Bilinear interpolation is a mathematical technique used for estimating unknown values that fall within a grid of known values. It extends linear interpolation to two dimensions, making it particularly useful in applications such as image processing, computer graphics, and geographical information systems.
The method works by taking a rectangular grid of four known points (corners of a rectangle), and estimating the value at an unknown point located within this rectangle. The process involves two steps of linear interpolation:
- First, linear interpolation is performed in one direction (usually the x-direction) to find intermediate values along the edges of the rectangle.
- Next, these intermediate values are interpolated in the other direction (y-direction) to arrive at the final estimated value at the desired point.
Mathematically, if we have four points: (x₁, y₁), (x₂, y₁), (x₁, y₂), and (x₂, y₂) with known values Z₁, Z₂, Z₃, and Z₄, the bilinear interpolation formula can be expressed as:
Z(x, y) = (Z₁(1 – a)(1 – b) + Z₂a(1 – b) + Z₃(1 – a)b + Z₄ab)
where:
- a = (x – x₁) / (x₂ – x₁)
- b = (y – y₁) / (y₂ – y₁)
Bilinear interpolation is widely used due to its simplicity and efficiency in generating smooth transitions between data points, making it a popular choice in various applications including image scaling, texture mapping in 3D graphics, and surface fitting in data analysis.