Entropy_**2009**, _11_, 513 - 528; doi:10.3390/e11030513  
_Artcle_  
**Scale-Based Gaussian Coverings: Combining Intra and Inter Mixture Models in Image Segmentation**  
**[PERSON] \({}^{1,2,\ast}\), [PERSON] \({}^{2}\) and [PERSON] \({}^{3,4}\)**  
\({}^{1}\) Science Foundation Ireland, Wilton Park House, Wilton Place, Dublin 2, Ireland  
\({}^{2}\) Department of Computer Science, Royal Holloway University of London, Egham TW20 0 EX, UK;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{3}\) CEA-Saclay, DAPNIA/SEDI-SAP, Service d'Astrophysique, 91191 Gif sur Yvette, France;  
E-Mail: [EMAIL_ADDRESS]  
\({}^{4}\) Laboratoire AIM (UMR 7158), CEA/DSM-CNRS, Universite Paris Diderot, Diderot, IRFU,  
SEDI-SAP, Service d'Astrophysique, Centre de Saclay, F-91191 Gif-Sur-Yvette cedex, France  
\({}^{\ast}\) Author to whom correspondence should be addressed: E-Mail: [EMAIL_ADDRESS];  
Tel. : +44 1784 443429; Fax: +44 1784 439786. _Received: 1 September 2009 / Accepted: 14 September 2009 / Published: 24 September 2009_  
**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 Renyi 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 Renyi 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; Renyi entropy; Shannon entropy Introduction  
We begin with some terminology used. Segments are contiguous clusters. In an imaging context, this means that clusters contain adjacent or contiguous pixels. For typical 2D (two-dimensional) images, we may also consider the 1D (one-dimensional) marginal which provides an empirical estimate of the pixel (probability) density function or PDF. For 3D (three-dimensional) images, we can consider 2D marginals, based on the voxels that constitute the 3D image volume, or also a 1D overall marginal. An image is representative of a signal. More loosely a signal is just data, mostly here with neccessary sequence or adjacency relationships. Often we will use interchangeably the terms image, image volume if relevant, signal and data. The word \"model\" is used, in general, in many senses-statistical [1], mathematical, physical models; mixture model; linear model; noise model; neural network model; sparse decomposition model; even, in different senses, data model. In practice, firstly and foremostly for algorithmic tractability, models of whatever persuasion tend to be homogeneous. In this article we wish to broaden the homogeneous mixture model framework in order to accommodate heterogeneity at least as relates to resolution scale. Our motivation is to have a rigorous model-based approach to data clustering or segmentation, that also and in addition encompasses resolution scale. Figure 1: Clusters of all morphologies are sought. Figure courtesy of [PERSON], Caltech. In Figure 1[2], the clustering task is portrayed in its full generality. One way to address it is to build up parametrized clusters, for example using a Gaussian mixture model (GMM), so that the cluster \"pods\" are approximated by the mixture made up of the cluster component \"peas\" (a viewpoint expressed by [PERSON], quoted in [3]). A step beyond a pure \"peas\" in a \"pod\" approach to clustering is a hierarchical approach.