Time-of-flight photon count data processing techniques

Histograms of photon counts versus time obtained by the TCSPC-based time-of-flight

depth imaging systems can be used to estimate the depth profiles of remote targets. In the experimental systems reported in this Thesis, the starting time (or “zero time”) for the histogram is taken as each repetitive synchronisation (“start”) electrical signal and the duration of the histogram time axis is equal to the period of the synchronisation signal. The minimum binning size is determined by hardware limitations and is equal to 1 ps for the HydraHarp 400. For example, Figure 2-54(a) shows the pixel-wise histogram of the scan on the plywood board at a stand-off distance of 4400 m using one of the depth imaging systems described in Chapter 4. The duration of the histogram axis corresponds to the period the synchronisation signal. The frequency of this signal is down-divided by a factor of 16 from the repetition rate of 40 MHz for the pulsed laser source. Thus, there are 16 repetitive pulse periods within the histogram. One of the period-wise histograms.is highlighted in Figure 2-54(b). Apart from positions (or depths) of the scattered returns from the remote target surfaces, their amplitudes (or photon counts) and/or number of returns/peaks can also be characterised depending on different time-of-flight data processing methods [2.136].

Figure 2-54 (a) shows the pixel-wise histogram of the scan on the plywood board at a stand-off distance of 4400 m using one of the depth imaging systems described in Chapter 4. The frequency of the synchronisation signal is down-divided by a factor of 16 from the repetition rate of 40 MHz for the pulsed laser source. A timing bin size of 16 ps was used to record the histogram. One of the 16 pulse periods in the histogram, as shown in (a), is highlighted in (b).

Given a pre-determined photon count threshold, locating the time/depth associated with a local maximum/peak in the photon histogram can be achieved using a peak-finder algorithm. However, as shown in Figure 2-55, a photon count histogram with reasonably high SNR is necessarily required to achieve a single return discrimination from the background noise. In addition, in order to effectively distinguish the signal, this method often strongly requires user to determine the photon count threshold.

Figure 2-55 The photon count histogram with 2 s acquisition time, 16 ps timing bin and SNR of ~10.2 is shown in (a) or (c). The photon count histogram with 100 ms acquisition time, 16 ps timing bin and SNR of ~2.6 is shown in (b) or (d). The insets in (a), (c) and (d) show the zoom-in potential peaks. Note that they are portion of Pixel 50 histograms of the scans on the plywood board at a stand-off distance of 4400 m using one of the depth imaging systems described in Chapter 4. The photon count threshold of 10 counts (i.e. Th1=10 counts) or 2 counts (i.e. Th2=2 counts) is chosen for the peak finding. Locating the time associated with a local maximum/peak in the photon histogram can be achieved well in (a) and (c). By contrast, the maximum counts in the histogram with the 100 ms acquisition time is less than Th1 so that no effective signal can be distinguished in (b). Even if Th2=2, two local maximum peaks are found, as shown in (d), so that it still cannot ensure that the time of flight associated with the depth of the target is located in the histogram.

By contrast, the cross-correlation method, which measures the degree of similarity between the photon count histogram and the reference instrumental response function (IRF) of the system, can be used to determine the depth profile in photon data with a

reduced SNR [2.137]. The essence of the cross-correlation method is similar to a convolution between two functions. It aims to estimate the similarity between the return signal (see Figure 2-56 (a)) and the time-shifted reference (see Figure 2-56(b)), and in turn to locate the position of the maximum cross correlation (see Figure 2-56(c)), which corresponds to the most likely location of the signal matching the reference. This can be expressed as:

= Equation (2.20)

where varies in [- , ℎ ] and is the time-bin-wise cross-correlation, which is performed between the acquired photon count histogram with ℎ time bins and the reference instrumental response function response with time bins. The highest cross-correlation can reveal the time bin associated with the target return. Therefore, the cross-correlation approach can be used effectively to estimate the time/depth of the return.

Figure 2-56 The cross-correlation technique aims to estimate the similarity between the return signal (a), which is the same to the signal with a SNR of ~2.6 shown in Figure 2-55(b) or (d), and the time-shifted reference (b), and in turn to locate the position of the maximum cross correlation, as shown in (c), which corresponds to the most likely location of the signal matching the reference. IRF: the reference instrumental response function of the system.

The cross-correlation method is a standard estimation method, which only covers the estimation procedure of the processing strategy, as shown in Figure 2-57. It offers a highly efficient approach to data processing and is not specific to the analysis of photon data. However, particularly, Poisson noise that typically occurs in photon counting techniques is not considered in the cross-correlation method. It can generally degrade the time/depth estimation performance due to the increased variance of the estimation and/or the introduction of a bias. In addition, the cross-correlation method does not take into account the background counts either. This can further degrade the time/depth estimation. If there is poor estimation of the depth position, then this will affect the estimation of the number of photon counts in the peak, if required. In addition, the cross-correlation method is not suitable to resolve multiple returns/peaks in a photon count histogram without some a-priori knowledge of the number or distribution of those returns/peaks.

Figure 2-57 Graphical model of a systematic strategy for (but not limited to) photon count data processing,

An alternative to the cross-correlation method is the family of Bayesian methods, which can be used to process single-photon data with reduced SNR while incorporating additional useful information (see Figure 2-57). Given that the observed data and the prior information are available, this is possible to derive a posterior distribution, which models more knowledge about the unknowns. This means that multiple parameters (e.g. number, positions, and amplitudes of returns/peaks) can be simultaneously estimated. A statistical model is involved to evaluate the likelihood for the data, for example the assumption of Poisson noise realisations in the data. Such methods can incorporate additional information through prior distributions, such as positive

amplitudes, smooth depth variation, background counts with low variation, distribution of the return/peak number and observation conditions. In order to obtain not only estimated parameters but also measures of the relevant uncertainty, Markov chain Monte Carlo (MCMC) techniques seem particularly appropriate methods. MCMC methods generally consist of generating a Markov chain of random variables that converges toward a distribution of interest (the posterior distribution shown in Figure 2-57) [2.136]. Standard MCMC methods are, however, not well adapted for problem where the number of unknowns (e.g. number of peaks) is undetermined [2.136]. In such cases, reversible jump MCMC (RJMCMC) methods can be used, which allow moves between different parameter spaces if the parameters correspond to individual state spaces [2.138].

Both cross-correlation-based and Bayesian-based data processing techniques are widely used to estimate depth images using individual pixel-wise photon count histograms in isolation (see for example [2.139-2.145]). This allows the 3D surface reconstruction of the remote target based on pixel-wise processing using these algorithms. Note that both pixel-wise cross-correlation methods and the RJMCMC algorithm are used to process the TCSPC-based photon data presented in the later Chapters of this Thesis.

Particularly, the photon data from a distributed target (e.g. tree) is processed by the RJMCMC algorithm, which is an adaptive approach (see Chapter 6). In addition, processing neighbouring pixels simultaneously using spatial-correlation approaches, which incorporated MCMC algorithms (see Chapter 7) [2.146] or convex optimisations (with knowledge of photon macro times [2.29]), has been proposed for efficient depth/intensity estimation from the photon data or even sparse photon data. By considering the relationships between adjacent pixels these approaches can optimise the analysis conducted by peak-finder algorithms to more efficiently generate depth/intensity profiles. For example, if a cluster of adjacent pixel from an approximate flat plane at a particular distance, it is unlikely that one further adjacent single pixel is likely to be at a distance that is widely separated from the plane.