Fig 1.
Markov on-off interrupted light model and intensity function.
State 1 indicates the light is on and switching occurs with mean exponential waiting time of . Photons are emitted with mean rate αx(t) with the model state x(t) ∈ {0, 1} and un-normalised intensity λ(t) ∈ {0, α}.
Fig 2.
Snyder inter-event intensity estimate for the interrupted model when all photon times are known.
The solution decays more rapidly relative to the switch time
as the mean number of photons per on state of the light model, β increases. The solution stays close to state 1 with increasing β, relative to the time to the next photon
. This indicates increased certainty as there is more information available.
Fig 3.
Spike metric optimisation shows a clear minimum when only photon counts are used.
Parameter q indicates the relative cost of shifting versus inserting/deleting photons. For small values of q (shifts are unimportant) a clear minimum appears at m = 1 indicating integrate-fire works best with a threshold set at the mean bump area.
Fig 4.
The integrate-fire-Snyder outperforms other machine learning and photon estimating schemes across intensity.
The integrate-fire was tested against Gaussian processes and optimised finite impulse response filters which used the bump data directly. It was also compared to other schemes that also estimated photons for processing with Snyder filters. These fired photons based on pure current thresholds or bump gradients. Data is shown for the interrupted model at γ = 20 for a given representative light model trajectory of 7000-8000 photons. The integrate-fire showed superior overall performance. Consistent results have also been obtained for γ = 5. This motivated the use of the integrate-fire-Snyder as a cascade estimator.
Fig 5.
General integrate-fire-Snyder scheme.
The standard Snyder filter gives the optimal (MMSE) intensity estimate given known photons. The random square wave intensity producing these photons delineates the on-off times in the interrupted light model. The MMSE estimate exactly characterises achievable performance at the front end of the cascade. The photons are converted into bumps by the cascade. The integrate-fire uses an optimised threshold to convert these bumps into estimated photons. These estimated photons are then Snyder filtered to obtain a non-MMSE estimator that upper bounds the deterioration introduced by the cascade.
Fig 6.
Suboptimal integrate-fire-Snyder on bumps outperforms MMSE linear filtering on known photons.
The integrate-fire-Snyder was run for γ = [5, 10, 20, 30] and results across 10 independent 8000 photon long light model trajectories given. The error-bars delineate the minimum to maximum of the MSE curves for each γ. Clearly there exist several γ over which the integrate-fire on all cascade noise does better than the linear MMSE on just photon noise, across a wide β range. Given the disparity, applying linear or continuity approximations can be a strongly misleading measure of the noise performance of discrete stochastic systems.
Fig 7.
The dominant noise transition from extrinsic to mean delay, with intensity β, is consistent for light models with differing relative flicker speed, γ.
Parameter γ indicates the number of bumps which can be received per Markov switch time of the light model. Higher γ therefore means a relatively slower flicker. At lower γ less information is available and hence the MSE higher. The ‘Snyder, delay’ curves are MMSE values obtained with the Snyder filter optimised for a deterministic delay set to the mean QB latency. These lower bound the noise introduced by the cascade. The ‘Integrate-fire, all noise’ ones are upper bounds as they involve processing the QBs from the fully stochastic Nikolic model. The closeness of both of these curve sets reflects the dominance of mean delay among cascade noise components. This hypothesis is further confirmed by the closeness of ‘Integrate-fire, all noise’ with the MSE generated by applying the integrate-fire-Snyder on the deterministic Nikolic model, which only featured the mean delay (‘Integrate-fire, delay’) and fixed QB shapes. The ‘Snyder, photon noise’ gives the noise floor since it is the MMSE achievable at the front of the cascade. Convergence with this at low β shows cascade noise is unimportant in that regime. All curves are averages over 10 independent 8000 photon-QB streams with error bars indicating the maximum and minimum MSE swings around this value.