  We present a grism extraction package (LINEAR) designed to reconstruct
one-dimensional spectra from a collection of slitless spectroscopic images,
ideally taken at a variety of orientations, dispersion directions, and/or
dither positions. Our approach is to enumerate every transformation between all
direct image positions (ie. a potential source) and the collection of grism
images at all relevant wavelengths. This leads to solving a large, sparse
system of linear equations, which we invert using the standard LSQR algorithm.
We implement a number of color and geometric corrections (such as flat field,
pixel-area map, source morphology, and spectral bandwidth), but assume many
effects have been calibrated out (such as basic reductions, background
subtraction, and astrometric refinement). We demonstrate the power of our
approach with several Monte Carlo simulations and the analysis of archival
data. The simulations include astrometric and photometric uncertainties,
sky-background estimation, and signal-to-noise calculations. The data are G141
observations obtained with the Wide-Field Camera 3 of the Hubble Ultra-Deep
Field, and show the power of our formalism by improving the spectral resolution
without sacrificing the signal-to-noise (a tradeoff that is often made by
current approaches). Additionally, our approach naturally accounts for source
contamination, which is only handled heuristically by present softwares. We
conclude with a discussion of various observations where our approach will
provide much improved spectral one-dimensional spectra, such as crowded fields
(star or galaxy clusters), spatially resolved spectroscopy, or surveys with
strict completeness requirements. At present our software is heavily geared for
Wide-Field Camera 3 IR, however we plan extend the codebase for additional
instruments.
