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Scale-based Gaussian coverings: combining intra and inter mixture models in image segmentation. (English) Zbl 1179.94015

Summary: By a “covering” we mean a Gaussian mixture model fit to observed data. Approximations of the Bayes factor can be availed of to judge model fit to the data within a given Gaussian mixture model. Between families of Gaussian mixture models, we propose the Rényi quadratic entropy as an excellent and tractable model comparison framework. We exemplify this using the segmentation of an MRI image volume, based (1) on a direct Gaussian mixture model applied to the marginal distribution function, and (2) Gaussian model fit through k-means applied to the 4D multivalued image volume furnished by the wavelet transform. Visual preference for one model over another is not immediate. The Rényi quadratic entropy allows us to show clearly that one of these modelings is superior to the other.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A17 Measures of information, entropy
94A11 Application of orthogonal and other special functions
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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[1] McCullagh, What is a statistical model?, Ann. Statist. 30 pp 1225– (2002) · Zbl 1039.62003 · doi:10.1214/aos/1035844977
[2] Djorgovski, Exploration of parameter spaces in a Virtual Observatory, Astronomical Data Analysis 4477 pp 43– (2001) · doi:10.1117/12.447189
[3] Silk, An astronomer’s perspective on SCMA III, Statistical Challenges in Modern Astronomy pp 387– (2003)
[4] DOI: 10.1109/PROC.1968.6414 · doi:10.1109/PROC.1968.6414
[5] DOI: 10.1016/j.imavis.2005.02.002 · doi:10.1016/j.imavis.2005.02.002
[6] Rissanen, Information and Complexity in Statistical Modeling (2007)
[7] DOI: 10.1198/016214501753168398 · Zbl 1017.62004 · doi:10.1198/016214501753168398
[8] DOI: 10.1093/comjnl/41.8.578 · Zbl 0920.68038 · doi:10.1093/comjnl/41.8.578
[9] Blum, On the approximation of correlated non-Gaussian noise PDFs using Gaussian mixture models, Conference on the Applications of Heavy Tailed Distributions in Economics, Engineering and Statistics (1999)
[10] DOI: 10.1109/89.943344 · doi:10.1109/89.943344
[11] DOI: 10.1016/S0167-8655(03)00038-2 · Zbl 01977718 · doi:10.1016/S0167-8655(03)00038-2
[12] Murtagh, Bayes factors for edge detection from wavelet product spaces, Opt. Eng. 42 pp 1375– (2003) · doi:10.1117/1.1564104
[14] Esteban, A summary of entropy statistics, Kybernetika 31 pp 337– (1995) · Zbl 0857.62002
[16] Starck, Astronomical Image and Data Analysis (2002)
[17] Starck, Sparse Image and Signal Processing: Wavelets, Curvelets, Morphological Diversity · Zbl 1196.94008
[18] DOI: 10.1093/comjnl/43.2.107 · doi:10.1093/comjnl/43.2.107
[19] DOI: 10.1016/S0031-3203(97)00115-5 · Zbl 05467706 · doi:10.1016/S0031-3203(97)00115-5
[20] DOI: 10.1109/82.718822 · Zbl 0999.94513 · doi:10.1109/82.718822
[21] DOI: 10.1002/j.1538-7305.1948.tb01338.x · Zbl 1154.94303 · doi:10.1002/j.1538-7305.1948.tb01338.x
[22] Frieden, Probability, Statistical Optics, and Data Testing: A Problem Solving Approach (1991)
[23] Burg, Multichannel maximum entropy spectral analysis, Multichannel Maximum Entropy Spectral Analysis, Annual Meeting International Society Exploratory Geophysics, reprinted in Modern Spectral Analysis pp 34– (1978)
[24] Frieden, Image enhancement and restoration, Topics in Applied Physics Vol. 6 pp 177– (1975) · doi:10.1007/978-3-662-41612-9_5
[25] Gull, MEMSYS5 Quantified Maximum Entropy User’s Manual (1991)
[26] DOI: 10.1109/34.982897 · doi:10.1109/34.982897
[27] MacKay, Information Theory, Inference, and Learning Theory (2003)
[29] Jenssen, Information theoretic learning and kernel methods, Information Theory and Statistical Learning (2009) · Zbl 1183.68480
[30] DOI: 10.2307/2412309 · doi:10.2307/2412309
[31] DOI: 10.1093/comjnl/28.1.82 · doi:10.1093/comjnl/28.1.82
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