# Deconvolution of Poissonian Images

(Not updated yet by 2016)

## Problem Statement

Images are often corrupted by noise and blurring during the acquisition process. In a variety of applications, ranging from astronomical imaging to biological microscopy, the predominant source of noise follows a Poisson distribution due to the quantum nature of the photon-counting process at the detectors. The observation model for a linear degradation caused by blurring and Poisson noise is given by

$\mathrm{y} = \alpha\mathcal{P}\Big(\frac{\mathbf{H}\mathrm{x}}{\alpha}\Big)$

where $$\mathrm{y}\in \mathbb{R}^N$$ denotes the distorted observation of the unknown true image $$\mathrm{x}\in \mathbb{R}_+^N$$. $$\mathbf{H}$$ is the convolution and $$\alpha$$ is the scaling factor, which controls the strength of noise. Specifically, larger values of $$\alpha$$ will lead to lower intensity images and thus higher Poisson noise.

Our objective is to find an estimate $$\hat{\mathrm{x}}$$ so that it is the closest possible to $$\mathrm{x}$$ in the minimum MSE sense. That is, ideally we would like to minimize

$MSE=\frac{1}{N}\mathcal{E}\left\{\|\hat{\mathrm{x}}-\mathrm{x}\|^2\right\}$

where $$\mathcal{E}\{\cdot\}$$ denotes the mathematical expectation operator.

## Related Works

• Richardson-Lucy (RL) algorithm

• Richardson, William Hadley. “Bayesian-Based Iterative Method of Image Restoration.” JOSA 62.1 (1972): 55-59.

• Lucy, Leon B. “An iterative technique for the rectification of observed distributions.” The astronomical journal 79 (1974): 745.

• Regularized RL algorithm

• Total variation

• Dey, Nicolas, et al. “Richardson–Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution.” Microscopy research and technique 69.4 (2006): 260-266.

• Harmany, Zachary T., Roummel F. Marcia, and Rebecca M. Willett. “This is SPIRAL-TAP: Sparse Poisson intensity reconstruction algorithms—theory and practice.” Image Processing, IEEE Transactions on 21.3 (2012): 1084-1096.

• Benfenati, Alessandro, and Valeria Ruggiero. “Inexact Bregman iteration with an application to Poisson data reconstruction.” Inverse Problems 29.6 (2013): 065016.

• Wavelet-based

• Starck, Jean-Luc, and Fionn Murtagh. “Image restoration with noise suppression using the wavelet transform.” Astronomy and Astrophysics 288 (1994): 342-348.

• Nowak, Robert D., and Michael J. Thul. “Wavelet-vaguelette restoration in photon-limited imaging.” Acoustics, Speech and Signal Processing, 1998. Proceedings of the 1998 IEEE International Conference on. Vol. 5. IEEE, 1998.

• Carlavan, Mikael, and Laure Blanc-Féraud. “Sparse Poisson noisy image deblurring.” Image Processing, IEEE Transactions on 21.4 (2012): 1834-1846.

• Accelerated

• Wang, Hongbin, and Paul C. Miller. “Scaled heavy-ball acceleration of the Richardson-Lucy algorithm for 3D microscopy image restoration.” Image Processing, IEEE Transactions on 23.2 (2014): 848-854.

• Harmany, Zachary T., Roummel F. Marcia, and Rebecca M. Willett. “This is SPIRAL-TAP: Sparse Poisson intensity reconstruction algorithms—theory and practice.” Image Processing, IEEE Transactions on 21.3 (2012): 1084-1096.

• Figueiredo, Mário AT, and José M. Bioucas-Dias. “Restoration of Poissonian images using alternating direction optimization.” Image Processing, IEEE Transactions on 19.12 (2010): 3133-3145.

• Setzer, Simon, Gabriele Steidl, and Tanja Teuber. “Deblurring Poissonian images by split Bregman techniques.” Journal of Visual Communication and Image Representation 21.3 (2010): 193-199.

• Pustelnik, Nelly, Caroline Chaux, and Jean-Christophe Pesquet. “Parallel proximal algorithm for image restoration using hybrid regularization.” Image Processing, IEEE Transactions on 20.9 (2011): 2450-2462.

• Chen, Dai-Qiang. “Regularized generalized inverse accelerating linearized alternating minimization algorithm for frame-based poissonian image deblurring.” SIAM Journal on Imaging Sciences 7.2 (2014): 716-739.

• Others

• Variance stabilizing transform

• Dupé, François-Xavier, Jalal M. Fadili, and Jean-Luc Starck. “A proximal iteration for deconvolving Poisson noisy images using sparse representations.” Image Processing, IEEE Transactions on 18.2 (2009): 310-321.

• Rond, Arie, Raja Giryes, and Michael Elad. “Poisson Inverse Problems by the Plug-and-Play scheme.” arXiv preprint arXiv:1511.02500 (2015).

• Second-order derivatative-based regularizer

• Lefkimmiatis, Stamatios, and Michael Unser. “Poisson image reconstruction with Hessian Schatten-norm regularization.” Image Processing, IEEE Transactions on 22.11 (2013): 4314-4327.

• Dictionary learning

• Ma, Liyan, et al. “A Dictionary learning approach for Poisson image deblurring.” Medical Imaging, IEEE Transactions on 32.7 (2013): 1277-1289.

## Resources

• Review papers

• Sarder, Pinaki, and Arye Nehorai. “Deconvolution methods for 3-D fluorescence microscopy images.” Signal Processing Magazine, IEEE 23.3 (2006): 32-45.

• Kervrann, Charles, et al. “A guided tour of selected image processing and analysis methods for fluorescence and electron microscopy.” IEEE J. Sel. Top. Sign. Process. 10.1 (2016): 6-30.

• Bertero, Mario, et al. “Image deblurring with Poisson data: from cells to galaxies.” Inverse Problems 25.12 (2009): 123006.