  Significance: Compressed sensing (CS) uses special measurement designs
combined with powerful mathematical algorithms to reduce the amount of data to
be collected while maintaining image quality. This is relevant to almost any
imaging modality, and in this paper we focus on CS in photoacoustic projection
imaging (PAPI) with integrating line detectors (ILDs).
  Aim: Our previous research involved rather general CS measurements, where
each ILD can contribute to any measurement. In the real world, however, the
design of CS measurements is subject to practical constraints. In this
research, we aim at a CS-PAPI system where each measurement involves only a
subset of ILDs, and which can be implemented in a cost-effective manner.
  Approach: We extend the existing PAPI with a self-developed CS unit. The
system provides structured CS matrices for which the existing recovery theory
cannot be applied directly. A random search strategy is applied to select the
CS measurement matrix within this class for which we obtain exact sparse
recovery.
  Results: We implement a CS PAPI system for a compression factor of $4:3$,
where specific measurements are made on separate groups of 16 ILDs. We
algorithmically design optimal CS measurements that have proven sparse CS
capabilities. Numerical experiments are used to support our results.
  Conclusions: CS with proven sparse recovery capabilities can be integrated
into PAPI, and numerical results support this setup. Future work will focus on
applying it to experimental data and utilizing data-driven approaches to
enhance the compression factor and generalize the signal class.
