Photon-counting statistics-based support vector machine with multi-mode photon illumination for quantum imaging

We propose a photon-counting-statistics-based imaging process for quantum imaging where background photon noise can be distinguished and eliminated by photon mode estimation from the multi-mode Bose–Einstein distribution. Photon-counting statistics show multi-mode behavior in a practical, low-cost single-photon-level quantum imaging system with a short coherence time and a long measurement time interval. Different mode numbers in photon-counting probability distributions from single-photon illumination and background photon noise can be classified by a machine learning technique such as a support vector machine (SVM). The proposed photon-counting statistics-based support vector machine (PSSVM) learns the difference in the photon-counting distribution of each pixel to distinguish between photons from the source and the background photon noise to improve the image quality. We demonstrated quantum imaging of a binary-image object with photon illumination from a spontaneous parametric down-conversion (SPDC) source. The experiment results show that the PSSVM applied quantum image improves a peak signal-to-noise ratio (PSNR) gain of 2.89dB and a structural similarity index measure (SSIM) gain of 27.7% compared to the conventional direct single-photon imaging.

This study shows that even if photon-counting statistics of both signal photon and background photon noise follows multi-mode Bose-Einstein distribution (mmBE), they can be distinguished if t he multi-mode values are different. However, there are only experimental attempts to distinguish those statistics for m=1152 and m=294, and the reference range for discrimination was not revealed. Supplementary Fig. 1 visualizes the Jensen-Shannon distance (JSD) according to the multi-mode values for the mmBE distributions through simulation and predicts the range of multi-mode values in which our research can operate correctly through the experimental results.
In order to check how two mmBE distributions are different, we calculated the distribution distance between those mmBE distributions using JSD 1 : where M = 1 2 (P + Q) and for two probability distributions P(x) and Q(x). We evaluate the JSD map for multi-mode values between 0 and 1500 for both P and Q. In our experiment, multi-mode values are m = 1152 and m = 294 for signal photons and background noise photons, respectively. Within this condition the JSD shows 3.6 × 10 −3 , as the red line of Supplementary Fig. 1. This suggests that in the limit of low JSD, our proposed method works properly.

Supplementary Note 2: Various experiment environment for PSSVM
The quantum imaging in this paper used an incandescent lamp to introduce a background photon noise, located at 22 cm away from the telescope. The PSSVM was trained to construct image against this photon noise. However, it is desired to confirm that it works for other background photon noises with various multi-mode values. We conducted an experiment to investigate how the PSSVM error changes by different multi-mode values corresponding to the different positions of the incandescent lamp. For this test, the location of the lamp was moved to {10 cm, 15 cm, 30 cm, 2 m} after training at 22 cm. Data were collected for 'Reflected' and 'Blocked' cases. In order to make the effect of the lamp position generalized, we estimated the JSDs of the photon statistics at these lamp locations with respect to that of the lamp location at 22 cm. The error rates of PSSVM with respect to the JSD are shown in Supplementary Fig. 2, for cases of 'Reflected' and 'Blocked'. As d jsd defined in Supplementary Fig. 2 increases, the error rates of PSSVM e tend to decrease. Supplementary Fig. 3 shows the error analyses for the cases of 'Reflected' and 'Blocked'. In the figure, the 'Reflected' case shows that the error decreases sharply as d jsd increases, and the coincidence data is more sensitive than the signal data. In the case of 'Blocked', d jsd shows little difference in error, but the signal data has much larger error than coincidence data. This leads to a significant improvement in quantum image quality when coincidence photon imaging is used rather than single photon imaging. A mathematical analysis of these phenomena is anticipated to show the possibility of expanding the engineering application of photon statistics as a future work.
the telescope.
(7) Suppl_testdata_200_Blocked.csv : Test data for 'Blocked' case where the lamp is located at 2 m away from the telescope.
(8) Suppl_testdata_200_Reflected.csv : Test data for 'Reflected' case where the lamp is located at 2 m away from the telescope.
(9) Suppl_traindata_22_Blocked.csv : Training data for 'Blocked' case where the lamp is located at 22 cm away from the telescope.
(10) Suppl_traindata_22_Reflected.csv : Training data for 'Reflected' case where the lamp is located at 22 cm away from the telescope.