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<p>In this work, we propose a new discretization for second-order total generalized variation (TGV) with some distinct properties compared to existing discrete formulations. The introduced model is based on the same design principles as Condat’s discrete total variation model (Condat in SIAM J Imaging Sci 10(3):1258-1290, 2017) and shares its benefits, particularly improved solution quality for imaging problems. We propose an algorithm for general discrete inverse problems with second-order TGV using the new discretization. Numerical results obtained with this algorithm for denoising and upscaling demonstrate the advantages of the discretization. Moreover, to assess the invariance properties of the new model, we compare the results of the proposed TGV and the classic discrete TGV for original data and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10851_2024_1224_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(90^{^{\circ }}\)</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mn>90</mn> <mrow /> <mmultiscripts> <mrow /> <mrow /> <mo>∘</mo> </mmultiscripts> </mmultiscripts> </math></EquationSource> </InlineEquation> rotated versions. Additionally, we provide an algorithm for calculating the TGV value with respect to the new discretization model.</p>

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A Second-Order TGV Discretization with \(90^{^{\circ }}\) Rotational Invariance Property

  • 期刊论文

  • 发表日期：

    2025年4月10日

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  • Alireza Hosseini,
  • Kristian Bredies

摘要

In this work, we propose a new discretization for second-order total generalized variation (TGV) with some distinct properties compared to existing discrete formulations. The introduced model is based on the same design principles as Condat’s discrete total variation model (Condat in SIAM J Imaging Sci 10(3):1258-1290, 2017) and shares its benefits, particularly improved solution quality for imaging problems. We propose an algorithm for general discrete inverse problems with second-order TGV using the new discretization. Numerical results obtained with this algorithm for denoising and upscaling demonstrate the advantages of the discretization. Moreover, to assess the invariance properties of the new model, we compare the results of the proposed TGV and the classic discrete TGV for original data and \(90^{^{\circ }}\) 90 ∘ rotated versions. Additionally, we provide an algorithm for calculating the TGV value with respect to the new discretization model.