Bicubic Interpolation

Image processing is one of the foundational pillars of modern technology and is an integral part of many applications we use daily, from photo editing software to artificial intelligence. One of the most commonly used methods in this field, bicubic interpolation, delivers high-quality results when resizing images or estimating missing pixels.

Bicubic interpolation is a method used to calculate new pixel values when resizing an image—for example, during scaling up or down. Known in Turkish as "çift kübik interpolasyon," this method derives its name from cubic (third-degree polynomial) mathematical functions and operates on a two-dimensional plane (x and y axes). Unlike simpler methods such as nearest-neighbor and bilinear interpolation, bicubic interpolation considers 16 surrounding pixels (a 4x4 matrix) when calculating a pixel’s value. This results in smoother and more natural-looking outcomes.

According to Cloudinary, "Bicubic interpolation is a method that calculates a pixel’s value based on its neighboring pixels." ^【1】 This method is preferred in situations requiring high image quality because it excels at preserving detail and reducing artifacts such as pixelation or blurriness.

In image processing, interpolation is typically used to change an image’s resolution or fill in missing data points. For instance, when enlarging a photograph, new pixels must be generated between the original ones. The values of these new pixels are estimated using interpolation algorithms. Bicubic interpolation performs these estimations in a more sophisticated manner, better preserving sharp edges and fine details.

^{Original image, differences in pixels produced by nearest-neighbor, bilinear, and bicubic interpolation (Credit: )}

The foundation of bicubic interpolation lies in cubic polynomials. This method does not only consider nearby pixels but also their gradients (rates of change). Mathematically, this process occurs in two stages: first interpolation along the x-axis, followed by interpolation along the y-axis.

Bicubic interpolation employs a cubic spline function, typically defined as:

This function assigns a weight to each surrounding pixel based on its distance. Wu et al. explained in their study: "Cubic interpolation provides a smoother transition by considering the derivatives of a continuous function."

To calculate a pixel’s value, a 4x4 pixel matrix surrounding the target pixel is selected. This matrix includes both horizontal and vertical neighbors. Each pixel’s value is then combined using cubic weighting. The mathematical expression can be summarized as:

$f(x,y)=\sum _{i=0}^3{a}_{i}j{x}^i{y}^i$

Here, a_ij are coefficients derived from the values and gradients of the 16 neighboring pixels. Although this process is computationally intensive, it significantly enhances result quality.

The key difference between bicubic and bilinear interpolation is that bicubic interpolation accounts not only for pixel values but also for their derivatives (gradients). This plays a critical role in preserving edges and details. Rowe et al. emphasized this in their presentation: "Gradients improve the accuracy of interpolation and reduce artifacts."

The step-by-step working mechanism of bicubic interpolation is as follows:

1. Target Pixel Identification: The position (x, y) of the new pixel is determined.
2. Neighboring Pixel Selection: A 4x4 pixel matrix closest to the target pixel is selected.
3. Weight Calculation: A weight is assigned to each neighboring pixel using a cubic function, based on distance.
4. Value Estimation: Weighted pixel values are summed to compute the new pixel’s value.
5. Two-Dimensional Application: This process is first applied along the x-axis, then along the y-axis.

^{Bicubic interpolation (Credit: )}

Like any method, bicubic interpolation has its strengths and weaknesses.

• High Quality: By considering 16 pixels, bicubic interpolation delivers smooth gradients and sharp details. In scaling operations, artifacts such as pixelation are minimized.
• Natural Appearance: It produces results that are visually more pleasing to the human eye, offering a significant advantage in photography and graphic design.
• Flexibility: It can be effectively applied to various types of images, such as natural scenes or digital graphics.

• Computational Cost: Due to the need to process more pixels and perform complex mathematical operations, bicubic interpolation is slower than nearest-neighbor or bilinear methods.
• Mild Blurring: In some cases, particularly with highly detailed images containing sharp edges, it may introduce slight blurring.
• Resource Intensity: Its use may be challenging on low-power devices or in real-time applications.

^{Comparison of Interpolation Methods (Generated by artificial intelligence.)}

Bicubic interpolation is not merely a theoretical concept but a widely used practical method.

• Image Editing Software: Professional software such as Photoshop favors bicubic interpolation for image enlargement, allowing users to scale photos without significant quality loss.
• Artificial Intelligence and Super-Resolution: Bicubic interpolation is often used as an initial step in deep learning-based super-resolution algorithms. A low-resolution image is first upscaled using bicubic interpolation, after which AI models further enhance the details.
• Computer Graphics and Gaming: In game engines and 3D rendering, bicubic interpolation is used to scale textures, resulting in more realistic visuals.
• Scientific Visualization: In the interpolation of scientific data—such as medical imaging or meteorological maps—the bicubic method plays a critical role in preserving detail.

In languages such as Python, this method can be implemented as follows:

1. The input image is represented as a matrix.
2. The coordinates of the target pixel are determined.
3. A 4x4 matrix of neighboring pixels around this pixel is selected.
4. A cubic weighting function is defined, assigning a weight to each neighboring pixel.
5. A weighted sum is calculated to determine the new pixel’s value.
6. This process is repeated for all new pixels.

Example pseudocode:

Detailed Code:

^{(1) Original image; (2) Image obtained by downsampling the original image at 4x scale using bicubic interpolation; (3) Image obtained by upsampling the downsampled image at 4x scale using bicubic interpolation. (Prepared and edited by Beyza Nur Türkü)}