Bilinear Interpolation

Visualize how pixel values are interpolated from four corner integer coordinates. The surface creates a hyperbolic paraboloid connecting the four values in 3D space.

Coordinates

Delta X (dx)0.50

Delta Y (dy)0.50

Interpolated Value

3.750

Corner Values (Z)

I(\xi, \eta+1)

I(\xi+1, \eta+1)

I(\xi, \eta)

I(\xi+1, \eta)

Modify the height (pixel value) of each corner to see how the surface adapts.

Formula

\xi = \lfloor x \rfloor

\eta = \lfloor y \rfloor

\Delta x = x - \xi

\Delta y = y - \eta

I(x, y) \approx

I(\xi, \eta)(1 - \Delta x)(1 - \Delta y) +

I(\xi + 1, \eta)\Delta x(1 - \Delta y) +

I(\xi, \eta + 1)(1 - \Delta x)\Delta y +

I(\xi + 1, \eta + 1)\Delta x \Delta y

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Key Concepts

2D Extension: Bilinear interpolation extends linear interpolation to a 2D grid. It performs linear interpolation first in one direction (e.g., X), and then interpolates those results in the other direction (Y).
Weighted Average: The estimated value $I(x, y)$ is a weighted average of the four nearest integer neighbors, where the weights are determined by the fractional distances $\Delta x$ and $\Delta y$ .
Hyperbolic Paraboloid: Unlike a flat plane, the resulting surface is curved (a hyperbolic paraboloid). This ensures that the surface passes smoothly through all four corner points, even if they are not coplanar.