*4.2. The Selection of the Size and Number of LSFRs*

The size and number of LSFRs determine the robustness of the proposed algorithm. Besides, the size of the constructed LSFRs determines the number of them. According to Equation (11), the size and number of constructed LSFRs in our experiment are determined by *k*1.

The 60 images from the database [52] are resized to 1024 × 1024 as experiment images here. We statistically analyzed the average number of constructed LSFRs with different *k*1. In theory, the larger and the greater the number of LSFRs, the better the robustness of the algorithm. Therefore, we also count the number of constructed LSFRs with side lengths of 240–300 and the number of constructed LSFRs with side lengths greater than 300, as shown in Table 1. When *k*<sup>1</sup> is set to 6 and 6.5, the most LSFRs with side lengths greater than 300 can be constructed, and the total number can also satisfy the requirements. Therefore, *k*<sup>1</sup> is set to 6.5 in our experiment.


**Table 1.** Average number of constructed LSFRs with a different *k*1.

#### *4.3. The Selection of the Threshold for Message Extraction*

According to Equation (17), *k*<sup>2</sup> determines the threshold for message extraction, which will affect the success rate and validity of watermark information extraction. We performed a statistical analysis of the extraction results of the 29 LSFRs constructed from the eight host images with and without watermarks to select the most appropriate threshold. The experiment was set at a shooting angle of 0, 15, and 30 degrees and a shooting distance from 40 to 110 cm at intervals of 10 cm. Therefore, each LSFR was captured 24 times with different shooting conditions.

Based on the extraction method in Section 3.3.3, a total of 696 results of watermarked LSFRs and 648 results of unwatermarked LSFRs were obtained. The average erroneous bits with a different *k*<sup>2</sup> is shown in Figure 12a. The extraction results of watermarked LSFRs achieve the minimum erroneous bits when *k*<sup>2</sup> is set to 1. The distributions of erroneous bits with *k*<sup>2</sup> = 1 are shown in Figure 12b. The average of detected erroneous bits of unwatermarked LSFRs is around nineteen which is independent of *k*2.

Therefore, *k*<sup>2</sup> is set to 1 in our experiment.

**Figure 12.** Erroneous bits corresponding to different message extraction thresholds.

#### *4.4. The Selection of the Threshold for Watermark Detection*

The selection of the threshold *T* determines the false-positive rate and the true-positive rate. *T* needs to be set low enough to ensure that the watermark can be detected from watermarked LSFRs and high enough to ensure that the watermark cannot be detected from unwatermarked LSFRs.

Messages extracted from an unwatermarked image can be considered as independent random variables [43]. Therefore, the probability that a single bit match is 0.5. The relationship between the false-positive rate of single detection *Pf* and the threshold *T* is:

$$P\_f = \sum\_{i=l-T}^{l} \left( 0.5 \right)^{l} \cdot \left( \frac{l!}{i!(l-i)!} \right) \tag{19}$$

As mentioned in Section 3.3.3, each LSFR will be iteratively detected at different radii and angles. The maximum number of iterations is 231 times. Suppose we complete all iterative detection, the false-positive rate of the detection of one LSFR *P*- *<sup>f</sup>* is:

$$P'\_f = 1 - \left(1 - P\_f\right)^{231} \tag{20}$$

The false-positive rate curve with different thresholds is shown in Figure 13a. In order to choose an appropriate threshold, we further analyzed the influence of different secret keys and different host images on the positive detection rate. The eight host images were all embedded with other three different keys: *K*2, *K*3, *K*4. Each watermarked image was captured 24 times with different shooting conditions. The experimental setting is the same as in Section 4.3.

Based on the 768 detection results of watermarked images with four different keys, eight different host images, and 24 different shooting conditions, we can calculate the true-positive rate with different thresholds. The true-positive rate curves between different keys and different host images are shown in Figure 13b,c. The true-positive rate can be seen to be stable for different embedding messages. However, different images have different variations during screen-cam, so the true-positive rate is also different when *T* is below 10.

According to the result, we set threshold *T* to 8, which means when the number of erroneous bits is below 8, the detection is successful. According to Formula 20, the false-positive rate of the detection of one LSFR is 8.86 × 10<sup>−</sup>8. The true-positive rate with *K*1, *K*2, *K*<sup>3</sup> and *K*<sup>4</sup> is 98.44%, 96.88%, 97.40%, and 96.35%, respectively. Furthermore, the true-positive rate of the eight host images is 100%, 92.71%, 97.92%, 100%, 94.79%, 100%, 93.75%, and 98.96%, respectively.

**Figure 13.** Influence of different thresholds on (**a**) false-positive rate of one LSFR and on true-positive rate between different (**b**) keys and (**c**) images.
