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.

Degradation model

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.