  The study of stellar surfaces can reveal information about the chemical
composition, interior structure, and magnetic properties of stars. It is also
critical to the detection and characterization of extrasolar planets, in
particular those targeted in extreme precision radial velocity (EPRV) searches,
which must contend with stellar variability that is often orders of magnitude
stronger than the planetary signal. One of the most successful methods to map
the surfaces of stars is Doppler imaging, in which the presence of
inhomogeneities is inferred from subtle line shape changes in high resolution
stellar spectra. In this paper, we present a novel, efficient, and closed-form
solution to the problem of Doppler imaging of stellar surfaces. Our model
explicitly allows for incomplete knowledge of the local (rest frame) stellar
spectrum, allowing one to learn differences from spectral templates while
simultaneously mapping the stellar surface. It therefore works on blended
lines, regions of the spectrum where line formation mechanisms are not well
understood, or stars whose spots have intrinsically different spectra from the
rest of the photosphere. We implement the model within the open source starry
framework, making it fast, differentiable, and easy to use in both optimization
and posterior inference settings. As a proof-of-concept, we use our model to
infer the surface map of the brown dwarf WISE 1049-5319B, finding close
agreement with the solution of Crossfield et al. (2014). We also discuss
Doppler imaging in the context of EPRV studies and describe an interpretable
spectral-temporal Gaussian process for stellar spectral variability that we
expect will be important for EPRV exoplanet searches.
