# Least squares fitting of digital polynomial segments

Conference paper

First Online:

## Abstract

It is proved that digital polynomial segments and their least squares polynomial fits are in one-to-one correspondence. This enables an efficient representation of digital polynomial segments by *n*+3 parameters, under the condition that an upper bound, say *n*, for the degrees of the digitized polynomials is assumed. One of such representations is (*x*_{1}, m, a_{n}, a_{n}−1,..., *a*_{0}), where *x*_{1} and *m* are the *x*-coordinate of the left endpoint and the number of digital points, respectively, while *a*_{n}, *a*_{n}−1,..., *a*_{0} are the coefficients of the least squares polynomial fit *Y=a*_{n}X^{n}+a_{n− 1}X^{n−1}+ ...+a_{0}, for a given digital polynomial segment.

## Key words

Image processing computer vision digital polynomial segment least squares fitting coding Download
to read the full conference paper text

## References

- [1]Aitken, A. C. (1956). Determinants and Matrices, Oliver and Boyd.Google Scholar
- [2]Arcelli, L. and Masarotti, A. (1978). On the parallel generation of straight digital lines, Computer Graphics and Image Processing, 7, 67–83.Google Scholar
- [3]Bongiovanni, G., Luccio, G. F. and Zurat A. (1975). The discrete equation of a straight line, IEEE Trans. Comput., 24, 310–313.Google Scholar
- [4]Boshernitzan, M. and Fraenkel, A. S. (1981). Nonhomogeneous spectra of numbers, Discrete Mathematics, 34, 325–327.Google Scholar
- [5]Boshernitzan, M. and Fraenkel, A. S. (1984). A linear algorithm for nonhomogeneous spectra of numbers, J. Algorithms, 5, 187–198.Google Scholar
- [6]Burr, I. W. (1974). Applied Statistical Methods, Academic Press.Google Scholar
- [7]Dorst, L. and Smeulders, A. W. M. (1984). Discrete representation of straight lines, IEEE Trans. Pattern Analysis and Machine Intelligence, 6, 450–463.Google Scholar
- [8]Graham, R. L., Lin. and Lin, C.-S. (1978). Spectra of numbers, Mathematics Magazine, 51, 174–176.Google Scholar
- [9]Kim, C. E. (1984). Digital disks, IEEE Trans. Pattern Analysis and Machine Intelligence, 6, 372–374.Google Scholar
- [10]Klassman, H. (1975). Some aspects of the accuracy of the approximated position of a straight line on a square grid, Computer Graphics and Image Processing, 4, 225–235.Google Scholar
- [11]Klette, R., Stojmenović, I. and Žunić J. (1996). A parametrization of digital planes by least square fits and generalizations, Graphical Models and Image Processing, 58, No. 3, 295–300.Google Scholar
- [12]Krechmar, V. A. (1974). A Problem Book in Algebra. Mir Publishers, Moscow.Google Scholar
- [13]Kovalevsky, V. A. (1990). New definition and fast recognition of digital straight segments and arcs, IEEE Proc. of Tenth Int. Conf. on Pattern Recognition, 10662, 31–34.Google Scholar
- [14]Lindenbaum, M. and Koplowitz, J. (1991). A new parametrization of digital straight lines, IEEE Trans. Pattern Analysis and Machine Intelligence, 13, 847–852.Google Scholar
- [15]Melter, R. A. and Rosenfeld, A. New views of linearity and connectedness in digital geometry, Pattern Recognition Letters, 10, 9–16.Google Scholar
- [16]Melter, R. A., Stojmenović, I. and Žunić, J. (1993). A new characterization of digital lines by least square fits, Pattern Recognition Letters, 14, 83–88.Google Scholar
- [17]Nakamura, A. and Aizawa, K. (1984). Digital circles, Computer Vision, Graphics and Image Processing, 26, 242–255.Google Scholar
- [18]Rosenfeld, A. (1974). Digital straight line segments, IEEE Trans. Comput., 23, 1264–1269.Google Scholar
- [19]Sauer, P. (1993). On the recognition of digital circles in linear time, Computational Geometry: Theory and Applications, 2, 287–302.Google Scholar
- [20]Žunić, J. (1995). A coding scheme for certain sets of digital curves, Pattern Recognition Letters, 16, 97–104.Google Scholar
- [21]Žunić, J. and Koplowitz, J. (1994). A representation of digital parabolas by least square fits, SPIE Proc., 2356, 71–78.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1996