The majority of methods that have been proposed for automated segmentation of brain tissues are based on statistical parametric models. The MPM-MAP (Maximizer of the posterior marginals- Maximum a posteriori) algorithm [1] exemplifies this approach. It implements Bayesian segmentation based on non-rigid registration of the atlas[AM1] . The algorithm uses Expectation-Maximization (EM) for estimation of model parameters and Hidden Markov Random Fields (HMRF) for spatial coherences. Two other examples are KVL (K. Van Leemput) [2] and CGMM (Constrained Gaussian mixture model) [3] which use maximum a posteriori probability (MAP) or maximum likelihood (ML) method for the estimation of model parameters. A drawback with these approaches is that it is difficult to integrate pixel spatial information with multi-valued pixel information (e.g. when several different MR scans have been acquired). This is because the HMRF is itself hard to implement[AM2] in high dimensional feature space.
Mean-shift (MS) segmentation overcomes this drawback. Mean-shift [4, 5] is a non-parametric technique used to estimate the modes of the multivariate distribution underlying a feature space. It does not require any prior information to initialize the position of the clusters and also does not constraint the shape of the clusters. Mean-shift segmentation involves concatenating both the spatial and range domains of an image and identifying modes in this multidimensional joint spatial-range feature space. The only free parameter is the kernel size (called the bandwidth). If the chosen value is too small then insignificant modes are detected (over-clustering). If it is too large then significant modes can be merged (under-clustering). Several methods [6] are available for the estimation of a single fixed bandwidth. However, the use of a single fixed bandwidth has the drawback that it can yield under- or over-clustering when the feature space has significantly different local characteristics across the space. Variable or adaptive bandwidth methods have been proposed [7] to overcome this drawback. Such methods involve determining the bandwidth value for each feature point by using the pilot density estimate[AM3] . In [8] it was shown that adaptive mean-shift clustering, in which the pilot density estimate is based on the nearest neighborhood, performs better than a single fixed bandwidth in high dimensional space. Nevertheless the bandwidth value defined in terms of the Euclidean distance to the farthest feature point from the kernel center can be biased by outliers [9].
References
[1] J. L. Marroquín, B. C. Vemuri, S. Botello, and F. Calderon, “An accurate and efficient Bayesian method for automatic segmentation of brain MRI,” IEEE Trans. Med. Imag., vol.21, no.8, Aug. 2002, pp. 934–944.
[2] K. Van Leemput, F. Maes, D. Vandeurmeulen, and P. Suetens, “Automated model-based tissue classification of MR images of the brain,” IEEE Trans. Med. Imag., vol.18, no.10 , 1999, pp. 897–908.
[3] H. Greenspan, A. Ruf and J. Goldberger, “Constrained Gaussian mixture model framework for automatic segmentation of MR brain images,” IEEE Trans. Med. Imag., vol. 25, no. 9, Sep. 2006, pp. 1233–1245.
[4] K. Fukunaga and L. Hostetler, “The estimation of the gradient of a density function, with applications in pattern recognition,” IEEE Trans Inf. Theory, vol.21, no.1, 1975, pp. 32-40.
[5] D. Comaniciu and P. Meer, “Mean Shift: A robust approach toward feature space analysis,” IEEE Trans. Pattern Analysis Machine Intell., vol. 24, no. 5, 2002, pp. 603-619.
[6] M. C. Jones, J.S Marron and S. J. Sheather, “A brief survey of bandwidth selection for density estimation,’’ Journal of the American Statistical Association, vol.91, no.433, pp.401-407.
[7] D. Comaniciu, V. Ramesh , P. Meer, “The variable bandwidth mean-shift and data-driven scale selection,” in ICCV, 2001, pp.438-445.
[8] B. Georgescu, I. Shimshoni, and P. Meer, “Mean-shift based clustering in high dimensions: A texture classification example,” in ICCV, 2003, pp. 456–463.
[9] A. G. Bors and N. Nasios, “Kernel bandwidth estimation for nonparametric modeling,” IEEE Trans.SMC, 2009, pp.1543 – 1555.