lifetime op• Data kinds: counts → table
• Call: import photoncount; photoncount.lifetime_fit(decay, bin_ps=100.0, background=None, min_counts=1.0, start_bin=None) (or opsphoton.get("lifetime_fit"))
Mono-exponential fluorescence lifetime from a TCSPC decay histogram.
Fits `I(t) = A*exp(-t/tau) + b` by a **Poisson-weighted log-linear least
squares**: the background is removed, the logarithm of the remaining counts
is linear in `t with slope -1/tau`, and each bin is weighted by its own
counts because `var(ln N) ~ 1/N` — which is exactly the Poisson error bar
:func:photon_uncertainty reports.
The fit starts at the peak bin by default (or at *start_bin* if given):
the rising edge before the peak is the instrument response convolved with the
decay, not the decay, and including it flattens the log slope and so biases
the lifetime long. Measured on a 2000 ps decay blurred by a 600 ps IRF
(256 bins x 100 ps): starting at the peak (bin 4) gives 2008.0 ps (+0.40%),
forcing `start_bin=0` gives 2100.7 ps (+5.0%) — a 12x worse bias from four
extra bins. Only bins with more than *min_counts* counts after background
removal take part (the
logarithm of 0 is `-inf`, and single-count tail bins carry almost no
information but huge log-scatter).
*background* is the flat pedestal per bin; `None` (default) estimates it as
the median of the last decile of bins, which for a decay is tail. Pass
`0.0` to state that the data are already background free.
Returns a dict: `lifetime_ps · amplitude (the fitted A at t=0`
of the fit window, in counts per bin) · `background` (the level used) ·
`start_bin · n_bins_used · r_squared` (of the weighted log fit).
Ground truth: on a noiseless exponential the recovery is exact —
`lifetime_ps came back as 2000.000000000 ps for tau = 2000 ps` (256
bins x 100 ps), a measured relative error of 0.0, with `r_squared` 1.0.
*That stays true when the histogram is built by integrating the exponential
over each bin* rather than sampling it, because bin integration multiplies
every bin by the same constant and so cannot change the slope.
With Poisson noise the log-linear estimator is biased high, and the size
of the bias is worth knowing: at 20000 total photons, seed 0,
`min_counts=1` gives 2058.8 ps (+2.9%) from 133 bins, and raising
`min_counts` to 10 gives 2047.3 ps (+2.4%) from 94 bins. Averaged over
seeds 0-19 at `min_counts=10` the mean is 2014.3 ps (**+0.72% systematic
bias**) with a 18.2 ps (0.9%) seed-to-seed spread — so seed 0 is a
2-sigma-high draw, and the bias, not the scatter, is the thing to remember.
It comes from `E[ln N] < ln E[N]` in the sparse tail; a full Poisson MLE
would remove it and is not what this op does.
Raises `ValueError`: negative, non-finite or non-1-D *decay*, a
non-positive *bin_ps*, a negative *background* / *min_counts*, a *start_bin*
outside the histogram, fewer than 2 usable bins after the background and
threshold cuts (a straight line needs two points), a degenerate fit (all
usable bins at the same time), and — instead of returning a negative
lifetime — a fitted slope that is zero or positive, i.e. a profile that does
not decay.
• photon_timeresolved family guide
• Sample-data catalog (download URLs / licences) — 2-D uses skimage.data (BSD/public domain) plus synthetic images; 3-D lists download URLs for real data sources (Stanford, PDS, …).
• Operator provenance and references — the sources of the research/methods this op family came from.
• The canonical algorithm (author, year) and its uses are named in the family usage guide above.
• photon_timeresolved — py -3.11 examples/photon_timeresolved.py
table as input)—
lifetime)*Provenance: photoncount.py — PHOTON operator registry. This per-op note is generated by tools/opdocs.py md (do not hand-edit).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.