Python 3D polynomial surface fit, order dependent

Question

I am currently working with astronomical data among which I have comet images. I would like to remove the background sky gradient in these images due to the time of capture

Answer 1

Griddata uses a spline fitting. A 3rd order spline is not the same thing as a 3rd order polynomial (instead, it's a different 3rd order polynomial at every point).

If you just want to fit a 2D, 3rd order polynomial to your data, then do something like the following to estimate the 16 coefficients using all of your data points.

import itertools
import numpy as np
import matplotlib.pyplot as plt

def main():
    # Generate Data...
    numdata = 100
    x = np.random.random(numdata)
    y = np.random.random(numdata)
    z = x**2 + y**2 + 3*x**3 + y + np.random.random(numdata)

    # Fit a 3rd order, 2d polynomial
    m = polyfit2d(x,y,z)

    # Evaluate it on a grid...
    nx, ny = 20, 20
    xx, yy = np.meshgrid(np.linspace(x.min(), x.max(), nx), 
                         np.linspace(y.min(), y.max(), ny))
    zz = polyval2d(xx, yy, m)

    # Plot
    plt.imshow(zz, extent=(x.min(), y.max(), x.max(), y.min()))
    plt.scatter(x, y, c=z)
    plt.show()

def polyfit2d(x, y, z, order=3):
    ncols = (order + 1)**2
    G = np.zeros((x.size, ncols))
    ij = itertools.product(range(order+1), range(order+1))
    for k, (i,j) in enumerate(ij):
        G[:,k] = x**i * y**j
    m, _, _, _ = np.linalg.lstsq(G, z)
    return m

def polyval2d(x, y, m):
    order = int(np.sqrt(len(m))) - 1
    ij = itertools.product(range(order+1), range(order+1))
    z = np.zeros_like(x)
    for a, (i,j) in zip(m, ij):
        z += a * x**i * y**j
    return z

main()