transform op• Data kinds: image2d → image2d
• Call: import fullseye as fs; fs.ledger.anscombe_inverse(values, gain=1.0, read_sigma=0.0, offset=0.0, mode='algebraic') (implementation को सीधे call करना हो तो import photoncount; photoncount.anscombe_inverse(values, gain=1.0, read_sigma=0.0, offset=0.0, mode='algebraic'), registry से लेना हो तो opsphoton.get("anscombe_inverse"))
anscombe_transform का inverse — algebraic रूप से, या bias के बिना।
> नीचे का विस्तृत विवरण मूल text में है — सारांश और headings का अनुवाद हो चुका है।
mode="algebraic" is the exact algebraic inverse of 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 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).
• photon_timeresolved family guide
• Sample data catalog (DL URL / license) — 2-D के लिए skimage.data (BSD/public) + synthetic, 3-D के लिए real data sources (Stanford/PDS आदि) के DL URL।
• Operators का स्रोत और संदर्भ — इस op family के मूल research/methods के स्रोत।
• Algorithm का मानक स्रोत (लेखक, वर्ष) और उपयोग ऊपर की family usage guide में दिए गए हैं।
• photon_timeresolved — py -3.11 examples/photon_timeresolved.py
image2d को input के रूप में लेते हैं)photon_sample · photon_statistics · photon_uncertainty · anscombe_transform
transform)*Provenance: photoncount.py — PHOTON operator registry. यह per-op note tools/opdocs.py md से अपने-आप generate होता है (हाथ से edit न करें)।*
© 2026 Kazufumi Furuse — Fullseye operator documentation. Licensed under Apache-2.0.