## 1 Introduction
It is also worth noting that even though the decay of delayed luminescence from an organism will reach a steady noise baseline, the noise baseline may not be random noise or may not contain random noise alone. Instead, the noise baseline could rest at the level of spontaneous ultraweak photon emission, which is challenging to measure using time-gated configurations that are conventional for acquiring the time trace of delayed luminescence. In practice, an imaging system for studying the luminescence of an organism may have to compromise between robustly acquiring the decay kinetics of the delayed luminescence and reliably acquiring the ultraweak spontaneous photon emission. In this work, we demonstrate an irradiation-acquisition interleaved time-integrated imaging approach that may enable the reliable acquisition of slow-decay delayed luminescence down to the level of steady-state ultraweak photon emission that is noise-laden. Repetitive irradiation was interleaved with gradually increased times of exposure. Such instrumental configuration, demonstrated with yeast as the sample, allows for the assessment of the decay kinetics of the delayed luminescence via the time differentiation of the time-integrated post-irradiation luminescence to reach the level of noise-laden ultraweak photon emission at the expense of the total time of acquisition. The principle that is intuitive to the proposed approach is illustrated in Figure 1 in the context of what contributes to photon emission post-stimulation, forming delayed luminescence. We denote the pure delayed photon emission (photon flux, or the number of photons per area per time) as \(R_{DL}(t)\)\([\mathrm{cm^{-2}\cdot s^{-1}}]\). A time integration of the delayed luminescence alone then leads to  
\[I_{DL}(0\Longleftrightarrow t)=\int_{0}^{t}R_{DL}(t)dt \tag{1}\]We note that \(I_{DL}(0\Longleftrightarrow t)\) corresponds to the total number of photons per area that are accumulated over a duration of \([0,t]\). Therefore \(I_{DL}(0\Longleftrightarrow t)\) has a unit of \([\text{cm}^{-2}]\). However, \(I_{DL}(0\Longleftrightarrow t)\) is also a function of time as it is related to the duration \([0,t]\) over which \(R_{DL}(t)\) is specified at each point of time \(t\). We denote the baseline noise of photon flux acquired by any photo-electronic detector as \(R_{noise}(t)\). The randomness of the electronic noise will be set to zero for the following time-ensembled value of the baseline noise throughout integration that is significantly longer than the characteristic temporal scale of its variance:  
\[\int_{t}^{t+AI}R_{noise}(t^{\prime})dt=\left\langle R_{noise}(t^{\prime}) \right\rangle=0 \tag{2}\]  
Since delayed luminescence is acquired in the presence of random noise, the acquired photon emission then becomes:  
\[R_{DL}^{noise}(t)=R_{DL}(t)+R_{noise}(t) \tag{3}\]  
and the time integration of the acquired photon emission of Equation (3) becomes:  
\[I_{DL}^{noise}(0\Longleftrightarrow t)=\int_{0}^{t}R_{DL}(t^{\prime})dt+ \left\langle R_{noise}(t)\right\rangle=\int_{0}^{t}R_{DL}(t^{\prime})dt\prime \tag{4}\]  
Now, let us assume the presence of an ultraweak photon emission \(R_{UPE}(t)\) that is not much different in magnitude than random noise. The ultraweak photon emission  
Figure 1: Conceptual illustration of the components of photon emission in measuring delayed luminescence. The solid trace refers to a kind of time-varying photon emission, and the dashed line marks the time integration of a kind of time-varying photon emission. (**A**) Pure delayed luminescence. (**B**) Random noise over a time duration much longer than its time of variation. (**C**) Delayed luminescence contaminated with random noise. (**D**) Delayed luminescence adding to the level of the baseline ultraweak photon emission. (**E**) Baseline ultraweak photon emission contaminated with random noise. (**F**) Delayed luminescence adding to the residual ultraweak photon emission contaminated with random noise.