B-spline Interpolation Example in Python

    Interpolation is a mathematical technique used to estimate or determine values between known data points. B-spline (Basis spline) interpolation is a generalization of cubic spline interpolation and it involves finding a B-spline curve that passes through each of the given data points.

    The SciPy API provides the BSpline class to implement B-spline fitting for a given data. In this tutorial, you'll learn how to implement B-spline interpolation using the BSpline class in Python. The tutorial covers:

  1. Understanding B-spline interpolation
  2. Implementing B-spline interpolation
  3. Source code listing

 We'll start by loading the required libraries.

 
from scipy import interpolate
import matplotlib.pyplot as plt
import numpy as np 
 

Understanding B-spline interpolation
 
    B-spline interpolation is a curve approximation technique utilizing specified coefficients. B-splines are represented as a combination of basis functions and control points, offering flexibility and smoothness in curve representation. 
    Essential parameters include;
  • Knots: set of values that define the parameter space over which the B-spline curve is defined.
  • Spline coefficients: also known as control points, are a set of values associated with each basis function in the B-spline representation.
  • Degree of the spline: indicates the order of the polynomials used in the basis functions.
    These parameters collectively define a B-spline curve. By adjusting the positions of knots, modifying spline coefficients, and choosing an appropriate degree, we can control the overall shape and behavior of the B-spline curve.
    B-splines provide a versatile and powerful tool for curve representation and approximation in various applications.

 
Implementation B-spline interpolation
 
    In this section, we'll delve into implementation of B-spline interpolation for given data. We'll prepare test data that contains x and y components.
 

y = [0, 1, 3, 4, 3, 5, 7, 5, 2, 3, 4, 8, 9, 8, 7]
n = len(y)
x = range(0, n) 
 
 
    Next, we'll visualize this data on a graph and proceed to apply linear interpolation. Linear interpolation involves connecting data points with straight lines on the graph.
  
 
plt.plot(x, y, 'ro', label="original")
plt.plot(x, y, 'b', label="linear interpolation")
plt.title("Target data")
plt.legend(loc='best', fancybox=True, shadow=True)
plt.grid()
plt.show() 

    To initiate B-spline interpolation, we must first obtain the necessary coefficients. This can be achieved using the 'splrep' function. The function returns a tuple (t, c, k) containing the vector of knots (t), B-spline coefficients (c), and the degree of the spline (k). These coefficients serve as the fundamental building blocks for our B-spline interpolation process. 

 
tck = interpolate.splrep(x, y, s=0, k=3) 
 

    Subsequently, we'll generate a new set of x data with an increased number of samples to facilitate the creation of a smoother curve. We will then proceed to construct the B-spline curve using this extended data.
 
 
x_new = np.linspace(min(x), max(x), 100)
y_fit = interpolate.BSpline(*tck)(x_new)

 
 
    In the last step, we can visualize the B-spline interpolated curve that we've constructed on a graph.
 
 
plt.title("BSpline curve fitting")
plt.plot(x, y, 'ro', label="original")
plt.plot(x_new, y_fit, '-c', label="B-spline")
plt.legend(loc='best', fancybox=True, shadow=True)
plt.grid()
plt.show() 
 
   
 
    In this tutorial, we've briefly learned how to implement B-spline interpolation by using SciPy BSpline class in Python. The full source code is listed below. 
 
 
Source code listing

 
from scipy import interpolate
import matplotlib.pyplot as plt
import numpy as np

y = [0,1,3,4,3,5,7,5,2,3,4,8,9,8,7]
n = len(y)
x = range(0, n)

plt.plot(x, y, 'ro', label="original")
plt.plot(x, y, 'b', label="linear interpolation")
plt.title("Target data")
plt.legend(loc='best', fancybox=True, shadow=True)
plt.grid()
plt.show()

tck = interpolate.splrep(x, y, s=0, k=3)
x_new = np.linspace(min(x), max(x), 100)
y_fit = interpolate.BSpline(*tck)(x_new)

plt.title("BSpline curve fitting")
plt.plot(x, y, 'ro', label="original")
plt.plot(x_new, y_fit, '-c', label="B-spline")
plt.legend(loc='best', fancybox=True, shadow=True)
plt.grid()
plt.show() 
 
 
 
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3 comments:

  1. AnonymousSeptember 29, 2023 at 6:17 AM

    How can I implement this to 3D curve? What should I do with the z variable

    ReplyDelete
    Replies
    1. DataTechNotesSeptember 29, 2023 at 7:57 PM

      You can use below example.

      from scipy import interpolate
      import matplotlib.pyplot as plt
      import numpy as np

      # Original data points in 3D
      x = [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
      y = [0, 1, 3, 4, 3, 5, 7, 5, 2, 3, 4, 8, 9, 8, 7]
      z = [0, 2, 1, 4, 3, 6, 8, 5, 3, 2, 4, 9, 7, 8, 6]

      # Create a parametric variable 't' based on the number of data points
      t = np.arange(len(x))

      # Number of points on the interpolated curve
      num_points = 100

      # Perform B-spline interpolation in 3D
      tck_x = interpolate.splrep(t, x, s=0, k=3)
      tck_y = interpolate.splrep(t, y, s=0, k=3)
      tck_z = interpolate.splrep(t, z, s=0, k=3)

      # Create a new parameter range
      t_new = np.linspace(t[0], t[-1], num_points)

      # Evaluate the B-spline curves at the new parameter values
      x_new = interpolate.BSpline(*tck_x)(t_new)
      y_new = interpolate.BSpline(*tck_y)(t_new)
      z_new = interpolate.BSpline(*tck_z)(t_new)

      # Plot the original and interpolated data points in 3D
      fig = plt.figure()
      ax = fig.add_subplot(111, projection='3d')
      ax.scatter(x, y, z, c='r', marker='o', label='Original Data')
      ax.plot(x_new, y_new, z_new, '-c', label='B-spline Interpolation')
      ax.set_xlabel('X-axis')
      ax.set_ylabel('Y-axis')
      ax.set_zlabel('Z-axis')
      plt.title('B-spline Interpolation in 3D')
      plt.legend(loc='best')
      plt.show()



      Delete
  2. AnonymousNovember 20, 2023 at 11:46 PM

    Amazing, was trying to translate some Maya code into python, thank you

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