The Discrete Moments of the Circles
Conference paper
First Online:
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
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
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.
$$
dm_{p,q} (C){\mathbf{ }} = {\mathbf{ }}\sum\limits_{\mathop {i,j{\mathbf{ }}are{\mathbf{ }}integers}\limits_{(i - a)^2 + (j - b)^2 \leqslant r^2 } } {i^p j^q } ;
$$
$$
m_{p,q} (C){\mathbf{ }} - {\mathbf{ }}dm_{p,q} (C){\mathbf{ }} = {\mathbf{ }}\int\limits_C {\int {x^p y^q dxdy} } {\mathbf{ }} - {\mathbf{ }}\sum\limits_{\mathop {i,j{\mathbf{ }}are{\mathbf{ }}integers}\limits_{(i - a)^2 + (j - b)^2 \leqslant r^2 } } {i^p j^q }
$$
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 Download
to read the full conference paper text
References
- 1.Huxley, M. N.: Exponential Sums and Lattice Points. Proc. London Math. Soc.3 (1990) 471–502.MathSciNetGoogle Scholar
- 2.Iwaniec, H., Mozzochi, C.J.: On the Divisor and Circles Problems. J. Number Theory29 (1988) 60–93.MATHCrossRefMathSciNetGoogle Scholar
- 3.Jiang, X., Bunke, H.: Simple and Fast Computation of Moments. Pattern Recognition24 (1991) 801–806.CrossRefGoogle Scholar
- 4.Worring, M., Smeulders, A.W.M.: Digitized Circular Arcs: Characterization and Parameter Esstimation. IEEE Trans. PAMI17 (1995) 587–598.Google Scholar
- 5.ŽuniĆ, J., N. Sladoje, N.: A Characterization of Digital Disks by Discrete Moments. Lecture Notes in Computer Science: Computer Analysis of Images and Patterns1296 (1997) 582–589.Google Scholar
Copyright information
© Springer-Verlag Berlin Heidelberg 1999