inverse_generalized_anscombe_transform

deepinv.models.inverse_generalized_anscombe_transform(x, gain, sigma)

Inverse Generalized Anscombe Transform (IGAT)

The transform converts an approximately Gaussian signal \(z\) (output of the generalized_anscombe_transform()) back to the original Poisson-Gaussian domain with gain \(\gamma\) and Gaussian noise standard deviation \(\sigma\), see Makitalo and Foi[1].

The transform is defined as the algebraic inverse of the generalized_anscombe_transform():

\[h^{-1}(x) = \frac{1}{4}x^2 + \frac{1}{4}\sqrt{\frac{3}{2}}\, x^{-1} - \frac{11}{8} x^{-2} + \frac{5}{8}\sqrt{\frac{3}{2}}\, x^{-3} - \frac{1}{8} - \frac{\sigma^2}{\gamma^2}\]

Note

The formula varies slightly from the one proposed in [1], as the library considers a normalized Poisson-Gaussian noise model, \(y = \gamma \mathcal{P}(x/\gamma) + \epsilon\), whereas the authors consider \(y = \gamma \mathcal{P}(x) + \epsilon\).

Parameters:

• x (torch.Tensor) – Anscombe-transformed tensor.

• gain (float | torch.Tensor) – Gain of the Poisson distribution \(\gamma\)

• sigma (float | torch.Tensor) – Standard deviation of the Gaussian noise \(\sigma\)

Return torch.Tensor:

Reconstructed measurements in the original domain

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References:

[1] (1,2)

Markku Makitalo and Alessandro Foi. Optimal inversion of the generalized anscombe transformation for poisson-gaussian noise. IEEE transactions on image processing, 22(1):91–103, 2012.