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Fast medical image mixture density clustering segmentation using stratification sampling and kernel density estimation

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Abstract

The Gaussian mixture models (GMMs) is a flexible and powerful density clustering tool. However, the application of it to medical image segmentation faces some difficulties. First, estimation of the number of components is still an open question. Second, the speed of it for large medical image is slow. Moreover, GMMs has the problem of noise sensitivity. In this paper, the kernel density estimation method is used to estimate the number of components K, and three strategies are proposed to improve the segmentation speed of GMMs. First, a histogram stratification sampling strategy is proposed to reduce the size of the training data. Second, a binning strategy is proposed to search the neighbor points of each center data to compute the approximate density function of the samples. Third, a hill-climbing algorithm with the dynamic step size is designed to find the local maxima of the density function. The kernel density estimation method and sampling technology reduce the effect of noise. Experimental results with the simulated brain images and real CT images show that the proposed algorithm has better performance in generating explainable segmentations with faster speed than the common GMMs algorithm.

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Abbreviations

X :

A medical image

{x 1, x 2, . . . x N }:

All pixels of X

N :

The number of pixels

N h :

The size of the hth stratum of X

n h :

Size of the hth stratum of SX

W h :

Proportion of the hth stratum of X

w h :

Proportion of the hth stratum of SX

N :

The Size of sample SX

K :

The number of components of mixture models

α r :

Is the weight of component r

μ r :

Is the mean of component r

\({\sum _r}\) :

The covariance of component r

\({\varphi _r =\{\alpha _r ,\mu _r ,\Sigma _r \}}\) :

All parameters of component r

\({\theta=(\alpha_1,\alpha_2,\ldots,\,\alpha_K,\mu_1,\mu_2,\ldots\mu_K,\,\sum _1 ,\sum _2 ,\ldots\sum _K ) }\) :

All parameters of mixture model

\({f_r (x|\mu _r ,\sum _r )}\) :

The density function of component r

log L(x|θ):

Log-likelihood function

SX :

The samples extracted from a medical image X

SY = {Y1 , . . . , Y n }:

2D data set converted from the 3D data set SX

G = {Lmin , Lmin + 1, . . . , Lmax}:

The gray level set of a medical image X

p :

The number of sampling strata

His:

The histogram of image X

SHis :

The smoothed histogram

\({\overline{Y}}\) :

The mean of image X

\({\overline{y}_h}\) :

The mean of the hth stratum of sample SX

α :

The precision

δ i :

The ith step optimum size of hill-climbing procedure

\({\tilde {f}(Y_i )}\) :

Approximate density

\({\nabla \tilde {f}(Y_i )}\) :

The gradient function

\({\nabla ^{2} \tilde {f}(Y_i )}\) :

Hessian matrix of density function

S :

Unit gradient vector

\({N_{p\cap g} (r)}\) :

The number of pixels classified by both the proposed method and the ground truth as model r

N p (r):

The number of pixels classified as model r by the proposed method

N g (r):

The number of pixels classified as model r by the ground truth

ξ :

Thresholding

σ :

Window width

xw :

The bin width of x-coordinate

yw :

The bin width of y-coordinate

Y :

The data in the same bin or neighbor bins of Y i

s :

The size of Y

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Authors and Affiliations

  1. School of Computer Science and Telecommunication, Jiangsu University, 212013, Zhenjiang, Jiangsu Province, Peoples’ Republic of China

    Cong-Hua Xie, Yu-Qing Song & Jian-Mei Chen

  2. School of Computer Science and Engineering, Changshu Institute of Technology, 215500, SuZhou, Jiangsu Province, Peoples’ Republic of China

    Cong-Hua Xie

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  1. Cong-Hua Xie
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  2. Yu-Qing Song
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Correspondence to Cong-Hua Xie.

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Xie, CH., Song, YQ. & Chen, JM. Fast medical image mixture density clustering segmentation using stratification sampling and kernel density estimation. SIViP 5, 257–267 (2011). https://doi.org/10.1007/s11760-010-0159-7

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