Elsevier

Neurocomputing

Volume 325, 24 January 2019, Pages 303-304
Neurocomputing

Short communication
Errata to: A 3D polar-radius-moment invariant as a shape circularity measure

https://doi.org/10.1016/j.neucom.2018.10.006Get rights and content

Abstract

In this paper we point out the theoretical results from [2] that were established and discussed earlier. All these statements were a part of results proven in [1], and do overlap partially with the results established in [3] and [4]. Several mistakes in [2] are pointed out, as well.

Introduction

A family of 3D circularity measures has been introduced recently in [2]. The authors have used, called therein, polar radial moments. A 3D polar radial moment, mp(S), is defined asmp(S)=Sr(x,y,z)pdVwhere r(x,y,z)=(xxc)2+(yxc)2+(zzc)2 is the distance of a given point (x, y, z) from the centroid (xc,yc,zc)=(1V·Sx·r·dV,1V·Sy·r·dV,1V·Sy·r·dV) of S, V the volume of S, and r=r(x,y,z) used for a short notation.

In their derivations [2], the authors have used the notation up(S), instead of mp(S). There are the following relations (equivalences) between notations used in [1] and [2].up(S)mp(S)S((xxc)2+(yxc)2+(zzc)2)p/2dxdydzu0(S)m0(S)Sdxdydzμ0,0,0(S)Area_of_S.

The authors of [2] give the following definition for a family ζPRC3D(S), of 3D shape measures.

Definition 1

(Definition 3.2. quoted from [2]) Let S be a given 3D shape the proposed 3D measure ζPRC3D(S) is defined as:ζPRC3D(S)=1p+3(3u0(S))p+331(4π)p3up(S).

Herein we point out that the family ζPRC3D(S) of 3D shape measures, as given above, is a subfamily of the family C23D(S,β) of 3D shape measures introduced in [1]1, by Definition 5.1 (see below). More precisely, each measure from the ζPRC3D(S) family coincides with a measure C23D(S,β), for a specific choice of β. At the same time, there are measures from C23D(S,β) that do not belong to the family ζPRC3D(S). We also point out relations to some of more specific work done in [3], [4].

Definition 2

(Definition 5.1. quoted from [1]) Given a shape S with its centroid located at the origin and a real number β such that 1<β and   β ≠ 0, the compactness measure C23D(S,β) is defined as:

C23D(S,β)={32β+3(34π)2β/3·μ0,0,0(S)(2β+3)/3S(x2+y2+z2)βdxdydzβ>02β+33(4π3)2β/3·S(x2+y2+z2)βdxdydzμ0,0,0(S)(2β+3)/31<β<0.

First, in the next note we clarify some imprecisions and errors from [2].

Note 1

Definition 1, as given above, is motivated by the following property of the function up(S)up(S)u0(S)p+331p+3·3p+33(4π)p3(the inequality (8) derived in the proof of Theorem 3.1. from [2]).

  • Remark-1. The inequality in (5), quoted from [2] (see Theorem 3.1. therein)u0(S)p+33up(S)1p+3·3p+33(4π)p3displays with a typo obviously, and consequently does not coincide with the correct inequality, given in (6) (see above), used to establish Definition 1.

  • Remark-2. The inequality in (6) is true for all the shape S and for all p ≥ 0, but not for p < 0. Indeed,

    • the quantity 1up(S) does not converge for all p < 0 and for all the shapes S (i.e. for shapes S that do include their centroid). Consequently, the quantity ζPRC3D(S) is not computable in such situations;

    • for p < 0 the statement in (6) is not true. Consequently, the quantity ζPRC3D(S) formally computed by using (4), as given in Definition 1, would not satisfy all the properties listed in Theorem 3.3., from [2].

      An alternative measure, ζalt, PRC3D(S), that would satisfy the properties from Theorem 3.3., from [2], for negative values of p should be modified accordingly to the formula in (5).

Now we give the statement which points out that each measure from the family ζPRC3D(S), from [2], coincides with a measure C23D(S,β) from [1], for a particular choice of β. Contrary is not true, i.e. due to Remark-2 not all measures from C23D(S,β) belong to the ζPRC3D(S) family.

Statement 1

Let a planar shape S and p > 0 be given. ThenζPRC3D(S)=C23D(S,β=p2).

Proof

The proof follows directly from the formulas in Definition 1, Definition 2, and notation equivalences listed in (2) and (3). 

Section snippets

Concluding remarks

We have shown that all the 3D shape measures, from [2], belong to a family of 3D shape measures defined much earlier in [1]. Surprisingly, the reference [1] has been quoted in [2]2, but such an essential overlap has not been noticed and mentioned. It is worth mentioning that the derivation techniques used in [1] and [2] coincide. The method is very generic. It also has been applied in the initial work [3], on this topic, were the measure C23D(S,β=1) has been

Acknowledgment

All the work by the second author is supported by the Serbian Ministry of Education, Science, and Technology.

Carlos obtained his BSc in computer engineering at La Salle University in Mexico City. At the University of Exeter, he obtained his MSc in applied artificial intelligence and Ph.D. on the topic of shape descriptors for image classification. Carlos has worked on various research projects at the University of Exeter and Plymouth University in collaboration with industrial partners such as C3 Resources Ltd, eCow Ltd and the Met Office. These projects involved scientific research in the areas of

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Carlos obtained his BSc in computer engineering at La Salle University in Mexico City. At the University of Exeter, he obtained his MSc in applied artificial intelligence and Ph.D. on the topic of shape descriptors for image classification. Carlos has worked on various research projects at the University of Exeter and Plymouth University in collaboration with industrial partners such as C3 Resources Ltd, eCow Ltd and the Met Office. These projects involved scientific research in the areas of video processing, machine learning and statistical modeling. The commercial outcomes of this research covered a wide range of applications such as automatic detection of abnormal energy consumption in buildings, video tracking of dairy cows, modeling high performance storage systems and segmentation of medical images. Before moving to the United Kingdom, he worked as a web developer and team leader at an eMarketing company, Interalia Digital, for different clients such as Coca-Cola Mexico, Coca-Cola Brazil and Nike Mexico.

Jovisa Zunic is a professor at the Mathematical Institute of the Serbian Academy of Sciences and Arts. His research interests are in pattern analysis, digital image analysis, digital and discrete geometry, shape representation and coding of digital objects, discrete mathematics, combinatorial optimisation, neural networks, and number theory.

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