Completion of Visible Contours

Shape Reconstruction from Apparent Contours

“…3, but made semi-transparent. Image taken from [14] Fig. 5 The bold graph represents the visible part of the apparent contour of the three-dimensional scene in Fig.…”

“…3; the whole graph represents its apparent contour. Image taken from [14] the singular points of vis.G † /, if any, are only of two types: terminal points and T-junctions. The example of Fig.…”

“…Following [14], in Chap. 4 we give a constructive solution to this problem (see Theorem 4.3.1, that we call the completion theorem), based on a suitable Morse description of K and G. We note here that the aim of the completion theorem is not to provide the "simplest" completion of K, whatever simplest could mean 7 ; the scope of the result is to show that the conditions 8 imposed on K are sharp, and allow us to construct at least one completion.…”

“…The problem of returning, as output of the completion, a "simple" graph G is related to points (v) and (vi), and will be addressed below. An improvement of the results of [14] is given by Corollary 4.5.1, based on the introduction of the background; this represents a further degree of freedom, which allows us to fix a priori the regions of G where f D 0. Here the function f is defined on R 2 n G; it can be obtained as a doubled winding number with respect to G, and has to satisfy various consistency relations with the labelling d .…”

“…Now, let us discuss the (huge) problems listed in points (vi) and (vii) and their consequences (for instance, the solution to point (viii)). To this aim, it is useful to start by recalling the well-known result [1, 21,64,71] of Reidemeister in knot theory, asserting that two link diagrams 14 represent ambient isotopic links if and only if they can be related by a finite number of local Reidemeister moves or their inverses. We are interested in a similar question for two-dimensional smooth closed manifolds M embedded in R 3 .…”