## 3 Proposed Work
### Interoperability Coefficient to Connect the Virtual and Real Environments  
We introduced the _interoperability coefficient_, which consists of determining a minimum number of real samples to connect the virtual and real domains using the GAN characteristics. This coefficient is composed of a correlation factor and the entropy generated by a GAN. One of the tools to assess the correlation between two images is the HOG [54]. This algorithm allows measuring the comparison of the authentic environment and its virtual representation. This tool obtains a characteristic vector for each of the samples and offers a coefficient that indicates the similarity level, whose hyperparameters are: orientation equal to 8, pixels per cell equal to 32 \(\times\) 32, and cells per block equal to 4 \(\times\) 4. For example, Figure 4 shows a real sample and its virtual representation with two different detail levels. We observed that the first variation has essential lighting, and the second has a more significant number of directional lighting sources and materials that give more realism to the virtual environment. Table 1 shows correlation measurements between 30 real-world samples and their virtual representation with two different detail levels. Since lights increase the detail level, the correlation coefficient of more detailed samples (lights and materials) is higher than essential light source samples. As the correlation coefficient between virtual samples created with video game engines and real examples is not high enough, the representation of the real world is inadequate. Figure 4: Real-world and virtual samples with different levels of details. (**a**) Samples in different domains, (**b**) the gradient generated by the HOG algorithm, and (**c**) the vector generated by the HOG algorithm. Likewise, the other element that composes this coefficient is the joint entropy [49] by the virtual representation of the real image and the predicted fake image described in Equation (8). The interoperability coefficient is composed of the HOG determined by the detail of the virtual representation multiplied by the entropy of a virtual sample with its generated samples. Let \(N_{real}\) be the number of real samples, \(x_{r}\) the real image, and \(x_{v}\) a virtual image to calculate the average HOG relation between both samples. Let \(N_{GAN}\) be the number of samples gendered by the GAN, \(y\) the real sample's virtual representation, \(y^{\prime}\) a fake sample, \(P(_{YY^{\prime}})\) the probability between a real and a fake sample, \(H(y)\) the entropy of the real sample's virtual representation, and \(H(y^{\prime})\) the fake sample's entropy to determine the average joint entropy for each step and the sum of the entropy of both samples. \[C_{interoperability}=\frac{\sum_{k}^{N_{real}}HOG(x_{{}_{R}},x_{{}_{D}})}{N_{real} }\cdot\left(1-\frac{\sum_{i}^{N_{GAN}}\frac{-\sum_{y_{i}}\sum_{j^{\prime}_{i}} P_{y^{\prime}_{i}y^{\prime}_{i}}(y_{i},y^{\prime}_{i})logP_{y^{\prime}_{i}y^{ \prime}_{i}}(y_{i},y^{\prime}_{i})}{H(y_{i})+H(y^{\prime}_{i})}}{N_{GAN}}\right) \tag{8}\]  
Thus, Table 2 describes the interoperability coefficient to define the number of minimum samples used to build a virtual representation. In this case, the HOG correlation is 0.5490, and the interoperability coefficient is 0.5047 in 43 real-world samples. For this reason, we recommend taking the number of samples when the interoperability coefficient is greater than 0.50. Hence, the details in the virtual representation are less than the authentic sample. It is essential to consider that the virtual representation must have enough information to allow deep learning [55] to use textures. Furthermore, the results showed that when the number of virtual representation samples increased their details in lights and materials, the interoperability coefficient must increase and the number of samples can be less. On the contrary, the joint entropy was low in the case that the dispersion was similar between the GAN architecture and virtual representation samples.