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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?

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  • $\begingroup$ “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. $\endgroup$ – Cris Luengo Jun 22 at 22:32
  • $\begingroup$ You can read the full paper here. Please check page 5, equation 11. $\endgroup$ – Nirmal Jun 23 at 8:40

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