Distortion region

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

ki≜(xiul,yiul,xilr,yilr)ki​≜(xiul​,yiul​,xilr​,yilr​)

where (xiul,yiul)(xiul​,yiul​) and (xilr,yilr)(xilr​,yilr​) 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≤i≤n1≤i≤n can be computed without rendering the scene GG.

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 kiki​, the higher information of wiwi​ can be restored then the higher robustness of the noise verification; but the lower the distortion kiki​, the higher fidelity of the image. Empirically, we use the bounds 4≤xilr−xiul, yilr−yiul≤74≤xilr​−xiul​, yilr​−yiul​≤7 for all 1≤k≤n1≤k≤n.

The figure above shows some distortion results of rendering watermarked scenes. From two original scenes, noise vectors of length 1212 with different distortion sizes are embedded, then different watermarked scenes are generated. When rendering the scenes containing noises whose distortion sizes are 77 or 88, the distortions are visible under the form of small rectangles dispersed in the rendered images. In contrast, when the sizes are 44 or 55, 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 512×512512×512 which are used for watermarking scenes shown in the figure above, their imprints are about 4×44×4. The sizes of the rendered images are much larger: 1080×10801080×1080 and 1920×10801920×1080.