Medial Axis and Singularities

Birbrair, Lev; Denkowski, Maciej P.

doi:10.1007/s12220-017-9763-x

Cited by 10 publications

(21 citation statements)

References 13 publications

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“…Let us notice that the proof of one of the main results in[50] was incorrect, so the question remained open. Very recently a partial confirmation of the claim of[50] was obtained in[13] 2. In contrast with Section 6.1, in[1] and here we make no probabilistic assumptions on the noise.…”

supporting

confidence: 71%

Accuracy of noisy Spike-Train Reconstruction: a Singularity Theory point of view

Goldman¹,

Salman²,

Yomdin³

2018

jsing

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This is a survey paper discussing one specific (and classical) system of algebraic equations -the so called "Prony system". We provide a short overview of its unusually wide connections with many different fields of Mathematics, stressing the role of Singularity Theory. We reformulate Prony System as the problem of reconstruction of "Spike-train" signals of the form F (x) = d j=1 ajδ(x − xj) from the noisy moment measurements. We provide an overview of some recent results of [1][2][3]7,8,10,11,29,53] on the "geometry of the error amplification" in the reconstruction process, in situations where the nodes xj near-collide. Some algebraic-geometric structures, underlying the error amplification, are described (Prony, Vieta, and Hankel mappings, Prony varieties), as well as their connection with Vandermonde mappings and varieties. Our main goal is to present some promising fields of possible applications of Singulary Theory.

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supporting

confidence: 71%

Accuracy of noisy Spike-Train Reconstruction: a Singularity Theory point of view

Goldman¹,

Salman²,

Yomdin³

2018

jsing

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“…Its geometrical origin is shown in the following easy lemma. Following [6] we will give here the correct version of [5] Proposition 3.24. It reads:…”

Section: 3mentioning

confidence: 99%

“…34 ([5] Theorem 3.27 -extended version,[6] Theorem 2.4). Assume that 0 ∈ M X ∩ X where X is a pure one-dimensional closed definable set in the plane.…”

mentioning

confidence: 99%

When the medial axis meets the singularities

Denkowski¹

2019

Analytic and Algebraic Geometry 3

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In this survey we present recent results in the study of the medial axes of sets definable in polynomially bounded o-minimal structures. We take the novel point of view of singularity theory. Indeed, it has been observed only recently that the medial axis -i.e. the set of points with more than one closest point to a given closed set X ⊂ R n (with respect to the Euclidean distance) -reaches some singular points of X bringing along some metric information about them.1991 Mathematics Subject Classification. 32B20, 54F99.We write B(x, r) = B n (x, r) ⊂ R n for the open Euclidean ball centred at x and with radius r > 0 and S(x, r) := ∂B(x, r) for the sphere. When r = d(x, X), we call the sphere or ball supporting. Note that X cannot enter a supporting ball.Remark 2.3. Any point x ∈ B(a, r) where a ∈ X has its distance d(x, X) realized in B(a, 2r). This is a mere student's exercise, but it plays an important role in many proofs.There are two other notions closely related to that of the medial axis. The first is the concept of the central set.The set C X consisting of of the centres of maximal balls for X is called the central set (formally: 'of R n \ X').Remark 2.5. If B(x, r) is a maximal ball, then r = d(x, X).The relation between M X and C X is considered folklore ( 1 ). This result has a practical consequence in that we often work with C X instead of M X (see e.g. the proof of Proposition 3.20). The closure M X is sometimes called cut locus.Theorem 2.6. There is always M X ⊂ C X ⊂ M X .Proof.[5] Theorem 2.25; see also [21] for a different proof. Both inclusions may be strict:Example 2.7. If X is the parabola y = x 2 , then M X = {0} × (1/2, +∞) whereas the focal point (0, 1/2) belongs to C X .

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“…Example 2.3. Let us observe after [1] that the equation y 3 = x 5 defines a plane analytic curve that is only C 1 -smooth at the origin, hence only a C 1 subanalytic leaf, though at the same time an analytic set.…”

Section: Condition (C) For Subanalytic Leavesmentioning

confidence: 99%

“…Now, applying the Curve Selecting Lemma, we see that we must have the inclusion of germs (M ) a ⊂ (N ) a (M has two branches at a which yield the two semi-lines building up T a M , see e.g. [1] where these observations are extensively used). This ends the proof.…”

Section: Proof It Suffices To Prove the Inequality Dimmentioning

confidence: 99%