  Signal retrieval from a series of indirect measurements is a common task in
many imaging, metrology and characterization platforms in science and
engineering. Because most of the indirect measurement processes are
well-described by physical models, signal retrieval can be solved with an
iterative optimization that enforces measurement consistency and prior
knowledge on the signal. These iterative processes are time-consuming and only
accommodate a linear measurement process and convex signal constraints.
Recently, neural networks have been widely adopted to supersede iterative
signal retrieval methods by approximating the inverse mapping of the
measurement model. However, networks with deterministic processes have failed
to distinguish signal ambiguities in an ill-posed measurement system, and
retrieved signals often lack consistency with the measurement. In this work we
introduce a variational generative model to capture the distribution of all
possible signals, given a particular measurement. By exploiting the known
measurement model in the variational generative framework, our signal retrieval
process resolves the ambiguity in the forward process, and learns to retrieve
signals that satisfy the measurement with high fidelity in a variety of linear
and nonlinear ill-posed systems, including ultrafast pulse retrieval, coded
aperture compressive video sensing and image retrieval from Fresnel hologram.
