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            <idno type="halRefHtml">&lt;i&gt;5th International Conference on Scale Space and Variational Methods in Computer Vision&lt;/i&gt;, Jun 2015, Lège-Cap Ferret, France. &lt;a target="_blank" href="https://dx.doi.org/10.1007/978-3-319-18461-6_7"&gt;&amp;#x27E8;10.1007/978-3-319-18461-6_7&amp;#x27E9;&lt;/a&gt;</idno>
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                <term xml:lang="en">Hamilton-Jacobi semigroup</term>
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              <p>The aim of this paper is to develop the theory, and to propose an algorithm, for morphological processing of images painted on point clouds, viewed as a length metric measure space $(X,d,\mu)$. In order to extend morphological operators to process point cloud supported images, one needs to define dilation and erosion as semigroup operators on $(X,d)$. That corresponds to a supremal convolution (and infimal convolution) using admissible structuring function on $(X,d)$. From a more theoretical perspective, we introduce the notion of abstract structuring functions formulated on length metric Maslov idempotent measurable spaces, which is the appropriate setting for $(X,d)$. In practice, computation of Maslov structuring function is approached by a random walks framework to estimate heat kernel on $(X,d,\mu)$, followed by the logarithmic trick.</p>
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