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Abstract
Optical sensing with single photon counting avalanche diode detectors has become a versatile approach for ranging and low light level imaging. In this paper, we compare time correlated and uncorrelated imaging of single photon events using an InGaAs single-photon-counting-avalanche-photo-diode (SPAD) sensor with a 32 × 32 focal plane array detector. We compare ranging, imaging and photon flux measurement capabilities at shortwave infrared wavelengths and determine the minimum number of photon event measurements to perform reliable scene reconstruction. With time-correlated-single-photon-counting (TCSPC), we obtained range images with centimeter resolution and determined the relative intensity. Using uncorrelated single photon counting (USPC), we demonstrated photon flux estimation with a high dynamic range from $\hat {\phi } = 2\times 10^4$ to 1.3 × 107 counts per second. Finally, we demonstrate imaging, ranging and photon flux measurements of a moving target from a few samples with a frame rate of 50 kHz.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Single photon counting avalanche diodes (SPAD) have become a versatile optical sensor with high sensitivity down to single photon detection [1], high timing precision [2] up to a few picoseconds and high sampling rates up to a few MHz. While CMOS SPAD detectors are based on Silicon and are used for sensing applications in the visible to near infrared spectral range [1], InGaAs SPAD are applied to sense light at shorwave infrared (SWIR) wavelengths [3–6].
In the SWIR regime, single detectors [7] or small size focal plane arrays (e.g. $4\times 4$) [8,9] are used in scanning light detection and ranging (LiDAR) systems for three dimensional imaging [10]. The advent of InGaAs SPAD focal plane array enable imaging and 3D mapping without scanners [11–15]. For instance, single photon imaging is applied for active computational imaging [16–18], the observation of light in flight [19–21] and non-line-of-sight sensing [22–26].
Further, active SPAD imaging was evaluated for long range 3D imaging [11,27,28] and imaging through obscurants such as camouflage nettings [29], foliage [28], fog and smoke [30], as well as through water [31]. Nevertheless, due to the binary character of photon counting and the impact of noise (dark current and ambient or scattered light), the reconstruction of intensity [32], and range [33,34] from photon counting data remains challenging.
As an alternative method to determine light intensity from single photon events, uncorrelated single photon counting (USPC) determine the photon flux of uncorrelated light by estimating the mean time between two photon events. Recently, USPC measurements were presented using a free running asynchronous single SPAD scanning the scene [35] or an CMOS SPAD focal plane array [36] with long exposure times.
In this paper, we investigate the application of a non-gated InGaAs SPAD camera consisting of a focal plane array detector with $32\times 32$ pixel with synchronous readout for time correlated single photon counting (TCSPC) range and intensity imaging as well as USPC measurements of the photon flux. In contrast to previous publications, we concentrate on an as low as possible number of measurements to determine the limiting factors and to reduce the overall exposure time. Finally, we demonstrate imaging of a dynamic moving target with a frame rate of 50 kHz.
2. Single photon counting
Beside photon multipliers or superconducting detectors, single photons can be detected by all-semiconductor photon counting avalanche diodes (SPAD). These devices operated in a meta-stable Geiger-mode at reverse bias voltages beyond the breakthrough voltage [2,11]. In these devices, a single photon can excite charge carriers through absorption. These carries are accelerated in a strong electric field causing a self sustaining avalanche process and a multiplication by an exponential gain. The event of a single photon detection can be measured with very high timing precision on a pico-second scale by using a time to digital conversion unit (either integrated in the readout circuit or as a stand alone timing unit). Due to the binary nature of the signal, photon events can be read out noise-free. Beside high sensitivity and timing capabilities, SPAD sensors are limited by a significant dead time $\tau$ due to the fact that the SPAD has to be refreshed (quenched) to stop the avalanche effect [2]. Further, SPAD sensors have a certain photon detection efficiency (PDE) $\eta$ and are limited to a certain wavelength range due to the physical properties of the active semiconductor material (e.g. InGaAs for shortwave infrared (SWIR) wavelengths). Until now, SPAD array detectors are limited to a few thousand sensor due to the need of complex read out and driver circuits.
Further, SPAD sensors can be used in an asynchronous or a synchronous operation mode. A continuous detection of impinging photons is performed by asynchronous operation which is limited by the dead time ($\tau$), only. Every event is tagged with a time code which can be use to correlate these events to e.g. laser pulse emissions. As illustrated in Fig. 1(a), in a synchronous operation, every SPAD sensor is synchronized to measure in a pre-defined measurement time window. Detected photons trigger the time-to-digital converter unit with a certain timing resolution. The measurement time window consists of $n$ time bins $b_i$ with a width of $\Delta t_{\textrm {bin}}$ and, thus, the total width of $n\times \Delta t_{\textrm {bin}}$. In this operation mode, the time bins are directly correlated to the synchronization signal. Furthermore, SPAD sensors could be gated by fast switching of the applied reverse bias voltage.
Fig. 1. The synchronous SPAD array detector (a) measures the time-of-incident relative to a common trigger signal. SPAD imager can use (a) time correlated active illumination and (b) uncorrelated ambient light.
Due to the dead time $\tau$ and the quenching process, every SPAD can detect only a single photon within a measurement cycle. The photon event time is recorded as a count in the corresponding time bin $b_i$. Further, no photon event could take place as well as (especially in non-gated operation) the SPAD can be triggered prior the measurement windows. These cases are recorded in specific memory bins $b_{\textrm {no~event}}$ and $b_{\textrm {prior}}$, respectively. After several measurement cycles, the count rate per time bins can be summarized in a histogram depicting for each time bin $b_i$ the number of counts $N(b_i)$.
2.1 Time correlated single photon counting with active illumination
Time correlated single photon counting (TCSPC) uses an active illumination by a pulsed laser source synchronized to the SPAD detector, as illustrated in Fig. 1(b). This method allows a precise direct measurement of the photon time of flight. The time bin with the highest count rate $b_{\textrm {peak}}$ indicates the time of flight $t_{\textrm {ToF}}$ and, thus, the target range $z_{\textrm {target}}$, as defined in Eq. (1). Here, $c$ is the speed of light and $\Delta t_{\textrm {bin}}$ the bin width. In a strict sense, the factor $\frac {c}{2}$ is valid for mono-static sensing [20], only, as it is typically used in direct time-of-flight imaging (such as LiDAR).
An estimated localization of the peak position $\hat {b}_{\textrm {peak}}$ can be carried out by analysis of the local maximum, weighted mean or through other methods (e.g. correlation etc.). Here, we will use a simple Kalman filter [37,38] to estimated this value from a limited number of measurements.
2.2 Uncorrelated single photon counting with ambient light illumination
Uncorrelated operation of SPAD sensors is typically discussed to describe dark current count rate or the parasitic impact of ambient light. In recent papers, uncorrelated single photon counting (USPC) was discussed to estimate and to map the photon flux impinging the SPAD detector by determine the mean time of arrival using a singel Si-SPAD in asynchronous (free-running) operation [35] or a Si-SPAD focal plane array [36].
The firing of SPAD detectors are following Poisson statistics when illuminated by incoherent or uncorrelated light [8]. Thus, the number of photons $N(b_i)$ detected in time bin $b_i$ can be described by an exponential function (Eq. (3)) [8,36,39].
3. Experimental results
In our experiments, we used an InGaAs SPAD camera from Princeton Lightwave Inc.,USA, with $32\times 32$ sensor elements with a spectral sensitivity range from 920 nm to 1620 nm and an average photon detection probability at 1550 nm of 22%. The single SPAD sensors have a diameter of 25 $\mu$m and are separated by a sensor pitch of 100 $\mu$m. Further, the focal plane array is equipped with micro-lenses to obtain a fill factor of 75%. The SPAD sensors are cooled by thermo-electric cooling (Peltier TEC) and have an average dark current rate (DCR) of $< 20$ kHz or $<$40 Hz/$\mu$m2 . The sensor array can be triggered and read out with a frame rate of 50 kHz (max. 80 kHz). In each frame, for each SPAD a single event is time tagged. These photon events were measured in a time window of 2 $\mu$s with 8000 time bins and a bin width of 250 ps. The camera was equipped with a SWIR lens from Optec SpA, Italy, with a focal length $f$ of 16 mm and an aperture of $f/D = 1.7$.
For time correlated single photon counting, we illuminate the scene with an Erbium-doped pulsed fiber laser (EPFL) from Keopsys, France, emitting laser pulses of 500 ps at a repetition rate of 50 kHz and a laser wavelength of 1550 nm. The laser source was driven with a maximum pulse energy of 3 $\mu$J which corresponds to a mean optical power of 150 mW. Further, the divergence of the laser source was broadened by an optical diffuser to illuminate the complete field of view of the camera. An image of both, laser source (without diffuser) and camera, is shown in Fig. 2(a).
Fig. 2. Experimental setup with (a) SPAD camera with EPFL laser source and targets: (b) fan and (c) a board with gray scale and letters.
Further, ambient light illumination was generated by light bulbs achieved by laboratory security lights. The two light bulbs were at the laboratory ceiling 3 m above the observed scene, separated by 4 m (i.e. 2 m left and right of the scene). The illumination by the light bulbs can be assumed to be homogeneous with regard to the dimensions of the scenes (d = 20 cm). The ambient light was dimmed to different low illumination levels of 14 to 1 $\frac {lm}{m^2}$ measured with a light meter placed in the scene before measurments. Figure 3 shows the two emission spectra of the light bulb at both illumination levels. They can be fitted by black body radiation [42] (dotted lines). Later, these illumination parameters are referred as high and low ambient light levels, respectively.
Fig. 3. The emission spectra of the ambient light illumination source (light bulb) and fit by black body radiation curves.
As shown in Figs. 2(b) and 2(c), in our experiments, we used two types of targets: (b) a fan and (c) a gray scale board. The fan was used to have a three dimensional scenario with high contrast between the bright foreground (white fan) and the dark background (black stage curtain). The fan was used as a static target or was rotating free running at an uncontrolled speed. On the other hand, in (c), the board target consists of letters and diffuse reflection plates. These six patches were coated with Permaflect [43] of different reflectivity from $94\%$ (upper left), $80\%$, $50\%$, $18\%$, $10\%$ to $5\%$ (bottom right).
3.1 TCSPC range and intensity imaging
TCSPC was used to sample range and intensity information of a static non-rotating fan. In total, datasets of each 100,000 frames were recorded which corresponds to a total acquisition time of 2 seconds per measurement. In Fig. 4(a), the two TCSPC histograms illustrate measurements recorded by two individual SPAD at the fan surface in the foreground (dark line) and the dark stage curtain in the background (red line). While the foreground signal shows a single very significant primary peak (around $b_i = 30$) with a full width half maximum (FWHM) of 3 bins, the background signal has a significant (primary) peak (around $b_i = 65$, FWHM = 5 bins) and an additional low broadened secondary signal (around $b_i = 37$, FWHM = 11 bins). In these histograms, the data sets were not corrected for the pile-up effect. [41]
Fig. 4. TCSPC of a non-rotating fan: (a) signals from foreground (black) and background (red). Imaging of (b) intensity (relative number of counts) and (c) range.
In both cases, the primary peak corresponds to the distance between TCSPC system and the target or the background (fan at 110 cm and curtain at 240 cm). The low secondary signature corresponds to a broadened and time delayed signal from the foreground. Its origin can be interpreted as an optical cross talk of photons reflected from the foreground target and the optical table into the lens’ wide entrance pupil.
Further, Fig. 4 shows maps of (b) the relative intensity and (c) the range of the scene. Both maps were calculated from the TCSPC data set of 100.000 measurement cycles using a simple Kalman filter (KF). In both cases, prior events were neglected. The relative intensity (b) was calculated using Eq. (2) for each cycle. Therefore, each measurement updates the KF with a "1" or a "0" for a valid or non-valid detection, respectively. In an analogous way, KF was used to estimate the scene range. Here, only valid detection were taken into account and the single photon range was calculated from Eq. (1). In the case of range calculation, the non-detected counts ($N_{\textrm {no~events}}$) are ignored. In later analysis, the presented intensity and range maps will be referred as ground truth data (see Sec. 3.3).
3.2 USPC photon flux imaging
In the uncorrelated sampling with ambient light illumination, the SPAD does not measure the photon round trip time of light traveling to the target and back to the sensing system. Furthermore, the SPAD sensor is waiting for a photon from the ambient light to incidentally impinge that sensor. In contrast to intensity measurements where we count the number of photons measure within a certain exposure time, we determine the waiting time until a signal event occurs. Although we do not distinguish photon and dark current events, this time is, in a first approximation, reciprocal to the photon flux that hits the detector. Similar to intensity measurements, at a low photon flux close to the dark current rate, the assumption starts failing. At very low photon flux level, the dark current rate is dominant (this is the lower limit). Further, the time is measured relative to the internal clock of the synchronous read-out.
In Fig. 5 typical results for measurements with 100000 measurement cycles at (a) high and (b) low ambient light levels are presented as: (left) the distribution of counts related to prior events ($N_{\textrm {prior}}$), (middle) histograms for sensor elements sensing photon flux from bright ($R = 95\%$, black line) and dark ($R = 5\%$, red line) surfaces and (right) the distribution of no events ($N_{\textrm {no~event}}$).
Fig. 5. Uncorrelated single photon counting at (a) high and (b) low ambient light levels. From left to right: Distribution of prior-events, histogram of single pixel signal from bright and dark surface and distribution of untriggered counts.
Under both illumination conditions, the distribution of the prior events (left) are higher in the center of the SPAD array than at the borders. The observed number of prior counts ranges from 200 to 1800 counts at low and 4000 to 27000 counts at high ambient light level which corresponds to to 0.2-1.8 % and 4-27 % of all measurement cycles, respectively. The distribution reflects the sensor’s distribution of the dark current count rate (DCR) and photon detection efficiency (PDE) [11]. This count numbers increase with the ambient light level, significant structures related to the observed scenario can hardly be identified. Thus, we assume that these prior events are related to dark current and diffused carriers.
Although the sensor is facing the scene the whole time, the distributions of the prior events do not reflect the intensity information. Further, we want to measure the waiting time until an event takes place. Therefore, prior events can be neglected in our analysis. In fact, prior events prevent the SPAD from being restored for the next measurement cycle and thus affect our measurements by reducing the number of usable measurement cycles.
In the histograms (Fig. 5(a) and 5(b), middle), the count numbers show the expected exponential behavior (Eq. (3)). In both histograms, the count rates are fitted by an exponential function. For high ambient light level (a), we observe a medium count rate for the dark surface signal within the whole measurement window. In contrast to this, the bright surface signal begins with very high count numbers and decays rapidly even to numbers below the dark surface signal. At low ambient light level (b), for both signal types, a medium count rate can be observed with significant higher amount for the bright surface signal.
In contrast to the prior events, the distributions of no-event counts (Fig. 5(a) and 5(b), right) show a significant structure correlated to the scene geometry. These distributions reflect the inverse absorbed photon distributions and are therefore indirect measures of the photon flux. Indeed, this distribution of no events correlates with the definition of the TCSPC intensity in Eq. (2).
From the uncorrelated photon event time measurements, we can estimate the photon flux impinging the SPAD array using the previously defined photon flux estimator $\hat {\phi }$, Eq. (6). In each measurement cycle, the SPAD camera records a bin value $b_i$, as given in Eq. (7). For a prior event and a valid photon event the bin value ranges from 0 to 8000, if no event was measured, the bin value is 8004. Fur(ther, as in Eq. (8), we count every event within the measurement window as "1" (valid event) and prior or no events as "0".
Fig. 6. Distribution of the estimated photon flux (upper row: linear scale, bottom row: logarithmic scale) illustrating the high dynamic range capability (here $10^4 - 10^7$ cps): gray scale board (a) (b) and non-rotating fan (c) (d) at low and high ambient light levels.
The high gray scale variation enable to distinguish different gray values and the imaging of various details, like different reflection properties of the diffuse reflective plates of the letters ("F","I","S" and "L"). Further, the use of a logarithmic scale is closer to the perception capabilities of the human eye and, thus, helps the reader to interpret the gray scale images. This is especially valid in the presence of specular reflections e.g. at the plastic surface of the fan (Fig. 6(c) and 6(d)).
3.3 Single photon high-speed imaging with kHz frame rate
Major advantages of SPAD sensors are, beside high sensibility and precise timing capability, the high sampling rate to read out photon events and sampling coverage (e.g. quasi-contineous sampling in free-running SPAD [35]). For instance, our camera is capable to sample the photon event time with a sampling rate of 50 kHz. While a large number of samples can be used for a static scenario, motion blur vanishes the contours of moving objects and limits the number of usable samples in dynamic scenarios. Therefore, to enable imaging of fast moving objects at kHz frame rate, we have to trade image quality and effective exposure time.
We analyzed the aforementioned data sets by using various number of samples to determine their minimum amount to attain a sufficient sensing quality. In Fig. 7(a), we compare the results for the estimated TCSPC range and intensity as well as the estimated photon flux at high and low ambient light condition using n = 2 to 250 samples per pixel. For this representation, we scaled the images of each line to the previously used gray scales, see Figs. 4 and 6. It is obvious, that in all cases the perception quality increases with the number of measurements. While the TCSPC data result in sufficient reconstruction of range and intensity with only 25 samples, the quality of the photon flux estimation depend dramatically on the ambient light level. Here, only high light level result in good images with 25 samples. At low ambient light level, more than 100 measurements are needed.
Fig. 7. Imaging with different number of samples of (a) range, intensity, and photon flux, and (b) estimation of the minimum number of samples from structual similarity (SSIM) analysis.
A more reliable analysis of the sensing quality is presented in Fig. 7(b). Here, the calculated images are compared to ground truths (see Figs. 4(b), 4(c) and 6) using the structural similarity index metric (SSIM) [44]. SSIM values close to "1" represents high similarity of the image and the ground truth. Analogous to the aforementioned observations, we can state that the use of a fewer samples result in less similarity: The more samples used the better the similarity becomes. Again, we observe that SSIM values better than 0.5 are obtained with $n > 10$ for TCSPC (range and intensity) and USPC at high ambient light conditions. At the lower illumination level, $n$ has to be greater than 100 samples.
In an additional experiment, we observed the fan rotating at about 35 rps using TCSPC and USPC with 50 kHz samping rate. To preserve the high sampling rate during the analysis and enable image reconstruction at frame rates in the kHz range, we have to limit the number of used samples. Further, we reconstruct the image from a buffer memory containing $n_{\textrm {buffer}}$ sampling frames. Therefore, we defined a buffer matrix with a size of $n_{\textrm {buffer}} \times m_{\textrm {array~size}}$ which was used as a cyclic memory. Each sample $n$ ($m_{\textrm {array~size}}$ elements) was written to the buffer using cyclic overwriting (write to buffer position $p_n = \mod (n,n_{\textrm {buffer}})$). With this analysis strategy it is possible to preserve the original frame rate while performing analysis of a sufficient number of measurement frames.
In Fig. 8, some exemplary frames are shown for $n_{\textrm {buffer}} = 25$, $50$ and $100$. The TCSPC range and intensity images as well as the USPC estimation of the photon flux at high ambient light level are presented in original $32 \times 32$ resolution and as up-sampled images using a 2D cubic spline interpolation [45] algorithm for better display (see Visualization1.avi and Visualization2.avi). Obviously, we observe that with increasing number of analyzed measurement cycles the noise level in the images decreases ($\sigma \propto 1/\sqrt {n_{\textrm {buffer}}}$). But, at the same time, the motion blur increases due to sampling of different object instances. This effect is more pronounced at the rotor tip, where larger displacement can be observed, and becomes dominant at a buffer of $n_{\textrm {buffer}} = 100$ samples. At a buffer size $n_{\textrm {buffer}} < 25$ samples (non presented), the noise would be dominant. Thus, for the investigated moving scenario, a good trade-off between noise and motion blur is attained at a buffer size of $n_{\textrm {buffer}} \in [25, 50]$ measurements. If the object would move slower or the camera could provide significant higher frame rate (e.g. free-running asynchronous sampling) the use of more samples could increase imaging quality due to better noise reduction.
Fig. 8. Single photon imaging of a rotating at a frame rate of 50kHz with $n_{\textrm {buffer}} =$ 25, 50 and 100 samples. The $32\times 32$ image frames are up-sampled to $320\times 320$ pixels using 2D spline interpolation. (Visualization 1: TCSPC and Visualization 2: USPC with $n_{\textrm {buffer}} =$ 25, 50 and 100)
4. Discussion and conclusion
We have investigated the capabilities of a synchronous non-gate SPAD array detector by time correlated and uncorrelated measurements. Further, we evaluated the minimum number of samples needed to obtain reliable reconstructions of range, intensity and photon flux. Finally, we derived theoretical descriptions of the applied estimators and carried out detailed experimental investigations which led to the reconstruction of scene properties (range, intensity and photon flux) at kHz-frame rate.
Time-correlated-single-photon-counting (TCSPC) uses its own pulsed illumination source and was applied to directly measure the photon time-of-flight. We reconstructed range and relative intensity from the single photon counting data using a Kalman filter. Further, from SSIM analysis, we could show that we need at least 10 measurements to obtain reliable reconstruction of range and intensity (SSIM > 0.5).
Uncorrelated photon counting (USPC) was used to estimate the photon flux impinging the detector by measuring the detection rate or mean time to photon event. USPC offer a complimentary measurement method to determine physical properties of the light field. We could demonstrate that the USPC photon flux measurements can handle high dynamic range and are a useful measure to determine light intensity. We could show the method is limited by the dark current (lower limt) to the sensor bandwidth (upper limit). Further, from SSIM analysis, we could show that the reconstruction quality depend on the ambient light level. Similar to TCSPC, at high levels, we need at least 10 measurements to obtain reliable results.
Finally, we realized imaging of a dynamic scene with 50 kHz frame rate. In our analysis, we used a buffer memory with cyclic overwriting procedure to analyze a sufficient number of measurements as well as to preserve the high frame rate of the cameras read out circuit. We observed that with rising buffer size, the scene motion starts to blur out object contours.
In conclusion, we state that TCSPC has advances in three dimensional imaging due to direct measurements of the photon time of flight. But, the TCSPC intensity measurements can be derived only from a relative number of detected photons. For most imaging application [8,27,34] this result is sufficient to receive a texture for the three-dimensional data model. A more precise estimation of the scene intensity is obtained by USPC photon flux measurements. Although, a range measurement is not possible with uncorrelated events, the USPC results convinces with high imaging quality and high dynamic range. Further, the use of a gated detector could reduce the impact of prior events, especially at high light level. On the other hand, an asynchronous SPAD array detector would be limited by the dead time $\tau$, only, and could deliver much higher frame rate even at high ambient light levels.
Due to the noise-free readout of photon event, the imaging quality of single photon counting sensors is mainly limited by the dark current rate. Therefore, the use of sensors with lower dark current rate would enable to use USPC imaging in even lower light level conditions. Today, significant lower dark current rate can be found in CMOS SPAD based on silicon. These sensors could be applied for low light level sensing in the visible to near infrared electromagnetic spectrum. To access the shortwave infrared or longer wavelength regime, new detectors has to developed with a focus on lower dark current. Today, very low dark current rates at infrared wavelengths as low as 1-100 Hz [46,47] are obtained by single pixel super-conduction nanowires single photon detectors (SNSPD). However, the application of USPC will make a trade off the detector prize and obtainable low dark count rate.
Funding
Defense Advanced Research Projects Agency (HR0011-16-C-0025); University of Wisconsin-Madison (693K125).
Acknowledgments
The author wants to thank A. Ingle, M. Gupta and A. Velten, from University of Wisconsin-Madison for fruitful discussions. Further, the author wants to acknowledge the contribution of Mr. Jean-Michel Poyet who helped to characterize the ambient light illumination.
Disclosures
The authors declare no conflicts of interest.
References
1. K. Kikuchi, T. Okoshi, and A. Hirose, “Achievement of shot-noise-limited sensitivity and 50 db dynamic range by photon-counting receiver using si avalanche photodiode,” Electron. Lett. 21(18), 801–802 (1985). [CrossRef]
2. S. Cova, M. Ghioni, A. Lacaita, C. Samori, and F. Zappa, “Avalanche photodiodes and quenching circuits for single-photon detection,” Appl. Opt. 35(12), 1956–1976 (1996). [CrossRef]
3. S. Pellegrini, R. E. Warburton, L. J. Tan, J. S. Ng, A. B. Krysa, K. Groom, J. P. David, S. Cova, M. J. Robertson, and G. S. Buller, “Design and performance of an ingaas-inp single-photon avalanche diode detector,” IEEE J. Quantum Electron. 42(4), 397–403 (2006). [CrossRef]
4. A. Tosi, A. Dalla Mora, F. Zappa, S. Cova, M. Itzler, and X. Jiang, “Ingaas/inp single-photon avalanche diodes show low dark counts and require moderate cooling,” in Quantum Sensing and Nanophotonic Devices VI, vol. 7222 (International Society for Optics and Photonics, 2009), p. 72221G.
5. A. Tosi, F. Acerbi, A. Dalla Mora, M. Itzler, and X. Jiang, “Active area uniformity of ingaas/inp single-photon avalanche diodes,” IEEE Photonics J. 3(1), 31–41 (2011). [CrossRef]
6. A. Tosi, A. Della Frera, A. Bahgat Shehata, and C. Scarcella, “Fully programmable single-photon detection module for ingaas/inp single-photon avalanche diodes with clean and sub-nanosecond gating transitions,” Rev. Sci. Instrum. 83(1), 013104 (2012). [CrossRef]
7. A. McCarthy, R. J. Collins, N. J. Krichel, V. Fernández, A. M. Wallace, and G. S. Buller, “Long-range time-of-flight scanning sensor based on high-speed time-correlated single-photon counting,” Appl. Opt. 48(32), 6241–6251 (2009). [CrossRef]
8. M. A. Albota, R. M. Heinrichs, D. G. Kocher, D. G. Fouche, B. E. Player, M. E. O’Brien, B. F. Aull, J. J. Zayhowski, J. Mooney, B. C. Willard, and R. R. Carlson, “Three-dimensional imaging laser radar with a photon-counting avalanche photodiode array and microchip laser,” Appl. Opt. 41(36), 7671–7678 (2002). [CrossRef]
9. M. A. Albota, B. F. Aull, D. G. Fouche, R. M. Heinrichs, D. G. Kocher, R. M. Marino, J. G. Mooney, N. R. Newbury, M. E. O’Brien, B. E. Player, B. C. Willard, and J. J. Zayhowski, “Three-dimensional imaging laser radars with geiger-mode avalanche photodiode arrays,” Lincoln Laboratory Journal 13, 351–370 (2002).
10. B. F. Aull, A. H. Loomis, D. J. Young, R. M. Heinrichs, B. J. Felton, P. J. Daniels, and D. J. Landers, “Geiger-mode avalanche photodiodes for three-dimensional imaging,” Linc. Lab. J. 13, 105 (2004). [CrossRef]
11. M. A. Itzler, M. Entwistle, M. Owens, K. Patel, X. Jiang, K. Slomkowski, S. Rangwala, P. F. Zalud, T. Senko, J. Tower, and J. Ferraro, “Geiger-mode avalanche photodiode focal plane arrays for three-dimensional imaging ladar,” in Infrared Remote Sensing and Instrumentation XVIII, vol. 7808 (International Society for Optics and Photonics, 2010), p. 78080C.
12. M. A. Itzler, M. Entwistle, M. Owens, X. Jiang, K. M. Patel, K. Slomkowski, T. Koch, S. Rangwala, P. F. Zalud, Y. Yu, J. Tower, and J. Ferraro, “Inp-based geiger-mode avalanche photodiode arrays for three-dimensional imaging at 1.06 um,” in Advanced Photon Counting Techniques III, vol. 7320 (International Society for Optics and Photonics, 2009), p. 73200O.
13. M. Entwistle, M. A. Itzler, J. Chen, M. Owens, K. Patel, X. Jiang, K. Slomkowski, and S. Rangwala, “Geiger-mode apd camera system for single-photon 3d ladar imaging,” in Advanced Photon Counting Techniques VI, vol. 8375 (International Society for Optics and Photonics, 2012), p. 83750D.
14. M. A. Itzler, M. Entwistle, M. Owens, K. Patel, X. Jiang, K. Slomkowski, S. Rangwala, P. F. Zalud, T. Senko, J. Tower, and J. Ferraro, “Comparison of 32 × 128 and 32 × 32 geiger-mode apd fpas for single photon 3d ladar imaging,” in Advanced Photon Counting Techniques V, vol. 8033 (International Society for Optics and Photonics, 2011), p. 80330G.
15. E. Theocharous, M. A. Itzler, J. Cheung, and C. J. Chunnilall, “Characterization of the linearity of response and spatial uniformity of response of two ingaasp/inp geiger-mode avalanche photodiodes,” IEEE J. Quantum Electron. 46(11), 1561–1567 (2010). [CrossRef]
16. Y. Altmann, S. McLaughlin, M. J. Padgett, V. K. Goyal, A. O. Hero, and D. Faccio, “Quantum-inspired computational imaging,” Science 361(6403), eaat2298 (2018). [CrossRef]
17. A. Kirmani, D. Venkatraman, D. Shin, A. Colaço, F. N. Wong, J. H. Shapiro, and V. K. Goyal, “First-photon imaging,” Science 343(6166), 58–61 (2014). [CrossRef]
18. D. Shin, F. Xu, D. Venkatraman, R. Lussana, F. Villa, F. Zappa, V. K. Goyal, F. N. Wong, and J. H. Shapiro, “Photon-efficient imaging with a single-photon camera,” Nat. Commun. 7(1), 12046 (2016). [CrossRef]
19. M. Laurenzis, J. Klein, E. Bacher, and N. Metzger, “Multiple-return single-photon counting of light in flight and sensing of non-line-of-sight objects at shortwave infrared wavelengths,” Opt. Lett. 40(20), 4815–4818 (2015). [CrossRef]
20. M. Laurenzis, J. Klein, and E. Bacher, “Relativistic effects in imaging of light in flight with arbitrary paths,” Opt. Lett. 41(9), 2001–2004 (2016). [CrossRef]
21. G. Gariepy, N. Krstajić, R. Henderson, C. Li, R. R. Thomson, G. S. Buller, B. Heshmat, R. Raskar, J. Leach, and D. Faccio, “Single-photon sensitive light-in-fight imaging,” Nat. Commun. 6(1), 6021 (2015). [CrossRef]
22. M. Laurenzis, F. Christnacher, J. Klein, M. B. Hullin, and A. Velten, “Study of single photon counting for non-line-of-sight vision,” in Advanced Photon Counting Techniques IX, vol. 9492 (International Society for Optics and Photonics, 2015), p. 94920K.
23. M. Laurenzis, A. Velten, and J. Klein, “Dual-mode optical sensing: three-dimensional imaging and seeing around a corner,” Opt. Eng. 56(3), 031202 (2016). [CrossRef]
24. M. O’Toole, D. B. Lindell, and G. Wetzstein, “Confocal non-line-of-sight imaging based on the light-cone transform,” Nature 555(7696), 338–341 (2018). [CrossRef]
25. M. Buttafava, J. Zeman, A. Tosi, K. Eliceiri, and A. Velten, “Non-line-of-sight imaging using a time-gated single photon avalanche diode,” Opt. Express 23(16), 20997–21011 (2015). [CrossRef]
26. G. Gariepy, F. Tonolini, R. Henderson, J. Leach, and D. Faccio, “Detection and tracking of moving objects hidden from view,” Nat. Photonics 10(1), 23–26 (2016). [CrossRef]
27. A. M. Pawlikowska, A. Halimi, R. A. Lamb, and G. S. Buller, “Single-photon three-dimensional imaging at up to 10 kilometers range,” Opt. Express 25(10), 11919–11931 (2017). [CrossRef]
28. M. Henriksson, H. Larsson, C. Grönwall, and G. Tolt, “Continuously scanning time-correlated single-photon-counting single-pixel 3-d lidar,” Opt. Eng. 56(3), 031204 (2016). [CrossRef]
29. R. Tobin, A. Halimi, A. McCarthy, X. Ren, K. J. McEwan, S. McLaughlin, and G. S. Buller, “Long-range depth profiling of camouflaged targets using single-photon detection,” Opt. Eng. 57(3), 031303 (2017). [CrossRef]
30. R. Tobin, A. Halimi, A. McCarthy, M. Laurenzis, F. Christnacher, and G. S. Buller, “Three-dimensional single-photon imaging through obscurants,” Opt. Express 27(4), 4590–4611 (2019). [CrossRef]
31. A. Maccarone, A. McCarthy, X. Ren, R. E. Warburton, A. M. Wallace, J. Moffat, Y. Petillot, and G. S. Buller, “Underwater depth imaging using time-correlated single-photon counting,” Opt. Express 23(26), 33911–33926 (2015). [CrossRef]
32. F. Yang, Y. M. Lu, L. Sbaiz, and M. Vetterli, “Bits from photons: Oversampled image acquisition using binary poisson statistics,” IEEE Trans. on Image Process. 21(4), 1421–1436 (2012). [CrossRef]
33. A. Halimi, Y. Altmann, A. McCarthy, X. Ren, R. Tobin, G. S. Buller, and S. McLaughlin, “Restoration of intensity and depth images constructed using sparse single-photon data,” in 2016 24th European Signal Processing Conference (EUSIPCO), (IEEE, 2016), pp. 86–90.
34. A. Halimi, A. Maccarone, A. McCarthy, S. McLaughlin, and G. S. Buller, “Object depth profile and reflectivity restoration from sparse single-photon data acquired in underwater environments,” IEEE Transactions on Comput. Imaging 3(3), 472–484 (2017). [CrossRef]
35. A. Ingle, A. Velten, and M. Gupta, “High flux passive imaging with single-photon sensors,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, (2019), pp. 6760–6769.
36. M. Zarghami, L. Gasparini, M. Perenzoni, and L. Pancheri, “High dynamic range imaging with tdc-based cmos spad arrays,” Instruments 3(3), 38 (2019). [CrossRef]
37. R. E. Kalman, “A new approach to linear filtering and prediction problems,” J. Basic Eng. 82(1), 35–45 (1960). [CrossRef]
38. G. Welch, “An introduction to the kalman filter,” in Proc. ACM SIGGRAPH, 2001, (2001).
39. G. M. Williams and A. S. Huntington, “Probabilistic analysis of linear mode vs. geiger mode apd fpas for advanced ladar enabled interceptors,” in Spaceborne Sensors III, vol. 6220 (International Society for Optics and Photonics, 2006), p. 622008.
40. E. Sarbazi, M. Safari, and H. Haas, “Statistical modeling of single-photon avalanche diode receivers for optical wireless communications,” IEEE Transactions on Commun. 66(9), 4043–4058 (2018). [CrossRef]
41. A. K. Pediredla, A. C. Sankaranarayanan, M. Buttafava, A. Tosi, and A. Veeraraghavan, “Signal processing based pile-up compensation for gated single-photon avalanche diodes,” arXiv preprint arXiv:1806.07437 (2018).
42. M. Planck, “Zur theorie der wärmestrahlung,” Ann. Phys. 336(4), 758–768 (1910). [CrossRef]
43. Permaflect® is a registered trademark of Labsphere Inc., North Sutton, NH, USA.
44. Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004). [CrossRef]
45. C. A. Hall and W. W. Meyer, “Optimal error bounds for cubic spline interpolation,” J. Approx. Theory 16(2), 105–122 (1976). [CrossRef]
46. R. H. Hadfield, M. J. Stevens, S. S. Gruber, A. J. Miller, R. E. Schwall, R. P. Mirin, and S. W. Nam, “Single photon source characterization with a superconducting single photon detector,” Opt. Express 13(26), 10846–10853 (2005). [CrossRef]
47. F. Marsili, V. B. Verma, J. A. Stern, S. Harrington, A. E. Lita, T. Gerrits, I. Vayshenker, B. Baek, M. D. Shaw, R. P. Mirin, and S. W. Nam, “Detecting single infrared photons with 93% system efficiency,” Nat. Photonics 7(3), 210–214 (2013). [CrossRef]
