# Gaussian kernel notation

In this paper on content-aware seam carving, they have given an equation:

$$w^{s}(k)=\frac{1}{n^{k}} \sum_{i=1}^{N}\left[\mathcal{N}\left( \|z_{i}-o| |_{2} \; | \; 0, \sigma^{2}\right) \cdot \delta\left[b\left(p_{i}\right)-C^{k}\right]\right]$$

where $$δ(.)$$ and $$o$$ represent the Kronecker delta function and the center of image, respectively. Gaussian kernel $$\mathcal{N}(.)$$ calculates the Euclidean distance between pixel $$z_i$$ (normalized location of the pixel $$p_i$$) and image center $$o$$. Variance $$σ^2$$ is the normalized radius of the image, and the normalization coefficient $$n^k$$ denotes the number of pixels in cluster $$C^k$$.

Pixel $$i$$ and its normalized location in the image are denoted by $$\left\{p_{i}\right\}_{i=1}^{N}$$ and $$\left\{z_{i}\right\}_{i=1}^{N}$$ respectively, $$N$$ is the total number of pixels. And the function $$b : \mathbb{R}^{2} \rightarrow\{1 \ldots K\}$$ associates the pixel $$p_i$$ and the cluster index $$b(p_i)$$.

What exactly does $$\mathcal{N}\left( \| z_{i}-o| |_{2} \; | \; 0, \sigma^{2}\right)$$ mean here?

My understanding is that $$\mathcal{N}\left(0, \sigma^{2}\right)$$ denotes a normal distribution with mean = $$0$$ and variance = $$\sigma^2$$. So how should the calculated $$L_2$$ norm be used here?

• “Gaussian kernel ... calculates the Euclidean distance between pixel 𝑧𝑖 ... and image center 𝑜.” Honestly, I don’t think it makes sense. The first argument to that function is the Euclidean distance. I don’t see how the Gaussian kernel would compute a distance. But it might transform that distance in a particular way (points close to the origin yield 1, further away they yield 0). The meaning I guess could be understood if more context were given. – Cris Luengo Jun 22 at 22:32
• You can read the full paper here. Please check page 5, equation 11. – Nirmal Jun 23 at 8:40