CN101661614A - Segmentation method of cell nucleolus and cell membrane based on mixed contour model - Google Patents

Abstract

The invention relates to a segmentation method of cell nucleolus and cell membrane based on a mixed contour model. The method comprises the following steps: 1) according to the mixed contour model, establishing an energy function of a cell image to be segmented, the form of which is: such as E<E>(Phi, c1, c2, f1, f2)=E<M>(Phi, c1, c2, f1, f2)+P(Phi)+L(Phi); and 2) constructing and using a steepestdescent method to minimize the formula, calculating the border of the cell nucleolus and cell membrane of the cells by utilizing the level set function Phi being equal to zero through an iterative approach of discretization of partial differential equations. The invention provides a segmentation method of cell nucleolus and cell membrane based on the mixed contour model, which has high precisionand excellent segmentation effect.

Description

Based on the entoblast of mixed contour model and the dividing method of cell membrane

Technical field

The present invention relates to Flame Image Process, biomedicine, computer vision, computing method, the especially dividing method of biological cell micro-image.

Background technology

In these years, biology invention and its are converted to the clinical practice treatment and develop rapidly.The virologist uses patient's histotomy pathological image, and detects at microscopically.When detecting this class image, the virologist usually changes with detected eucaryotic cell structure variation of tissue image and the ratio of entoblast in tenuigenin and comes the assess lesion degree.Automatically detect the profile of cell and the method for its kernel and seem particularly important.

In the cell image process field, a large amount of achievements in research has been arranged.But also some challenge is present in these researchs simultaneously.Noise pollution is the highest class problem of the frequency of occurrences.This class problem results from histiocytic dyeing course, therefore occurs the problem of dye distribution inequality when handling dyeing pigment through regular meeting.Except noise, hatching effect often appears at and makes the unsmooth of cell image performance in the image.

Before cell analysis,, need cell segmentation accurately such as cellular morphology and cell behavior.Active contour model becomes one of model of success.Technology based on active contour model has the potentiality of estimating cellular morphology better.Existing active contour model can be divided into two classes: based on the skeleton pattern on border with based on the skeleton pattern in zone.On the one hand, directly using shade of gray information to order about outline line based on the model on border moves towards object boundary.What therefore this class model showed when handling the weak boundary subject is bad.This is because cell image is because thereby near the low contrast cell membrane shows obscurity boundary.On the other hand, by certain class region description, be intended to distinguish out the zone of different gray scales based on the model in zone.It is ordering about the motion of profile, to some degree, and also insensitive to the initial profile line position.This model is more suitable for cell segmentation with respect to preceding a kind of model.

The IEEE Flame Image Process journal the 10th that T.Chan and L.Vese published in calendar year 2001 is rolled up the article Active contours without edges (need not the active profile on border) that delivers in the 266-277 page or leaf and has been proposed a kind of popular active contour model based on the zone.This model successfully has been used for image segmentation.What but area-of-interest often showed on statistics in the image is unsmooth, and therefore this model can not be directly applied for cell image.People such as the Li nonhomogeneous property of gray scale that Implicit active contours drivenby local binary fitting energy (the implicit expression active contour model of local two-value match energy drives) mentions in the 1-7 page or leaf in IEEE computer vision in 2007 and pattern-recognition meeting, this is a problem maximum in low SNR images, often appears in histotomy figure and the medical image.

Summary of the invention

The deficiency of, segmentation effect difference low for the precision that overcomes existing cell image dividing method, the invention provides a kind of precision height, segmentation effect good based on the entoblast of mixed contour model and the dividing method of cell membrane.

The technical solution adopted for the present invention to solve the technical problems is:

A kind of based on the entoblast of mixed contour model and the dividing method of cell membrane, described dividing method may further comprise the steps:

1), cell image to be split is set up energy function according to mixed contour model, form as:

E ^E(φ，c ₁，c ₂，f ₁，f ₂)＝E ^M(φ，c ₁，c ₂，f ₁，f ₂)+P(φ)+L(φ) (11)；

Wherein, E ^M(φ, c ₁, c ₂, f ₁, f ₂) represent energy function based on regional model and local two-value model of fit, its computing formula is:

E ^M(φ，c ₁，c ₂，f ₁，f ₂)＝(1-λ _g)E ^LBF(φ，f ₁，f ₂)

+λ _g(x，y)E ^G(φ，c ₁，c ₂) (7)；

λ wherein _gBe tactful weight parameter, E ^LBF(φ, f ₁, f ₂) as definition in the formula (6), E ^G(φ, c ₁, c ₂) as defining in the formula (5):

E ^G(φ，c ₁，c ₂)＝∫(c ₁-c ₂)(I(x)-c ₁p ₂-c ₂p ₁)H(φ(x))dx -.；

Wherein sheet is the Heaviside function, and φ () is the symbolic distance function, p ₁Represented A _C1With the ratio of A, p ₂Represented A _C2Ratio with A;

This strategy weight parameter λ _g(x y) is defined as follows:

${\mathrm{\&lambda;}}_g(x,y)=\frac{1}{(1+\mathrm{\&alpha;}|\&dtri;I(x,y)\left|\right)}----\left(8\right);$

Wherein

Shade of gray letter among the presentation video I, α is a positive constant;

Inhibition level set function φ departs from the gauged distance sign function and comes the regularization level set function, and this is defined as follows,

$P\left(\mathrm{\&phi;}\right)=\&Integral;\frac{1}{2}{\left(\right|\&dtri;\mathrm{\&phi;}\left(x\right)|-1)}^2\mathrm{dx}---\left(9\right);$

The length constraint item is used for level and smooth zero level collection profile,

$L\left(\mathrm{\&phi;}\right)=\&Integral;|\&dtri;H\left(\mathrm{\&phi;}\left(x\right)\right)|\mathrm{dx}---\left(10\right);$

Fitting function f ₁And f ₂It is as follows,

$f_1\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left(H\left(\mathrm{\&phi;}\right)I\right)}{K_{\mathrm{\&sigma;}}*H\left(\mathrm{\&phi;}\right)},---\left(12\right)$

$f_2\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left[(1-H\left(\mathrm{\&phi;}\right))I\right]}{K_{\mathrm{\&sigma;}}*[1-H\left(\mathrm{\&phi;}\right)]},---\left(13\right)$

Constant c ₁And c ₂

2), adopt method of steepest descent to minimize formula (11), as follows:

$\frac{\&PartialD;\mathrm{\&phi;}}{\&PartialD;t}={\mathrm{\&lambda;}}_g\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)(c_1-c_2)(I-c_1{p}_2-c_2{p}_1)$

$+(1-{\mathrm{\&lambda;}}_g)\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)[{\mathrm{\&lambda;}}_1{e}_1-{\mathrm{\&lambda;}}_2{e}_2]+\mathrm{v\&delta;}\left(\mathrm{\&phi;}\right)\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right)$

$+\mathrm{\&mu;}({\&dtri;}^2\mathrm{\&phi;}-\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right))---\left(14\right)$

λ wherein ₁, λ ₂, v and μ are constants, δ () is the Dirac function; e ₁And e ₂As follows:

e _i(x)＝∫K _σ(y-x)|I(x)-f _i(y)| ²dy，i＝1，2.

Obtain iterative formula by discretize partial differential equation (14), level set function φ o'clock calculates the border of the entoblast and the cell membrane of cell in φ=0.

Technical conceive of the present invention is: according to the mutual difference maximization principle in zone, this model can successfully be surveyed local fuzzy object boundary and be partitioned into the main difference zone.The energy function of this model definition has comprised two dominant terms: local fit item and overall match item.The local fit item produces powerful gravitational attraction border and profile is parked on the object boundary, although object boundary is very not clear even some fuzzy.Overall situation match item has guaranteed that under the differentiation in different regions maximization principle curve can extract the main difference part in the image.In addition, utilize gradient of image and gray scale information to introduce a kind of tactful weight parameter, it makes above-mentioned two match items act on together, thereby has constituted mixed contour model.This parameter model has been introduced a class by the local and global information mixing force as driving.

Beneficial effect of the present invention mainly shows: precision height, segmentation effect are good.

Description of drawings

Fig. 1 is the result schematic diagram after a kind of cutting apart.

Fig. 2 is the result schematic diagram after another kind is cut apart.

Embodiment

Below in conjunction with accompanying drawing the present invention is further described.

A kind of based on the entoblast of mixed contour model and the dividing method of cell membrane, described dividing method may further comprise the steps:

1), cell image to be split is set up energy function according to mixed contour model, form as:

E ^E(φ，c ₁，c ₂，f ₁，f ₂)＝E ^M(φ，c ₁，c ₂，f ₁，f ₂)+P(φ)+L(φ) (11)；

Wherein, E ^M(φ, c ₁, c ₂, f ₁, f ₂) represent energy function based on regional model and local two-value model of fit, its computing formula is:

E ^M(φ，c ₁，c ₂，f ₁，f ₂)＝(1-λ _g)E ^LBF(φ，f ₁，f ₂)

+λ _g(x，y)E ^G(φ，c ₁，c ₂) (7)；

λ wherein _gBe tactful weight parameter, E ^LBF(φ, f ₁, f ₂) as definition in the formula (6), E ^G(φ, c ₁, c ₂) as defining in the formula (5):

E ^G(φ，c ₁，c ₂)＝∫(c ₁-c ₂)(I(x)-c ₁p ₂-c ₂p ₁)H(φ(x))dx-.；

Wherein H is the Heaviside function, and φ () is the symbolic distance function, p ₁Represented A _C1With the ratio of A, p ₂Represented A _C2Ratio with A;

This strategy weight parameter λ _g(x y) is defined as follows:

${\mathrm{\&lambda;}}_g(x,y)=\frac{1}{(1+\mathrm{\&alpha;}|\&dtri;I(x,y)\left|\right)}----\left(8\right);$

Wherein

Shade of gray letter among the presentation video I, α is a positive constant;

Inhibition level set function φ departs from the gauged distance sign function and comes the regularization level set function, and this is defined as follows,

$P\left(\mathrm{\&phi;}\right)=\&Integral;\frac{1}{2}{\left(\right|\&dtri;\mathrm{\&phi;}\left(x\right)|-1)}^2\mathrm{dx}---\left(9\right);$

The length constraint item is used for level and smooth zero level collection profile,

$L\left(\mathrm{\&phi;}\right)=\&Integral;|\&dtri;H\left(\mathrm{\&phi;}\left(x\right)\right)|\mathrm{dx}---\left(10\right);$

Fitting function f ₁And f ₂It is as follows,

$f_1\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left(H\left(\mathrm{\&phi;}\right)I\right)}{K_{\mathrm{\&sigma;}}*H\left(\mathrm{\&phi;}\right)},---\left(12\right)$

$f_2\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left[(1-H\left(\mathrm{\&phi;}\right))I\right]}{K_{\mathrm{\&sigma;}}*[1-H\left(\mathrm{\&phi;}\right)]},---\left(13\right)$

Constant c ₁And c ₂

2), adopt method of steepest descent to minimize formula (11), as follows:

$\frac{\&PartialD;\mathrm{\&phi;}}{\&PartialD;t}={\mathrm{\&lambda;}}_g\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)(c_1-c_2)(I-c_1{p}_2-c_2{p}_1)$

$+(1-{\mathrm{\&lambda;}}_g)\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)[{\mathrm{\&lambda;}}_1{e}_1-{\mathrm{\&lambda;}}_2{e}_2]+\mathrm{v\&delta;}\left(\mathrm{\&phi;}\right)\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right)$

$+\mathrm{\&mu;}({\&dtri;}^2\mathrm{\&phi;}-\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right))---\left(14\right)$

λ wherein ₁, λ ₂, v and μ are constants, δ () is the Dirac function; e ₁And e ₂As follows:

e _i(x)＝∫K _σ(y-x)|I(x)-f _i(y)| ²dy，i＝1，2.

Obtain iterative formula by discretize partial differential equation (14), level set function φ o'clock calculates the border of the entoblast and the cell membrane of cell in φ=0.

The active contour model based on the zone that Chan and Vese mention by minimizing the Mumford-Shah functional of simplification, can be the set of two classes with a kind of image segmentation.This model basic thought is as follows.Suppose image-region:

And I:

Be given image.Mumford and Shah are thought of as the boundary profile C that seeks a best with image segmentation problem, with image region segmentation for being similar to the regional u of segmentation gray scale constant _iAnd u _oC then represents the border.So global data match item is defined as follows in the Chan-Vese model:

$E^{\mathrm{CV}}(c_1,c_2)={\&Integral;}_{\stackrel{\&OverBar;}{\mathrm{\&Omega;}}}{(I-c_1)}^2\mathrm{dxdy}+{\&Integral;}_{\mathrm{\&Omega;}}{(I-c_2)}^2\mathrm{dxdy}---\left(1\right)$

Wherein Ω and Ω represent respectively outside outline line C and outline line C in the zone, c ₁And c ₂Be outer and interior two the match constants of outline line of outline line.

This model thinks that the pixel that coexists in the zone has maximum similarity, thereby has remedied the shortcoming of Boundary Detection.When outline line was caught object boundary accurately, two match items made energy function match value minimum.In each cut zone, the pixel average of polymerization equals c respectively ₁And c ₂Therefore be with parameter c ₁And c ₂The match item be under the homogeneous property of interior zone criterion, drive the profile curvilinear motion.

Because differentiation in different regions is to instruct principle, so the distinctiveness between zones of different should be counted as the driving of model in image segmentation, and is as follows:

$E=-\frac{1}{2}{(c_1-c_2)}^2.---\left(2\right)$

This class defines according to interregional difference maximization based on the active contour model energy function in zone.The energy function that is minimized in definition in the formula (2) is equal to the interregional difference of maximization.Formula (2) has defined overall guidance item.It is as follows that this paper author revises this guidance item,

$E=-\frac{A}{2}{(c_1-c_2)}^2---\left(3\right)$

Middle A has represented the entire image zone.The curve evolvement energy function of outline line C is defined as follows:

$E^G(c_1,c_2)=A(c_1-c_2)(\frac{I-c_1}{A_{c1}}+\frac{I-c_2}{A_{c2}})N---\left(4\right)$

A wherein _C1=∫ _ΩDxdy and A _C2=∫ _ΩThe dxdy item has defined the outside and inner region area of outline line respectively, and N represents the outer normal direction of outline line C.When the outline line divided area is too small, in order to allow above-mentioned energy term (4) that value is stably arranged, rather than show and shake the value of dispersing, that is to say and avoid that denominator term goes to zero in (4) formula, with E ^GItem is amended as follows:

E ^G(φ，c ₁，c ₂)＝∫(c ₁-c ₂)(I(x)-c ₁p ₂-c ₂p ₁)H(φ(x))dx (5)

Wherein H is the Heaviside function, and φ () is the symbolic distance function, p ₁Represented A _C1With the ratio of A, p ₂Represented A _C2Ratio with A.

Though the energy function E of definition in formula (5) ^GDistinctiveness between consideration of regional is as the rule of curve evolvement, but this regional model has no idea to handle nonhomogeneous property of the gray scale that produces and hatching effect in imaging process.Because based on the model in zone with global disparity as driving force, and ignored local detail information, so this model needs a local fit energy term come the performance of improved model in above-mentioned situation.

Local two-value model of fit has been proposed.This important parameter that wherein is called as kernel function is introduced into the scope of regulation regional area of being used for.In addition, f ₁And f ₂Be two fitting function items, be used for approaching local gray level with spatial variations.In the LBF model, this local data's match item is defined as follows:

E ^LBF(φ，f ₁，f ₂)＝

λ ₁∫[∫K _σ(x-y)|I(y)-f ₁(x)| ²H(φ(y))dy]dx+

λ ₂∫[∫K _σ(x-y)|I(y)-f ₂(x)| ²(1-H(φ(y)))dy]dx (6)

Wherein H is the Heaviside function, K _σBe gaussian kernel function, when | K when x| increases _σ(x) reduce and trend towards zero.Function f (x) has been calculated near the fitting degree the some x, and some x can be regarded as the central point of regional area.This LBF model is exactly that it captures the target local detail more accurately than Chan-Vese model than the advantage of Chan-Vese model maximum.

The model that this paper proposes is introduced the overall situation and local half-tone information item, has made full use of the advantage of mentioning in the formula (5) based on regional model and local two-value model of fit.Whole model energy function definition is as follows,

E ^M(φ，c ₁，c ₂，f ₁，f ₂)＝(1-λ _g)E ^LBF(φ，f ₁，f ₂)

+λ _gE ^G(φ，c ₁，c ₂) (7)

λ wherein _gBe tactful weight parameter, E ^LBF(φ, f ₁, f ₂) as definition in the formula (6), E ^G(φ, c ₁, c ₂) as definition in the formula (5).

This strategy weight parameter λ _gBe defined as follows:

${\mathrm{\&lambda;}}_g(x,y)=\frac{1}{(1+\mathrm{\&alpha;}|\&dtri;I(x,y)\left|\right)}----\left(8\right)$

Wherein,

Shade of gray letter among the presentation video I, α is a positive constant.

This tactful weight parameter has realized the weighting factor thought in the mixture model.Under the effect of mixed tensor bound term, the driving force of curve performance have a tactic.When the image local zone is tending towards smooth or the wide border,

Value less relatively, so the value of the right-hand component of formula (8) approaches one.Mixture model is directed into the interregional difference model of maximization.When point (x, when y) falling in the gray scale nonhomogeneous region at place, border,

Value will be very big, so λ _g(x, value y) levels off to zero.Mixture model has the potential ability of extracting local boundary.It combines the advantage of local fit model and global area difference model, and utilizes shade of gray information to draw its policing parameter.

Minimization of energy function (7) means and separates partial differential equation.This paper adopts the gradient method of steepest descent, provides the numerical solution of this partial differential equation.

In addition, in order to obtain more accurate curve evolvement result, this paper has also introduced a match item of mentioning in the document [6], departs from the gauged distance sign function by inhibition level set function φ and comes the regularization level set function.This is defined as follows,

$P\left(\mathrm{\&phi;}\right)=\&Integral;\frac{1}{2}{\left(\right|\&dtri;\mathrm{\&phi;}\left(x\right)|-1)}^2\mathrm{dx}---\left(9\right)$

In addition, this paper is introduced in the length constraint item that uses in the Chan-Vese model and is used for level and smooth zero level collection profile:

$L\left(\mathrm{\&phi;}\right)=\&Integral;|\&dtri;H\left(\mathrm{\&phi;}\left(x\right)\right)|\mathrm{dx}---\left(10\right)$

So far, the whole model definition of this paper introduction is as follows:

E ^E(φ，c ₁，c ₂，f ₁，f ₂)＝E ^M(φ，c ₁，c ₂，f ₁，f ₂)+P(φ)+L(φ) (11)

Fitting function f ₁And f ₂It is as follows,

$f_1\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left(H\left(\mathrm{\&phi;}\right)I\right)}{K_{\mathrm{\&sigma;}}*H\left(\mathrm{\&phi;}\right)},---\left(12\right)$

$f_2\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left[(1-H\left(\mathrm{\&phi;}\right))I\right]}{K_{\mathrm{\&sigma;}}*[1-H\left(\mathrm{\&phi;}\right)]},---\left(13\right)$

Constant c ₁And c ₂Definition and document [13] in identical.

In order to minimize the energy function about φ of definition in the formula (11), this paper has provided the solution that adopts method of steepest descent, and is as follows:

$\frac{\&PartialD;\mathrm{\&phi;}}{\&PartialD;t}={\mathrm{\&lambda;}}_g\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)(c_1-c_2)(I-c_1{p}_2-c_2{p}_1)$

$+(1-{\mathrm{\&lambda;}}_g)\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)[{\mathrm{\&lambda;}}_1{e}_1-{\mathrm{\&lambda;}}_2{e}_2]+\mathrm{v\&delta;}\left(\mathrm{\&phi;}\right)\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right)$

$.+\mathrm{\&mu;}{({\&dtri;}^2\mathrm{\&phi;}-\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right)\_)}^1---\left(14\right)$

λ wherein ₁, λ ₂, v and μ are constants, δ () is Dirac function .e ₁And e ₂As follows:

e _i(x)＝∫K _σ(y-x)|I(x)-f _i(y)| ²dy，i＝1，2. (15)

Obtain iterative formula by discretize partial differential equation (14).Level set function φ o'clock has provided the border of target in φ=0.Further, at expression formula λ _g(x, y) in, the author provides λ ₁=λ ₂=1, μ=1, v=60, α equals three.Gaussian kernel function K _σIn σ equal two.These parameters are different constants according to different image setting.

The model that present embodiment proposes can detect the weak boundary of neutrophil cell, exactly kernel is distinguished from tenuigenin simultaneously.Global drive power has been guaranteed the border that model can obtain target by utilizing maximization global area difference.

Adopt the lympha tumour cell as the experiment picture, cutting apart phenomenon can handle elimination by follow-up morphology.At first, φ＜0 zone is set to zero, and φ＞0 zone is set to one, and φ=0 zone then is the border of object.Thus, can obtain as depicted in figs. 1 and 2 bianry image.

Secondly, provide subsequent operation for Fig. 1.A certain threshold value is set, area in the neutrophil cell nuclear zone in the image is filled up less than the part of this threshold value, the result as shown in Figure 3.

Claims (1)

1, a kind of based on the entoblast of mixed contour model and the dividing method of cell membrane, described dividing method may further comprise the steps:

1), cell image to be split is set up energy function according to mixed contour model, form as:

E ^E(φ，c ₁，c ₂，f ₁，f ₂)＝E ^M(φ，c ₁，c ₂，f ₁，f ₂)+P(φ)+L(φ) (11)；

Wherein, E ^M(φ, c ₁, c ₂, f ₁, f ₂) represent energy function based on regional model and local two-value model of fit, its computing formula is:

E ^M(φ，c ₁，c ₂，f ₁，f ₂)＝(1-λ _g)E ^LBF(φ，f ₁，f ₂)

+λ _g(x，y)E ^G(φ，c ₁，c ₂) (7)；

λ wherein _gBe tactful weight parameter, E ^LBF(φ, f ₁, f ₂) as definition in the formula (6), E ^G(φ, c ₁, c ₂) as defining in the formula (5):

E ^G(φ，c ₁，c ₂)＝∫(c ₁-c ₂)(I(x)-c ₁p ₂-c ₂p ₁)H(φ(x))dx (5)；

Wherein H is the Heaviside function, and φ () is the symbolic distance function, p ₁Represented A _C1With the ratio of A, p ₂Represented A _C2Ratio with A;

This strategy weight parameter λ _g(x y) is defined as follows:

${\mathrm{\&lambda;}}_g(x,y)=\frac{1}{(1+\mathrm{\&alpha;}|\&dtri;I(x,y)\left|\right)}---\left(8\right);$

Wherein

Shade of gray letter among the presentation video I, α is a positive constant;

Inhibition level set function φ departs from the gauged distance sign function and comes the regularization level set function, and this is defined as follows,

$P\left(\mathrm{\&phi;}\right)=\&Integral;\frac{1}{2}{\left(\right|\&dtri;\mathrm{\&phi;}\left(x\right)|-1)}^2\mathrm{dx}---\left(9\right);$

The length constraint item is used for level and smooth zero level collection profile,

$L\left(\mathrm{\&phi;}\right)=\&Integral;|\&dtri;H\left(\mathrm{\&phi;}\left(x\right)\right)|\mathrm{dx}---\left(10\right);$

Fitting function f ₁And f ₂It is as follows,

$f_1\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left(H\left(\mathrm{\&phi;}\right)I\right)}{K_{\mathrm{\&sigma;}}*H\left(\mathrm{\&phi;}\right)},---\left(12\right)$

$f_2\left(x\right)=\frac{K_{\mathrm{\&sigma;}}*\left[(1-H\left(\mathrm{\&phi;}\right))I\right]}{K_{\mathrm{\&sigma;}}*[1-H\left(\mathrm{\&phi;}\right)]},---\left(13\right)$

Constant c ₁And c ₂

2), adopt method of steepest descent to minimize formula (11), as follows:

$\frac{\&PartialD;\mathrm{\&phi;}}{\&PartialD;t}={\mathrm{\&lambda;}}_g\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)(c_1-c_2)(I-c_1{p}_2-c_2{p}_1)$

$+(1-{\mathrm{\&lambda;}}_g)\mathrm{\&delta;}\left(\mathrm{\&phi;}\right)[{\mathrm{\&lambda;}}_1{e}_1-{\mathrm{\&lambda;}}_2{e}_2]+\mathrm{v\&delta;}\left(\mathrm{\&phi;}\right)\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right)$

$+\mathrm{\&mu;}({\&dtri;}^2\mathrm{\&phi;}-\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right))---\left(14\right)$

λ wherein ₁, λ ₂, v and μ are constants, δ () is the Dirac function; e ₁And e ₂As follows:

e _i(x)＝∫K _σ(y-x)|I(x)-f _i(y)| ²dy，i＝1，2. (15)

Obtain iterative formula by discretize partial differential equation (14), level set function φ o'clock calculates the border of the entoblast and the cell membrane of cell in φ=0.

Images