Bilinear Interpolation Calculator

What is Bilinear Interpolation?

Bilinear interpolation is a two-dimensional interpolation method that estimates a value inside a rectangular grid cell by performing linear interpolation along one axis and then again along the other axis. It uses four known corner values (Q11, Q21, Q12, and Q22) to calculate an interpolated result at any point (x, y) within the rectangle. This technique creates a smooth surface across the grid and is widely used in image processing, computer graphics, GIS terrain analysis, and engineering lookup tables.

How to Use the Bilinear Interpolation Calculator

1. Set the grid bounds - Enter the x-axis range (x1 and x2) and y-axis range (y1 and y2) that define your rectangular grid cell.
2. Enter the target point - Input the x and y coordinates where you want to estimate the interpolated value.
3. Provide corner values - Fill in the four corner values: Q11 (bottom-left), Q21 (bottom-right), Q12 (top-left), and Q22 (top-right).
4. View the result - The interpolated value P(x, y) is calculated instantly along with intermediate values R1 (lower edge) and R2 (upper edge).

Bilinear Interpolation Formula

The bilinear interpolation formula calculates the value at point (x, y) using a weighted average of the four corner values:

Where:

• \( R_1 = \frac{x_2 - x}{x_2 - x_1} \times Q_{11} + \frac{x - x_1}{x_2 - x_1} \times Q_{21} \)
• \( R_2 = \frac{x_2 - x}{x_2 - x_1} \times Q_{12} + \frac{x - x_1}{x_2 - x_1} \times Q_{22} \)

In matrix form, this can be expressed as:

Applications of Bilinear Interpolation

• Image Processing - Used when resizing or transforming images to blend neighboring pixel values smoothly on a surface.
• GIS and Terrain Analysis - Estimates elevation, temperature, or rainfall at locations between sampled grid points on a digital elevation model.
• Engineering Tables - Reads intermediate values from two-variable design charts without needing to solve complex equations manually.
• Texture Mapping - Samples colors from textures at non-integer coordinates in 3D rendering for a smooth appearance.