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<p>To better reconstruct the piecewise constant optical coefficients in diffuse optical tomography, nonconvex and nonsmooth approximation of weak Mumford–Shah functional is considered in this paper. We theoretically analyze the existence of minimizers in piecewise constant finite element space and constrained finite dimensional subsets. However, optimizing such minimization problems presents a computational challenge due to the nonconvex and non-differentiable properties. To overcome these difficulties, we propose a fast graduated nonconvex alternative directional multiplier method to solve this numerical problem. Compared with graduated nonconvex Gaussian–Newton, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(TV^q\)</EquationSource> </InlineEquation> Gaussian–Newton, and TV Gaussian–Newton, our simulations show that the proposed GNC-ADMM can well keep edges and values of the anomaly with fewer iteration steps and fewer measurements.</p>

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Efficient parameter identification using nonsmooth and nonconvex regularization for inverse problem of diffuse optical tomography

  • 期刊论文

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  • 发表日期：

    2025年7月18日

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  • Jinping Tang,
  • Bo Bi

摘要

To better reconstruct the piecewise constant optical coefficients in diffuse optical tomography, nonconvex and nonsmooth approximation of weak Mumford–Shah functional is considered in this paper. We theoretically analyze the existence of minimizers in piecewise constant finite element space and constrained finite dimensional subsets. However, optimizing such minimization problems presents a computational challenge due to the nonconvex and non-differentiable properties. To overcome these difficulties, we propose a fast graduated nonconvex alternative directional multiplier method to solve this numerical problem. Compared with graduated nonconvex Gaussian–Newton, \(TV^q\) Gaussian–Newton, and TV Gaussian–Newton, our simulations show that the proposed GNC-ADMM can well keep edges and values of the anomaly with fewer iteration steps and fewer measurements.