  Image-to-image translation is a subset of computer vision and pattern
recognition problems where our goal is to learn a mapping between input images
of domain $\mathbf{X}_1$ and output images of domain $\mathbf{X}_2$. Current
methods use neural networks with an encoder-decoder structure to learn a
mapping $G:\mathbf{X}_1 \to\mathbf{X}_2$ such that the distribution of images
from $\mathbf{X}_2$ and $G(\mathbf{X}_1)$ are identical, where $G(\mathbf{X}_1)
= d_G (f_G (\mathbf{X}_1))$ and $f_G (\cdot)$ is referred as the encoder and
$d_G(\cdot)$ is referred to as the decoder. Currently, such methods which also
compute an inverse mapping $F:\mathbf{X}_2 \to \mathbf{X}_1$ use a separate
encoder-decoder pair $d_F (f_F (\mathbf{X}_2))$ or at least a separate decoder
$d_F (\cdot)$ to do so. Here we introduce a method to perform cross domain
image-to-image translation across multiple domains using a single
encoder-decoder architecture. We use an auto-encoder network which given an
input image $\mathbf{X}_1$, first computes a latent domain encoding $Z_d = f_d
(\mathbf{X}_1)$ and a latent content encoding $Z_c = f_c (\mathbf{X}_1)$, where
the domain encoding $Z_d$ and content encoding $Z_c$ are independent. And then
a decoder network $g(Z_d,Z_c)$ creates a reconstruction of the original image
$\mathbf{\widehat{X}}_1=g(Z_d,Z_c )\approx \mathbf{X}_1$. Ideally, the domain
encoding $Z_d$ contains no information regarding the content of the image and
the content encoding $Z_c$ contains no information regarding the domain of the
image. We use this property of the encodings to find the mapping across domains
$G: X\to Y$ by simply changing the domain encoding $Z_d$ of the decoder's
input. $G(\mathbf{X}_1 )=d(f_d (\mathbf{x}_2^i ),f_c (\mathbf{X}_1))$ where
$\mathbf{x}_2^i$ is the $i^{th}$ observation of $\mathbf{X}_2$.
