The mature concept of compressed sensing (CS) is being transferred to the application level as a means to save the physical resources spent in the analog-to-digital interface of challenging signal and image acquisition tasks, i.e., when the underlying sensing process requires a critical amount of time, power, sensor area and cost. For most structured signals, this method amounts to applying a dimensionality-reducing random matrix followed by an accurate, yet computationally expensive recovery algorithm that is capable of producing full-resolution signal recoveries from such low-dimensional measurements.