if ndim <= k: raise ValueError(f"There are {ndim} points in dimension {i}," f" but method {method} requires at least " f" {k+1} points per dimension.") def _check_points(self, points): return _check_points(points) def _check_values(self, values): if not cp.issubdtype(values.dtype, cp.inexact): values = values.astype(float) return values def _check_fill_value(self, values, fill_value): if fill_value is not None: fill_value_dtype = cp.asarray(fill_value).dtype if (hasattr(values, 'dtype') and not cp.can_cast(fill_value_dtype, values.dtype, casting='same_kind')): raise ValueError("fill_value must be either 'None' or " "of a type compatible with values") return fill_value def __call__(self, xi, method=None): """ Interpolation at coordinates. Parameters ---------- xi : cupy.ndarray of shape (..., ndim) The coordinates to evaluate the interpolator at. method : str, optional The method of interpolation to perform. Supported are "linear" and "nearest". Default is the method chosen when the interpolator was created. Returns ------- values_x : cupy.ndarray, shape xi.shape[:-1] + values.shape[ndim:] Interpolated values at `xi`. See notes for behaviour when ``xi.ndim == 1``. Notes ----- In the case that ``xi.ndim == 1`` a new axis is inserted into the 0 position of the returned array, values_x, so its shape is instead ``(1,) + values.shape[ndim:]``. Examples -------- Here we define a nearest-neighbor interpolator of a simple function >>> import cupy as cp >>> x, y = cp.array([0, 1, 2]), cp.array([1, 3, 7]) >>> def f(x, y): ... return x**2 + y**2 >>> data = f(*cp.meshgrid(x, y, indexing='ij', sparse=True)) >>> from cupyx.scipy.interpolate import RegularGridInterpolator >>> interp = RegularGridInterpolator((x, y), data, method='nearest') By construction, the interpolator uses the nearest-neighbor interpolation >>> interp([[1.5, 1.3], [0.3, 4.5]]) array([2., 9.]) We can however evaluate the linear interpolant by overriding the `method` parameter >>> interp([[1.5, 1.3], [0.3, 4.5]], method='linear') array([ 4.7, 24.3]) """ is_method_changed = self.method != method method = self.method if method is None else method if method not in self._ALL_METHODS: raise ValueError("Method '%s' is not defined" % method) xi, xi_shape, ndim, nans, out_of_bounds = self._prepare_xi(xi) if method == "linear": indices, norm_distances = self._find_indices(xi.T) result = self._evaluate_linear(indices, norm_distances) elif method == "nearest": indices, norm_distances = self._find_indices(xi.T) result = self._evaluate_nearest(indices, norm_distances) elif method in self._SPLINE_METHODS: if is_method_changed: self._validate_grid_dimensions(self.grid, method) result = self._evaluate_spline(xi, method) if not self.bounds_error and self.fill_value is not None: result[out_of_bounds] = self.fill_value if nans.ndim < result.ndim: nans = nans[..., None] result = cp.where(nans, cp.nan, result) return result.reshape(xi_shape[:-1] + self.values.shape[ndim:]) def _prepare_xi(self, xi): ndim = len(self.grid) xi = _ndim_coords_from_arrays(xi, ndim=ndim) if xi.shape[-1] != len(self.grid): raise ValueError("The requested sample points xi have dimension " f"{xi.shape[-1]} but this " f"RegularGridInterpolator has dimension {ndim}") xi_shape = xi.shape xi = xi.reshape(-1, xi_shape[-1]) xi = cp.asarray(xi, dtype=float) # find nans in input is_nans = cp.isnan(xi).T nans = is_nans[0].copy() for is_nan in is_nans[1:]: cp.logical_or(nans, is_nan, nans) if self.bounds_error: for i, p in enumerate(xi.T): if not cp.logical_and(cp.all(self.grid[i][0] <= p), cp.all(p <= self.grid[i][-1])): raise ValueError("One of the requested xi is out of bounds" " in dimension %d" % i) out_of_bounds = None else: out_of_bounds = self._find_out_of_bounds(xi.T) return xi, xi_shape, ndim, nans, out_of_bounds def _evaluate_linear(self, indices, norm_distances): # slice for broadcasting over trailing dimensions in self.values vslice = (slice(None),) + (None,)*(self.values.ndim - len(indices)) # Compute shifting up front before zipping everything together shift_norm_distances = [1 - yi for yi in norm_distances] shift_indices = [i + 1 for i in indices] # The formula for linear interpolation in 2d takes the form: # values = self.values[(i0, i1)] * (1 - y0) * (1 - y1) + \ # self.values[(i0, i1 + 1)] * (1 - y0) * y1 + \ # self.values[(i0 + 1, i1)] * y0 * (1 - y1) + \ # self.values[(i0 + 1, i1 + 1)] * y0 * y1 # We pair i with 1 - yi (zipped1) and i + 1 with yi (zipped2) zipped1 = zip(indices, shift_norm_distances) zipped2 = zip(shift_indices, norm_distances) # Take all products of zipped1 and zipped2 and iterate over them # to get the terms in the above formula. This corresponds to iterating # over the vertices of a hypercube. hypercube = itertools.product(*zip(zipped1, zipped2)) value = cp.array([0.]) for h in hypercube: edge_indices, weights = zip(*h) term = cp.asarray(self.values[edge_indices]) for w in weights: term *= w[vslice] value = value + term # cannot use += because broadcasting return value def _evaluate_nearest(self, indices, norm_distances): idx_res = [cp.where(yi <= .5, i, i + 1) for i, yi in zip(indices, norm_distances)] return self.values[tuple(idx_res)] def _evaluate_spline(self, xi, method): # ensure xi is 2D list of points to evaluate (`m` is the number of # points and `n` is the number of interpolation dimensions, # ``n == len(self.grid)``.) if xi.ndim == 1: xi = xi.reshape((1, xi.size)) m, n = xi.shape # Reorder the axes: n-dimensional process iterates over the # interpolation axes from the last axis downwards: E.g. for a 4D grid # the order of axes is 3, 2, 1, 0. Each 1D interpolation works along # the 0th axis of its argument array (for 1D routine it's its ``y`` # array). Thus permute the interpolation axes of `values` *and keep # trailing dimensions trailing*. axes = tuple(range(self.values.ndim)) axx = axes[:n][::-1] + axes[n:] values = self.values.transpose(axx) if method == 'pchip': _eval_func = self._do_pchip else: _eval_func = self._do_spline_fit k = self._SPLINE_DEGREE_MAP[method] # Non-stationary procedure: difficult to vectorize this part entirely # into numpy-level operations. Unfortunately this requires explicit # looping over each point in xi. # can at least vectorize the first pass across all points in the # last variable of xi. last_dim = n - 1 first_values = _eval_func(self.grid[last_dim], values, xi[:, last_dim], k) # the rest of the dimensions have to be on a per point-in-xi basis shape = (m, *self.values.shape[n:]) result = cp.empty(shape, dtype=self.values.dtype) for j in range(m): # Main process: Apply 1D interpolate in each dimension # sequentially, starting with the last dimension. # These are then "folded" into the next dimension in-place. folded_values = first_values[j, ...] for i in range(last_dim-1, -1, -1): # Interpolate for each 1D from the last dimensions. # This collapses each 1D sequence into a scalar. folded_values = _eval_func(self.grid[i], folded_values, xi[j, i], k) result[j, ...] = folded_values return result @staticmethod def _do_spline_fit(x, y, pt, k): local_interp = make_interp_spline(x, y, k=k, axis=0) values = local_interp(pt) return values @staticmethod def _do_pchip(x, y, pt, k): local_interp = PchipInterpolator(x, y, axis=0) values = local_interp(pt) return values def _find_indices(self, xi): # find relevant edges between which xi are situated indices = [] # compute distance to lower edge in unity units norm_distances = [] # iterate through dimensions for x, grid in zip(xi, self.grid): i = cp.searchsorted(grid, x) - 1 cp.clip(i, 0, grid.size - 2, i) indices.append(i) # compute norm_distances, incl length-1 grids, # where `grid[i+1] == grid[i]` denom = grid[i + 1] - grid[i] norm_dist = cp.where(denom != 0, (x - grid[i]) / denom, 0) norm_distances.append(norm_dist) return indices, norm_distances def _find_out_of_bounds(self, xi): # check for out of bounds xi out_of_bounds = cp.zeros((xi.shape[1]), dtype=bool) # iterate through dimensions for x, grid in zip(xi, self.grid): out_of_bounds += x < grid[0] out_of_bounds += x > grid[-1] return out_of_bounds def interpn(points, values, xi, method="linear", bounds_error=True, fill_value=cp.nan): """ Multidimensional interpolation on regular or rectilinear grids. Strictly speaking, not all regular grids are supported - this function works on *rectilinear* grids, that is, a rectangular grid with even or uneven spacing. Parameters ---------- points : tuple of cupy.ndarray of float, with shapes (m1, ), ..., (mn, ) The points defining the regular grid in n dimensions. The points in each dimension (i.e. every elements of the points tuple) must be strictly ascending or descending. values : cupy.ndarray of shape (m1, ..., mn, ...) The data on the regular grid in n dimensions. Complex data can be acceptable. xi : cupy.ndarray of shape (..., ndim) The coordinates to sample the gridded data at method : str, optional The method of interpolation to perform. Supported are "linear", "nearest", "slinear", "cubic", "quintic" and "pchip". bounds_error : bool, optional If True, when interpolated values are requested outside of the domain of the input data, a ValueError is raised. If False, then `fill_value` is used. fill_value : number, optional If provided, the value to use for points outside of the interpolation domain. If None, values outside the domain are extrapolated. Returns ------- values_x : ndarray, shape xi.shape[:-1] + values.shape[ndim:] Interpolated values at `xi`. See notes for behaviour when ``xi.ndim == 1``. Notes ----- In the case that ``xi.ndim == 1`` a new axis is inserted into the 0 position of the returned array, values_x, so its shape is instead ``(1,) + values.shape[ndim:]``. If the input data is such that input dimensions have incommensurate units and differ by many orders of magnitude, the interpolant may have numerical artifacts. Consider rescaling the data before interpolation. Examples -------- Evaluate a simple example function on the points of a regular 3-D grid: >>> import cupy as cp >>> from cupyx.scipy.interpolate import interpn >>> def value_func_3d(x, y, z): ... return 2 * x + 3 * y - z >>> x = cp.linspace(0, 4, 5) >>> y = cp.linspace(0, 5, 6) >>> z = cp.linspace(0, 6, 7) >>> points = (x, y, z) >>> values = value_func_3d(*cp.meshgrid(*points, indexing='ij')) Evaluate the interpolating function at a point >>> point = cp.array([2.21, 3.12, 1.15]) >>> print(interpn(points, values, point)) [12.63] See Also -------- RegularGridInterpolator : interpolation on a regular or rectilinear grid in arbitrary dimensions (`interpn` wraps this class). cupyx.scipy.ndimage.map_coordinates : interpolation on grids with equal spacing (suitable for e.g., N-D image resampling) """ # sanity check 'method' kwarg if method not in ["linear", "nearest", "slinear", "cubic", "quintic", "pchip"]: raise ValueError( "interpn only understands the methods 'linear', 'nearest', " "'slinear', 'cubic', 'quintic' and 'pchip'. " "You provided {method}.") ndim = values.ndim # sanity check consistency of input dimensions if len(points) > ndim: raise ValueError("There are %d point arrays, but values has %d " "dimensions" % (len(points), ndim)) grid, descending_dimensions = _check_points(points) _check_dimensionality(grid, values) # sanity check requested xi xi = _ndim_coords_from_arrays(xi, ndim=len(grid)) if xi.shape[-1] != len(grid): raise ValueError("The requested sample points xi have dimension " "%d, but this RegularGridInterpolator has " "dimension %d" % (xi.shape[-1], len(grid))) if bounds_error: for i, p in enumerate(xi.T): if not cp.logical_and(cp.all(grid[i][0] <= p), cp.all(p <= grid[i][-1])): raise ValueError("One of the requested xi is out of bounds " "in dimension %d" % i) # perform interpolation if method in ["linear", "nearest", "slinear", "cubic", "quintic", "pchip"]: interp = RegularGridInterpolator(points, values, method=method, bounds_error=bounds_error, fill_value=fill_value) return interp(xi)