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Entropy 2009, 11(3), 513-528; doi:10.3390/e11030513
Article

Scale-Based Gaussian Coverings: Combining Intra and Inter Mixture Models in Image Segmentation

1,2,* , 2
 and 3,4
1 Science Foundation Ireland, Wilton Park House, Wilton Place, Dublin 2, Ireland 2 Department of Computer Science, Royal Holloway University of London, Egham TW20 0EX, UK 3 CEA-Saclay, DAPNIA/SEDI-SAP, Service d’Astrophysique, 91191 Gif sur Yvette, France 4 Laboratoire AIM (UMR 7158), CEA/DSM-CNRS, Université Paris Diderot, Diderot, IRFU, SEDI-SAP, Service d'Astrophysique, Centre de Saclay, F-91191 Gif-Sur-Yvette cedex, France
* Author to whom correspondence should be addressed.
Received: 1 September 2009 / Accepted: 14 September 2009 / Published: 24 September 2009
(This article belongs to the Special Issue Entropy and Information)
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Abstract

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.
Keywords: image segmentation; clustering; model selection; minimum description length; Bayes factor; Rényi entropy; Shannon entropy image segmentation; clustering; model selection; minimum description length; Bayes factor; Rényi entropy; Shannon entropy
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Murtagh, F.; Contreras, P.; Starck, J.-L. Scale-Based Gaussian Coverings: Combining Intra and Inter Mixture Models in Image Segmentation. Entropy 2009, 11, 513-528.

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