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Parametric PSF estimation based on predicted-SURE with \(\ell _1\)-penalized sparse deconvolution

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Abstract

Point spread function (PSF) estimation plays an important role in blind image deconvolution. It has been shown that incorporating Wiener filter, minimization of the predicted Stein’s unbiased risk estimate (p-SURE)—unbiased estimate of predicted mean squared error—could yield an accurate PSF estimate. In this paper, we provide a theoretical analysis for the PSF estimation error, which shows that the better deconvolution leads to more accurate PSF estimate. It motivates us to incorporate an \(\ell _1\)-penalized sparse deconvolution into the p-SURE minimization, instead of the Wiener-type filtering. In particular, based on FISTA—one of the most popular iterative \(\ell _1\)-solvers, we evaluate the p-SURE for each update, by Jacobian recursion and Monte Carlo simulation. Numerical results of both synthetic and real experiments demonstrate the improvements in PSF estimate, and therefore, deconvolution performance.

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Notes

  1. See [13, 25] for a few parametric examples.

  2. The optimal solution \(\mathbf {s}^\star \) may not be unique. The uniqueness of the solution depends on the specific parametric form of PSF.

  3. This similar phenomenon is also frequently encountered in non-blind sparse deconvolution (where \(\mathbf {H}\) is exactly known), especially when the regularization parameter \(\lambda \) is very small. Refer to Fig. 2-(1) of [26], Fig. 2-(2) of [29] and Fig. 2-(2) of [28] for the typical examples with \(\ell _1\) or total variation as regularizer, we can also see that the predicted error increases after a few iterations, since the iterative algorithm used aims at minimizing a given functional, not the predicted or reconstructed error. That is also why there is generally no need to set a very strict stopping criterion for convergence in practice, since the exact solution at the convergence usually does not have the good restoration quality, especially when \(\lambda \) is small [3, 18].

  4. We cannot compute p-MSE, since the original image \(\mathbf {x}_0\) and true PSF \(\mathbf {H}_0\) are unknown in the real experiments.

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Authors and Affiliations

  1. National Key Laboratory of Science and Technology on Test Physics and Numerical Mathematics, Beijing, 100076, China

    Feng Xue, Jiaqi Liu & Xia Ai

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  1. Feng Xue
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  2. Jiaqi Liu
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  3. Xia Ai
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Correspondence to Feng Xue.

Additional information

This work was supported by the National Natural Science Foundation of China (No. 61401013)

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Xue, F., Liu, J. & Ai, X. Parametric PSF estimation based on predicted-SURE with \(\ell _1\)-penalized sparse deconvolution. SIViP 13, 635–642 (2019). https://doi.org/10.1007/s11760-018-1391-9

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