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Gaussian Kernel Generation

Implement a function to generate a normalized 2D Gaussian kernel, a fundamental component in image processing and computer vision. Given a kernel size, an odd integer, and standard deviation $\sigma$ , create a 2D kernel where each element is computed based on its distance from the center.

The Gaussian distribution is a continuous probability distribution, commonly used to model noise in images. In the context of a 2D Gaussian kernel, the value at each position $(i, j)$ is determined by its distance from the center $(c, c)$ , where $c = \text{size} // 2$ .

Compute the distance of each position from the center.
Calculate the Gaussian value at each position using the formula. The main equation for the Gaussian value $G(i, j)$ is

G(i, j) = \exp\left(-\frac{(i - c)^2 + (j - c)^2}{2\sigma^2}\right)

This technique is widely used in image filtering.

Example:

Input:

kernel_size = 3, sigma = 1.0

Output:

[[0.075114, 0.123841, 0.075114], [0.123841, 0.20418, 0.123841], [0.075114, 0.123841, 0.075114]]

Reasoning:

The center of the kernel $c$ is calculated as $\text{size} // 2 = 3 // 2 = 1$ , so the kernel is centered at $(1, 1)$ .
For each position $(i, j)$ , the Gaussian value $G(i, j)$ is computed using the formula: $G(i, j) = \exp\left(-\frac{(i - 1)^2 + (j - 1)^2}{2\cdot1.0^2}\right)$ , resulting in a 3x3 matrix of unnormalized values.
The unnormalized values are then normalized by dividing each value by the sum of all values in the matrix, so that the sum of all values equals 1.
The normalized values are rounded to 6 decimal places, yielding the final output: [[0.075114, 0.123841, 0.075114], [0.123841, 0.20418, 0.123841], [0.075114, 0.123841, 0.075114]].

Constraints:

kernel_size is a positive odd integer
sigma is a positive float
Return 2D list normalized to sum to 1, rounded to 6 decimal places

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.