Nonasymptotic densities for shape reconstruction. (English) Zbl 1472.49062

Summary: In this work, we study the problem of reconstructing shapes from simple nonasymptotic densities measured only along shape boundaries. The particular density we study is also known as the integral area invariant and corresponds to the area of a disk centered on the boundary that is also inside the shape. It is easy to show uniqueness when these densities are known for all radii in a neighborhood of \(r = 0\), but much less straightforward when we assume that we only know the area invariant and its derivatives for only one \(r > 0\). We present variations of uniqueness results for reconstruction (modulo translation and rotation) of polygons and (a dense set of) smooth curves under certain regularity conditions.


49Q10 Optimization of shapes other than minimal surfaces
51M04 Elementary problems in Euclidean geometries
51M15 Geometric constructions in real or complex geometry
52A10 Convex sets in \(2\) dimensions (including convex curves)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry


Full Text: DOI arXiv


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