Channeled linear imaging polarimeters measure the two-dimensional distribution of the linear Stokes parameters. A key aspect of this technique is to accurately reconstruct the Stokes parameters from a snapshot, modulated measurement of the channeled linear imaging polarimeter. The state-of-the-art reconstruction takes the Fourier transform of the measurement to separate the Stokes parameters into channels. While straightforward, this approach is sensitive to channel cross-talk and imposes bandwidth limitations that cut off high frequency details. To overcome these drawbacks, we present a reconstruction method called compressed channeled linear imaging polarimetry. In this framework, reconstruction in channeled linear imaging polarimetry is an underdetermined problem, where we measure N pixels and recover 3N Stokes parameters. We formulate an optimization problem by creating a mathematical model of the channeled linear imaging polarimeter with inspiration from compressed sensing. Through simulations, we show that our approach mitigates artifacts seen in Fourier reconstruction, including image blurring and degradation and ringing artifacts caused by windowing and channel cross-talk. By demonstrating more accurate reconstructions, we push performance to the native resolution of the sensor, allowing more information to be recovered from a single measurement of a channeled linear imaging polarimeter.