The Discrete Moments of the Circles

Abstract

The moment of (p, q)-order, m _{p, q}(C), of a circle C given by (x − a)² + (y − b)² ≤ r 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.

Research supported by Mathematical Institute, SANU, Beograd, under the project 04M02