Interpolation is a technique in Statistics to determine an approximation of a mathematical statement by using any intermediate value for the independent variable. This technique is often used by mathematicians and engineers in picture scaling to determine the location of the next pixel depending on the provided positions of pixels in an image. In this article, we will learn in detail about bilinear interpolation.

What Is Bilinear Interpolation?

A technique for two-dimensional interpolation on a rectangle is called bilinear interpolation. An interpolation strategy enables you to estimate a function's value at any location inside a rectangle if its value is known at each of the rectangle's four corners. The bilinear interpolation process uses the average of the data at each corner of the rectangle. The weights are based on the separation between the point and the corners for a (x,y) position inside the rectangle. The weight of the corner increases the closer it is to the tip.

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Bilinear Interpolation Formula

Simply transfer your rectangle onto the unit square and do the interpolation there to interpolate on any other rectangle. Because the weighted average only depends on a point's relative position in relation to the rectangle's corners, this approach works for bilinear interpolation. The transformation (x, y) (u, v) can be used to transfer the lower-left corner of a rectangle (x0, y0) to the upper-right corner (x1, y1).

Bilinear Interpolation Properties

  • The weighted average of the values at the four corners of the rectangle, Q11, Q12, Q21, and Q22, is known as bilinear interpolation. The weights are based on the distance between the corners and the point (x, y). A corner gains more weight the nearer it is to (x, y).
  • Regarding the unknown function's Q11, Q12, Q21, and Q22 values at the corners of the rectangle, it is linear. Every horizontal line (where x changes and y stays the same) and every vertical line (where x stays the same and y changes) are linear; however, the interpolation point position, which is a function of both x and y, is quadratic.

Examples of Bilinear Interpolation

Let’s check out an example.

Consider the following values of an unknown function: 12 at (0, 1), -4 at (0, 3), 0 at (4, 1), and 8 at (4, 3). We want to calculate this function's value at (1, 2). Let's start by recording some information.

The rectangle's corners are given by the equations x1 = 0, x2 = 4, y1 = 1, and y2 = 3.

These are the corresponding function values: Q11 = 12, Q21 = -4, Q12 = 0, and Q22 = 8.

The bilinear interpolation is done at the following points: x = 1, y = 2.

Let's figure out the components of the bilinear interpolation formula for P:

(x₂ - x₁) * (y₂ - y₁) = (4 - 0) * (3 - 1) = 8

(x₂ - x) * (y₂ - y) = (4 - 1) * (3 - 2) = 3

(x - x₁) * (y₂ - y) = (1 - 0) * (3 - 2) = 1

(x₂ - x) * (y - y₁) = (4 - 1) * (2 - 1) = 3

(x - x₁) * (y - y₁) = (1 - 0) * (2 - 1) = 1

Let's combine those data to create the P formula. We discover:

P = 3/8 * Q₁₁ + 1/8 * Q₂₁ + 3/8 * Q₁₂ + 1/8 * Q₂₂

It's time to change the values for Q11, Q21, Q12, and Q22:

P = 3/8 * 12 + 1/8 * (-4) + 3/8 * 0 + 1/8 * 8

In the end, we have:

P = 9/2 - 1/2 + 1 = 5

When to Use Bilinear Interpolation

Bilinear interpolation is advised for continuous data sets without clear boundaries. The nearest points must be connected and the surface must be continuous. The procedure produces a smoother surface when you run it, although it is not as severe as cubic convolution, which employs 16 nearby cells.

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Benefits of Bilinear Interpolation

From a small sample size of data points, bilinear interpolation makes predictions about the values of the cells in a raster. Any geographic point data, such as elevation, rainfall, chemical concentrations, noise levels, etc., can be utilized to forecast unknown values.

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Conclusion

Interpolation is a great way to determine the locations of pixels in a data set making it a useful tool in imaging or data visualization. With Data becoming one of the hottest trends in today’s age and age, having information and knowledge about these topics is a great way to upskill yourself for the future. 

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FAQs

1. What is the bilinear interpolation method?

The resampling technique known as bilinear interpolation estimates a new pixel value using the distance-weighted average of the four closest pixels. The four input raster cell centers that are closest to the output processing cell center will be weighted and averaged depending on the distance.

2. When should you use bilinear interpolation?

For continuous data sets with no discernible boundaries, bilinear interpolation is recommended. The surface must be continuous, and the points nearest to each other must be connected. When the process is run, it produces a smoother surface, but not as severe as cubic convolution, which employs 16 neighboring cells.

3. How do you solve bilinear interpolation?

  • Begin with two linear interpolations in the x-direction (horizontal): first at (x, y1), then at (x, y2).
  • Linear interpolation in the y-direction (vertical) is then performed: utilize the interpolated values at (x, y1) and (x, y2) to derive the interpolation at the end location (x, y).

4. What is the difference between bilinear and bicubic interpolation?

In contrast to bilinear interpolation, which only examines four pixels (2*2), bicubic interpolation considers sixteen pixels (4*4). Depending on the b and c values used, images resampled with bicubic interpolation might exhibit varying interpolation artifacts.

5. Which interpolation method is most accurate?

Radial Basis Function interpolation is a broad category of data interpolation techniques. Many people believe that the Multiquadric approach is the best for fitting your data and producing a smooth surface.

6. Why do we interpolate signals?

The primary reason for interpolation is to increase the sampling rate at the output of one system so that the signal can be input to another system operating at a higher sampling rate.

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