> For the complete documentation index, see [llms.txt](https://docs.inferix.io/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.inferix.io/inferix-whitepaper/high-level-description/noise-generation.md).

# Noise generation

In practice, a scene may contain multiple frames, each task of this scene contains some range of frames to be rendered, consequently each worker may render only a subset of these frames. For the simplification purpose, we assume in this section that a scene has only one frame, so the output image is determined uniquely by the scene.

<figure><img src="/files/MEO14drDsG6lowYOlbwW" alt=""><figcaption><p>Noise generation</p></figcaption></figure>

Let $$R$$ denote the rendering process, for each input scene $$G$$, the result of the rendering is an image:

$$
I = \mathcal{R} \left(G\right)
$$

It is important to note that $$I$$ is actually ***never*** computed, neither by the *manager* in the noise embedding (see also the discussion about [frame sampling](/inferix-whitepaper/implementation/adaptive-noise-spreading.md)) nor by *workers* in the frame rendering. The equation above represents only equality.

Similar with invisible watermark schemes in the literature [\[2\]](/inferix-whitepaper/references.md#2), [\[3\]](/inferix-whitepaper/references.md#3), [\[9\]](/inferix-whitepaper/references.md#9) a noise $$W$$ consists in a random vector of atomic watermarks:

$$
W \triangleq \left(w\_1, \dots, w\_n \right)
$$

where $$w\_i , \left(1 \leq i \leq n\right)$$ is independently chosen from some normal probability distribution $$\mathcal{N}\left(\mu, \sigma^2\right)$$. Furthermore, $$w\_i$$ has a special structure depending on where it is introduced in the scene $$G$$. The number $$n$$ of atomic watermark signals is chosen around an experimental trade-off between human perception threshold about the image distortion and the false positive ratio of the noise verification.

Using a uniformly generated task identification number $$J\_{\mathtt{id}}$$, we calculate a verification key which is a vector of the same size as the noise vector $$W$$:

$$
K\_{\mathtt{verif}} \left(S, W, J\_{\mathtt{id}}\right) \triangleq \left( k\_1,\dots,k\_n \right)
$$

that will be used later for the noise verification.

We have discussed that embedding watermarks into $$I$$ cannot help the authentication, then the noise $$W$$ is not embedded into the image $$I$$ but into the scene $$G$$. Let $$\mathcal{E}$$ denote the embedding function, we now create a watermarked scene:

$$
\hat{G} = \mathcal{E} \left(G, W\right)
$$

Finally, $$\hat{G}$$ is sent to *workers* for rendering, that results in a rendered image:

$$
\hat{I} = \mathcal{R} (\hat{G})
$$

If got accepted, namely $$\hat{I}$$ passes the noise verification which will be presented hereafter, this is the image sent back to the user (recall that $$I$$ in the rendering equation is not computed). The encoding function $$\mathcal{E}$$ and the noise $$W$$ are designed so that the distortion of $$\hat{I}$$ against $$I$$ is imperceptible [\[10\]](/inferix-whitepaper/references.md#10), [\[11\]](/inferix-whitepaper/references.md#11) then $$\hat{I}$$ can be authentically used as a result of the graphics rendering.