1. Introduction

Fluorescence lifetime imaging (FLIM) is a frequent spectrometric analysis in biology [1,2], chemistry [3], and materials engineering [4]. There are various techniques for measuring or evaluating photoluminescence (PL) dynamics, such as time-correlated single photon counting [5], gated PL counting [6], streak camera [7], time-domain or frequency-domain analog recording technique [8,9]. Another available method for PL decay measurement is based on random excitation of the measured sample - the RATS method (RAndom Temporal Signals) [10], which could also be set in a single-pixel camera (SPC) configuration [11] and carry out the FLIM experiment. We abbreviate this measurement as 2D-RATS. At the current state, the method can characterize morphology in the order of microseconds or sub-micro seconds (hundreds of nanoseconds), including halide perovskite samples, solid-phase luminophores, or Si nanocrystals [12–14]. However, ongoing research shows that it will be realistic to measure even at the nanosecond scale.

The 2D-RATS method combines the SPC technique with the excitation of the PL with random patterns (masks) blinking randomly in time. Each mask leads to a specific PL decay, which is measured for a set of uncorrelated masks. Then, using algorithms for undetermined systems, it is possible to reconstruct PL decays in each individual pixel of the 2D scene. The quality of reconstruction is affected by noise, which is inevitably present in all measured data. Therefore, the noise analysis of the method is an essential piece of knowledge that allows optimization of the method and identifies the most sensitive aspects of the setup. Moreover, many results valid for the RATS method are also relevant for the PL measurement, where the deconvolution step is used [15].

In this article, we carry out a detailed study of various noise effects on the 2D-RATS measurement and their impact on the resulting FLIM information. Our analysis is based on extensive simulations and a set of synthetic data faithfully following the real experimental conditions for the SPC experiment carried with a digital micro-mirror device (binary illumination mask). A digital micro-mirror device is part of our latest 2D-RATS setup, where it replaced an optical diffuser (greyscale mask) [11]. Nevertheless, both binary and greyscale masks lead to the same trends with respect to the noise level, and the presented results are generally valid for both random mask implementations.

We analyze the benefits of various approaches to noise reduction. Besides the generally used option to prolong the acquisition time [16], the deconvolution step in the RATS method allows us to optimize the regularization parameter of the deconvolution [17,18]. Due to the random nature of the excitation signal in RATS, we avoid the use of any mathematical filters which can be apparently optimized for one type of signal, but due to the possibility of choosing a different signal frequency and sampling, such a filter can be difficult to transfer to a general case.

Our article provides guidelines for optimizing experiments for PL decay measurement and FLIM, where the deconvolution-based retrieval of PL dynamics is used. We demonstrate that the noise present in the PL dataset is the most critical factor. At the same time, we show that the resulting noise level in the FLIM dataset can be highly decreased by using a longer non-periodic excitation signal or by optimizing the regularization parameter. The parameter features an optimum value for a given noise level. We point out the issues connected to a periodic excitation signal.

In particular, we demonstrate that with proper choice of regularization parameter and acquisition time, the RATS method can attain results that are not distorted at all and can accurately map a 2D scene even with a relatively high noise level (3%).

2. RATS method

We will first introduce the principles of the single-spot 0D-RATS method and imaging 2D-RATS measurement. The cornerstone of the RATS method is sample excitation with a random signal I_EXC. The random signal I_EXC could be generated using a generator (lens, rotating diffuser, aperture) described in our previous work [10]. However, there are many different ways to generate a random signal. In the experiment should be detected I_EXC (photodiode) and PL signal I_PL (photomultiplier), which is given as convolution of I_EXC and PL decay I_D (Eq. (1)).

This fact can be used to recover the PL decay from I_PL by using the Fourier transform and the convolution theorem. The deconvolution is attainable only for the frequencies where the excitation signal has a non-zero amplitude. Nevertheless, due to random excitation, a single measurement contains a broad range of frequencies in a single dataset. PL decay I_D could then be determined via deconvolution (Eq. (2)), wherein the denominator is regularization parameter ɛ adding a part of the average power of the spectrum due to it is possible to solve ill-conditioned problems.

The 2D-RATS measurement is carried out in an SPC configuration, as we described in our previous article [11]. We point the reader to numerous articles summarizing the SPC experiment for more details [19–22]. In our implementation of the SPC experiment, the sample was illuminated with a set of random excitation patterns (masks). The excitation masks were “blinking” with the same waveform I_EXC(t), and the whole PL from each illuminated point i is collected to a single-pixel detector. The time t for which a single mask illuminates the sample following the intensity fluctuation of I_EXC(t) is stated as acquisition time and is marked as t_acq. For the 0D-RATS, the acquisition time is equal to the total duration of the experiment. For the 2D-RATS experiment, the measurement over the acquisition time is repeated for each random mask.

The photoluminescence I_PL is given as a summation of I_PL(i) signals. Then Eq. (1) can be rewritten for the total PL intensity as:

The number of excitation masks M is set by the number of image pixels N and the so-called compression ratio k = M/N. Because masks are not coherent, i.e., one pattern is not correlated with each other, each mask illuminates a different combination of the sample pixels, and, therefore, each mask represents its own I_PL, which can be used to retrieve the PL decay I_DA:

PL snapshot m(t) can be reconstructed in each delay after excitation t using the calculated I_DA(t) curves. For the given delay and each excitation mask, the I_DA(t) provides a dataset I_SPC describing intensity fluctuation of M values – see Fig. 1(C), where each curve corresponds to a single I_SPC dataset. Then PL snapshot m(t) is then retrieved from the underdetermined dataset by using a standard SPC retrieval:

The sensing matrix A is formed from vectorized excitation patterns, TV stands for the total variation calculation.

 

Fig. 1. (A) Scheme of 2D-RATS approach, where the sample is illuminated with a set of random patterns (masks) blinking according to I_EXC. The total PL intensity waveform I_PL is detected for each mask with a single-pixel detector. (B) Example of a set of calculated I_DA's for corresponding masks – see Eq. (4). (C) Intensity fluctuation for each delay after excitation of I_DA. Knowing the fluctuation (I_SPC signal) and set of masks makes it possible to determine the PL map m at a given delay after excitation of I_DA using compressed sensing algorithms. (D) An example of reconstructed PL maps in delays t₁ and t₂. Plotted I_D’s corresponding to pixel [14 10] (area “a”) and to pixel [16 25] (area “b”).

Download Full Size | PDF

By reconstructing a temporal snapshot for each delay t, we get a 3D datacube, which contains PL decay in every i-th pixel of the sample I_D(i,t). The whole concept is illustrated and summarized in Fig. 1.

3. Noise effect simulation

A small contribution of noise can be expected in each measured signal in the experiment. In order to maintain stable conditions, the role of the noise was explored through simulations. A primary signal I_EXC was simulated using temporal speckles patterns [23], which are cornerstones of a random analog signal generator presented in our previous work [10]. The length of the simulated I_EXC signal was 0.1 s with an impulse response function of FWHM of 2.07µs. The PL decay I_D was considered with the lifetime $\tau = 20$ µs. Used noise levels throughout the article are consistent with the noise that can be expected in a realistic experiment [15]. In the current optical setup, an amplified signal from a photomultiplier is used for I_PL detection, and this sensitive part is the main source of the noise. The noise originates from scattered excitation photons and the noise induced in the detector itself, which is based on shot noise and standard electronic noise. The experimentally observed noise can be very well described with a flat frequency dependence, i.e., white noise.

The excitation signal I_EXC was simulated as a non-periodic random signal. The importance of a non-periodic signal will be explained in section 4.1. It is worth noting that with the exception of section 4.3, the regularization parameter ɛ (Eq. (2) and Eq. (4), respectively) is considered as ɛ = 0.1.

3.1 0D-RATS

To demonstrate the basic steps, we shall start with the non-imaging RATS, i.e., 0D-RATS. We use two datasets of I_EXC and I_PL, which are subsequently used to retrieve the PL decay via Eq. (2). Therefore, we initially simulated the effect of noise present in these two datasets.

Firstly, we added white noise to I_EXC, while I_PL was kept absolutely noiseless (Fig. 2(A)) and vice versa (Fig. 2(B)). In both datasets in Fig. 2, the signal-to-noise ratio (SNR) was 15.2 dB which corresponds to 3% percent of noise in the system (see Eq. (6)).

 

Fig. 2. (A) Reconstructed I_D with a noise level of 3% in I_EXC, which corresponds with SNR about 15.2 dB. I_PL was assumed noiseless. (B) Reconstructed I_D with a noise level of 3% in I_PL, corresponding with SNR about 15.2 dB. I_EXC was assumed noiseless.

Download Full Size | PDF

For both retrieved decays in Fig. 2 (red lines), the noise had an effect on reconstructed I_D, but the investigated PL dynamics were not biased, i.e., the overall decay shape was not altered, and it was in perfect agreement with the expected output (black lines). Nevertheless, we observed that the noise in the I_PL dataset had a significantly more pronounced effect on the resulting noise in the retrieved PL decay (Fig. 2(B)).

We can quantify the noise level by the root-mean-square error, which reaches RMSE_PL = 12.9·10⁻³ for the noise in the PL signal and RMSE_EXC = 2.8·10⁻³ for the noise in the excitation signal. That suggests that regularization in the denominator of Eq. (2) has a more pronounced effect on I_EXC than I_PL, which becomes the dominating source of noise in the retrieved decay.

3.2 2D-RATS

As the next step, we also simulated the effect of noise in the I_PL and I_EXC signal on the 2D-RATS experiment. Here, the situation is more complex because the noise present in the I_PL and I_EXC datasets is first transposed into the noise of the retrieved PL decay curve I_DA(t) from Eq. (4). The noise present in these curves is then propagating into the SPC signal I_SPC, which is then used for the retrieval of the set of PL snapshots for each delay in Eq. (5).

All simulated reconstructions were performed using binary masks. We used the reconstruction algorithm based on the sparsity of total variation TVAL3 [24,25]. The main parameters of the TVAL3 algorithm were: mu (2¹¹), beta (2⁷). As an investigated sample, we used Matlab predefined image Phantom, where the PL decay with a single lifetime of $\tau = 20$ µs was set to be constant all over the Phantom's “body”. Acquisition time and other conditions simulating real experiments were kept the same as for the 0D-RATS simulations above.

We carried out the same calculation as we presented in Section 3.1, and we show in Fig. 3 the retrieved PL snapshots m(t) in the peak PL intensity, i.e., t = 0 µs, where the contrast of the PL signal is the highest. The case was studied for three different SPC compression ratios k = 0.4, 0.6, 0.8 and four signal-to-noise ratios SNR = 15.2, 18.2, 20, 23 dB, which corresponds to 3%, 1.5%, 1%, 0.5% level of noise in the signal.

 

Fig. 3. Left part: reconstructions of PL map m in delay with maximal intensity (t = 0 µs) in case of noise in I_EXC. Right part: reconstructions of PL map m in delay with maximal intensity (t = 0 µs) in case of noise in the I_PL. The noise parameters are for both SNR = 15.2, 18.2, 20, 25 dB (rows) and the compression ratios are k = 0.4, 0.6, 0.8 (columns).

Download Full Size | PDF

The results are summarized in Fig. 3 for the noise introduced in the I_EXC signal (left-hand side) and the noise introduced in the I_PL signal (right-hand side), and the corresponding resulting noise levels are presented in Table 1.

Table 1. Quality of the retrieved image (R, lower number = higher quality) and SPC signal (SPC-SNR, higher number = higher quality) corresponding to Fig. 3 for noise introduced via IEXC and IPL signal.

View Table

Analogously to the 0D-RATS, the noise in I_PL has a significantly worse effect on the image retrieval than the noise in the I_EXC signal. By comparing different noise levels (compare lines in Fig. 3), one can see that the SNR of the I_PL is the main limiting factor in the snapshot retrieval. The effect of compression ratio (compare columns in Fig. 3) does not play any significant role for the values above 0.4. This means that it is not efficient to compensate for higher noise in the measured signal by simply increasing the number of measured excitation masks.

To quantify the effect of each parameter, we introduced three different measures. Firstly, we can judge the image reconstruction quality, where we compare the retrieved snapshot m (in delay t = 0 µs) with the actual image used in simulations U by using Frobenius norm of the two:

We can also focus on the PL dynamics for each pixel i, where the reconstructed decay curve I_DREC(i) is normalized and compared with the actual decay curve I_D(i), which was used to simulate the data. The comparison was made from t₀ = 0 µs to t_Τ = 120 µs, which is a sufficient temporal range for the decay curve with a lifetime of 20 µs. We denote this error as the decay deviation σ:

Finally, we also determined the SNR of the I_SPC signal, which is used for the image retrieval m in delay t = 0 µs. Note that the I_SPC signal indicates the fluctuation of I_DA curves corresponding to the respective delay. The fluctuations (I_SPC) are affected by the noise in both I_EXC and I_PL. We extract the I_SPC noise level to provide a comparison to other SPC experiments. SPC-SNR_EXC denotes the signal-to-noise level in I_SPC when the noise was added to I_EXC and SPC-SNR_PL indicates the case when the noise was added to the I_PL. Analogously, the indices “PL” and “EXC” have the same meanings for R_EXC and R_PL or σ_EXC and σ_PL.

4. Noise level optimization

In Section 3, we studied the effect of the noise level in the I_EXC respective I_PL simulated dataset on the retrieval of the PL decay curve and PL snapshot. The noise levels are determined by the properties of an optical setup, which features a certain photon budget (number of detectable photons), types of detectors, and samples. For an optimized optical setup, such characteristics can not be easily improved.

On the other hand, it is possible to enhance the quality of the retrieved PL decay or FLIM on the expenses of the acquisition time. Therefore, we studied the benefits connected to a longer acquisition time. Due to the fact that the RATS method is based on signal deconvolution, an important factor is the periodicity of the signal, as we discuss in the following subsection.

4.1 Periodic extension of acquisition time

A straightforward approach to improve the quality of the PL decay I_D retrieval via Eq. (2) is to repeat the same measurement for the same excitation waveform. Hence, we attain a periodic I_EXC and I_PL signal. However, it is worth stressing that the use of a periodic signal I_EXC will create unwanted artefacts in the retrieved data. It follows from the nature of deconvolution in Eq. (2) that such periodic excitation waveform leads to a periodic I_D signal with a lower amplitude. We will demonstrate this effect on a 0D-RATS simulation.

We compared the retrieved PL decay (I_D curve) in the case of a non-periodic I_EXC signal with a duration of 0.1 s and a periodic I_EXC signal (7 periods) with the same total duration of 0.1 s. This is illustrated by Fig. 5. For completeness, we present in Fig. 4 the non-periodic signal used and the periodic signal counting seven periods. Both signals were simulated based on simulations of randomly changing speckle patterns [23], which corresponds with excitation signals generated via a random signal generator (lens-rotary diffuser-aperture) presented in our previous work [10]. Due to the rotating diffuser, a periodic signal can be expected, which due to small vibrations may be rather quasi-periodic. However, a perfectly periodic and non-periodic signal can be achieved via a suitable laser modulation.

 

Fig. 4. Used excitation non-periodic signal (left panel) and periodic signal with 7 periods (right panel). Both signals were simulated based on simulations of randomly changing speckle patterns [23], which corresponds with excitation signals generated via generator (lens-rotary diffuser-aperture) presented in our previous work [10].

Download Full Size | PDF

 

Fig. 5. Results of I_D reconstruction (deconvolution), when non-periodic (panel A) and periodic (panel B) I_EXC was used for sample excitation with lifetime Τ = 20 µs.

Download Full Size | PDF

The entire retrieved PL dataset is depicted in Fig. 5. The comparison between the periodic and non-periodic excitation – see right and left-hand side, respectively – shows that the amplitude of the retrieved PL for the periodic I_EXC signal is about seven times smaller compared to the non-periodic case and the PL signal has multiple replicas.

When we zoom in Fig. 5(B) into the PL dynamics during the first hundreds of microseconds, i.e., one of the PL decay replicas, we get after normalization the correct I_D following the actual decay – see Fig. 6. However, the resulting SNR is decreased. We tested in Fig. 6 the noise introduced via the excitation signal I_EXC (panel A) and PL signal I_PL (panel B). The root-mean-square error for curves in Fig. 6 reaches RMSE_PL = 29.7·10⁻³ and RMSE_EXC = 7.8·10⁻³ for the noise introduced in the I_PL and I_EXC signal, respectively. The noise level in Fig. 6 can be directly compared to Fig. 2, where we used a non-periodic signal and where RMSE_PL = 12.9·10⁻³ and RMSE_EXC = 2.8·10⁻³. For the non-periodic signal, the resulting noise level was more than 2-times lower.

 

Fig. 6. Zoomed results of I_D reconstruction (Τ = 20 µs.), when I_EXC was used as a periodic signal (7 periods). Redline is reconstructed data, and the black line is reference data. (A) Reconstruction with an amount of noise of 3% in I_EXC (correspond with SNR about 15.2 dB). (B) Reconstruction with an amount of noise of 3% in I_PL (correspond with SNR about 15.2 dB).

Download Full Size | PDF

To further illustrate this behavior, we carried out a set of simulations, where we kept a constant total acquisition time of 0.1 s, while the I_EXC signal was set to be periodic with up to 16 periods – see Fig. 7. The root-mean-square error increases with the number of periods for both situations – noise in the I_EXC and I_PL signal. Therefore, for the sake of optimum signal reconstruction in the RATS method, it is worth avoiding any periodicity in the I_EXC signal.

 

Fig. 7. RMSE of I_D for a periodical I_EXC with a constant total acquisition time of t_acq = 0.1 s. The number of periods in the I_EXC signal is varied. I_PL respective I_EXC noise level: SNR 15.2 dB.

Download Full Size | PDF

4.2 Non-periodic acquistion time extension

The possible way to suppress the noise effect is to extend the acquisition time so that we avoid any I_EXC signal periodicity. This approach favors frequencies representing the true signal in the Fourier spectrum and suppresses the contribution of white noise. We carried out simulations corresponding to the 2D-RATS measurement presented in Fig. 3. Nevertheless, here we used a set of acquisition times t_acq = 0.1 s, t_acq = 0.2 s, t_acq = 0.4 s, t_acq = 1 s and t_acq = 2 s. Other conditions, such as reconstruction parameters, regularization parameter ɛ, and signal properties, were kept the same as in Section 3.

As in Section 3.2, we focus on the quality of the retrieved image (R_EXC, R_PL), noise introduced into the SPC signal (SPC-SNR_EXC, SPC-SNR_PL) in delay t = 0 µs. Moreover, this section also focuses on the deviation of the whole reconstructed PL decay in each pixel of the sample (σ_EXC, σ_PL).

Dependences of each noise characteristics on the acquisition time are presented in Fig. 8–10. In these figures, the line/symbol color indicates the given SNR (red: 15.2dB, blue: 18.2dB, black: 20dB and magenta: 23dB); line/symbol type represent the compression ratio k (cross: 0.8, circle: 0.6, full line: 0.4).

 

Fig. 8. Reconstruction error R_EXC (left part), R_PL (right part) at the delay t = 0 µs dependence on sensing time for different SNR: 15.2 dB (red), 18.2 dB (blue), 20 dB (black), 23 dB (magenta) and three different compression ratio k = 0.4 (full line), k = 0.6 (circle), k = 0.8 (cross).

Download Full Size | PDF

 

Fig. 9. Noise dependence in SPC signal (random fluctuation in intensity of I_D in delay t = 0 µs, because of changing illumination masks) on sensing time according to added noise to I_EXC (left part) and I_PL (right part) with SNR: 15.2 dB (red), 18.2 dB (blue), 20 dB (black), 23 dB (magenta) and three different compression ratio k = 0.4 (full line), k = 0.6 (circle), k = 0.8 (cross).

Download Full Size | PDF

 

Fig. 10. Acquisition time dependance of σ_EXC – noise added to I_EXC (left part) and σ_PL – noise added to I_PL (right part). SNR: 15.2 dB (red), 18.2 dB (blue), 20 dB (black), 23 dB (magenta) and three different compression ratio k = 0.4 (full line), k = 0.6 (circle), k = 0.8 (cross).

Download Full Size | PDF

All the results in Fig. 8–10 confirm that the effect of noise level (varying color) is much more pronounced than the compression ratio (varying line/symbol type), i.e., increasing the number of excitation patterns above 40% compared to the number of the pixel has a negligible effect on the image quality.

When we focus on the noise introduced via I_EXC signal, it has a low effect on the resulting image reconstruction – see Fig. 3, left-hand side. For this reason, the change in the acquisition time has almost no effect on the resulting noise level regarding the image quality and SPC signal error (see Fig. 8,9). A subtle peak in Fig. 8 for the acquisition time 0.2 s and a decrease in Fig. 9 for the same acquisition time are connected. They both indicate an increased amount of noise in the I_SPC signal for the given case of PL map reconstruction at delay t = 0 µs, which is unexpected in light of the other results. However, we should consider that this result is connected just with only one delay point (t = 0 µs). If we take into account the whole decay curve I_D (signal-to-noise ratio in all SPC signals connected with delay points of the curve I_D), which is shown in Fig. 10, it can be observed that this irregularity is averaged out.

On the contrary, for the noise introduced via the I_PL signal, we can highly improve the quality of the retrieved image by the increased acquisition time. This image quality enhancement is much more pronounced for higher noise levels. In the case where the noise level is 15.2 dB (3%), the R_PL will decrease by 50%, while when the noise level is 23 dB (0.5%), the R_PL will decrease by only 25%.

The results described in Fig. 8 are linked to the results presented in Fig. 9. As expected, the highest SPC-SNR_EXC and SPC-SNR_PL are reached for the lowest noise level in the system - SNR = 23 dB (0.5%). On the other hand, the SPC-SNR_PL increases as the acquisition time increases, which results in a decrease in the R_PL. An interesting finding is that when the acquisition time is extended to 2s, the SPC-SNR_PL increases even above the SPC-SNR_EXC value corresponding to this acquisition time. Consequently, the same trend is observed for the R_PL and R_EXC values at a scanning time of 2 s.

Although it was stated that R_EXC and SPC-SNR_EXC do not change significantly with the acquisition time, it is worth noting that these values represent only a single PL snapshot at the selected delay t = 0 µs. The effect of acquisition time is notable for the decay curve reconstruction, which is highly improved both for the noise in I_EXC and I_PL – see σ in Fig. 10. Decay deviation σ decreases with the increasing acquisition time. Although both errors are in a different order of magnitude, by increasing the acquisition time from 0.1 s to 2 s, both σ errors decreased by the same ratio.

4.3. Regularization parameter effect

Another way to suppress the effect of noise is to change the regularization parameter ɛ in Eq. (2) and Eq. (4). The regularization parameter makes it possible to solve the ill-conditioned problem, where “division by zero” could occur for frequencies with very low amplitude in the I_EXC signal [26]. The regularization parameter adds a part of the averaged power of the spectrum to the denominator of the Eq. (2) respective Eq. (4) and thus suppresses the influence of less frequent frequencies in the signal (white noise). As a result, the calculated I_D or I_DA is smoothed.

As in the previous section, the influence of the parameter ɛ was investigated regarding the quality of the reconstructed PL snapshot at the zero delay after excitation, which corresponds to the maximum PL intensity. All simulations were carried with the acquisition time of 0.1 s and a range of ɛ from 0.05 to 1. Reconstruction parameters and signal properties were held the same as in Section 3. Figure 11–13 follow the same color scheme as in the previous section with respect to SNR (color) and compression ratio (line/symbol type).

 

Fig. 11. Reconstruction error R_EXC (left part), R_PL (right part) at the delay t = 0 µs dependence on ɛ - different SNR: 15.2 dB (red), 18.2 dB (blue), 20 dB (black), 23 dB (magenta) and three different compression ratio k = 0.4 (full line), k = 0.6 (circle), k = 0.8 (cross).

Download Full Size | PDF

 

Fig. 12. Noise dependence in SPC signal (random fluctuation in intensity of I_D in delay t = 0 µs, because of changing illumination masks) on ɛ according to added noise to I_EXC (left part) and I_PL (right part) with SNR: 15.2 dB (red), 18.2 dB (blue), 20 dB (black), 23 dB (magenta) and three different compression ratio k = 0.4 (full line), k = 0.6 (circle), k = 0.8 (cross).

Download Full Size | PDF

 

Fig. 13. ɛ dependance of σ_EXC – noise added to I_EXC (left part) and σ_PL – noise added to I_PL (right part). SNR: 15.2 dB (red), 18.2 dB (blue), 20 dB (black), 23 dB (magenta) and three different compression ratio k = 0.4 (full line), k = 0.6 (circle), k = 0.8 (cross).

Download Full Size | PDF

In accordance with the previous results, the effect of the added noise is much more pronounced than the compression ratio used. We will first focus on the influence of regularization parameter ɛ on the quality of the retrieved image, i.e., R_EXC and R_PL – see Fig. 11. In these results, we see an interplay of two effects: the smoothing of the I_DA and I_D, respectively, and the bias of the reconstructed I_DA respective I_D.

Since the noise level in the I_EXC affects the I_DA reconstruction less than in I_PL, the effect of biased reconstruction of I_DA prevails and the R_EXC error increase with ɛ. Whereas in the case of R_PL, the noise smoothing effect prevails in I_DA for the low ɛ values and the R_PL decreases. However, with increasing ɛ, the bias effect becomes dominant and the image quality is decreased again.

This is confirmed by Fig. 12, which shows the amount of noise in the individual SPC-SNR_EXC and SPC-SNR_PL. This implies that for the given noise level in the I_PL signal, there is an optimum regularization parameter providing the best results and this provides a guideline of the optimum ɛ value.

Figure 13 focuses on σ, which evaluates the fidelity of the reconstructed decay curves. Here, with increasing ɛ, both σ_EXC and σ_PL decrease despite the bias effect, i.e., a higher ɛ value provides a better estimate of the decay curve. We ascribe it to the fact that with decreasing intensity of I_DA (the following delay after excitation), the noise level in the SPC signal increases, and thus the smoothing effect of the regularization parameter prevails over the bias effect.

Both the dependence of σ_EXC and σ_PL on ɛ are influenced by the amount of noise in the system. Especially for the high-noise PL signal (15.2 dB, 3%), the increase of ɛ from 0.05 to 1 reduced the σ_PL by 79%. Analogously to the previous results, the increase in regularization parameter has a lower effect on the noise introduced via I_EXC signal – see σ_EXC, which decreases by 32% for the same noise level and ɛ change.

5. Conclusion

In this paper, we carried out an extensive analysis of the effect of noise within the RATS technique – a novel method for the PL decay reconstruction and FLIM. As we mentioned in the introduction, the method can now be used mainly in materials engineering due to the possibility of detecting PL decays at the microsecond or sub-microsecond level. However, ongoing research opens the possibility of measuring in units of nanoseconds, which could open up further possibilities for the characterization of a range of optical materials, chemical substances, as well as biological samples. Since this is a new method, the findings presented are important for the further direction of the method. The problem was investigated using simulations for cases with noise levels reaching 0.5–3% (SNR 23–15.2dB) of the signal. To disentangle the effect of each noise source, the noise was added to I_EXC only and I_PL was left noiseless and vice versa.

We consistently observed that the same noise level has a significantly greater effect when present in the I_PL, which we ascribe to the fact that the regularization parameter ɛ is associated with the I_EXC in the denominator of the relationship Eq. (2) and Eq. (4). Hence, the most efficient strategy to improve the data quality is to focus on the I_PL intensity level and the connected noise.

Secondly, all our data simulations revealed that the effect of noise level is incomparably more important than the compression ratio k, i.e., the number of measured excitation masks in the single-pixel experiment. In other words, for k > 0.4, it is not possible to compensate for a higher noise level by increasing the number of excitation masks.

It is also essential to avoid using periodic excitation signals, which can cause periodicity in the retrieved decay and increase the resulting I_D relative noise level. It is worth noting that the RATS method was originally based on an analog random signal generator, the main component of which is a rotary diffuser. Therefore, we attained a quasi-periodic signal, which played a significant role in the method. Using a non-periodic signal, for instance, by laser modulation, a greater signal-to-noise ratio will be obtained.

As expected, we observed that increasing the acquisition time of a single decay measurement (single excitation mask) can highly reduce the resulting noise and effectively compensate for the noise present in the PL measurement. The choice of acquisition time is also connected with the IRF of the signal and the expected PL lifetimes. Moreover, it fundamentally manages total sensing time during which the entire data set is acquired from all masks. Here we used lifetime Τ = 20 µs, so that I_EXC signal with IRF = 2.07 µs was selected with data sampling 0.99 µs. For other signal parameters, different results than those presented here can be achieved after the same signal length extension. If we focus on shorter lifetimes, a faster signal (smaller IRF) is needed, and the total scanning time is also reduced in a given ratio. However, the simulated trends for R, SPC-SNR, and σ can be applied without compromising generality.

In order to suppress signal noise, we have deliberately avoided any mathematical signal filtering techniques so that we do not lose important information due to the random nature of the signal. Mathematical signal filtering could be optimized on the expenses of the general applicability of the results. Therefore, we explored instead the possibility to optimize the regularization parameter ɛ used in the PL decay retrieval.

The parameter ɛ was changed in the range 0.05 to 1. We observed a trade-off between the two effects. On the one hand, a higher value of ɛ smoothes the I_DA curve, while, on the other hand, it distorts the I_DA because some frequencies of the Fourier spectrum are favored. Therefore, we observed that there is an optimum value of ɛ connected to the noise level in the I_PL signal. As a rule of thumb, for a system with a low noise level, we recommend keeping the parameter ɛ at 0.1 or 0.2. For a system with a high noise level, it is possible to increase the regularization parameter ɛ more significantly.