  NASA's Kepler, K2 and TESS missions employ Simple Aperture Photometry (SAP)
to derive time-series photometry, where an aperture is estimated for each star,
and pixels containing each star are summed to create a single light curve. This
method is simple, but in crowded fields the derived time-series can be highly
contaminated. The alternate method of fitting a Point Spread Function (PSF) to
the data is able to account for crowding, but is computationally expensive. In
this paper, we present a new approach to extracting photometry from these
time-series missions, which fits the PSF directly, but makes simplifying
assumptions in order to greatly reduce the computation expense. Our method
fixes the scene of the field in each image, estimates the PSF shape of the
instrument with a linear model, and allows only source flux and position to
vary. We demonstrate that our method is able to separate the photometry from
blended targets in the Kepler dataset that are separated by less than a pixel.
Our method is fast to compute, and fully accounts for uncertainties from
degeneracies due to crowded fields. We name the method described in this work
Linearized Field Deblending (LFD). We demonstrate our method on the false
positive Kepler target \koi. We are able to separate the photometry of the two
sources in the data, and demonstrate the contaminating transiting signal is
consistent with a small, sub-stellar companion with a radius of $2.67R_{jup}$
($0.27R_{sol}$). Our method is equally applicable to extracting photometry from
NASA's TESS mission.
