3D points curve fitting

cubePCA.ipynb

curve_fitting.py

#!/usr/bin/evn python

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,-0.5,0.8], [-0.5,1.1,0.0], [0.8,0.0,1.0]])
data = np.random.multivariate_normal(mean, cov, 50)

# regular grid covering the domain of the data
X,Y = np.meshgrid(np.arange(-3.0, 3.0, 0.5), np.arange(-3.0, 3.0, 0.5))
XX = X.flatten()
YY = Y.flatten()

order = 1    # 1: linear, 2: quadratic
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)

# 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')
plt.show()

pca_labe_fit.ipynb

plane_fitting.py

#!/usr/bin/env python3

import numpy as np

from sklearn.linear_model import LinearRegression

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,-0.5,0.8], [-0.5,1.1,0.0], [0.8,0.0,1.0]])
data = np.random.multivariate_normal(mean, cov, 50)

# regular grid covering the domain of the data
X,Y = np.meshgrid(np.arange(-3.0, 3.0, 0.5), np.arange(-3.0, 3.0, 0.5))
XX = X.flatten()
YY = Y.flatten()

# best-fit linear plane
model = LinearRegression()
model.fit(data[:,:2], data[:,2])

# evaluate it on grid
Z = model.coef_[0]*X + model.coef_[1]*Y + model.intercept_
print("Model score: ", model.score(data[:,:2], data[:,2]))

# 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')
plt.show()

plot_pca_3d.ipynb

plot_pca_3d.py

#!/usr/bin/python
# -*- coding: utf-8 -*-

"""
=========================================================
Principal components analysis (PCA)
=========================================================

These figures aid in illustrating how a point cloud
can be very flat in one direction--which is where PCA
comes in to choose a direction that is not flat.

"""
print(__doc__)

# Authors: Gael Varoquaux
#          Jaques Grobler
#          Kevin Hughes
# License: BSD 3 clause

from sklearn.decomposition import PCA

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


# #############################################################################
# Create the data

e = np.exp(1)
np.random.seed(4)


def pdf(x):
    return 0.5 * (stats.norm(scale=0.25 / e).pdf(x)
                  + stats.norm(scale=4 / e).pdf(x))

y = np.random.normal(scale=0.5, size=(30000))
x = np.random.normal(scale=0.5, size=(30000))
z = np.random.normal(scale=0.1, size=len(x))

density = pdf(x) * pdf(y)
pdf_z = pdf(5 * z)

density *= pdf_z

a = x + y
b = 2 * y
c = a - b + z

norm = np.sqrt(a.var() + b.var())
a /= norm
b /= norm


# #############################################################################
# Plot the figures
def plot_figs(fig_num, elev, azim):
    fig = plt.figure(fig_num, figsize=(4, 3))
    plt.clf()
    ax = Axes3D(fig, rect=[0, 0, .95, 1], elev=elev, azim=azim)

    ax.scatter(a[::10], b[::10], c[::10], c=density[::10], marker='+', alpha=.4)
    Y = np.c_[a, b, c]

    # Using SciPy's SVD, this would be:
    # _, pca_score, V = scipy.linalg.svd(Y, full_matrices=False)

    pca = PCA(n_components=3)
    pca.fit(Y)
    pca_score = pca.explained_variance_ratio_
    V = pca.components_

    x_pca_axis, y_pca_axis, z_pca_axis = 3 * V.T
    x_pca_plane = np.r_[x_pca_axis[:2], - x_pca_axis[1::-1]]
    y_pca_plane = np.r_[y_pca_axis[:2], - y_pca_axis[1::-1]]
    z_pca_plane = np.r_[z_pca_axis[:2], - z_pca_axis[1::-1]]
    x_pca_plane.shape = (2, 2)
    y_pca_plane.shape = (2, 2)
    z_pca_plane.shape = (2, 2)
    ax.plot_surface(x_pca_plane, y_pca_plane, z_pca_plane)
    ax.w_xaxis.set_ticklabels([])
    ax.w_yaxis.set_ticklabels([])
    ax.w_zaxis.set_ticklabels([])


elev = -40
azim = -80
plot_figs(1, elev, azim)

elev = 30
azim = 20
plot_figs(2, elev, azim)

plt.show()