Bilinear Interpolation

[JiantaoFu]

Bilinear interpolation is a method used to estimate the value of a function at a point within a 2D grid based on the values of the function at the grid's surrounding points. It is a straightforward extension of linear interpolation to two dimensions.

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How It Works

Suppose you have a rectangular grid with known values at four corners, and you want to find the interpolated value at a point inside this rectangle.

1. Grid and Points:

   • Let the four known points on the grid be $(x_1, y_1), (x_2, y_1), (x_1, y_2), (x_2, y_2)$, with values $Q_{11}, Q_{21}, Q_{12}, Q_{22}$, respectively.
   • The point where you want to interpolate the value is $(x, y)$, where $x_1 \leq x \leq x_2 ) and ( y_1 \leq y \leq y_2$.

2. Two-Step Process:

   • Step 1: Interpolate along the x-direction at fixed $ y_1$ and $y_2$:
     • At $y_1$: Interpolate between $Q_{11}$ and $Q_{21}$:

$$ [ Q_{x1} = \frac{(x_2 - x)}{(x_2 - x_1)} Q_{11} + \frac{(x - x_1)}{(x_2 - x_1)} Q_{21} ] $$

 - At $y_2$: Interpolate between $Q_{12}$ and $Q_{22}$:

$$ [ Q_{x2} = \frac{(x_2 - x)}{(x_2 - x_1)} Q_{12} + \frac{(x - x_1)}{(x_2 - x_1)} Q_{22} ] $$

• Step 2: Interpolate along the y-direction between $Q_{x1}$ and $Q_{x2}$:

$$ [ Q(x, y) = \frac{(y_2 - y)}{(y_2 - y_1)} Q_{x1} + \frac{(y - y_1)}{(y_2 - y_1)} Q_{x2} ] $$

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Key Features

• Linear Interpolation in Two Directions: The method performs linear interpolation first in one direction (x) and then in the other direction (y).
• Smooth Transition: Bilinear interpolation gives a smooth result, as it is based on weighted averages of nearby points.
• Grid Constraints: It assumes a rectangular grid structure and requires the function values at four corners of the rectangle enclosing the interpolation point.

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Example

If you have the grid:

$$ \begin{array}{c|c|c} & x_1 = 0 & x_2 = 1 \\ \hline y_1 = 0 & Q_{11} = 10 & Q_{21} = 20 \\ y_2 = 1 & Q_{12} = 30 & Q_{22} = 40 \\ \end{array} $$

To find the value at $(x, y) = (0.5, 0.5)$:

1. Interpolate along $x$ for $y = 0$: $Q_{x1} = (10 + 20)/2 = 15$.
2. Interpolate along $x$ for $y = 1$: $Q_{x2} = (30 + 40)/2 = 35$.
3. Interpolate along $y$ : $Q(0.5, 0.5) = (15 + 35)/2 = 25$.

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Bilinear interpolation is widely used in image processing, computer graphics, and numerical simulations for tasks like resizing images or filling missing data.