This paper concerns the reconstruction of a temporally-varying scene from a video sequence of noisy linear projections. Assuming that each video frame is sparse or compressible in some basis, this inverse problem can be formulated as an ℓ₂-ℓ₁ minimization problem, which can be solved efficiently using gradient projection. Since the signal of interest corresponds to nonnegative pixel intensities, additional nonnegativity constraints are included in the minimization problem, rendering the optimization problem more difficult to solve but with a greater potential for more accurate reconstructions. In this paper, we propose a method for reconstructing a video sequence that incorporates nonnegativity constraints and exploits inter-frame correlations to improve upon the naive approach of solving each frame independently. We present numerical experiments to demonstrate the effectiveness of this approach.