2.4.2. Latent Matching loss

Given two multi-spectral images sampled from the same data distribution, the latent code should be matched after encoding. In previous work, auto-encoders and GANs use KLD loss, and adversarial loss [16,33] or implicitly constrain [29] the latent domain distribution. The present model uses the calculation of L1 Loss across domains to strongly constrain different domains to encode in the same space.

$$L\_{LM}^{\mathbf{x}\_{i},\mathbf{x}\_{j}} = \mathbb{E}\_{\mathbf{x}\_{i},\mathbf{x}\_{j},\mathbf{r} \sim p(\mathbf{x}\_{i},\mathbf{x}\_{j},\mathbf{r})} \left[ \left\| \left| E\_{i}(\mathbf{x}\_{i},\mathbf{r}) - E\_{j}(\mathbf{x}\_{j},\mathbf{r}) \right\|\_{1} \right] \tag{4}$$

$$L\_{\text{Match}} = \sum\_{i=1}^{N} \sum\_{j=i+1}^{N} L\_{LM}^{x\_i, x\_j} \tag{5}$$

where *Lxi*,*xj LM* is the latent matching loss that depicts the L1 loss between the latent of the *i*-domain and *j*-domain images under the joint probability distribution.
