Free Bilinear Interpolation Calculator

Understanding Bilinear Interpolation

Bilinear interpolation is a key technique in two-dimensional interpolation. It estimates an unknown function's value at a point inside a rectangle using the values at the rectangle's four corners. This method is essential in computer vision, image processing, and many other fields where rectilinear data needs to be approximated.

The Four Corner Points

Assume we know an unknown function's values at the corners of a rectangle defined by coordinates $(x_1, y_1)$, $(x_1, y_2)$, $(x_2, y_1)$, and $(x_2, y_2)$. The known values are:

• $Q_{11}$ at $(x_1, y_1)$
• $Q_{12}$ at $(x_1, y_2)$
• $Q_{21}$ at $(x_2, y_1)$
• $Q_{22}$ at $(x_2, y_2)$

We want to compute the interpolated value $P$ at a location $(x, y)$ inside this rectangle.

How the Formula Works

We obtain $P$ through three successive linear interpolations. First, interpolate along the horizontal direction at the top and bottom y-levels:

$$R_1 = \frac{x_2 - x}{x_2 - x_1} Q_{11} + \frac{x - x_1}{x_2 - x_1} Q_{21}$$ $$R_2 = \frac{x_2 - x}{x_2 - x_1} Q_{12} + \frac{x - x_1}{x_2 - x_1} Q_{22}$$

Then interpolate vertically between $R_1$ and $R_2$:

$$P = \frac{y_2 - y}{y_2 - y_1} R_1 + \frac{y - y_1}{y_2 - y_1} R_2$$

Replacing $R_1$ and $R_2$ gives the complete bilinear interpolation formula:

$$P = \frac{(x_2 - x)(y_2 - y)}{(x_2 - x_1)(y_2 - y_1)} Q_{11} + \frac{(x - x_1)(y_2 - y)}{(x_2 - x_1)(y_2 - y_1)} Q_{21} + \frac{(x_2 - x)(y - y_1)}{(x_2 - x_1)(y_2 - y_1)} Q_{12} + \frac{(x - x_1)(y - y_1)}{(x_2 - x_1)(y_2 - y_1)} Q_{22}$$

A matrix representation is also available for those familiar with linear algebra.

Using the Bilinear Interpolation Calculator

To use the tool, simply input:

• The rectangle corner coordinates $x_1, y_1, x_2, y_2$
• The four known values $Q_{11}, Q_{12}, Q_{21}, Q_{22}$
• The target point coordinates $x, y$

The calculator automatically applies the bilinear interpolation formula and returns the estimated value $P$. It's a quick way to perform two-dimensional interpolation without manually computing the weights.

Step-by-Step Example

Suppose we have the following data:

• $Q_{11}=12$ at $(0,1)$
• $Q_{12}=-4$ at $(0,3)$
• $Q_{21}=0$ at $(4,1)$
• $Q_{22}=8$ at $(4,3)$

We want the value at $(1,2)$.

Set $x_1=0$, $x_2=4$, $y_1=1$, $y_2=3$. Compute the necessary products:

$$(x_2 - x_1)(y_2 - y_1) = 4 \times 2 = 8$$ $$(x_2 - x)(y_2 - y) = 3 \times 1 = 3$$ $$(x - x_1)(y_2 - y) = 1 \times 1 = 1$$ $$(x_2 - x)(y - y_1) = 3 \times 1 = 3$$ $$(x - x_1)(y - y_1) = 1 \times 1 = 1$$

Now insert into the formula:

$$P = \frac{3}{8} \cdot 12 + \frac{1}{8} \cdot 0 + \frac{3}{8} \cdot (-4) + \frac{1}{8} \cdot 8$$ $$P = \frac{36}{8} - \frac{12}{8} + \frac{8}{8} = \frac{32}{8} = 4$$

Therefore, the interpolated value at $(1,2)$ is 4.

Important Characteristics

• Weighted average: The result is a weighted sum of the corner values. The weight for each corner is proportional to the area of the opposite sub-rectangle, meaning closer corners have more influence.
• Linearity: The formula is linear with respect to the corner values $Q_{11}$ through $Q_{22}$. It is also linear along any row (constant $y$) or column (constant $x$).
• Overall quadratic behavior: When considered as a function of both $x$ and $y$, the interpolation is quadratic due to the presence of the $xy$ bilinear term.

These properties make bilinear interpolation a widely used compromise between simplicity and smoothness in two-dimensional tasks like image scaling and data interpolation on rectangular grids.