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<p>The paper settles a previous analysis by Lucchetti and Sansò (A class of sets where convergence in Hausdorff sense and in measure coincide, 2020) of families of compact sets arriving at the definition of a metric space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\overline{K}_h\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation> of so-called kernels. <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\overline{K}_h\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mover> <mi>K</mi> <mo>¯</mo> </mover> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation> has the useful property of being complete and compact, with respect to its metric <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\overline{d}\)</EquationSource> <EquationSource Format="MATHML"><math> <mover> <mi>d</mi> <mo>¯</mo> </mover> </math></EquationSource> </InlineEquation>. This allows the definition of a space of quasi-partitions of the basic set <i>B</i> that we want to segment, in view of the optimization of a suitably modified Mumford–Shah (M.S.) functional. Since the space <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(Q_h\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>Q</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation> of quasi-partitions of <i>B</i> is itself complete and compact, the theorem of existence of the minimum of the (M.S.) functional runs quite smoothly by applying standard techniques of variational calculus.</p>

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A new family of compact topological spaces of closed sets with an application to the segmentation problem

  • 期刊论文

  • 发表日期：

    2025年6月9日

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  • Fernando Sansò

摘要

The paper settles a previous analysis by Lucchetti and Sansò (A class of sets where convergence in Hausdorff sense and in measure coincide, 2020) of families of compact sets arriving at the definition of a metric space \(\overline{K}_h\) K ¯ h of so-called kernels. \(\overline{K}_h\) K ¯ h has the useful property of being complete and compact, with respect to its metric \(\overline{d}\) d ¯ . This allows the definition of a space of quasi-partitions of the basic set B that we want to segment, in view of the optimization of a suitably modified Mumford–Shah (M.S.) functional. Since the space \(Q_h\) Q h of quasi-partitions of B is itself complete and compact, the theorem of existence of the minimum of the (M.S.) functional runs quite smoothly by applying standard techniques of variational calculus.