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7 - Probability density estimation by linear models

from Part II - Stochastic models

Published online by Cambridge University Press:  05 November 2014

Aly A. Farag
Aly A. Farag
Affiliation:
University of Louisville, Kentucky
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Summary

Probability density estimation is a crucial step in stochastic system identification. In the random field models studied in Chapter 6 as well as the applications to follow in image analysis, probability density estimation is a fundamental component. The purpose of this chapter is to study approaches for density estimation that are local, i.e. may be identified using empirical measurements, which are assumed realizations of a random process or random field of a certain statistical experiment. Of interest to us are density models that have manageable numerical implementation and may be used at various levels of image analysis.

Introduction

Numerical methods for estimating the probability density function (PDF) of a random variable X (a random vector in general) are important in various signal and image analysis applications. Such estimates form the basis of optimal filtering, synthesis, and segmentation of an image or a signal. Indeed, PDF estimation is fundamental in Bayesian statistics and in a huge number of machine-learning applications [7.1].

Type
Chapter
Information
Biomedical Image Analysis
Statistical and Variational Methods
 , pp. 163 - 180
Publisher: Cambridge University Press
Print publication year: 2014

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References

Duda, R. O., Hart, P. E. and Stork, D. G., Pattern Classification. New York: Wiley (2001).Google Scholar
Vapnik, V. N., Density Estimation for Statistics and Data Analysis. Chapman and Hall (1986).Google Scholar
Vapnik, V. N., Statistical Learning Theory. New York: Wiley (1998).Google Scholar
Fukunaga, K. and Hayes, R. R., The reduced Parzen classifier. IEEE Trans. Pattern Anal. Machine Intel. 11 (1989) 423–425.CrossRefGoogle Scholar
Silverman, B. W., Algorithm, AS176. Kernel density estimation using the fast Fourier transform. Appl. Stat. 31 (1982) 93–97.CrossRefGoogle Scholar
Jeon, B. W. and Landgrebe, D. A., Fast Parzen density-estimation using clustering-based branch-and-bound. IEEE Trans. Pattern Anal. Machine Intel. 16(9) (1994) 950–954.CrossRefGoogle Scholar
Girolami, M. and He, C., Probability density estimation from optimally condensed data samples. IEEE Trans. Pattern Anal. Machine Intel. 25(10) (2003) 1253–1264.CrossRefGoogle Scholar
Moon, T., The expectation-maximization algorithm. IEEE Signal Proc. Mag. 11 (1996) 47–60.CrossRefGoogle Scholar
Dempster, A. P., Laird, N. M. and Rubin, D. B., Maximum likelihood from incomplete data via the em algorithm. J. Roy. Stat. Soc. 39B (1977) 1–38.Google Scholar
Sorenson, H. W. and Alspach, D. L., Recursive Bayesian estimation using Gaussian sums. Automatica 7 (1971) 465–479.CrossRefGoogle Scholar
Poggio, T. and Girosi, F., Networks for approximation and learning, Proc. IEEE 78 (1990) 1481–1497.CrossRefGoogle Scholar
Goshtasby, A. and O’Neill, W. D., Curve fitting by a sum of Gaussians. CVGIP Graph. Model Image Proc. 56 (1999) 281–288.CrossRefGoogle Scholar
Farag, A. A., El-Baz, A. and Gimel’farb, G., Density estimation using modified expectation maximization for a linear combination of Gaussians. Proc. ICIP 3 (2004) 1871–1874.Google Scholar
Ali, A. M. and Farag, A. A., Density estimation using a new AIC-type criterion and the EM algorithm for a linear combination of Gaussians. Proc. IEEE Int. Conf. Image Proc. (ICIP08) (2008) 3024–3027.Google Scholar
Wand, M. P. and Jonse, M., Kernel Smoothing. London: Chapman and Hall, (1995).CrossRefGoogle Scholar
Pal, N. R. and Pal, S. K., A review on image segmentation techniques. Patt. Rec. 26 (1993) 1277–1294CrossRefGoogle Scholar
Dubes, R. C. and Jain, A. K., Random field models in image analysis. J. Appl. Stat. 16 (1989) 131–164.CrossRefGoogle Scholar

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