transform op• Datenarten: image2d → image2d
• Aufruf: import photoncount; photoncount.anscombe_inverse(values, gain=1.0, read_sigma=0.0, offset=0.0, mode='algebraic') (oder opsphoton.get("anscombe_inverse"))
Invertiert :func:anscombe_transform — algebraisch oder erwartungstreu.
> Die ausführliche Beschreibung unten ist der Originaltext — Zusammenfassung und Überschriften sind übersetzt.
`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).
• Leitfaden zur Familie photon_timeresolved
• Katalog der Beispieldaten (Download-URLs / Lizenzen) — 2-D nutzt skimage.data (BSD/Public Domain) plus synthetische Bilder, 3-D nennt Download-URLs echter Datenquellen (Stanford, PDS, …).
• Herkunft und Literatur der Operatoren — die Quellen der Forschung/Verfahren, auf denen diese Operatorfamilie beruht.
• Der kanonische Algorithmus (Autor, Jahr) und seine Anwendungen stehen im Familienleitfaden oben.
• photon_timeresolved — py -3.11 examples/photon_timeresolved.py
image2d als Eingabe)photon_sample · photon_statistics · photon_uncertainty · anscombe_transform
transform)*Provenance: photoncount.py — PHOTON Operator-Registry. Diese Notiz wird von tools/opdocs.py md erzeugt (nicht von Hand bearbeiten).*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.