Structure of noise

As introduced in the previous section, a noise $W$ is a random vector $\left(w_i\right)_{1 \leq i \leq n}$ where each element is independently chosen from a normal distribution. This atomic watermark is constructed as a rectangular image of periodic patterns as follows:

• let fix some values $M,N$ for the width and the height of the rectangle, and

• let $\mathcal{X}\_i, \mathcal{Y}\_i$ be independent and identically distributed normal random variables:

  $$\mathcal{X}_i \sim \mathcal{Y}_i \sim \mathcal{N}\left(\mu, \sigma^2\right)$$

  for some $\mu$ and $\sigma$, then take $X_i, Y_i$ be respectively some samples of $\mathcal{X}_i, \mathcal{Y}_i$.

The complex atomic signal $w_i$ is defined by:

$$w_i(x,y) = A e^{2i\pi\left(\frac{x}{X_i} + \frac{y}{Y_i}\right)} \ \left(0 \leq x < M, 0 \leq y < N \right)$$

for some amplitude $A$. We observe that $w_i\left(x,y\right) = w_i\left(x + X_i, y\right) = w_i\left(x, y + Y_i\right) \, \forall x,y$ then $X_i, Y_i$ are actually the horizontal and the vertical periods.

Remark: $\mathcal{X}_i$ and $\mathcal{Y}_i$ are elements of a set $\left\{ \mathcal{X}_i, \mathcal{Y}_i \mid 1 \leq i \leq n\right\}$ of independent and identically distributed normal random variables $\mathcal{N}\left(\mu, \sigma^2\right)$. The parameters $\mu$ and $\sigma$ are chosen by analyzing the input scene that is discussed in~\cref{subsec:noise_spreading}.

Proposition 1. (Fourier transform of complex atomic signals)

$$F_i\left(u,v\right) = \frac{A}{M \times N} \frac{\left(1 - e^{2i\pi \frac{M}{X_i}}\right) \left(1 - e^{2i\pi \frac{N}{Y_i}}\right)}{\left(1 - e^{2i\pi\left(\frac{1}{X_i} + \frac{u}{M}\right)}\right) \left(1 - e^{2i\pi\left(\frac{1}{Y_i} + \frac{v}{N}\right)}\right)}$$

Proof: Direct calculation (for details, see Appendix 1).

The given structure of noise has two folds: we empirically find that this form of signal makes the wrapping graphical objects (discussed in the noise insertion) persistent in the rendering of graphics scenes. Furthermore, the distortion raised by any atomic watermark is easily controlled thanks to the simple form of the signal amplitude given in the proposition.

The spectrums of atomic signals play a crucial role in the noise verification since they help to distinguish embedded noises from the original image signals. They are also completely determined by the periods $X_i, Y_i$ given fixed $M,N$ since the discrete Fourier transform in the proposition. In turn, these periods statistically rely on the expectation $\mu$, we will discuss how to choose this value in the noise spreading.

The length $n$ of the noise vector is one of the principal factors which decides the robustness of noise: the higher the value $n$, the lower the false positive of noise verification. But this size influences the quality of the rendered image: the lower value $n$, the higher fidelity of the rendered images. Consequently, the value $n$ is a trade-off between the robustness of the embedded noise and the fidelity of the rendered image, it is empirically chosen to be about 8 to 15.

Figure 4:

A random vector of 5 atomic watermarks

Figure 4 shows a noise as a vector of $5$ atomic watermarks and the Fourier transforms showing the corresponding frequency characteristics. The vital frequencies of energy are clearly shown in the spectrums. For illustration purpose, we take $X_i = Y_i \sim \mathcal{N}\left(25,5\right) \ \left(1 \leq i \leq 5\right)$, and $M = N = 512$.