Bilinear Interpolation Calculator

Bilinear interpolation extends linear interpolation into a two-dimensional grid. Given a target point located at coordinates (x, y) surrounded by four known grid nodes—(x1, y1) with value Q11, (x1, y2) with value Q12, (x2, y1) with value Q21, and (x2, y2) with value Q22—the calculator applies a weighted average formula. The mathematical equation is expressed as P = [Q11(x2-x)(y2-y) + Q21(x-x1)(y2-y) + Q12(x2-x)(y-y1) + Q22(x-x1)(y-y1)] / [(x2-x1)(y2-y1)]. This method performs a linear interpolation first along the x-axis, and then uses those intermediate results to interpolate along the y-axis.

Imagine you have a grid where the bottom-left point (x1=0, y1=0) has a value Q11 = 10, the top-left point (x1=0, y2=10) has Q12 = 20, the bottom-right point (x2=10, y1=0) has Q21 = 30, and the top-right point (x2=10, y2=10) has Q22 = 40. You want to find the interpolated value P at coordinates x = 3 and y = 7. Substituting these numbers into the formula: denominator = (10 - 0) * (10 - 0) = 100. The numerator becomes [10*(7)*(3) + 30*(3)*(3) + 20*(7)*(7) + 40*(3)*(7)] = [210 + 270 + 980 + 840] = 2300. Finally, dividing 2300 by 100 yields an estimated value P = 23.

Always ensure your target coordinate (x, y) falls strictly within the bounding box defined by x1, x2, y1, and y2 to avoid extrapolation artifacts. Check that your grid interval coordinates are properly ordered such that x2 is greater than x1 and y2 is greater than y1. Keep in mind that while bilinear interpolation is computationally fast and smooth, it can occasionally soften sharp edges in image processing applications.