Create centered & normalized 1D gaussian kernel aka normal distribution function aka probability density function

1d_gauss.h

template <typename T>
T norm_pdf(T x, T mu, T sigma)
{
	static const T inv_sqrt_2pi = static_cast<T>(0.3989422804014327);   // Precomputed 1/sqrt(2*PI) to avoid sqrt()
	T a = (x - mu) / sigma;

	return inv_sqrt_2pi / sigma * std::exp(-T(0.5) * a * a);
}

template <typename T>
void generateCenteredNormPdf(std::vector<T>& density)
{
	size_t n = density.size();    // Number of steps
	T mu = static_cast<T>(0.0);   // Mu is always 0 centered

  T half_size = static_cast<T>(floor(n / 2.0));     // Half the size of our array
	T sigma = static_cast<T>(half_size / 3.0);        // Adjust our sigma dependent on the vector size to always stay within +/- 3 sigma
	T sum = 0.0;                                      // For normalization
	T step_size = static_cast<T>(1.0);                // How fast we run away from the center

	for (size_t i = 0; i < n; i++)
	{
		density[i] = norm_pdf(static_cast<T>(step_size * (i - half_size)), mu, sigma);
		sum += density[i];
	}

	// Make up for any energy loss during calculation (sum should already be very close to 1.0)
	for (size_t j = 0; j < n; j++)
	{
		density[j] /= sum;
	}
}