Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography
Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography
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DOI：
10.1088/1361-6420/ab80d7
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发表时间：
2020-06-01
期刊：
影响因子：
2.1
通讯作者：
Khatri, Ratna
中科院分区：
文献类型：
--
作者：
Antil, Harbir;Di, Zichao Wendy;Khatri, Ratna
在这项工作中，我们考虑一个用于解决逆问题的广义双层优化框架。我们引入分数拉普拉斯作为正则化器来提高重建质量，并将其与全变分正则化进行比较。我们强调，使用分数拉普拉斯算子作为正则化器的关键优点是它会产生线性算子，而不是导致非线性简并算子的全变分正则化。受残差神经网络的启发，为了学习正则化的最佳强度和分数拉普拉斯指数，我们开发了一种专用的具有可变深度的双层优化神经网络，用于解决一般正则化逆问题。我们说明了如何将各种正则化器选择合并到我们提出的网络中。例如，我们将断层扫描重建视为模型问题，并通过分数拉普拉斯正则化显示重建质量的改进，特别是对于有限的数据。我们通过我们提出的双层优化神经网络成功地学习了正则化强度和分数指数。我们观察到分数拉普拉斯正则化优于全变分正则化。在数据有限且嘈杂的情况下，这是特别令人鼓舞且重要的。
In this work we consider a generalized bilevel optimization framework for solving inverse problems. We introduce fractional Laplacian as a regularizer to improve the reconstruction quality, and compare it with the total variation regularization. We emphasize that the key advantage of using fractional Laplacian as a regularizer is that it leads to a linear operator, as opposed to the total variation regularization which results in a nonlinear degenerate operator. Inspired by residual neural networks, to learn the optimal strength of regularization and the exponent of fractional Laplacian, we develop a dedicated bilevel optimization neural network with a variable depth for a general regularized inverse problem. We illustrate how to incorporate various regularizer choices into our proposed network. As an example, we consider tomographic reconstruction as a model problem and show an improvement in reconstruction quality, especially for limited data, via fractional Laplacian regularization. We successfully learn the regularization strength and the fractional exponent via our proposed bilevel optimization neural network. We observe that the fractional Laplacian regularization outperforms total variation regularization. This is specially encouraging, and important, in the case of limited and noisy data.