transform opimage2d → image2dimport fullseye as fs; fs.ledger.anscombe_inverse(values, gain=1.0, read_sigma=0.0, offset=0.0, mode='algebraic') (実装を直接呼ぶなら import photoncount; photoncount.anscombe_inverse(values, gain=1.0, read_sigma=0.0, offset=0.0, mode='algebraic')、台帳から引くなら opsphoton.get("anscombe_inverse"))Invert :func:anscombe_transform — algebraically, or without bias.
mode="algebraic" is the exact algebraic inverse of
:func:anscombe_transform::
x = ((g*A/2)^2 - (3/8)*g^2 - sigma_r^2)/g + offset
so anscombe_inverse(anscombe_transform(x)) == x to machine precision
(measured over 100001 values of x spanning [0, 1e4]: max absolute
error 2.7e-12, max relative error 3.7e-16 for x > 1). It is, however,
biased: E[A(X)] != A(E[X]), so applying it to a denoised (i.e.
averaged) Anscombe image underestimates the intensity.
mode="unbiased" is the closed-form exact unbiased inverse of Makitalo &
Foi (IEEE TIP 2011)::
x = D^2/4 + (1/4)*sqrt(3/2)/D - (11/8)/D^2 + (5/8)*sqrt(3/2)/D^3 - 1/8
which is defined for the classical transform only (gain=1,
read_sigma=0, offset=0) — passing generalised parameters with this
mode raises rather than returning a formula that does not apply.
Measured bias — apply the inverse to the ideal denoised value
D = E[A(X)], X ~ Poisson(lambda), and compare with lambda. Both
computed exactly from the Poisson pmf (no sampling):
======== ================== ================== lambda algebraic bias unbiased bias ======== ================== ================== 1 -0.179361 -0.003668 2 -0.231074 -0.006374 4 -0.249688 +0.003779 10 -0.250227 +0.016904 30 -0.250019 +0.017041 100 -0.250002 +0.011960 ======== ================== ==================
The algebraic inverse converges to a constant -1/4 photon offset, which
at 1 photon/pixel is a 18% error; the closed form keeps the worst case to
0.017 photons (a 49x reduction at lambda = 1, 15x at its own worst point
near lambda = 10-30). It is a closed-form approximation of the exact
unbiased inverse, so it is not bias-free — those +0.017 are the honest
residual, not round-off.
The result is clipped at 0 — stated here rather than done quietly — but the
clip essentially never fires: the closed form’s positive root is
exactly A(0) = 2*sqrt(3/8) = 1.2247448714 (measured: the root and
A(0) agree to 0.0), which is also the smallest value
:func:anscombe_transform can produce, so over the whole valid domain the
formula is non-negative to round-off (measured minimum -1.11e-16 over
D in [A(0), 6] on 500001 samples). Below A(0) it does go
genuinely negative (-0.0217 at D = 1.20), which is why values there are
refused outright rather than clipped.
Returns a float64 array of the same shape as values.
Raises ValueError: non-finite values, an unknown mode, a
non-positive gain, a negative read_sigma, mode="unbiased" combined
with any non-default generalised parameter, and — instead of dividing by
zero — mode="unbiased" with any value <= 0 (the 1/D^3 term).
py -3.11 examples/photon_timeresolved.pyimage2d を入力に取れる)photon_sample · photon_statistics · photon_uncertainty · anscombe_transform
transform)Provenance: photoncount.py — PHOTON operator registry. この per-op ノートは tools/opdocs.py md が自動生成(手編集しない)。
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.