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Compressed sensing (CS) theory realizes sparse signal sampling at a sub-Nyquist frequency to reduce the high computational cost of signal processing systems. As one of the main signal processing modules of the CS system, the signal reconstruction algorithm determines the signal processing capability. Although orthogonal matching pursuit (OMP) is conventionally selected in hardware implementations for its low complexity and high regularity, its low recovery performance is still a problem. In this paper, based on the projection-based atom selection orthogonal matching pursuit (POMP) algorithm, a threshold projection orthogonal matching pursuit (TPOMP) algorithm is proposed to improve the reconstruction accuracy and reduce the computational complexity. TPOMP algorithm expands the final support set dynamically and in parallel by setting threshold standards. The reconstruction simulation results show that the TPOMP algorithm exhibits a higher reconstruction success rate and higher reconstruction efficiency than the original POMP. The proposed reconstruction processor architecture is designed and implemented using Virtex UltraScale+ HBM VCU 128 FPGA under the configuration of $M=256$ , $N=1024$ , and $K=36$ . The reconstruction results show that the proposed processor achieves a higher reconstruction signal-to-noise ratio (RSNR) of 37.69 dB with 18bits data width. The processor can run at a clock frequency of 224.32 MHz and the reconstruction time is $149.6~\mu \text{s}$ .