# The Discrete Moments of the Circles

• Joviša ŽuniĆ
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1568)

## Abstract

The moment of (p, q)-order, m p, q(C), of a circle C given by (xa)2 + (yb)2r 2; is defined to be $$\int\limits_C {\int {x^p y^q dxdy} }$$. It is naturally to assume that the discrete moments dm p, q(C), defined as
$$dm_{p,q} (C){\mathbf{ }} = {\mathbf{ }}\sum\limits_{\mathop {i,j{\mathbf{ }}are{\mathbf{ }}integers}\limits_{(i - a)^2 + (j - b)^2 \leqslant r^2 } } {i^p j^q } ;$$
can be a good approximation for m p, q(C). This paper gives an answer what is the order of magnitude for the difference between a real moment m p, q(C) and its approximation dm p, q(C), calculated from the corresponding digital picture. Namely, we estimate
$$m_{p,q} (C){\mathbf{ }} - {\mathbf{ }}dm_{p,q} (C){\mathbf{ }} = {\mathbf{ }}\int\limits_C {\int {x^p y^q dxdy} } {\mathbf{ }} - {\mathbf{ }}\sum\limits_{\mathop {i,j{\mathbf{ }}are{\mathbf{ }}integers}\limits_{(i - a)^2 + (j - b)^2 \leqslant r^2 } } {i^p j^q }$$
in function of the size of the considered circle C and its center position if p and q are assumed to be integers. These differences are upper bounded with $$\mathcal{O}\left( {a^p \cdot b^q \cdot r^{\tfrac{7} {{11}} + \varepsilon } } \right)$$, where ε is an arbitrary small positive number.

The established upper bound can be understood as very sharp.

The result has a practical importance, especially in the area of image processing and pattern recognition, because it shows what the picture resolution should be used in order to obtain a required precision in the parameter estimation from the digital data taken from the corresponded binary picture.

## Keywords

Digital geometry discrete shapes parameter estimation

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