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Image Analysis through Shifted Orthogonal Polynomial Moments
Rajarshi Biswas
1
and Sambhunath Biswas
2
1
Department of Computer Science, Saarland University, Saarbrucken, 66123, Germany
2
Machine Intelligence Unit, Indian Statistical Institute, 203, B. T. Road, Kolkata, 700108, India
Keywords:
Rotational Invariance, Discrete Disc, Illumination, Noise.
Abstract:
Image analysis is significant from the standpoint of image description. A well described image has merits
in different research areas, e.g., image compression, machine learning, computer vision etc. This paper is an
attempt to analyze graylevel images through shifted orthogonal polynomial moments, computed on a discrete
disc. This removes the difficulty of computing the moments on an analytic disc. Excellent rotational invariance
as well as illumination invariance is observed.
1 INTRODUCTION
Image analysis through moments has recently gained
a good amount of attention during the last two
decades in the community of image processing, com-
puter vision and pattern recognition, though its initi-
ation was made in 1962 when Hu (Hu, 1962) did his
pioneering work on moment invariants. Afterwards,
various works based on both non-orthogonal and or-
thogonal moments were carried out. Among the non-
orthogonal moments, some of the reported works
can be found in (Prokop and Reeves, 1992), (Reddi,
1981), (Abu-Mostafa and Psaltis, 1984). Similarly,
works based on orthogonal moments can be found
in (Teague, 1980), (Teh and Chin, 1988), (Z.L. Ping
and Sheng, 2002), (H. Ren and Sheng, 2003) and
(T. Xia and Luo, 2007). Attempts on discrete orthog-
onal moments using Chebyshev moments were made
by (P.T. Yap and Ong, 2003) and (R. Mukundan and
Lee, 2001), while Zhu et al. (H.Q. Zhu and Coatrieux,
2007) introduced a kind of orthogonal polynomials
defined on non-uniform lattice, knownas Racah poly-
nomials. A good survey of works on moments can be
found in the article of Shu et al. (H. Shu and Coa-
trieux, 2007) and (Jan Flusser and Zitova, 2009). In
the present paper,moment-basedrotational invariance
using orthogonal shifted polynomials on discrete disc
(Biswas and Chaudhuri, 1985) is proposed. Shifting
function bijectively maps the interval [0,1] to the in-
terval [−1,1]. Shifted polynomials are, therefore, or-
thogonal on [0,1], i.e.,on the unit disc. It should be
noted that it is difficult to use analytic disc because of
the pixel mapping problem on the analytic disc. On
the other hand, using discrete disc has many advan-
tages. The mapping is unique and straightforward be-
cause of the mathematical description of the discrete
disc. This makes the algorithms straightforward. Re-
sults show excellent behavior of invariance under ro-
tation and different conditions of illumination. This
facilitates significant image description through or-
thogonal shifted polynomial image moments.
Below in Section 2, we briefly discuss discrete
circles, rings and discs to help readers understand
the mapping on discrete disc. Section 3 describes
the proposed three different methods, while Section 4
demonstrates results and discussion. Finally, in Sec-
tion 5 we present our conclusion.
2 DISCRETE CIRCLE, RING AND
DISC
Consider a 2-dimensional discrete array space of m×
n points or pixels so that any point or pixel (x, y),
0 ≤ x ≤ m −1 , 0 ≤ y ≤ n −1. x, y,m,n ∈ I (set of
integers) can be mapped to the continuous real plane
by a unit square about the center point (x±
1
2
, y±
1
2
).
Also, for simplicity and convenience, let the radii of
the discrete circle, ring and disc be integer valued with
center of the unit squares.
Discrete Circle (dc)
A dc is a discrete space approximation to the circle
defined in Euclidean geometry. In the present scheme
of generation, a dc is defined as follows
411
Biswas R. and Biswas S..
Image Analysis through Shifted Orthogonal Polynomial Moments.
DOI: 10.5220/0004648004110416
In Proceedings of the 9th International Conference on Computer Vision Theory and Applications (VISAPP-2014), pages 411-416
ISBN: 978-989-758-003-1
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)