Bilinear interpolation

Author

Abhishek Kumar Pandey
March 28, 2024

Bilinear interpolation

Easy:

Imagine you have a big grid of squares, and each square has a number on it. Now, you want to find out what number would be in a square that’s exactly in the middle of two squares. Bilinear interpolation is like a magic trick that helps you guess what number would be in that middle square.

Here’s how it works:

  1. Find the Nearest Squares: First, you look at the four squares that are closest to the middle square. Let’s say these are squares A, B, C, and D.

  2. Count the Numbers: Next, you look at the numbers on these squares. Let’s say A has 1, B has 2, C has 3, and D has 4.

  3. Figure Out How Close Each Square Is: You decide how close each of these squares is to the middle square. The closer a square is, the more it should influence the number in the middle square.

  4. Mix the Numbers: Finally, you mix the numbers from squares A, B, C, and D together, using the distances you figured out in the previous step. This mix gives you the number that would be in the middle square.

So, bilinear interpolation is like a magic trick that helps you guess what number would be in the middle square by mixing the numbers from the closest squares.

Another easy example:

Bilinear interpolation is like a magic trick we use to make pictures bigger or smaller without making them look blurry. Imagine you have a small picture, and you want to make it bigger, but you don’t want it to look fuzzy or stretched. Bilinear interpolation helps us do that by looking at the colors of the pixels in the picture and figuring out how to make the picture larger while keeping it clear and sharp.

It’s a bit like connecting the dots in a coloring book. When we want to make the picture bigger, bilinear interpolation helps us fill in the new spaces between the dots with colors that make the picture look just right. So, it’s like a clever way of making pictures bigger or smaller while keeping them looking nice and not all blurry.

Moderate:

Bilinear interpolation is a technique used in computer graphics and image processing to estimate values between two sets of values. It’s particularly useful when you need to resize images or when you’re working with data that has a grid structure, such as height maps in 3D graphics.

How Bilinear Interpolation Works

  1. Grid Structure: Bilinear interpolation assumes that the data points are arranged in a grid. Each point in the grid is associated with a value.

  2. Interpolation: When you want to find the value at a point that doesn’t exactly match any of the grid points, bilinear interpolation estimates the value by taking a weighted average of the four nearest grid points.

  3. Weights: The weights are determined by the distances from the point of interest to the four nearest grid points. The closer a grid point is to the point of interest, the more weight it has in the final value.

Example

Imagine you have a 2x2 grid of points with values as follows:

- Point A: (0,0) with value 1

- Point B: (1,0) with value 2

- Point C: (0,1) with value 3

- Point D: (1,1) with value 4

And you want to find the value at point (0.5, 0.5), which is exactly in the middle of the grid.

  1. Identify the Nearest Grid Points: The nearest grid points to (0.5, 0.5) are A, B, C, and D.

  2. Calculate the Weights: The weights are determined by the distances from (0.5, 0.5) to each of the grid points. For simplicity, let’s assume the weights are proportional to the inverse of the square of the distance.

  3. Calculate the Value: The value at (0.5, 0.5) is the weighted average of the values at A, B, C, and D.

Advantages

  1. Smoothness: Bilinear interpolation provides smoother results compared to nearest-neighbor interpolation, especially when resizing images or working with continuous data.

  2. Efficiency: It’s computationally efficient, especially for 2D data, as it only requires a constant amount of computation for each interpolation.

Disadvantages

  1. Assumptions: It assumes that the data is smooth and continuous, which might not always be the case.

  2. Limited to 2D: Bilinear interpolation is limited to 2D data. For 3D data, trilinear interpolation is used.

In summary, bilinear interpolation is a powerful technique for estimating values in a grid structure, providing a balance between accuracy and computational efficiency.

Hard:

Bilinear interpolation is a method used to estimate the value of a function at a point within a grid of known values. It’s particularly useful in computer graphics and image processing to scale or transform images while maintaining smoothness and avoiding jagged edges or artifacts.

The key idea behind bilinear interpolation is to interpolate both horizontally and vertically between the values of a 2D grid, such as the pixels of an image. Here’s a step-by-step breakdown:

1. Imagine a 2D grid of points, like the pixels in an image. Each point has a value associated with it, such as color or intensity.

2. Choose a point within this grid for which you want to estimate a value. This could be a new pixel location you’re inserting into the image.

3. Define four points that surround the target point: left (L), right (R), top (T), and bottom (B). The target point is at some fractional position between these four points.

4. Calculate the horizontal interpolated value at the current row by linearly interpolating between the left and right points. This is often written as:

V = (1 — α) L + α R

Where α is the fractional position of the target point between L and R. α would be 0 at the left point, and 1 at the right point.

5. Repeat this process vertically by interpolating between the top and bottom points to get a final value for the target point:

final value = (1 — β) (1 — α) LT + (1 — β) α RT + β (1 — α) LB + β α RB

Here, β is the fractional position of the target point between the top and bottom rows.

6. This weighted average of the surrounding points’ values is the estimate of the function at the target point.

Bilinear interpolation is a simple and fast method for upscaling or transforming images, making it a popular choice in applications where computational efficiency is important. It assumes that the changes in the function across the grid are linear, leading to reasonably good results for smooth functions, but it may not be as accurate for complex or rapidly changing functions. More advanced techniques, such as bicubic interpolation, can be used for higher accuracy but are more computationally intensive.

A few books on deep learning that I am reading: