Bilinear Interpolation

Introduction

• Bilinear interpolation: A mathematical method for interpolating functions of two variables, typically on a 2D rectilinear grid.
• Generalizable to functions on vertices of arbitrary convex quadrilaterals.

Process

• Steps:
  1. Linear interpolation in one direction.
  2. Linear interpolation in the other direction.
• Result is not strictly linear but quadratic in location.

Applications

• Widely used in computer vision and image processing as bilinear filtering or texture mapping.

Mathematical Explanation

• Given function f values at four points _Q_11, _Q_12, _Q_21, _Q_22.
• Interpolation involves:
  • Repeated linear interpolation in x and y directions.
  • Solution also expressible as a multilinear polynomial or weighted mean.

Alternative Matrix Form

• Can use a simplified coordinate system:
  • Points: (0,0), (0,1), (1,0), (1,1).
• Formula involves matrix operations.
• Interpolant is not linear but quadratic along lines not parallel to axes.

Properties

• Result independent of interpolation order (x or y direction first).
• Interpolant is a bilinear polynomial, harmonic function satisfying Laplace's equation.

Inverse and Generalization

• Interpolation generally non-invertible, but invertible when applied to vector fields under certain conditions.
• Can generalize to any convex quadrilateral leading to bilinear transformation or distortion.
• Trilinear interpolation: Extension to three dimensions.

Image Processing

• Used for resampling images, involving mapping screen pixels to texture maps.
• Important for image scaling, especially with non-integral scale factors.
• Reduces visual distortions compared to nearest-neighbor interpolation.

Simplification

• Standard calculation involves multiple operations; can be reduced by using temporary variables.
• Simplifying terms reduces computational load.

Related Concepts

• Bicubic interpolation
• Trilinear interpolation
• Spline interpolation
• Lanczos resampling
• Stairstep interpolation
• Barycentric coordinates

References

• Numerical recipes and various academic articles on bilinear interpolation and its applications.