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Bilevel optimization, deep learning and fractional Laplacian regularization with applications in tomography

Journal Article · · Inverse Problems
 [1];  [2];  [1]
  1. George Mason Univ., Fairfax, VA (United States)
  2. Argonne National Lab. (ANL), Argonne, IL (United States)
+ Show Author Affiliations
Here 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.
Research Organization:
Argonne National Laboratory (ANL), Argonne, IL (United States)
Sponsoring Organization:
National Science Foundation (NSF); US Air Force Office of Scientific Research (AFOSR); US Department of the Navy; USDOE Office of Science
Grant/Contract Number:
AC02-06CH11357
OSTI ID:
1660712
Alternate ID(s):
OSTI ID: 23093960
Journal Information:
Inverse Problems, Journal Name: Inverse Problems Journal Issue: 6 Vol. 36; ISSN 0266-5611
Publisher:
IOPscienceCopyright Statement
Country of Publication:
United States
Language:
English

References (49)

Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing journal August 2008
Recent Advances in CT Image Reconstruction journal January 2013
A survey of truncated-Newton methods journal December 2000
Extension Problem and Harnack's Inequality for Some Fractional Operators journal October 2010
Can one use total variation prior for edge-preserving Bayesian inversion? journal August 2004
The curvelet transform for image denoising journal May 2002
On the determination of functions from their integral values along certain manifolds journal December 1986
Sobolev Spaces with Non-Muckenhoupt Weights, Fractional Elliptic Operators, and Applications journal January 2019
Spectral Approximation of Fractional PDEs in Image Processing and Phase Field Modeling journal October 2017
Learning a variational network for reconstruction of accelerated MRI data: Learning a Variational Network for Reconstruction of Accelerated MRI Data journal November 2017
The Elements of Statistical Learning book January 2009
Regularization,GSVD and truncatedGSVD journal September 1989
REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems journal March 1994
Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part II: Algorithm, Its Analysis and Numerical Tests journal June 2017
Optimal Selection of the Regularization Function in a Weighted Total Variation Model. Part I: Modelling and Theory journal July 2017
Deep Neural Networks Motivated by Partial Differential Equations journal September 2019
Deep residual learning for image steganalysis journal February 2017
Nonlinear total variation based noise removal algorithms journal November 1992
VoxResNet: Deep voxelwise residual networks for brain segmentation from 3D MR images journal April 2018
Overtraining, Regularization, and Searching for Minimum in Neural Networks journal July 1992
Competitive performance of a modularized deep neural network compared to commercial algorithms for low-dose CT image reconstruction journal June 2019
External optimal control of fractional parabolic PDEs journal January 2020
An Extension Problem Related to the Fractional Laplacian journal August 2007
Non-convergence of the L-curve regularization parameter selection method journal August 1996
External optimal control of nonlocal PDEs journal July 2019
Machine Learning: Mathematical Theory and Scientific Applications journal December 2019
Generalized gradients and applications journal January 1975
Fractional diffusion models of cardiac electrical propagation: role of structural heterogeneity in dispersion of repolarization journal August 2014
Learning Deep CNN Denoiser Prior for Image Restoration conference July 2017
Deep Residual Learning for Accelerated MRI Using Magnitude and Phase Networks journal September 2018
The curvelet transform for image denoising journal May 2002
Convolutional Neural Networks for Inverse Problems in Imaging: A Review journal November 2017
Using Deep Neural Networks for Inverse Problems in Imaging: Beyond Analytical Methods journal January 2018
$p$ -Laplacian Regularization for Scene Recognition journal August 2019
Missing Data Reconstruction in Remote Sensing Image With a Unified Spatial–Temporal–Spectral Deep Convolutional Neural Network journal August 2018
Deep Convolutional Neural Network for Inverse Problems in Imaging journal September 2017
Principles of Computerized Tomographic Imaging journal January 2002
Optimal Regularized Reconstruction in Computerized Tomography journal November 1987
The Use of the L-Curve in the Regularization of Discrete Ill-Posed Problems journal November 1993
Iterative Methods for Optimization book January 1999
Sparse Tomography journal January 2013
Optimization-Based Approach for Joint X-Ray Fluorescence and Transmission Tomographic Inversion journal January 2016
Multiresolution Parameter Choice Method for Total Variation Regularized Tomography journal January 2016
Optimization Methods for Large-Scale Machine Learning journal January 2018
Deep Learning: An Introduction for Applied Mathematicians journal January 2019
Simultaneous Sensing Error Recovery and Tomographic Inversion Using an Optimization-Based Approach journal January 2019
Mathematical Models for Local Nontexture Inpaintings journal January 2002
A survey of machine learning for big data processing journal May 2016
Fractional operators with inhomogeneous boundary conditions: analysis, control, and discretization journal January 2018

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Related Subjects

97 MATHEMATICS AND COMPUTING
bilevel optimization neural network
deep residual learning
fractional Laplacian regularization
imaging science
inverse problems
tomographic reconstruction