3D polynomial surface fit

polyN3.py

#!/usr/local/bin/python3

# Make a 3d point cloud and fit a surface to it


import numpy as np
import scipy.linalg
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt

# some 3-dim points
mean = np.array([0.0,0.0,0.0])
cov = np.array([[1.0,-0.2,0.8], [-0.2,1.1,0.0], [0.8,0.0,1.0]])
data = np.random.multivariate_normal(mean, cov, 90)
data[:,1] = data[:,1]**2

# regular grid covering the domain of the data
r=4
X,Y = np.meshgrid(np.arange(-r, r, 0.5), np.arange(-r, r*2, 0.5))
XX = X.flatten()
YY = Y.flatten()

order = 1   # 1: linear, 2: quadratic, 3: cubic
if order == 1:
    # best-fit linear plane
    A = np.c_[data[:,0], data[:,1], np.ones(data.shape[0])]
    C,_,_,_ = scipy.linalg.lstsq(A, data[:,2])    # coefficients
    
    # evaluate it on grid
    Z = C[0]*X + C[1]*Y + C[2]
    
    # or expressed using matrix/vector product
    #Z = np.dot(np.c_[XX, YY, np.ones(XX.shape)], C).reshape(X.shape)

elif order == 2:
    # best-fit quadratic curve
    A = np.c_[np.ones(data.shape[0]), data[:,:2], np.prod(data[:,:2], axis=1), data[:,:2]**2]
    C,_,_,_ = scipy.linalg.lstsq(A, data[:,2])
    
    # evaluate it on a grid
    Z = np.dot(np.c_[np.ones(XX.shape), XX, YY, XX*YY, XX**2, YY**2], C).reshape(X.shape)

elif order == 3:
    # best-fit cubic curve
    A = np.c_[np.ones(data.shape[0]), data[:,:2], np.prod(data[:,:2], axis=1), data[:,:2]**2, data[:,:2]**3]
    C,_,_,_ = scipy.linalg.lstsq(A, data[:,2])
    
    # evaluate it on a grid
    Z = np.dot(np.c_[np.ones(XX.shape), XX, YY, XX*YY, XX**2, YY**2, XX**3, YY**3], C).reshape(X.shape)


# plot points and fitted surface
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot_surface(X, Y, Z, rstride=1, cstride=1, alpha=0.2)
ax.scatter(data[:,0], data[:,1], data[:,2], c='r', s=50)
plt.xlabel('X')
plt.ylabel('Y')
ax.set_zlabel('Z')
ax.axis('equal')
ax.axis('tight')
plt.show()