Bilinear Interpolation

Bilinear interpolation is a method for Interpolation in two dimensions that performs Linear Interpolation sequentially in each direction.

Problem Setup

Given four points forming the corners of a rectangle:

• $(x_1,y_1,f_{11})$ where $f_{11}=f(x_1,y_1)$
• $(x_1,y_2,f_{12})$ where $f_{12}=f(x_1,y_2)$
• $(x_2,y_1,f_{21})$ where $f_{21}=f(x_2,y_1)$
• $(x_2,y_2,f_{22})$ where $f_{22}=f(x_2,y_2)$

We want to find $f(x,y)$ for a point $(x,y)$ inside the rectangle, where $x_1\le x\le x_2$ and $y_1\le y\le y_2$.

Method

Bilinear interpolation is performed in two steps:

Step 1: Interpolate in the X-direction

First, interpolate at $y=y_1$ and $y=y_2$:

$$f(x,y_1)=\frac{x_2-x}{x_2-x_1}f(x_1,y_1)+\frac{x-x_1}{x_2-x_1}f(x_2,y_1)$$ $$f(x,y_2)=\frac{x_2-x}{x_2-x_1}f(x_1,y_2)+\frac{x-x_1}{x_2-x_1}f(x_2,y_2)$$

Step 2: Interpolate in the Y-direction

Then interpolate between $f(x,y_1)$ and $f(x,y_2)$:

$$f(x,y)=\frac{y_2-y}{y_2-y_1}f(x,y_1)+\frac{y-y_1}{y_2-y_1}f(x,y_2)$$

Combined Formula

Substituting Step 1 into Step 2 gives the full bilinear interpolation formula:

$$f(x,y)=\frac{y_2-y}{y_2-y_1}\left[\frac{x_2-x}{x_2-x_1}f(x_1,y_1)+\frac{x-x_1}{x_2-x_1}f(x_2,y_1)\right]$$ $$+\frac{y-y_1}{y_2-y_1}\left[\frac{x_2-x}{x_2-x_1}f(x_1,y_2)+\frac{x-x_1}{x_2-x_1}f(x_2,y_2)\right]$$

This can be expanded and written in matrix form:

$$f(x,y)=\left[\begin{array}{cc}\frac{x_2-x}{x_2-x_1}& \frac{x-x_1}{x_2-x_1}\end{array}\right]\left[\begin{array}{cc}f(x_1,y_1)& f(x_1,y_2)\\ f(x_2,y_1)& f(x_2,y_2)\end{array}\right]\left[\begin{array}{c}\frac{y_2-y}{y_2-y_1}\\ \frac{y-y_1}{y_2-y_1}\end{array}\right]$$

Order of Interpolation

The order doesn’t matter - interpolating first in $y$ then in $x$ gives the same result. This is because:

• Linear interpolation is a linear operation
• The order of linear operations doesn’t affect the result

Properties

1. Bilinearity: The function is linear in each variable when the other is held constant
2. Exactness: For bilinear functions $f(x,y)=a+bx+cy+dxy$, bilinear interpolation is exact
3. Continuity: Result is continuous across rectangle boundaries
4. Not smooth: Generally not differentiable at grid points

Special Case: Unit Square

For a unit square with corners at $(0,0),(0,1),(1,0),(1,1)$, the formula simplifies to:

$$f(x,y)=(1-x)(1-y)f(0,0)+x(1-y)f(1,0)+(1-x)yf(0,1)+xyf(1,1)$$

This shows the weighted average interpretation, where weights are the areas of opposite rectangles.

Advantages

• Simple and fast: Only requires four function evaluations
• Smooth: Produces continuous (but not necessarily differentiable) results
• Natural extension: Direct generalization of Linear Interpolation to 2D
• Exact for bilinear functions: Perfect for functions of form $f(x,y)=a+bx+cy+dxy$

Limitations

• Limited accuracy: Only captures linear and bilinear behaviour
• Non-smooth: Derivatives discontinuous at grid boundaries
• Rectangular grid required: Needs data on a regular grid
• Poor for highly curved surfaces: Cannot capture complex curvature

Applications

1. Image processing:

   • Resizing/scaling images
   • Texture mapping in computer graphics
   • Digital zoom

2. Scientific computing:

   • Temperature/pressure field interpolation
   • Geographical data interpolation
   • Finite element analysis

3. Computer graphics:

   • Texture mapping
   • Anti-aliasing
   • Mipmap generation

Python Implementation

def bilinear_interpolation(x_target, y_target, x1, x2, y1, y2, 
                          f_x1_y1, f_x1_y2, f_x2_y1, f_x2_y2):
    """
    Perform bilinear interpolation on a rectangular grid.
    
    Parameters
    ----------
    x_target : float
        Target x-coordinate for interpolation
    y_target : float
        Target y-coordinate for interpolation
    x1, x2 : float
        Lower and upper x-coordinates of rectangle
    y1, y2 : float
        Lower and upper y-coordinates of rectangle
    f_x1_y1, f_x1_y2, f_x2_y1, f_x2_y2 : float
        Function values at the four corners
        
    Returns
    -------
    float
        Interpolated value at (x_target, y_target)
    """
    # Interpolate in x direction at y=y1 and y=y2
    f_x_y1 = ((x2 - x_target) / (x2 - x1)) * f_x1_y1 + \
             ((x_target - x1) / (x2 - x1)) * f_x2_y1
    f_x_y2 = ((x2 - x_target) / (x2 - x1)) * f_x1_y2 + \
             ((x_target - x1) / (x2 - x1)) * f_x2_y2
    
    # Interpolate in y direction
    result = ((y2 - y_target) / (y2 - y1)) * f_x_y1 + \
             ((y_target - y1) / (y2 - y1)) * f_x_y2
    
    return result

Extensions

• Trilinear interpolation: Extension to 3D
• Bicubic interpolation: Uses cubic polynomials for smoother results
• Higher-order: Biquadratic, biquartic, etc.

Related Methods

• Linear Interpolation: 1D equivalent
• Spline Interpolation: Smoother alternative
• Interpolation: General concept