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Bilinear Interpolation

Bilinear interpolation estimates the value at a point (x,y) inside a rectangle using the four corner values Qโ‚โ‚, Qโ‚‚โ‚, Qโ‚โ‚‚, Qโ‚‚โ‚‚. It performs linear interpolation in x, then y. Used in image resizing, texture mapping, and GIS.

Concept Fundamentals
f = (1โˆ’tโ‚“)(1โˆ’tแตง)Qโ‚โ‚ + tโ‚“(1โˆ’tแตง)Qโ‚‚โ‚ + ...
Formula
Normalized coords in [0,1]
tโ‚“, tแตง
At corners; linear on edges
Exact
Images, GIS, textures
Uses

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At corners, result equals the corner value exactly. Along edges, bilinear reduces to 1D linear interpolation. Trilinear extends this to 3D using eight corner values.

Key quantities
f = (1โˆ’tโ‚“)(1โˆ’tแตง)Qโ‚โ‚ + tโ‚“(1โˆ’tแตง)Qโ‚‚โ‚ + ...
Formula
Key relation
Normalized coords in [0,1]
tโ‚“, tแตง
Key relation
At corners; linear on edges
Exact
Key relation
Images, GIS, textures
Uses
Key relation

Ready to run the numbers?

Why: Bilinear interpolation fills values between grid points in 2D. Essential for image resizing, texture mapping in 3D graphics, elevation data in GIS, and numerical methods.

How: Compute tโ‚“=(xโˆ’xโ‚)/(xโ‚‚โˆ’xโ‚), tแตง=(yโˆ’yโ‚)/(yโ‚‚โˆ’yโ‚). Then f = (1โˆ’tโ‚“)(1โˆ’tแตง)Qโ‚โ‚ + tโ‚“(1โˆ’tแตง)Qโ‚‚โ‚ + (1โˆ’tโ‚“)tแตงQโ‚โ‚‚ + tโ‚“tแตงQโ‚‚โ‚‚. Point must be inside the rectangle.

At corners, result equals the corner value exactly.Along edges, bilinear reduces to 1D linear interpolation.
Sources:Wolfram MathWorld โ€” Bilinear InterpolationWikipedia โ€” Bilinear Interpolation

Run the calculator when you are ready.

Interpolate ValueEnter rectangle corners and four corner values

Rectangle and Corner Values

Rectangle Corners

Corner Values

For educational and informational purposes only. Verify with a qualified professional.

๐Ÿงฎ Fascinating Math Facts

โŠž

f(x,y) = weighted average of 4 corners.

โ€” Numerical Methods

๐Ÿ–ผ

Image resize: apply to each RGB channel.

โ€” Graphics

Key Takeaways

  • โ€ข Bilinear interpolation extends linear interpolation to two dimensions.
  • โ€ข It uses four corner values and produces a smooth surface inside the rectangle.
  • โ€ข The result is exact at the four corners and linear along edges.
  • โ€ข Widely used in image resizing, texture mapping, and GIS.
  • โ€ข For higher accuracy, use bicubic or spline interpolation.

Did You Know?

Image Processing

When you resize an image, bilinear interpolation fills in pixel values between the original grid points.

Texture Mapping

3D graphics use bilinear interpolation to sample textures at non-integer UV coordinates.

GIS

Elevation and temperature data are often interpolated between measurement stations using bilinear methods.

Weather

Weather models interpolate measurements between stations to create continuous field maps.

CFD

Computational fluid dynamics interpolates field values between grid points.

Trilinear

Trilinear interpolation extends this to 3D using eight corner values.

Understanding Bilinear Interpolation

Bilinear interpolation estimates the value at a point (x, y) inside a rectangle using the values at the four corners. It performs linear interpolation in x, then in y (or vice versa).

f(x,y)=(1โˆ’tx)(1โˆ’ty)Q11+tx(1โˆ’ty)Q21+(1โˆ’tx)tyQ12+txtyQ22f(x,y) = (1-t_x)(1-t_y)Q_{11} + t_x(1-t_y)Q_{21} + (1-t_x)t_y Q_{12} + t_x t_y Q_{22}

where tx=fracxโˆ’x1x2โˆ’x1t_x = \\frac{x-x_1}{x_2-x_1} and ty=fracyโˆ’y1y2โˆ’y1t_y = \\frac{y-y_1}{y_2-y_1}

Expert Tips

Corner Order

Q11 at (x1,y1), Q21 at (x2,y1), Q12 at (x1,y2), Q22 at (x2,y2). Ensure consistent ordering.

Inside Rectangle

The point (x,y) must lie strictly inside the rectangle for valid interpolation.

Smoothness

Bilinear gives Cโฐ continuity (continuous but not smooth derivatives at edges).

Performance

For real-time graphics, bilinear is preferred over bicubic for speed.

Frequently Asked Questions

What is bilinear interpolation?

A method to estimate the value at a point inside a rectangle using the values at the four corners. It performs linear interpolation in both x and y.

When is it exact?

At the four corners. Along edges it reduces to linear interpolation.

What if the point is outside the rectangle?

That would be extrapolation. This calculator requires the point to be inside.

How does it differ from bicubic?

Bicubic uses 16 points and gives smoother results with continuous derivatives.

Where is it used?

Image resizing, texture mapping, GIS, weather models, and numerical methods.

What is trilinear interpolation?

Extension to 3D using eight corner values of a cube.

Can I use it for color images?

Yes. Apply bilinear interpolation to each color channel (R, G, B) separately.

How to Use This Calculator

  1. Enter the rectangle corners (x1,y1) and (x2,y2).
  2. Enter the four corner values Q11, Q12, Q21, Q22.
  3. Enter the target point (x, y) inside the rectangle.
  4. Click "Calculate" to get the interpolated value.

Disclaimer: The point must be inside the rectangle. For highly nonlinear functions, higher-order methods may be more accurate.

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