Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods.
Regularization parameter selection for nonlinear iterative image restoration and MRI reconstruction using GCV and SURE-based methods.
复制标题
使用基于 GCV 和 SURE 的方法进行非线性迭代图像恢复和 MRI 重建的正则化参数选择。
DOI：
10.1109/tip.2012.2195015
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发表时间：
2012-08
期刊：
影响因子：
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通讯作者：
Fessler JA
中科院分区：
文献类型：
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作者：
Ramani S;Liu Z;Rosen J;Nielsen JF;Fessler JA
成像逆问题的正则化迭代重建算法需要选择适当的正则化参数值。我们专注于具有挑战性的问题，调整正则化参数的非线性算法的情况下，加性（可能复杂）高斯噪声。广义交叉验证（GCV）和（加权）均方误差（MSE）方法（基于Stein无偏风险估计- SURE）需要相对于数据的非线性重建算子（代表迭代算法）的雅可比矩阵。我们推导出所需的雅可比矩阵的两种类型的非线性迭代算法：标准的迭代加权最小二乘法和当代分裂Bregman算法，这两者都可以容纳各种各样的分析和合成型正则化的快速变种。所提出的方法迭代计算两个加权SURE类型的措施：预测SURE和投影SURE（需要噪声方差σ2的知识），和GCV（不需要σ2），这些算法。我们将该方法应用于图像恢复和磁共振图像（MRI）重建，使用全变分（TV）和分析型正则化。我们证明，通过模拟和实验与真实的数据，最大限度地减少预测SURE和投影SURE一致导致接近MSE最优重建。我们还观察到，最小化GCV产生的重建结果对于图像恢复接近MSE最优，对于MRI稍微次优。在这项工作中，与雅可比矩阵评估相关的理论推导原则上可以扩展到其他类型的正则化器和重建算法。
Regularized iterative reconstruction algorithms for imaging inverse problems require selection of appropriate regularization parameter values. We focus on the challenging problem of tuning regularization parameters for nonlinear algorithms for the case of additive (possibly complex) Gaussian noise. Generalized cross-validation (GCV) and (weighted) mean-squared error (MSE) approaches (based on Stein's Unbiased Risk Estimate— SURE) need the Jacobian matrix of the nonlinear reconstruction operator (representative of the iterative algorithm) with respect to the data. We derive the desired Jacobian matrix for two types of nonlinear iterative algorithms: a fast variant of the standard iterative reweighted least-squares method and the contemporary split-Bregman algorithm, both of which can accommodate a wide variety of analysis- and synthesis-type regularizers. The proposed approach iteratively computes two weighted SURE-type measures: Predicted-SURE and Projected-SURE (that require knowledge of noise variance σ2), and GCV (that does not need σ2) for these algorithms. We apply the methods to image restoration and to magnetic resonance image (MRI) reconstruction using total variation (TV) and an analysis-type ℓ1-regularization. We demonstrate through simulations and experiments with real data that minimizing Predicted-SURE and Projected-SURE consistently lead to near-MSE-optimal reconstructions. We also observed that minimizing GCV yields reconstruction results that are near-MSE-optimal for image restoration and slightly sub-optimal for MRI. Theoretical derivations in this work related to Jacobian matrix evaluations can be extended, in principle, to other types of regularizers and reconstruction algorithms.