کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
415613 681217 2013 18 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes
موضوعات مرتبط
مهندسی و علوم پایه مهندسی کامپیوتر نظریه محاسباتی و ریاضیات
پیش نمایش صفحه اول مقاله
Vietoris–Rips complexes also provide topologically correct reconstructions of sampled shapes
چکیده انگلیسی

Given a point set that samples a shape, we formulate conditions under which the Rips complex of the point set at some scale reflects the homotopy type of the shape. For this, we associate with each compact set X   of RnRn two real-valued functions cXcX and hXhX defined on R+R+ which provide two measures of how much the set X fails to be convex at a given scale. First, we show that, when P   is a finite point set, an upper bound on cP(t)cP(t) entails that the Rips complex of P at scale r collapses to the Čech complex of P at scale r for some suitable values of the parameters t and r. Second, we prove that, when P samples a compact set X  , an upper bound on hXhX over some interval guarantees a topologically correct reconstruction of the shape X either with a Čech complex of P or with a Rips complex of P. Regarding the reconstruction with Čech complexes, our work compares well with previous approaches when X is a smooth set and surprisingly enough, even improves constants when X has a positive μ-reach. Most importantly, our work shows that Rips complexes can also be used to provide shape reconstructions having the correct homotopy type. This may be of some computational interest in high dimensions.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Computational Geometry - Volume 46, Issue 4, May 2013, Pages 448–465
نویسندگان
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