The digitization of our most common appliances has led to a literal data deluge, some- times referred to as Big Data. The ever increasing volume of data we generate, coupled with our desire to exploit it ever faster, forces us to come up with innovative data pro- cessing techniques. Interestingly, the information we often look for has a very specific structure that distinguishes it from pure clutter. In this thesis, we explore the use of structured representations to propose new sensing techniques that severely limit the data throughput necessary to recover meaningful information. In particular, we exploit the intrinsic low-dimensionality of light field videos using tensor low-rank and sparse constraints to recover light field views from a single coded image per video frame. As opposed to conventional methods, our scheme neither alters the spatial resolution for angular resolution nor requires computationally extensive learning stage but rather depends on the intrinsic structures of light fields. In the second part of this thesis, we propose a novel algorithm to estimate depth from light fields. This method is based on representation of each patch in a light field view as a linear combination of patches from other views for a set of depth hypotheses. The structure in this representation is deployed to estimate accurate depth values. Finally, we introduce a low-power multi-channel cortical signal acquisition based on compressive sampling theory as an alternative to Nyquist-Shannon sampling theorem. Our scheme exploits the strong correlations between cortical signals to recover neural signals from a few compressive measurements.