Penalized least squares are widely used in signal and image processing. Yet, it suffers from a major limitation since it requires fine-tuning of the regularization parameters. Under assumptions on the noise probability distribution, Stein-based approaches provide unbiased estimator of the quadratic risk. The Generalized Stein Unbiased Risk Estimator is revisited to handle correlated Gaussian noise without requiring to invert the covariance matrix. Then, in order to avoid expansive grid search, it is necessary to design algorithmic scheme minimizing the quadratic risk with respect to regularization parameters. This work extends the Stein’s Unbiased GrAdient estimator of the Risk of Deledalle et al. (SIAM J Imaging Sci 7(4):2448–2487, 2014) to the case of correlated Gaussian noise, deriving a general automatic tuning of regularization parameters. First, the theoretical asymptotic unbiasedness of the gradient estimator is demonstrated in the case of general correlated Gaussian noise. Then, the proposed parameter selection strategy is particularized to fractal texture segmentation, where problem formulation naturally entails inter-scale and spatially correlated noise. Numerical assessment is provided, as well as discussion of the practical issues.

惩罚最小二乘广泛用于信号和图像处理。然而，它受到一个主要限制，因为它需要对正则化参数进行微调。在噪声概率分布的假设下，基于 Stein 的方法提供了二次风险的无偏估计。重新使用广义 Stein 无偏风险估计器来处理相关的高斯噪声，而无需反转协方差矩阵。然后，为了避免扩展网格搜索，有必要设计算法方案以最小化正则化参数的二次风险。这项工作扩展了 Deledalle 等人风险的 Stein 无偏梯度估计量。(SIAM J Imaging Sci 7(4):2448–2487, 2014) 到相关高斯噪声的情况，推导出正则化参数的一般自动调整。首先，在一般相关高斯噪声的情况下证明了梯度估计器的理论渐近无偏性。然后，所提出的参数选择策略特别适用于分形纹理分割，其中问题的制定自然需要尺度间和空间相关的噪声。提供了数值评估，以及对实际问题的讨论。