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
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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.zbMATHCrossRefMathSciNetGoogle 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
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