In contrast with imaging using position-resolving cameras, single-pixel imaging uses a bucket detector along with spatially structured illumination for image recovery. This emerging imaging technique is a promising candidate for a broad range of applications due to high signal-to-noise ratio (SNR) and sensitivity, and applicability in a wide range of frequency bands. Here, inspired by single-pixel imaging in the spatial domain, we demonstrate a temporal single-pixel imaging (TSPI) system that covers frequency bands including both terahertz (THz) and near-infrared (NIR) region. By implementing a programmable temporal fan-out (TFO) gate based on a digital micromirror device (DMD), we can deterministically prepare temporally structured pulses with a temporal sampling size down to 16.00$pm$0.01 fs. By inheriting the advantages in detection efficiency and sensitivity from spatial single-pixel imaging, TSPI enables the compressive recovery of a 5 fJ THz pulse and two NIR pulses with over 97$%$ fidelity. We demonstrate that the TSPI is robust against temporal distortions in the probe pulse train as well. As a direct application, we apply TSPI to machine-learning-aided THz spectroscopy and demonstrate a high sample identification accuracy (97.5$%$) even under low SNRs (SNR $sim$ 10).
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