Determining τ from two gated measurements

Fluorescence is emitted from a single fluorophore with a lifetime of τ.
The idealized exponential decay is indicated by the dottedline in Figure
1. However, you have a detector with a signifcant exposure time ∆t,
such that the fluorescence level changes during the exposure. Your setup
allows you to trigger two such exposures D1 and D2, indicated in the
figure. As an experimenter, you know the two measurements began (were
triggered) at 0.8 and 2.5 nsec, but you do not know ∆t, the exposure
time (the areas in the figure are just for illustration).

You have a linear array of 51 such detectors. On the website, you can
download a file

called FLIMdata.dat, a 2-by-51 matrix that represents D1 and D2
measurements at each

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τ , the fluorescence lifetime, versus detector position (which would
correspond to, for example, a linear cut across a region of cells).

This will require some open-ended thinking about how to extract τ from
the measured

data. You should find that the calculated lifetime is much more
spatially uniform than

the noisy intensity data, with a single well-defined region of spatial
contrast. What is the

uniform background lifetime, in nanoseconds, and what is the lifetime in
the contrasting

region?