CN102236898A - Image segmentation method based on t mixed model with infinite component number - Google Patents

Abstract

The invention discloses an image segmentation method based on a t mixed model with an infinite component number. In the method, two aspects of advantages of an infinite mixed component structure and t distribution are used fully, so that a better image segmentation effect can be achieved. The image segmentation method comprises the following steps of: firstly, extracting characteristic information from an image to be segmented; secondly, performing parameter estimation on the t mixed model with the infinite component number by using a vector of the extracted characteristic information of each pixel point to acquire the probability of each pixel point generated from each category to be classified after the estimation is finished; and finally, judging and taking a sequence number corresponding to the maximum value of the probability value of each pixel point related to each category as a final distributed category of the pixel point so as to finish the image segmentation process. By the method, the image segmentation effect can be enhanced effectively and a wild point in the image has higher robustness, so the segmented images have higher smoothness; furthermore, by the method, the mixed component number of the model can be self-adaptively adjusted according to the image to be segmented, so that the phenomena of excessive fitting and sufficient fitting which easily occurs in the conventional Gaussian mixed model-based image segmentation method due to improper setting of the mixed component number can be avoided.

Description

Image partition method based on the t mixture model of unlimited one-tenth mark

Technical field

The present invention relates to Flame Image Process and machine learning field, relate generally to a kind of image partition method of the t mixture model based on unlimited one-tenth mark.

Background technology

Image segmentation is one of gordian technique in the Digital Image Processing process.The task of image segmentation is that input picture is divided into some distinct area, makes the same area have identical attribute, and makes zones of different have different attributes.Image segmentation is further to carry out image recognition, and the basis of analyzing and understanding all obtains people and payes attention to widely in theory research and practical application.For image segmentation problem, people have proposed a lot of methods, but kind is many, data volume big, change characteristics such as many in view of image has, also do not have a kind of method of image segmentation to be applicable to all situations up to now, the quality of segmentation result also needs to go to estimate according to concrete occasion and requirement in addition.Therefore, image segmentation remains one of present research focus.

In existing image partition method, based on the image partition method of statistical model use quite extensive.These class methods usually select for use certain statistical model to describe the distribution of image pixel value to be split, estimate the structure and the relevant parameters of statistical model by certain training and learning process, and obtain the size of each pixel, at last the class that the pairing class of maximum probability is divided into as current pixel point about the probability of each class of desiring to mark off.Corresponding with such flow process is unsupervised learning process in the machine learning field.In the known image partition method based on statistical model, the most common also is that the model that is most widely used is exactly gauss hybrid models (GMM).But owing to gather or image itself, can exist the value difference of the fragmentary pixel of minority and most of pixels not bigger in practice, such pixel usually is called as outlier.Because each blending constituent Gaussian distributed among the GMM, the afterbody of its probability density function falls short of, so relatively poor to the robust performance of outlier.In addition, in GMM, need specify the number of blending constituent in advance, in case after this number was specified, the structure of models of this GMM was definite substantially, and the pairing blending constituent number of distribution of the eigenwert of actual image pixel can't obtain, therefore, adopt GMM when describing the distribution of eigenwert, meeting is set the improper GMM of generation over-fitting (this number is set excessive) owing to the blending constituent number or is owed the phenomenon of match (this number is set too small), thereby has reduced the effect of image segmentation.Exist above-mentioned two problems just because of existing image partition method,, further improve the effect and the performance of image segmentation system so need to improve existing method based on GMM.

Summary of the invention

Purpose of the present invention just is to address the deficiencies of the prior art, and designs, studies the image partition method based on the t mixture model of unlimited one-tenth mark.

Technical scheme of the present invention is:

Image partition method based on the t mixture model of unlimited one-tenth mark is characterized in that may further comprise the steps:

(1) extract the characteristic information of image to be split: with the pixel value of each pixel in the image to be split from the RGB coordinate conversion to the LUV coordinate, thereby obtained a 3-D data set X,

Wherein N is the number of pixel, x _nCharacteristic information data vector for each pixel;

(2) the t mixture model with unlimited one-tenth mark is carried out parameter estimation; After finishing this estimation procedure, for the characteristic information data vector x of each pixel _n, can obtain relative hidden variable z _nDistribution, in this distributes, q (z _Nj=1), j=1 ..., L represents that current pixel point n is the probability that is produced by j composition in the t mixture model with unlimited one-tenth mark, j=1 ..., L;

(3) judgement: q (z that will be relevant with each pixel n _Nj=1), j=1 ..., the pairing sequence number of the maximal value among the L is as this pixel x _nThe class C that is finally allocated to _n, promptly

$C_n=\{i=\underset{j}{\arg \max }q(z_{\mathrm{nj}}=1)\},$

Thereby image segmentation is become to have the class of like attribute, obtain cutting apart the image of finishing.

In the image partition method of described t mixture model based on unlimited one-tenth mark, described L is an approximate bigger number representing ∞ in the actual mechanical process, and its span is the positive integer between 10～30.

In the image partition method of described t mixture model based on unlimited one-tenth mark, the step that described t mixture model with unlimited one-tenth mark is carried out parameter estimation is as follows:

(1) produces equally distributed random integers on N obedience [1, the L] interval, add up the probability that each integer occurs on this interval; That is, if produced N _jIndividual integer j, δ so _j=N _j/ N; For each x _n, corresponding hidden variable z _nInitial distribution be

$q\left(z_n\right)=\underset{j=1}{\overset{L}{\mathrm{\Π}}}q(z_{\mathrm{nj}}=1)=\underset{j=1}{\overset{L}{\mathrm{\Π}}}{\mathrm{\δ}}_j$

(2) set super parameter

Initial value; For all j (j=1 ..., L), m _j=0, λ _j=1, ρ _jCan get any number between 3～20, W _j=10I, I are unit matrix, v _jCan get any number between 1～100, α can get any number between 1～10; In addition, iterations counting variable k=1;

(3) upgrade hidden variable

Distribution, that is, Its super parameter

More new formula be:

${\tilde{v}}_{\mathrm{<figure-callout\; id="1"\; label="nj"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">nj</figure-callout>}1}=\frac{1}{2}[q(z_{\mathrm{nj}}=1)\&CenterDot;3+v_j],$

${\tilde{v}}_{\mathrm{nj}2}=\frac{1}{2}[q(z_{\mathrm{nj}}=1)\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>+v_j],$

Wherein

$<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>=\frac{3}{{\widetilde{\mathrm{\&lambda;}}}_j}+{\widetilde{\mathrm{\&rho;}}}_j{(x_n-{\tilde{m}}_j)}^T{\tilde{W}}_j(x_n-{\tilde{m}}_j);$

Calculating＜(x when iteration first _n-μ _j) ^TΛ _j(x _n-μ _j) time,

(4) upgrade stochastic variable

Distribution, that is, $q({\mathrm{\&mu;}}_j,{\mathrm{\&Lambda;}}_j)=N\left({\mathrm{\&mu;}}_j\right|{\tilde{m}}_j,{\widetilde{\mathrm{\&lambda;}}}_j{\mathrm{\&Lambda;}}_j)W\left({\mathrm{\&Lambda;}}_j\right|{\tilde{W}}_j,{\widetilde{\mathrm{\&rho;}}}_j),$ Corresponding super parameter ${\{m_j,{\mathrm{\&lambda;}}_j,W_j,{\mathrm{\&rho;}}_j\}}_{j=1}^L$ More new formula as follows:

${\widetilde{\mathrm{\&lambda;}}}_j={\mathrm{\&lambda;}}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>,$

${\tilde{m}}_j=\frac{1}{{\widetilde{\mathrm{\&lambda;}}}_j}({\mathrm{\&lambda;}}_j{m}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>\&CenterDot;x_n),$

${\widetilde{\mathrm{\&rho;}}}_j={\mathrm{\&rho;}}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1),$

${\tilde{W}}_j^{-1}=W_j^{-1}+{\mathrm{\&lambda;}}_j{m}_j{m}_j^T+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>\&CenterDot;x_n\&CenterDot;x_n^T-{\widetilde{\mathrm{\&lambda;}}}_j{\tilde{m}}_j{\tilde{m}}_j^T,$

Wherein, $<u_{\mathrm{nj}}>={\tilde{v}}_{\mathrm{<figure-callout\; id="1"\; label="nj"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">nj</figure-callout>}1}/{\tilde{v}}_{\mathrm{nj}2};$

(5) upgrade stochastic variable Distribution, that is,

Corresponding super parameter

More new formula be:

${\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}=1+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1),$

${\widetilde{\mathrm{\&beta;}}}_{j2}=\mathrm{\&alpha;}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=j+1}{\overset{L}{\mathrm{\&Sigma;}}}q(z_{\mathrm{ni}}=1);$

(6) upgrade hidden variable

Distribution

$q\left(z_n\right)=\underset{j=1}{\overset{L}{\mathrm{\&Pi;}}}{\left(\frac{{\widetilde{\mathrm{\&gamma;}}}_{\mathrm{nj}}}{{\mathrm{\&Sigma;}}_{j^{\&prime;}=1}^L{\widetilde{\mathrm{\&gamma;}}}_{{\mathrm{nj}}^{\&prime;}}}\right)}^{z_{\mathrm{nj}}}$

Wherein

${\widetilde{\mathrm{\&gamma;}}}_{\mathrm{nj}}=\exp \{\underset{i=1}{\overset{j-1}{\mathrm{\&Sigma;}}}<\log (1-V_i)+<\log V_i>+[\frac{1}{2}<\log \left|{\mathrm{\&Lambda;}}_j\right|-\frac{3}{2}<\log u_{\mathrm{nj}}>-\frac{1}{2}<u_{\mathrm{nj}}>\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>\left]\right\},$

In following formula, every expectation＜computing formula as follows:

$<\log V_i>=\frac{\mathrm{\&Gamma;}{\left({\widetilde{\mathrm{\&beta;}}}_{j1}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\widetilde{\mathrm{\&beta;}}}_{j1}\right)}-\frac{\mathrm{\&Gamma;}{({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})}^{\&prime;}}{\mathrm{\&Gamma;}({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})},$

$<\log (1-V_i)>=\frac{\mathrm{\&Gamma;}{\left({\widetilde{\mathrm{\&beta;}}}_{j2}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\widetilde{\mathrm{\&beta;}}}_{j2}\right)}-\frac{\mathrm{\&Gamma;}{({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})}^{\&prime;}}{\mathrm{\&Gamma;}({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})},$

$<\log u_{\mathrm{nj}}>=\frac{\mathrm{\&Gamma;}{\left({\tilde{v}}_{\mathrm{nj}1}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\tilde{v}}_{\mathrm{nj}1}\right)}-\log {\tilde{v}}_{\mathrm{nj}2},$

$<\log \left|{\mathrm{\&Lambda;}}_j\right|>=\underset{d=1}{\overset{3}{\mathrm{\&Sigma;}}}\mathrm{\&Gamma;}{\left(\frac{{\widetilde{\mathrm{\&rho;}}}_j+1-d}{2}\right)}^{\&prime;}/\mathrm{\&Gamma;}\left(\frac{{\widetilde{\mathrm{\&rho;}}}_j+1-d}{2}\right)+\log \left|{\tilde{W}}_j\right|+3\log 2,$

Wherein Γ () is the gamma function of standard, and Γ () ' is a standard gamma function derivative; In addition,＜(x _n-μ _j) ^TΛ _j(x _n-μ _j) and＜u _NjComputing method provide in step (3) and step (4) respectively;

(7) upgrade the degree of freedom parameter

That is, separate the following v of containing _jEquation:

$1+\frac{1}{{\mathrm{\&Sigma;}}_{n=1}^{N}q(z_{\mathrm{nj}}=1)}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)[<\log u_{\mathrm{nj}}>-<u_{\mathrm{nj}}>]+\log \left(\frac{v_j}{2}\right)-\frac{{\mathrm{\&Gamma;}}^{\&prime;}(v_j/2)}{\mathrm{\&Gamma;}(v_j/2)}=0,$

Can select numerical computation method commonly used for use,, obtain the v that separates of this equation apace as Newton method _j

(8) the likelihood value LIK after the calculating current iteration _k, k is current iterations:

${\mathrm{LIK}}_k=\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{L}{\mathrm{\&Sigma;}}}\{q(z_{\mathrm{nj}}=1)\&CenterDot;[\underset{i=1}{\overset{j-1}{\mathrm{\&Sigma;}}}<\log (1-V_i)>+<\log V_i>+(\frac{1}{2}<\log |{\mathrm{\&Lambda;}}_j|>$

$-\frac{3}{2}<\log u_{\mathrm{nj}}>-\frac{1}{2}<u_{\mathrm{nj}}>\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>\left)\right]\}$

(9) calculate after the current iteration with last iteration after the difference DELTA LIK=LIK of likelihood value _k-LIK _K-1If Δ LIK≤δ, parameter estimation procedure finishes so, otherwise forwards step (3) to, and the value of k increases by 1, proceeds iteration next time; The span of threshold value δ is 10 ^-5～10 ^-4

Advantage of the present invention and effect are:

1. the t mixture model of the unlimited one-tenth mark that is adopted among the present invention has and infinitely is mixed into mark, this structure makes this model have very strong dirigibility, can automatically regulate the optimum structure of model according to the concrete distribution of image pixel value to be split, thereby determine the appropriate ingredients number automatically.Solved and be mixed in traditional image partition method that mark is fixed and uncontrollable shortcoming, and model over-fitting that causes thus or the problem of owing fitting data, thereby improved the effect and the quality of image segmentation based on gauss hybrid models.

2. what the distribution function of each composition adopted in the t mixture model of the unlimited one-tenth mark that is adopted among the present invention is that t distributes modeling, its advantage is, compare with the gauss of distribution function that adopts in the traditional model, t distributes the outlier that occurs easily in noise in the image and the data acquisition is had stronger robustness, makes that the flatness of the image after cutting apart is better.

Other advantages of the present invention and effect will continue to describe below.

Description of drawings

Fig. 1---method flow diagram of the present invention.

Fig. 2---have the structural drawing of the t mixture model (itMM) of unlimited one-tenth mark.

Fig. 3---the contrast of image segmentation effect

Fig. 4---adopt the objective evaluation result of the image segmentation of Probability Rand index

Fig. 5---two kinds of dividing method contrasts of the number of definite blending constituent automatically

Embodiment

Below in conjunction with drawings and Examples, technical solutions according to the invention are further elaborated.Fig. 1 is a method flow diagram of the present invention, and it was three steps that method of the present invention is divided into.

The first step: the characteristic information that extracts image to be split

Because the pixel value of most of in actual applications images to be split is represented with the three-dimensional coordinate in the rgb space, and in the image segmentation task, it is general that what adopt is that three-dimensional coordinate in the LUV space is represented mode, because the coordinate in the LUV space can carry out cluster with similar pixel value better, therefore, in feature extraction of the present invention, need be coordinate under the LUV by the coordinate transformation under the RGB with the pixel value of image.

Detailed process is as follows:

(1) with the coordinate (R of current pixel point n _n, G _n, B _n) ^TBe transformed into XYZ space from rgb space, obtain (X _n, Y _n, Z _n) ^T

$\left(\begin{array}{c}{X}_n\\ Y_n\\ Z_n\end{array}\right)=\frac{1}{0.17697}\left(\begin{array}{ccc}0.49& 0.31& 0.20\\ 0.17697& 0.81240& 0.01063\\ 0.00& 0.01& 0.99\end{array}\right)\&CenterDot;\left(\begin{array}{c}{R}_n\\ G_n\\ B_n\end{array}\right)---\left(1\right)$

(2) try to achieve U ' and V ' by following formula,

$U^{\&prime;}=\frac{4X_n}{X_n+15Y_n+2Z_n},$ $V^{\&prime;}=\frac{9Y_n}{X_n+15Y_n+3Z_n}---\left(2\right)$

(3) with (X _n, Y _n, Z _n) ^TBe transformed into the LUV space, obtain the LUV coordinate (L of current pixel point n _n, U _n, V _n) ^TThereby, finish the leaching process of characteristic information.Concrete computing formula is as follows:

$L_n=\left\{\begin{array}{cc}116\&CenterDot;{\left(\frac{Y_n}{Y_c}\right)}^{\frac{1}{3}}-16,& \frac{Y_n}{Y_c}>\left(\frac{6}{29}\right)\\ {\left(\frac{29}{3}\right)}^3\&CenterDot;\frac{Y_n}{Y_c},& \frac{Y_n}{Y_c}\&le;{\left(\frac{6}{29}\right)}^3\end{array}\right.$

U _n＝13L _n·(U′-U _c)

(3)

V _n＝13L _n·(V′-V _c)

Y wherein _c=1, U _c=0.20116, V _c=0.460806.

Each pixel for the treatment of in the split image according to said process carries out feature extraction, thereby has obtained a 3-D data set X,

Wherein N is the number of the pixel of this image, and the characteristic information data vector of each pixel is x _n=(L _n, U _n, V _n) ^T

Second step: the t mixture model with unlimited one-tenth mark is carried out parameter estimation

Relatively poor in order to solve based on the robustness to the outlier that exists in the image segmentation process of GMM, and being mixed into mark and need preestablishing among the GMM, and in actual applications, the optimal value of this one-tenth mark is difficult to problems such as acquisition, and what adopt here is the t mixture model (itMM) with unlimited one-tenth mark.Compare with GMM, itMM has two remarkable different characteristics: at first, itMM has unlimited blending constituent number, and secondly, each composition is obeyed the t distribution among the itMM.For data set X, adopt the expression formula that its probability density is described of itMM as follows:

$p\left(X\right)=\underset{n=1}{\overset{N}{\mathrm{\&Pi;}}}\underset{j=1}{\overset{\&infin;}{\mathrm{\&Sigma;}}}{\mathrm{\&pi;}}_j\left(V\right)\&CenterDot;\left(x_n\right|{\mathrm{\&mu;}}_j,{\mathrm{\&Lambda;}}_j,v_j)---\left(4\right)$

Wherein, π _j(V), μ _j, Λ _j, v _jThe weights of representing j blending constituent respectively, average, covariance matrix and degree of freedom parameter.T (x _n| μ _j, Λ _j, v _j) being the probability density function that t distributes, it can be expressed as

$t\left(x_n\right|{\mathrm{\&mu;}}_j,{\mathrm{\&Lambda;}}_j,v_j)={\&Integral;}_0^{\&infin;}N\left(x_n\right|{\mathrm{\&mu;}}_j,u_{\mathrm{ij}}{\mathrm{\&Lambda;}}_j)\mathrm{Gam}\left(u_{\mathrm{nj}}\right|v_j/2,v_j/2){\mathrm{du}}_{\mathrm{nj}}---\left(5\right)$

Wherein N () and Gam () represent Gaussian distribution function and Gamma distribution function, u respectively _NjFor with x _nWith j the hidden variable that blending constituent is relevant.Weights π _j(V) satisfy

Its expression formula is:

${\mathrm{\&pi;}}_j\left(V\right)=V_j\&CenterDot;\underset{i=1}{\overset{j-1}{\mathrm{\&Pi;}}}(1-V_i)---\left(6\right)$

Variable V in the following formula _jObey Beta and distribute, be i.e. p (V _j)=Beta (V _j| 1, α), the super parameter that α distributes for this Beta.In addition, μ _j, Λ _jObey associating Gaussian-Wishart distribution (being the product that Gaussian distribution and Wishart distribute, N () W ()):

p(μ _j，Λ _j)＝N(μ _j|m _j，κ _jΛ _j)W(Λ _j|W _j，ρ _j) (7)

Wherein

Super parameter for this associating Gaussian-Wishart distribution.m _jBe three-dimensional column vector, λ _jAnd ρ _jBe scalar, W _jIt is the matrix of (3 * 3).Also need to introduce a hidden variable

Z wherein _nIndicate current data x _nBe to produce by which composition in the t mixture model.Work as x _nBe when producing by j blending constituent, z _Nj=1.Based on the above, the structure of itMM as shown in Figure 2, wherein stochastic variable is represented with open circles, super parameter represents that with black circle the super parameter of whole model is:

Because infinitely great " ∞ " can't accurately represent when calculating, be similar to bigger several L usually in actual mechanical process and represent ∞.The value of L is comparatively flexible, and L gets greater than 10 usually less than 30 integer in the present invention.

Under the itMM of above-mentioned definition, the step that the parameter of this model is estimated is as follows:

(1) produces equally distributed random integers on N obedience [1, the L] interval, add up the probability that each integer occurs on this interval.That is, if produced N _jIndividual integer j, δ so _j=N _j/ N.For each x _n, corresponding hidden variable z _nInitial distribution be

$q\left(z_n\right)=\underset{j=1}{\overset{L}{\mathrm{\&Pi;}}}q(z_{\mathrm{nj}}=1)=\underset{j=1}{\overset{L}{\mathrm{\&Pi;}}}{\mathrm{\&delta;}}_j---\left(8\right)$

(2) set super parameter

Initial value, iterations counting variable k=1;

Particularly, for all blending constituent j (j=1 ..., L), m _j=0 (0 is three-dimensional zero vector), λ _j=1, ρ _jCan get any number between 3～20, W _j=10I (I is a unit matrix), v _jCan get any number between 1～100, α can get any number between 1～10.

(3) upgrade hidden variable Distribution, it is still obeyed Gamma and distributes, promptly

Its super parameter

More new formula be:

${\tilde{v}}_{\mathrm{nj}1}=\frac{1}{2}[q(z_{\mathrm{nj}}=1)\&CenterDot;3+v_j],$ (9)

${\tilde{v}}_{\mathrm{nj}2}=\frac{1}{2}[q(z_{\mathrm{nj}}=1)\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>+v_j],$

Wherein

$<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>=\frac{3}{{\widetilde{\mathrm{\&lambda;}}}_j}+{\widetilde{\mathrm{\&rho;}}}_j{(x_n-{\tilde{m}}_j)}^T{\tilde{W}}_j(x_n-{\tilde{m}}_j)---\left(10\right)$

When it should be noted that in iteration (k=1) first calculating formula (10),

(4) upgrade stochastic variable Distribution, it is still obeyed associating Gaussian-Wishart and distributes, promptly $q({\mathrm{\&mu;}}_j,{\mathrm{\&Lambda;}}_j)=N\left({\mathrm{\&mu;}}_j\right|{\tilde{m}}_j,{\widetilde{\mathrm{\&lambda;}}}_j{\mathrm{\&Lambda;}}_j)W\left({\mathrm{\&Lambda;}}_j\right|{\tilde{W}}_j,{\widetilde{\mathrm{\&rho;}}}_j),$ Corresponding super parameter

More new formula as follows:

${\widetilde{\mathrm{\&lambda;}}}_j={\mathrm{\&lambda;}}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>,$

${\tilde{m}}_j=\frac{1}{{\widetilde{\mathrm{\&lambda;}}}_j}({\mathrm{\&lambda;}}_j{m}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>\&CenterDot;x_n),---\left(11\right)$

${\widetilde{\mathrm{\&rho;}}}_j={\mathrm{\&rho;}}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1),$

${\tilde{W}}_j^{-1}=W_j^{-1}+{\mathrm{\&lambda;}}_j{m}_j{m}_j^T+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>\&CenterDot;x_n\&CenterDot;x_n^T-{\widetilde{\mathrm{\&lambda;}}}_j{\tilde{m}}_j{\tilde{m}}_j^T,$

Wherein

$<u_{\mathrm{nj}}>={\tilde{v}}_{\mathrm{nj}1}/{\tilde{v}}_{\mathrm{nj}2}---\left(12\right)$

(5) upgrade stochastic variable

Distribution, it is still obeyed Beta and distributes, promptly Corresponding super parameter

More new formula be:

${\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}=1+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1),$ (13)

${\widetilde{\mathrm{\&beta;}}}_{j2}=\mathrm{\&alpha;}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=j+1}{\overset{L}{\mathrm{\&Sigma;}}}q(z_{\mathrm{ni}}=1);$

(6) upgrade hidden variable Distribution

$q\left(z_n\right)=\underset{j=1}{\overset{L}{\mathrm{\&Pi;}}}{\left(\frac{{\widetilde{\mathrm{\&gamma;}}}_{\mathrm{nj}}}{{\mathrm{\&Sigma;}}_{j^{\&prime;}=1}^L{\widetilde{\mathrm{\&gamma;}}}_{{\mathrm{nj}}^{\&prime;}}}\right)}^{z_{\mathrm{nj}}}$

Wherein

${\widetilde{\mathrm{\&gamma;}}}_{\mathrm{nj}}=\exp \{\underset{i=1}{\overset{j-1}{\mathrm{\&Sigma;}}}<\log (1-V_i)+<\log V_i>+[\frac{1}{2}<\log \left|{\mathrm{\&Lambda;}}_j\right|-\frac{3}{2}<\log u_{\mathrm{nj}}>-\frac{1}{2}<u_{\mathrm{nj}}>\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>\left]\right\},$

In formula (15), every expectation＜computing formula as follows:

$<\log V_i>=\frac{\mathrm{\&Gamma;}{\left({\widetilde{\mathrm{\&beta;}}}_{j1}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\widetilde{\mathrm{\&beta;}}}_{j1}\right)}-\frac{\mathrm{\&Gamma;}{({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})}^{\&prime;}}{\mathrm{\&Gamma;}({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})},$

$<\log (1-V_i)>=\frac{\mathrm{\&Gamma;}{\left({\widetilde{\mathrm{\&beta;}}}_{j2}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\widetilde{\mathrm{\&beta;}}}_{j2}\right)}-\frac{\mathrm{\&Gamma;}{({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})}^{\&prime;}}{\mathrm{\&Gamma;}({\widetilde{\mathrm{\&beta;}}}_{\mathrm{<figure-callout\; id="1"\; label="j"\; filenames="HSA00000555271300021.png"\; state="\{\{state\}\}">j</figure-callout>}1}+{\widetilde{\mathrm{\&beta;}}}_{j2})},$ (16)

$<\log u_{\mathrm{nj}}>=\frac{\mathrm{\&Gamma;}{\left({\tilde{v}}_{\mathrm{nj}1}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\tilde{v}}_{\mathrm{nj}1}\right)}-\log {\tilde{v}}_{\mathrm{nj}2},$

$<\log \left|{\mathrm{\&Lambda;}}_j\right|>=\underset{d=1}{\overset{3}{\mathrm{\&Sigma;}}}\mathrm{\&Gamma;}{\left(\frac{{\widetilde{\mathrm{\&rho;}}}_j+1-d}{2}\right)}^{\&prime;}/\mathrm{\&Gamma;}\left(\frac{{\widetilde{\mathrm{\&rho;}}}_j+1-d}{2}\right)+\log \left|{\tilde{W}}_j\right|+3\log 2,$

Wherein Γ () is a standard gamma function, and Γ () ' is a standard gamma function derivative.In addition＜u _NjAnd＜(x _n-μ _j) ^TΛ _j(x _n-μ _j) computing method provide by formula (10) and formula (12).

(7) upgrade the degree of freedom parameter

Specifically, separate the following v of containing _jEquation.

$1+\frac{1}{{\mathrm{\&Sigma;}}_{n=1}^{N}q(z_{\mathrm{nj}}=1)}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)[<\log u_{\mathrm{nj}}>-<u_{\mathrm{nj}}>]+\log \left(\frac{v_j}{2}\right)-\frac{{\mathrm{\&Gamma;}}^{\&prime;}(v_j/2)}{\mathrm{\&Gamma;}(v_j/2)}=0---\left(17\right)$

Wherein＜u _NjAnd＜logu _NjSee formula (12) and formula (16) respectively.Here can select for use typical number value calculating method (as Newton method) to obtain the v that separates of this equation _j

(8) the likelihood value LIK after the calculating current iteration _k(k is current iterations):

${\mathrm{LIK}}_k=\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{L}{\mathrm{\&Sigma;}}}\{q(z_{\mathrm{nj}}=1)\&CenterDot;[\underset{i=1}{\overset{j-1}{\mathrm{\&Sigma;}}}<\log (1-V_i)>+<\log V_i>+(\frac{1}{2}<\log |{\mathrm{\&Lambda;}}_j|>$ (18)

$-\frac{3}{2}<\log u_{\mathrm{nj}}>-\frac{1}{2}<u_{\mathrm{nj}}>\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>\left)\right]\}$

Involved expectation in the formula (18)＜calculating see formula (10), formula (12) and formula (16).

(9) calculate after the current iteration with last iteration after the difference DELTA LIK=LIK of likelihood value _k-LIK _K-1If Δ LIK≤δ, parameter estimation procedure stops so, otherwise forwards step (3) to, and the value of k increases by 1, proceeds iteration next time.The span of threshold value δ is 10 ^-5～10 ^-4

The above-mentioned parameter estimation procedure is shown in square frame maximum among Fig. 1.Need annotatedly be, the Gaussian distribution N () that is mentioned in this step, Gamma distribution Gam (), Beta distribution Beta (), Wishart distribution W () and gamma function gamma () all are the functions with canonical form, the expression formula that these functions are all arranged in most probability statistics books and the documents and materials, they also all are the functions that this area scientific and technical personnel know and often need to use, implementing only need to consult corresponding probability statistics teaching material when of the present invention or relevant encyclopaedia introduction can obtain easily, providing its concrete form herein no longer one by one.

The 3rd step: judgement, each pixel is assigned in the class with similar features attribute, obtain cutting apart the image of finishing.

For each pixel n, the judgement of the class under it is as follows:

$C_n=\{i=\underset{j}{\arg \mathrm{mx}}q(z_{\mathrm{nj}}=1)\}---\left(19\right)$

That is to say that the class at its place is all q (z _Nj=1), j=1 ..., the pairing sequence number of the maximal value among the L (convenience in order to describe supposes that this sequence number is i here).

Performance evaluation

In order to verify the segmentation effect of the image partition method that has adopted the t mixture model (itMM) based on unlimited one-tenth mark of the present invention, with its with based on traditional gauss hybrid models (GMM), t mixture model (tMM) and infinitely become the resulting effect of image partition method of the gauss hybrid models (iGMM) of mark to contrast.What select for use here is that famous Berkeley image segmentation database is opposed than test.In this database, for each image to be split the corresponding result of manually cutting apart is arranged all, so better the comparative evaluation adopts the degree of agreement of segmentation effect that obtains behind the image partition method and the result of manually cutting apart.Test mainly comprises dual mode, that is, and and the subjective assessment mode of Direct observation segmentation effect and and try to achieve the objective evaluation mode of cutting apart index accordingly by calculating.Carry out image segmentation property comparison and evaluation with this dual mode to method of the present invention with based on the method for other three kinds of models.

Fig. 3 provided the result manually cut apart in the Berkeley image segmentation database, based on the segmentation result of the image partition method of iGMM and the segmentation result that the present invention relates to based on the image partition method of itMM, adopt method of the present invention as can be seen, segmentation effect is more level and smooth than iGMM, method of the present invention has stronger robustness to the outlier that occurs on the image, and more approaching with the result of manually cutting apart.Except above-mentioned subjective direct evaluation, Fig. 4 has provided the objective evaluation result based on Probability Rand index.The ProbabilityRand index is to be used for weighing having adopted segmentation result that method of the present invention obtains and artificial segmentation result in similarity, and the span of this index be [0,1], and value is big more, and to show that segmentation result and the result of manually cutting apart coincide good more.The numbering of the first tabulation diagrammatic sketch picture in the Berkeley storehouse of Fig. 4, the test here is to select 30 its performances of width of cloth picture appraisal from this storehouse arbitrarily.The result who provides from Fig. 4 as can be seen, for each width of cloth image, t mixture model itMM based on unlimited one-tenth mark of the present invention) image partition method is all bigger than the Probability Rand index of other three kinds of methods, so it has optimum segmentation effect.Fig. 5 has provided the image partition method of the t mixture model (itMM) based on unlimited one-tenth mark of the present invention and the iGMM method number through the blending constituent determined automatically after the parameter estimation, the composition number that can significantly find out method of the present invention is equal to or less than the iGMM method, this further illustrate method synthesis of the present invention t distribute to the outlier in the image have robustness with and the unlimited blending constituent that has can determine advantages such as proper model structure adaptively, thereby can obtain good image segmentation effect.

The scope that the present invention asks for protection is not limited only to the description of this embodiment.

Claims (3)

1. based on the image partition method of the t mixture model of unlimited one-tenth mark, it is characterized in that may further comprise the steps:

(1) extract the characteristic information of image to be split: with the pixel value of each pixel in the image to be split from the RGB coordinate conversion to the LUV coordinate, thereby obtained a 3-D data set X,

Wherein N is the number of pixel, x _nCharacteristic information data vector for each pixel;

(2) the t mixture model with unlimited one-tenth mark is carried out parameter estimation; After finishing this estimation procedure, for the characteristic information data vector x of each pixel _n, can obtain relative hidden variable z _nDistribution, in this distributes, q (z _Nj=1), j=1 .., L represent that current pixel point n is the probability that is produced by j composition in the t mixture model with unlimited one-tenth mark, j=1 ..., L;

(3) judgement: q (z that will be relevant with each pixel n _Nj=1), j=1 ..., the pairing sequence number of the maximal value among the L is as this pixel x _nThe class C that is finally allocated to _n, promptly

$C_n=\{i=\underset{j}{\arg \max }q(z_{\mathrm{nj}}=1)\},$

Thereby image segmentation is become to have the class of like attribute, obtain cutting apart the image of finishing.

2. the image partition method of the t mixture model based on unlimited one-tenth mark according to claim 1 is characterized in that, L is an approximate bigger number representing ∞ in the actual mechanical process, and it can get certain any positive integer between 10～30.

3. the image partition method of the t mixture model based on unlimited one-tenth mark according to claim 1 is characterized in that the step of the t mixture model with unlimited one-tenth mark being carried out parameter estimation is as follows:

(1) produce equally distributed random integers on N obedience [1, the L] interval, add up each integer j on this interval (j=1 ..., L) the probability δ of Chu Xianing _jThat is, if produced N _jIndividual integer j, δ so _j=N _j/ N; For each x _n, corresponding hidden variable z _nInitial distribution be

$q\left(z_n\right)=\underset{j=1}{\overset{L}{\mathrm{\&Pi;}}}q(z_{\mathrm{nj}}=1)=\underset{j=1}{\overset{L}{\mathrm{\&Pi;}}}{\mathrm{\&delta;}}_j$

(2) set super parameter

Initial value; For all j (j=1 ..., L), m _j=0, λ _j=1, ρ _jCan get any number between 3～20, W _j=10I, I are unit matrix, v _jCan get any number between 1～100, α can get any number between 1～10; In addition, iterations counting variable k=1;

(3) upgrade hidden variable

Distribution, that is, Its super parameter

More new formula be:

${\tilde{v}}_{\mathrm{nj}1}=\frac{1}{2}[q(z_{\mathrm{nj}}=1)\&CenterDot;3+v_j],$

${\tilde{v}}_{\mathrm{nj}2}=\frac{1}{2}[q(z_{\mathrm{nj}}=1)\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>+v_j],$

Wherein

$<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>=\frac{3}{{\widetilde{\mathrm{\&lambda;}}}_j}+{\widetilde{\mathrm{\&rho;}}}_j{(x_n-{\tilde{m}}_j)}^T{\tilde{W}}_j(x_n-{\tilde{m}}_j);$

Calculating＜(x in iteration first _n-μ _j) ^TΛ _j(x _n-μ _j) time,

(4) upgrade stochastic variable

Distribution, that is, $q({\mathrm{\&mu;}}_j,{\mathrm{\&Lambda;}}_j)=N\left({\mathrm{\&mu;}}_j\right|{\tilde{m}}_j,{\widetilde{\mathrm{\&lambda;}}}_j{\mathrm{\&Lambda;}}_j)W\left({\mathrm{\&Lambda;}}_j\right|{\tilde{W}}_j,{\widetilde{\mathrm{\&rho;}}}_j),$ Corresponding super parameter

More new formula as follows:

${\widetilde{\mathrm{\&lambda;}}}_j={\mathrm{\&lambda;}}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>,$

${\tilde{m}}_j=\frac{1}{{\widetilde{\mathrm{\&lambda;}}}_j}({\mathrm{\&lambda;}}_j{m}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>\&CenterDot;x_n),$

${\widetilde{\mathrm{\&rho;}}}_j={\mathrm{\&rho;}}_j+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1),$

${\tilde{W}}_j^{-1}=W_j^{-1}+{\mathrm{\&lambda;}}_j{m}_j{m}_j^T+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)\&CenterDot;<u_{\mathrm{nj}}>\&CenterDot;x_n\&CenterDot;x_n^T-{\widetilde{\mathrm{\&lambda;}}}_j{\tilde{m}}_j{\tilde{m}}_j^T,$

Wherein, $<u_{\mathrm{nj}}>={\tilde{v}}_{\mathrm{nj}1}/{\tilde{v}}_{\mathrm{nj}2};$

(5) upgrade stochastic variable

Distribution, that is,

Corresponding super parameter

More new formula be:

${\widetilde{\mathrm{\&beta;}}}_{j1}=1+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1),$

${\widetilde{\mathrm{\&beta;}}}_{j2}=\mathrm{\&alpha;}+\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{i=j+1}{\overset{L}{\mathrm{\&Sigma;}}}q(z_{\mathrm{ni}}=1);$

(6) upgrade hidden variable

Distribution

$q\left(z_n\right)=\underset{j=1}{\overset{L}{\mathrm{\&Pi;}}}{\left(\frac{{\widetilde{\mathrm{\&gamma;}}}_{\mathrm{nj}}}{{\mathrm{\&Sigma;}}_{j^{\&prime;}=1}^L{\widetilde{\mathrm{\&gamma;}}}_{{\mathrm{nj}}^{\&prime;}}}\right)}^{z_{\mathrm{nj}}}$

Wherein

${\widetilde{\mathrm{\&gamma;}}}_{\mathrm{nj}}=\exp \{\underset{i=1}{\overset{j-1}{\mathrm{\&Sigma;}}}<\log (1-V_i)+<\log V_i>+[\frac{1}{2}<\log \left|{\mathrm{\&Lambda;}}_j\right|-\frac{3}{2}<\log u_{\mathrm{nj}}>-\frac{1}{2}<u_{\mathrm{nj}}>\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>\left]\right\},$

In following formula, every expectation＜computing formula as follows:

$<\log V_i>=\frac{\mathrm{\&Gamma;}{\left({\widetilde{\mathrm{\&beta;}}}_{j1}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\widetilde{\mathrm{\&beta;}}}_{j1}\right)}-\frac{\mathrm{\&Gamma;}{({\widetilde{\mathrm{\&beta;}}}_{j1}+{\widetilde{\mathrm{\&beta;}}}_{j2})}^{\&prime;}}{\mathrm{\&Gamma;}({\widetilde{\mathrm{\&beta;}}}_{j1}+{\widetilde{\mathrm{\&beta;}}}_{j2})},$

$<\log (1-V_i)>=\frac{\mathrm{\&Gamma;}{\left({\widetilde{\mathrm{\&beta;}}}_{j2}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\widetilde{\mathrm{\&beta;}}}_{j2}\right)}-\frac{\mathrm{\&Gamma;}{({\widetilde{\mathrm{\&beta;}}}_{j1}+{\widetilde{\mathrm{\&beta;}}}_{j2})}^{\&prime;}}{\mathrm{\&Gamma;}({\widetilde{\mathrm{\&beta;}}}_{j1}+{\widetilde{\mathrm{\&beta;}}}_{j2})},$

$<\log u_{\mathrm{nj}}>=\frac{\mathrm{\&Gamma;}{\left({\tilde{v}}_{\mathrm{nj}1}\right)}^{\&prime;}}{\mathrm{\&Gamma;}\left({\tilde{v}}_{\mathrm{nj}1}\right)}-\log {\tilde{v}}_{\mathrm{nj}2},$

$<\log \left|{\mathrm{\&Lambda;}}_j\right|>=\underset{d=1}{\overset{3}{\mathrm{\&Sigma;}}}\mathrm{\&Gamma;}{\left(\frac{{\widetilde{\mathrm{\&rho;}}}_j+1-d}{2}\right)}^{\&prime;}/\mathrm{\&Gamma;}\left(\frac{{\widetilde{\mathrm{\&rho;}}}_j+1-d}{2}\right)+\log \left|{\tilde{W}}_j\right|+3\log 2,$

Wherein Γ () is the gamma function of standard, and Γ () ' is a standard gamma function derivative; In addition,＜(x _n-μ _j) ^TΛ _j(x _n-μ _j) and＜μ _NjComputing method provide in step (3) and step (4) respectively;

(7) upgrade the degree of freedom parameter

That is, separate the following v of containing _jEquation:

$1+\frac{1}{{\mathrm{\&Sigma;}}_{n=1}^{N}q(z_{\mathrm{nj}}=1)}\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}q(z_{\mathrm{nj}}=1)[<\log u_{\mathrm{nj}}>-<u_{\mathrm{nj}}>]+\log \left(\frac{v_j}{2}\right)-\frac{{\mathrm{\&Gamma;}}^{\&prime;}(v_j/2)}{\mathrm{\&Gamma;}(v_j/2)}=0,$

Can select numerical computation method commonly used for use,, obtain the v that separates of this equation apace as Newton method _j

(8) the likelihood value LIK after the calculating current iteration _k, k is current iterations:

${\mathrm{LIK}}_k=\underset{n=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{L}{\mathrm{\&Sigma;}}}\{q(z_{\mathrm{nj}}=1)\&CenterDot;[\underset{i=1}{\overset{j-1}{\mathrm{\&Sigma;}}}<\log (1-V_i)>+<\log V_i>+(\frac{1}{2}<\log |{\mathrm{\&Lambda;}}_j|>$

$-\frac{3}{2}<\log u_{\mathrm{nj}}>-\frac{1}{2}<u_{\mathrm{nj}}>\&CenterDot;<{(x_n-{\mathrm{\&mu;}}_j)}^T{\mathrm{\&Lambda;}}_j(x_n-{\mathrm{\&mu;}}_j)>\left)\right]\}$

(9) calculate after the current iteration with last iteration after the difference DELTA LIK=LIK of likelihood value _k-LIK _K-1If Δ LIK≤δ, parameter estimation procedure finishes so, otherwise forwards step (3) to, and the value of k increases by 1, proceeds iteration next time; The span of threshold value δ is 10 ^-5～10 ^-4

Images