How to create a Gaussian kernel of arbitrary width?

Question

How to create a Gaussian kernel by only specifying its width w (3,5,7,9...), and without specifying its variance sigma?

In other word, how to adapt sigma so that the Gaussian distribution 'fits well' w?

I would be interested in a C++ implementation:

void create_gaussian_kernel(int w, std::vector<std::vector<float>>& kernel)
{
    kernel = std::vector<std::vector<float>>(w, std::vector<float>(w, 0.f)); // 2D array of size w x w 
    const Scalar sigma = 1.0; // how to adapt sigma to w ???
    const int hw = (w-1)/2; // half width

    for(int di = -hw; di <= +hw; ++di)
    {
        const int i = hw + di;
        for(int dj = -hw; dj <= +hw; ++dj)
        {
            const int j = hw + dj;
            kernel[i][j] = gauss2D(di, dj, sigma);
        }
    } 
}

Everything I see on the Internet use a fixed size w and a fixed variance sigma :

`w` and `sigma` are independent of each other. You can't calculate `sigma` from `w`. — Thomas Sablik, Feb 11 '21 at 10:41
Generally, the inverse is done. For a given sigma, as the gaussian function is decreasing rapidly, we apply for example the 3-sigma rule and consider it is useless to consider input values greater than `3*sigma` (0r `2*sigma` or `2.5*sigma`: it is very arbitrary). — Damien, Feb 11 '21 at 10:47
I understand, but I am working with images and specifying the smoothing kernel radius in pixels makes sens to me — T.L, Feb 15 '21 at 10:13
Related question: https://stackoverflow.com/questions/8204645/implementing-gaussian-blur-how-to-calculate-convolution-matrix-kernel. My answer there https://stackoverflow.com/a/71968356/2505186 contains a calculation that uses the desired kernel size, but the kernel is not modified. — Tobias Knauss, Apr 22 '22 at 12:30

score 0 · Accepted Answer · answered Feb 15 '21 at 10:09

I found a simple (arbitrary) relation between sigma and w.

I want the next value outside the kernel (along one axis) below a very small value epsilon:

exp( - (half_width + 1)^2 / (2 * sigma^2) ) < epsilon

with half_width the kernel 'half width'.

The result is

sigma^2 = - (half_width + 1)^2 / (2 * log(epsilon))

I use the following c++ code:

#include <vector>
#include <cmath>
#include <cassert>

using Matrix = std::vector<std::vector<float>>;

// compute sigma^2 that 'fit' the kernel half width 
float compute_squared_variance(int half_width, float epsilon = 0.001)
{
    assert(0 < epsilon && epsilon < 1); // small value required
    return - (half_width + 1.0) * (half_width + 1.0) / 2.0 / std::log(epsilon);
}

float gaussian_exp(float y, float x, float sigma2)
{
    assert(0 < sigma2);
    return std::exp( - (x*x + y*y) / (2 * sigma2) );
}

// create a Gaussian kernel of size 2*half_width+1 x 2*half_width+1
Matrix make_gaussian_kernel(int half_width)
{
    if(half_width <= 0)
    {
        // kernel of size 1 x 1
        Matrix kernel(1, std::vector<float>(1, 1.0));
        return kernel;
    }

    Matrix kernel(2*half_width+1, std::vector<float>(2*half_width+1, 0.0));

    const float sigma2 = compute_squared_variance(half_width, 0.1);

    float sum = 0;
    for(int di = -half_width; di <= +half_width; ++di)
    {
        const int i = half_width + di;
        for(int dj = -half_width; dj <= +half_width; ++dj)
        {
            const int j = half_width + dj;
            kernel[i][j] = gaussian_exp(di, dj, sigma2);
            sum += kernel[i][j];
        }
    }

    assert(0 < sum);

    // normalize 
    for(int i=0; i<2*half_width+1; ++i)
    {
        for(int j=0; j<2*half_width+1; ++j)
        {
            kernel[i][j] /= sum;
        }
    }

    return kernel;
}

How to create a Gaussian kernel of arbitrary width?

1 Answers1