Distortion region

Under constraints about position and direction of noise objects, the imprint of $\omega_i$ on the rendered image is a rectangular region denoted by:

$$k_i \triangleq \left(x_i^{\mathtt{ul}}, y_i^{\mathtt{ul}}, x_i^{\mathtt{lr}}, y_i^{\mathtt{lr}} \right)$$

where $\left(x_i^{\mathtt{ul}}, y_i^{\mathtt{ul}}\right)$ and $\left(x_i^{\mathtt{lr}}, y_i^{\mathtt{lr}}\right)$ are respectively the upper left and lower right positions in the image coordinate system. It is important to note that $k_i$ for all $1 \leq i \leq n$ can be computed without rendering the scene $G$.

For the size of distortion regions, similar with the length of the noise random vector, there is a compromise between the robustness of the embedded noise and the fidelity of the rendered frame. The larger the distortion $k_i$, the higher information of $w_i$ can be restored then the higher robustness of the noise verification; but the lower the distortion $k_i$, the higher fidelity of the image. Empirically, we use the bounds $4 \leq x^{\mathtt{lr}}_{i} - x^{\mathtt{ul}}_{i},\ y^{\mathtt{lr}}_{i} - y^{\mathtt{ul}}_{i} \leq 7$ for all $1 \leq k \leq n$.

Rendered watermarked scenes (random vector length 12)

The figure above shows some distortion results of rendering watermarked scenes. From two original scenes, noise vectors of length $12$ with different distortion sizes are embedded, then different watermarked scenes are generated. When rendering the scenes containing noises whose distortion sizes are $7$ or $8$, the distortions are visible under the form of small rectangles dispersed in the rendered images. In contrast, when the sizes are $4$ or $5$, the distortions are imperceptible.

Remark: While the atomic watermarks are quite large, the distortions made by them on rendered images are constrained relatively small. The figure in the previous section shows atomic watermarks of size $512 \times 512$ which are used for watermarking scenes shown in the figure above, their imprints are about $4 \times 4$. The sizes of the rendered images are much larger: $1080 \times 1080$ and $1920 \times 1080$.