A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations

@article{Dup2008API,
  title={A Proximal Iteration for Deconvolving Poisson Noisy Images Using Sparse Representations},
  author={François-Xavier Dup{\'e} and Mohamed-Jalal Fadili and Jean-Luc Starck},
  journal={IEEE Transactions on Image Processing},
  year={2008},
  volume={18},
  pages={310-321}
}
We propose an image deconvolution algorithm when the data is contaminated by Poisson noise. The image to restore is assumed to be sparsely represented in a dictionary of waveforms such as the wavelet or curvelet transforms. Our key contributions are as follows. First, we handle the Poisson noise properly by using the Anscombe variance stabilizing transform leading to a nonlinear degradation equation with additive Gaussian noise. Second, the deconvolution problem is formulated as the… 
View on IEEE
arxiv.org
220 Citations
Highly Influential Citations
14
Background Citations
64
Methods Citations
67
Results Citations
6

Figures and Tables from this paper

220 Citations

Deconvolution under Poisson noise using exact data fidelity and synthesis or analysis sparsity priors

A primal-dual proximal splitting approach for restoring data corrupted with poisson-gaussian noise

This work proposes a convex optimization algorithm for the reconstruction of signals degraded by a linear operator and corrupted with mixed Poisson-Gaussian noise, and derives a primal-dual iterative scheme for minimizing the associated penalized criterion.

PURE-LET Image Deconvolution

Simulation experiments indicate that the proposed non-iterative image deconvolution algorithm for data corrupted by Poisson or mixed Poisson-Gaussian noise outperforms the state-of-the-art techniques, in terms of both restoration quality and computational complexity.

A Convex Approach for Image Restoration with Exact Poisson-Gaussian Likelihood

This work proposes a convex optimization strategy for the reconstruction of images degraded by a linear operator and corrupted with a mixed Poisson-Gaussian noise, and shows that in a variational framework the Shifted Poisson and Exponential approximations lead to very good restoration results.

Poisson Image Reconstruction With Hessian Schatten-Norm Regularization

This paper proposes an efficient framework for Poisson image reconstruction, under a regularization approach, which depends on matrix-valued regularization operators and derives a link that relates the proximal map of an lp norm with the proximate map of a Schatten matrix norm of order p.

Inverse problems with poisson noise: Primal and primal-dual splitting

The well-posedness of the optimization problem, characterize the corresponding minimizers, and solve it by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory.

DECONVOLUTION OF CONFOCAL MICROSCOPY IMAGES USING PROXIMAL ITERATION AND SPARSE REPRESENTATIONS

The results on simulated microscopy images of neurons and cells are confronted to some state-of-the-art algorithms and show that the approach is very competitive, and as expected, the importance of the non-linearity due to Poisson noise is more salient at low and medium intensities.

DECONVOLUTION OF CONFOCAL MICROSCOPY IMAGES USING PROXIMAL ITERATION AND SPARSE REPRESENTATIONS F.-X. Dup´

The results on simulated microscopy images of neurons and cells are confronted to some state-of-the-art algorithms and show that the approach is very competitive, and as expected the importance of the non-linearity due to Poisson noise is more salient at low and medium intensities.

Linear inverse problems with various noise models and mixed regularizations

Two algorithms for solving linear inverse problems when the observations are corrupted by noise are proposed by means of primal and primal-dual proximal splitting algorithms originating from the field of non-smooth convex optimization theory.

Linear inverse problems with noise: primal and primal-dual splitting

In this paper, we propose two algorithms for solving linear inverse problems when the observations are corrupted by noise. A proper data fidelity term (log-likelihood) is introduced to reflect the…
...

References

SHOWING 1-10 OF 77 REFERENCES

Sparse Representation-based Image Deconvolution by iterative Thresholding

Recent advances in fast solution algorithms of deconvolution problems, i.e. Stagewise Iterative Thresholding, are exploited to solve the optimization problem and provide fast and good image recovery results.

An EM algorithm for wavelet-based image restoration

An expectation-maximization (EM) algorithm for image restoration (deconvolution) based on a penalized likelihood formulated in the wavelet domain is introduced, and it is shown that under mild conditions the algorithm converges to a globally optimal restoration.

Sparse signal reconstruction from limited data using FOCUSS: a re-weighted minimum norm algorithm

A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided and Mathematical results on conditions for uniqueness of sparse solutions are also given.

A fast iterative thresholding algorithm for wavelet-regularized deconvolution

One of the first applications of wavelet-regularized deconvolution to 3D fluorescence microscopy is presented, and it is shown that the subband-dependent parameters allow for a substantial convergence acceleration compared to the existing optimization method.

FAST WAVELET-REGULARIZED IMAGE DECONVOLUTION

  • C. VoneschM. Unser
  • Computer Science
    2007 4th IEEE International Symposium on Biomedical Imaging: From Nano to Macro
  • 2007
A modified version of the deconvolution algorithm introduced by Figueiredo and Nowak is presented, which leads to a substantial acceleration at high SNR levels, suggesting that wavelet-domain l1-regularization may become tractable for the deconVolution of large datasets, e.g. in fluorescence microscopy.

Signal Recovery by Proximal Forward-Backward Splitting

We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation…

Image restoration with noise suppression using a multiresolution support.

Nonlinear Solution of Linear Inverse Problems by Wavelet–Vaguelette Decomposition

We describe the wavelet–vaguelette decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of…

Multiscale Poisson data smoothing

A framework for non‐linear multiscale decompositions of Poisson data that have piecewise smooth intensity curves is introduced, which combines the advantages of the Haar–Fisz transform with wavelet smoothing and (Bayesian) multiscales likelihood models, with additional benefits, such as extendability towards arbitrary wavelet families.

An iterative thresholding algorithm for linear inverse problems with a sparsity constraint

It is proved that replacing the usual quadratic regularizing penalties by weighted 𝓁p‐penalized penalties on the coefficients of such expansions, with 1 ≤ p ≤ 2, still regularizes the problem.
...