Rapid reconstruction of depth images from sparsely sampled data is important for many applications in machine perception, including robot or vehicle assistance or autonomy. Approximate computing techniques have been widely adopted to reduce resource consumption and increase efficiency in energy and resource constrained systems, especially targeted at FPGA and solid state implementation. Whereas previous work has focused on approximate, but static, representation of data in LiDAR systems, in this paper we show how the flexibility of an arbitrary precision accelerator with fine-grain tuning allows a better trade-off between algorithmic performance and implementation efficiency. A mixed precision framework of \(\ell _1\) solvers is presented, with compact ADMM and PGD, for the lasso problem, enabling compressive depth reconstruction by varying the precision scaling in single bit granularity during the iterative optimization process. Implementing mixed precision \(\ell _1\) solvers on an FPGA with a pipelined architecture for depth image reconstruction across various sensing scenarios, over 74% savings in hardware resources and 60% in power are achieved with only minor reductions in reconstructed depth image quality when compared to single float precision, while over 10% saving in hardware resources and power is achieved compared to relative consistently reduced precision solutions.