Gaussian Kernel Generation

easyPROConvolution and FilteringComputer Vision 2

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

Inputs: size $= 3$ means a $3 \times 3$ grid, and sigma $= 1.0$ sets how fast weights fall off from the middle. The center is $c = \lfloor 3/2 \rfloor = 1$ , so position $(1, 1)$ sits in the middle.
Squared distances: the center has $0$ , the four edge cells have $1$ , and the four corners have $1^2 + 1^2 = 2$ .
Unnormalized weights: with $2\sigma^2 = 2$ , the center gets $e^{0} = 1$ , the edges get $e^{-1/2} \approx 0.606531$ , and the corners get $e^{-1} \approx 0.367879$ .
Normalize: the total is $1 + 4(0.606531) + 4(0.367879) \approx 4.897640$ . Dividing every weight by it gives $1/4.897640 \approx 0.20418$ in the middle, $0.123841$ on the edges and $0.075114$ in the corners, each rounded to 6 decimal places.
Output: a $3 \times 3$ nested list whose values add up to 1. Notice that it is symmetric under flips and rotations, because only the distance from the center matters. A smaller $\sigma$ piles more weight onto the middle: with $\sigma = 0.5$ the center rises to $0.619347$ .

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

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Probability & StatisticsFoundations

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Background Knowledge

The Gaussian kernel, also known as the Gaussian filter or Gaussian blur, is a fundamental concept in image and signal processing. It is a 2D distribution of values that are used to blur or smooth an image. The kernel is centered at a specific point, and the values in the kernel decrease as the distance from the center increases. The rate of this decrease is controlled by the standard deviation, denoted by $\sigma$ . In the context of this problem, we are tasked with generating a 2D Gaussian kernel given a kernel size and $\sigma$ .

The formula provided, $G(i, j) = \exp\left(-\frac{(i - c)^2 + (j - c)^2}{2\sigma^2}\right)$ , represents the Gaussian distribution, where $c$ is the center of the kernel, and $(i, j)$ are the coordinates of each position in the kernel. This formula calculates the value at each position based on its distance from the center. The resulting values will form a bell-shaped curve, which is characteristic of the Gaussian distribution.

Understanding the concept of normalization is also crucial. Normalization ensures that the sum of all values in the kernel equals 1. This is important because it allows the kernel to be used as a weighting function, where the weights sum to 1. In the context of image processing, this means that the kernel will not change the overall brightness of the image.

Algorithm/Approach

The general approach to solving this problem involves using nested loops to iterate over each position in the kernel, calculating the value at each position using the provided formula, and then normalizing the values so they sum to 1. This can be achieved by first calculating the sum of all values in the kernel and then dividing each value by this sum.

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