2.4.3. Adversarial loss

This study employs Patch-GANs to distinguish between the real images or the images translated from the latent space in the target domain.

$$\begin{array}{lcl} \mathcal{L}\_{\text{GAN}}^{x\_{i}} = \mathbb{E}\_{\begin{subarray}{c} \mathbf{x}\_{i}, r \sim p(\mathbf{x}\_{i}, r) \end{subarray}} \left[ \log \left( 1 - D\_{i} (G\_{i} (E\_{i} (\mathbf{x}\_{i}, r))) \right) \right] \\ + \quad \mathbb{E}\_{\begin{subarray}{c} \mathbf{x}\_{i}, r \sim p(\mathbf{x}\_{i}, r) \end{subarray}} \left[ \log \left( D\_{i} (\mathbf{x}\_{i}) \right) \right] \end{array} \tag{6}$$

$$L\_{\text{GAN}}^{x\_{\bar{j}\rightarrow i}} = \mathbb{E}\_{\mathbf{x}\_{i}, \mathbf{x}\_{\bar{j}}, r \sim p(\mathbf{x}\_{i}, \mathbf{x}\_{\bar{j}}, r)} \left[ \log \left( 1 - D\_{i} \left( G\_{i} \left( E\_{\bar{j}} \left( \mathbf{x}\_{\bar{j}}, r \right) \right) \right) \right) \right] \tag{7}$$

$$L\_{\rm GAN} = \lambda\_{\rm cross} \sum\_{i=1}^{N} \sum\_{j=i+1}^{N} L\_{\rmGAN}^{x\_{j \to i}} + \lambda\_{\rm within} \sum\_{i=1}^{N} L\_{\rmGAN}^{x\_{i}} \tag{8}$$

where *Lxi* GAN is the within-domain GAN loss of the images sampled from domain *i*; *L xj*→*<sup>i</sup>* GAN is the cross-domain GAN loss that depicts the GAN loss of spectral image translation from domain *j* to domain *i*, and *LGAN* is the sum of multi-domain GAN loss.
