🔢Numerical Analysis I Unit 9 – Interpolation – Spline Interpolation

What's Interpolation Again?

• Interpolation estimates unknown values between known data points
• Involves constructing new data points within the range of a discrete set of known data points
• Commonly used in numerical analysis to approximate complex functions or missing data
• Differs from extrapolation, which estimates values beyond the range of known data points
• Various interpolation methods exist, including linear, polynomial, and spline interpolation
  • Linear interpolation connects two data points with a straight line
  • Polynomial interpolation fits a polynomial curve through the data points
• Interpolation is useful when the underlying function is unknown or difficult to evaluate
• Accuracy of interpolation depends on the number and distribution of known data points

Enter Spline Interpolation

• Spline interpolation is a type of interpolation that uses piecewise polynomials called splines
• Splines are smooth curves that pass through each of the known data points
• Provides a more flexible and accurate approach compared to simple polynomial interpolation
• Spline interpolation divides the interval into smaller subintervals and fits a separate polynomial for each subinterval
  • Polynomials are chosen to ensure continuity and smoothness at the subinterval boundaries
• Degree of the polynomials used in spline interpolation can vary (linear, quadratic, cubic, etc.)
• Cubic spline interpolation is the most common type, offering a good balance between accuracy and computational efficiency
• Spline interpolation produces a piecewise continuous function that is differentiable at the known data points

Types of Spline Interpolation

• Linear spline interpolation
  • Uses first-degree polynomials (straight lines) to connect the data points
  • Simplest form of spline interpolation, but may result in a non-smooth curve
• Quadratic spline interpolation
  • Uses second-degree polynomials (parabolas) to fit the data points
  • Provides better smoothness compared to linear splines, but may still have discontinuities in the first derivative
• Cubic spline interpolation
  • Uses third-degree polynomials to interpolate between data points
  • Ensures continuity of the function and its first and second derivatives at the subinterval boundaries
  • Most widely used type of spline interpolation due to its smoothness and accuracy
• Higher-degree spline interpolation (quartic, quintic, etc.)
  • Uses polynomials of degree four or higher for interpolation
  • Offers increased flexibility but may introduce oscillations and increased computational complexity
• Natural spline interpolation
  • A type of cubic spline interpolation with additional boundary conditions
  • Assumes zero second derivatives at the endpoints, resulting in a more natural-looking curve
• Periodic spline interpolation
  • Used when the data points exhibit periodic behavior
  • Ensures continuity and smoothness across the periodic boundaries

How Spline Interpolation Works

• Given a set of $n+1$ data points $(x_0, y_0), (x_1, y_1), \ldots, (x_n, y_n)$, where $x_0 < x_1 < \ldots < x_n$
• Divide the interval $[x_0, x_n]$ into $n$ subintervals $[x_i, x_{i+1}]$ for $i = 0, 1, \ldots, n-1$
• For each subinterval, construct a polynomial $S_i(x)$ of degree $k$ that satisfies:
  • $S_i(x_i) = y_i$ and $S_i(x_{i+1}) = y_{i+1}$ (interpolation conditions)
  • $S_i^{(j)}(x_i) = S_{i-1}^{(j)}(x_i)$ for $j = 1, 2, \ldots, k-1$ and $i = 1, 2, \ldots, n-1$ (continuity conditions)
• The resulting spline interpolant $S(x)$ is defined as:
  • $S(x) = S_i(x)$ for $x \in [x_i, x_{i+1}]$, $i = 0, 1, \ldots, n-1$
• Solve a system of linear equations to determine the coefficients of the polynomials $S_i(x)$
• The choice of boundary conditions (natural, clamped, or periodic) affects the behavior of the spline at the endpoints

Pros and Cons of Spline Interpolation

Pros:

• Produces smooth and visually pleasing curves that pass through all the data points
• Provides higher accuracy compared to simple polynomial interpolation
• Offers flexibility in choosing the degree of the polynomials (linear, quadratic, cubic, etc.)
• Cubic spline interpolation ensures continuity of the function and its first and second derivatives
• Computationally efficient, especially for large datasets with many subintervals
• Suitable for interpolating data with local variations or irregularities

Cons:

• Requires solving a system of linear equations, which can be computationally expensive for high-degree splines or large datasets
• May introduce oscillations or overshooting, particularly with higher-degree splines or unevenly spaced data points
• Sensitive to the choice of boundary conditions, which can affect the behavior of the spline at the endpoints
• Requires the data points to be sorted in ascending order of the independent variable
• May not be suitable for extrapolation beyond the range of known data points
• Assumes the data is free from noise or measurement errors, which can impact the interpolation accuracy

Real-World Applications

• Computer graphics and animation
  • Spline interpolation is used to create smooth curves and surfaces for 2D and 3D modeling
  • Bézier curves and B-splines are commonly used in vector graphics and computer-aided design (CAD) software
• Data visualization
  • Spline interpolation helps create visually appealing and smooth graphs from discrete data points
  • Used in creating charts, plots, and infographics to represent data trends and patterns
• Image and signal processing
  • Spline interpolation is used for image resizing, zooming, and reconstruction
  • Helps in smoothing and denoising digital signals and images
• Numerical analysis and scientific computing
  • Spline interpolation is employed in solving differential equations, numerical integration, and approximation of functions
  • Used in finite element analysis (FEA) for approximating solutions to partial differential equations
• Robotics and motion planning
  • Spline interpolation generates smooth trajectories for robot motion and path planning
  • Ensures smooth and continuous movements of robotic arms and autonomous vehicles
• Geospatial analysis and terrain modeling
  • Spline interpolation creates continuous surfaces from discrete elevation or geospatial data points
  • Used in generating digital elevation models (DEMs) and contour maps

Coding It Up

• Most programming languages have libraries or functions for spline interpolation
  • Python:

    scipy.interpolate

    module provides classes like

    InterpolatedUnivariateSpline

    and

    CubicSpline

  • MATLAB:

    spline

    and

    interp1

    functions for spline interpolation
  • C++: GNU Scientific Library (GSL) offers spline interpolation routines
• Basic steps for implementing spline interpolation:
  1. Define the input data points as arrays or vectors
  2. Create an instance of the spline interpolation class or function
  3. Set the boundary conditions (natural, clamped, or periodic) if applicable
  4. Evaluate the spline interpolant at desired points using the interpolation object
• Example code snippet in Python using

  scipy.interpolate

  :

  from scipy.interpolate import CubicSpline
  
  x = [0, 1, 2, 3, 4]
  y = [1, 2, 1, 3, 2]
  
  cs = CubicSpline(x, y)
  x_new = np.linspace(0, 4, 100)
  y_new = cs(x_new)
  

• Spline interpolation functions typically handle the construction and solving of the linear system internally
• It's important to handle any potential errors or exceptions, such as non-increasing input data or invalid boundary conditions

Key Takeaways and Tips

• Spline interpolation is a powerful technique for estimating values between known data points
• It offers a smooth and accurate approximation compared to simple polynomial interpolation
• Cubic spline interpolation is the most commonly used type, providing a good balance between smoothness and computational efficiency
• Choose the appropriate degree of the spline based on the desired smoothness and computational constraints
• Be cautious of oscillations or overshooting, especially with higher-degree splines or unevenly spaced data points
• Consider the boundary conditions (natural, clamped, or periodic) based on the behavior of the data at the endpoints
• Spline interpolation assumes the data is noise-free, so preprocess the data if necessary to remove outliers or smooth the measurements
• Utilize built-in libraries or functions in your programming language for efficient implementation of spline interpolation
• Visualize the interpolated curve along with the original data points to assess the quality and reasonableness of the interpolation
• Remember that spline interpolation is suitable for interpolation within the range of known data points, but extrapolation beyond the range may be unreliable