如何在Pytorch中进行三次花键插值和积分？

Question

在Pytorch中，是否有类似于立方花键插值法的 Scipy's 给定一维输入时序 x 和 y我想通过这些点进行插值，并在以下地点进行评估。xs 以获得 ys. 此外，我还想用一个积分函数来找到 Ys的花键插值的积分，从 x[0] 到 xs.

Answer 1

这是我用以下方法做的一个要点。立方型Hermite花键 在Pytorch中高效地实现，并且支持autograd。

为了方便，我也把代码放在这里。

import matplotlib.pylab as P
import torch as T

def h_poly_helper(tt):
  A = T.tensor([
      [1, 0, -3, 2],
      [0, 1, -2, 1],
      [0, 0, 3, -2],
      [0, 0, -1, 1]
      ], dtype=tt[-1].dtype)
  return [
    sum( A[i, j]*tt[j] for j in range(4) )
    for i in range(4) ]

def h_poly(t):
  tt = [ None for _ in range(4) ]
  tt[0] = 1
  for i in range(1, 4):
    tt[i] = tt[i-1]*t
  return h_poly_helper(tt)

def H_poly(t):
  tt = [ None for _ in range(4) ]
  tt[0] = t
  for i in range(1, 4):
    tt[i] = tt[i-1]*t*i/(i+1)
  return h_poly_helper(tt)

def interp(x, y, xs):
  m = (y[1:] - y[:-1])/(x[1:] - x[:-1])
  m = T.cat([m[[0]], (m[1:] + m[:-1])/2, m[[-1]]])
  I = P.searchsorted(x[1:], xs)
  dx = (x[I+1]-x[I])
  hh = h_poly((xs-x[I])/dx)
  return hh[0]*y[I] + hh[1]*m[I]*dx + hh[2]*y[I+1] + hh[3]*m[I+1]*dx

def integ(x, y, xs):
  m = (y[1:] - y[:-1])/(x[1:] - x[:-1])
  m = T.cat([m[[0]], (m[1:] + m[:-1])/2, m[[-1]]])
  I = P.searchsorted(x[1:], xs)
  Y = T.zeros_like(y)
  Y[1:] = (x[1:]-x[:-1])*(
      (y[:-1]+y[1:])/2 + (m[:-1] - m[1:])*(x[1:]-x[:-1])/12
      )
  Y = Y.cumsum(0)
  dx = (x[I+1]-x[I])
  hh = H_poly((xs-x[I])/dx)
  return Y[I] + dx*(
      hh[0]*y[I] + hh[1]*m[I]*dx + hh[2]*y[I+1] + hh[3]*m[I+1]*dx
      )

# Example
if __name__ == "__main__":
  x = T.linspace(0, 6, 7)
  y = x.sin()
  xs = T.linspace(0, 6, 101)
  ys = interp(x, y, xs)
  Ys = integ(x, y, xs)
  P.scatter(x, y, label='Samples', color='purple')
  P.plot(xs, ys, label='Interpolated curve')
  P.plot(xs, xs.sin(), '--', label='True Curve')
  P.plot(xs, Ys, label='Spline Integral')
  P.plot(xs, 1-xs.cos(), '--', label='True Integral')
  P.legend()
  P.show()