ADR shape descriptor – Distance between shape centroids versus shape diameter
, Joviša Z
The Department of Computer Science, The University of Auckland, Auckland 1142, New Zealand
Department of Computer Science, University of Exeter, Exeter EX4 4QF, UK
Mathematical Institute, Serbian Academy of Arts and Sciences, Belgrade, Serbia
Received 21 October 2011
Accepted 2 February 2012
Available online 10 February 2012
In this paper we study the ADR shape descriptor
(S), where ADR is short for ‘‘asymmetries in the distri-
bution of roughness’’. This descriptor was deﬁned in 1998 as the ratio of the squared distance between
two different shape centroids (namely of area and frontier) to the squared shape diameter. After known
for more than ten years, the behavior of
(S) was not well understood till today, thus hindering its appli-
cation. Two very basic questions remained unanswered so far:
– What is the range for
(S), if S is any bounded compact shape?
– How do shapes look like having a large
This paper answers both questions. We show that
(S) ranges over the interval [0, 1). We show that the
established upper bound 1 is the best possible by constructing shapes whose
(S) values are arbitrary close
to 1. In experiments we provide examples to indicate the kind of shapes that have relatively large
Ó 2012 Elsevier Inc. All rights reserved.
Shape is an object characteristic that is most suitable for recog-
nition, classiﬁcation, or identiﬁcation tasks. This is because shape
allows various numerical characterizations which are particularly
suitable when performing computer-based data analysis. These
numerical characteristics are used as components of feature vec-
tors allowing object comparisons, as necessary for pattern or image
data analysis. It is widely accepted that comparing the objects in
feature space is often more efﬁcient than object comparison in
the original object space.
A broad variety of shape descriptors has been developed so far.
Some of them are very generic, such as Fourier descriptors ,
Zernike moments  or moment invariants [2,6], while some
other relate to speciﬁc object or shape characteristics: circularity
[17,31], convexity [18,21,30], rectangularity [19,27], sigmoidality
, orientability , elongation , tortuosity , and so forth.
Notice that due to the diversity of shapes, we cannot expect that a
particular shape descriptor would dominate others in all situa-
tions, even for a class of objects in a given application only. For this
reason, different methods are developed to measure shape proper-
ties (e.g., convexity, circularity, rectangularity, and so forth).
In order to match a spectrum of speciﬁc demands for efﬁcient
object recognition, identiﬁcation or classiﬁcation systems, different
techniques have been used to describe the shape: algebraic invari-
ants , integral invariants , Fourier analysis , statistics
, wavelets , fractals , curvature [8,12], integral transfor-
mations , computational geometry , and so forth.
Also note that requests vary when shape descriptors were cre-
ated. A common request is that they should be invariant with re-
spect to translation, rotation and scaling transformations, and
sometimes an invariance with respect to afﬁne transformations
in general is also required. Robustness of created descriptors
(e.g., required when working with low quality data) is usually en-
sured by considering area-based descriptors. On the other hand,
high sensitivity (e.g., required when performing high-precision
tasks) is usually satisﬁed by deciding for frontier-based descrip-
tors. Among many other desirable properties, we just mention
the tuning possibility (e.g., a possibility to control the behavior of
the descriptor by changing its parameters).
One of the properties, that is not so much studied in literature,
is the geometric interpretation of designed shape descriptors. For
example, the Hu moment invariants  were introduced almost
50 years ago but just recently analyzed with respect to their geo-
metric interpretation. They had been deﬁned to satisfy invariance
under similarity transforms. As such, they were used in a wide
spectra of applications. But, to be able to predict (or understand)
1077-3142/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved.
This paper has been recommended for acceptance by Albert C.S. Chung.
Corresponding author at: The Department of Computer Science, The University
of Auckland, Auckland 1142, New Zealand.
E-mail addresses: email@example.com (R. Klette), J.Zunic@ex.ac.uk (J. Z
URLs: http://www.cs.auckland.ac.nz/~rklette (R. Klette), http://empslocal.ex.
ac.uk/people/staff/jz205/ (J. Z
Computer Vision and Image Understanding 116 (2012) 690–697
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Computer Vision and Image Understanding
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