Optimal inversion of the Anscombe and Generalized Anscombe variance-stabilizing transformations

Efficient Denoising and Deblurring of Low-Count Poisson Images Using Off-the-Shelf Gaussian Filters


Abstract
The removal of Poisson or Poisson-Gaussian noise is often performed through the following three-step procedure. First, the noise variance is stabilized by applying either the Anscombe or the Generalized Anscombe root transformation (also called Anscombe transform) to the data, producing a signal in which the noise can be treated as additive Gaussian noise with unitary variance. Second, the noise is removed using a conventional denoising algorithm for additive white Gaussian noise. Third, an inverse transformation is applied to the denoised signal, obtaining the estimate of the signal of interest.
The choice of the proper inverse transformation is crucial in order to minimize the bias error which arises when the nonlinear forward transformation is applied.
We introduce the exact unbiased inverses of the Anscombe and Generalized Anscombe transformations and show that they play an integral part in ensuring accurate denoising results, particularly at the low-count regime, leading to state-of-the-art photon-limited imaging without any notable increase in the computational complexity compared to the other inverses. We also show that these inverses are optimal in the sense that they can be interpreted as maximum-likelihood inverses. Moreover, we thoroughly analyze the behaviour of the proposed inverses, which enables us to derive their closed-form approximations.
Our recent work on iterative filtering of combinations of the noisy image with a progressively refined estimate leads to very efficient denoising and deblurring of even extremely low-count images (less than one count per pixel) using off-the-shelf Gaussian filters.




SOFTWARE PEOPLE LINKS REFERENCES


Software new
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Denoising software for Poisson and Poisson-Gaussian data
for Matlab (ver. 7 or later)
download zip package

5-Mbyte zip-file
includes functions implementing the exact unbiased inverse in the iterative framework for Poisson denoising via VST

v3.00, released March 16, 2016





Read me
Information and instructions



Iterative Poisson image denoising software

for Matlab (ver. 7 or later)
download zip package

3.3-Mbyte zip-file
Stand-alone package: includes all required invansc and BM3D components.

v1.00, released March 16, 2016





Read me
Information and instructions



Iterative Poisson image deblurring software
new

for Matlab (ver. 7 or later)
download zip package

1.5-Mbyte zip-file
Stand-alone package: includes all required invansc and BM3D components.

v1.00new, released April 18, 2017






Any unauthorized use of the provided software and files for industrial or profit-oriented activities is expressively prohibited. By downloading any of the files contained in this site, you implicitly agree to all the terms of the TUT limited license. Please read the TUT limited license PDF before you proceed with downloading any of the files.



People People

  Lucio Azzari
  Markku Mäkitalo
  Alessandro Foi



Links Links

  Block-matching and 3D filtering (BM3D) algorithm (with Matlab software)



References References

L. Azzari and A. Foi, “Variance Stabilization in Poisson Image Deblurring”, Proc. 2017 IEEE Int. Sym. Biomedical Imaging, Melbourne, Australia, April 18-21, 2017.

L. Azzari and A. Foi, “Variance Stabilization for Noisy+Estimate Combination in Iterative Poisson Denoising”, IEEE Signal Processing Letters, vol. 23, no. 8, pp. 1086-1090, August 2016.  DOIhttp://doi.org/10.1109/LSP.2016.2580600 Supplementary material

M. Mäkitalo and A. Foi, “Noise parameter mismatch in variance stabilization, with an application to Poisson-Gaussian noise estimationIEEE Trans. Image Process., vol. 23, no. 12, pp. 5348-5359, December 2014.  DOIhttp://doi.org/10.1109/TIP.2014.2363735

PDFM. Mäkitalo and A. Foi, “Optimal inversion of the generalized Anscombe transformation for Poisson-Gaussian noise”, IEEE Trans. Image Process., vol. 22, no. 1, pp. 91-103, January 2013.  DOI doi:10.1109/TIP.2012.2202675

PDFM. Mäkitalo and A. Foi, “Poisson-Gaussian denoising using the exact unbiased inverse of the generalized Anscombe transformation”, Proc. 2012 IEEE Int. Conf. Acoustics, Speech, Signal Process. (ICASSP 2012), pp. 1081-1084, Kyoto, Japan, March 2012.

PDFM. Mäkitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising”, IEEE Trans. Image Process., vol. 20, no. 1, pp. 99-109, January 2011. DOIdoi:10.1109/TIP.2010.2056693

PDFM. Mäkitalo and A. Foi, “A closed-form approximation of the exact unbiased inverse of the Anscombe variance-stabilizing transformation”, IEEE Trans. Image Process., vol. 20, no. 9, pp. 2697-2698, September 2011. DOI doi:10.1109/TIP.2011.2121085

PDFM. Mäkitalo and A. Foi, “On the inversion of the Anscombe transformation in low-count Poisson image denoising”, Proc. Int. Workshop on Local and Non-Local Approx. in Image Process., LNLA 2009, Tuusula, Finland, pp. 26-32, August 2009. DOIdoi:10.1109/LNLA.2009.5278406



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