We study a variational formulation for reconstructing nonlinearly distorted signals corrupted with a Poisson-Gaussian noise. In this situation, the data fidelity term consists of a sum of a weighted least squares term and a logarithmic one. Both of them are precomposed by a nonlinearity, modelling a clipping effect, which is assumed to be rational. A regularization term, being a piecewise rational approximation of the $\ell _0$ function provides a suitable sparsity measure with respect to a preset linear operator. We propose a global optimization approach for such a problem. More specifically, it is first transformed into a generalized moment problem by introducing some auxiliary variables. Then, a hierarchy of semidefinite programming relaxations is built. Numerical examples show the good performance of the proposed approach.

中文翻译：

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恢复被泊松-高斯噪声破坏的限幅信号的全局优化

我们研究了一种用于重建被泊松-高斯噪声破坏的非线​​性失真信号的变分公式。在这种情况下，数据保真度项由加权最小二乘项和对数项之和组成。它们都由非线性预先构成，对剪裁效应进行建模，假设是合理的。作为 $\ell_0$ 函数的分段有理近似的正则化项提供了关于预设线性算子的合适稀疏度量。我们为这样的问题提出了一种全局优化方法。更具体地说，首先通过引入一些辅助变量将其转化为广义矩问题。然后，建立半定规划松弛的层次结构。数值例子显示了所提出方法的良好性能。