  This paper introduces a novel generative encoder (GE) model for generative
imaging and image processing with applications in compressed sensing and
imaging, image compression, denoising, inpainting, deblurring, and
super-resolution. The GE model consists of a pre-training phase and a solving
phase. In the pre-training phase, we separately train two deep neural networks:
a generative adversarial network (GAN) with a generator $\G$ that captures the
data distribution of a given image set, and an auto-encoder (AE) network with
an encoder $\EN$ that compresses images following the estimated distribution by
GAN. In the solving phase, given a noisy image $x=\mathcal{P}(x^*)$, where
$x^*$ is the target unknown image, $\mathcal{P}$ is an operator adding an
addictive, or multiplicative, or convolutional noise, or equivalently given
such an image $x$ in the compressed domain, i.e., given $m=\EN(x)$, we solve
the optimization problem
  \[
  z^*=\underset{z}{\mathrm{argmin}} \|\EN(\G(z))-m\|_2^2+\lambda\|z\|_2^2
  \] to recover the image $x^*$ in a generative way via
$\hat{x}:=\G(z^*)\approx x^*$, where $\lambda>0$ is a hyperparameter. The GE
model unifies the generative capacity of GANs and the stability of AEs in an
optimization framework above instead of stacking GANs and AEs into a single
network or combining their loss functions into one as in existing literature.
Numerical experiments show that the proposed model outperforms several
state-of-the-art algorithms.
