CN102542556B - Method for automatically extracting ultrasonic breast tumor image - Google Patents

Abstract

The invention relates to the field of signal processing of biomedicine, in particular to a method for automatically extracting an ultrasonic breast tumor image. The method comprises the following steps of: 1, selecting a target image, namely marking a square frame by a user through a mouse so as to contain the tumor in the square frame; 2, automatically extracting an edge of the ultrasonic tumor image, namely suppressing noise of a cut image through a speckle reducing anisotropic diffusion (SRAD) algorithm; 3, obtaining an image I1 subjected to noise suppression through the step (2), and automatically segmenting the tumor image by utilizing an improved geometrical active contour model, namely 3a, pre-processing the image, 3b, optimizing the model, 3c, extracting the edge of the image, 3d, updating the model, and 3e, converging the image. A novel energy function is provided to improve the original model, so that the model is more suitable for lesion extraction of medical ultrasonic tumor image, and the accuracy and practicality of the algorithm are improved further.

Description

Ultrasonic Breast Tumor Image extraction method

Technical field

The invention belongs to processing of biomedical signals field, be specifically related to tumour ultrasonoscopy extraction method.

Background technology

Breast cancer has become women's No.1 killer, and its morbidity number significantly rises with the speed of average annual 3%-5%, and has increasingly serious trend.Research shows, if can check in time in early days, cancer can be cured, and cure rate is up to more than 92%.Visible, the early detection of tumor of breast is to the vital effect that healed the sick.Detection technique based on ultrasonoscopy is one of with the fastest developing speed in medical science, tumor disease detection technique that be most widely used.But due to the special mechanism of its imaging, Ultrasonic Breast Tumor Image focus checks that problem is not also resolved so far, is one of current main study hotspot, is also a classic problem.For this problem, researchist has proposed a large amount of partitioning algorithms so far both at home and abroad, but the shortcoming that these algorithms all exist and are of limited application, limitation is stronger, does not up to the present also exist a kind of general focus detection method.

In recent years, the movable contour model method (as snake model) based on edge is cut apart in field and is showed and enliven at image, is all widely used in many aspects.But they have following shortcoming conventionally:

(1) more responsive to noise and clutter;

(2) easily there is border leakage phenomenon in weak edge;

(3) harsher to the requirement of starting condition.

Chan and Vese are at article " Chan, T.F.; Vese, L.A.; Activecontours without edges.Image Processing, IEEE Transactions, Vol.10, pp.266 – 277,2001 ", in; proposed a kind of geometric active contour model-Chan-Vese model based on region of classics, it has greatly improved the above-mentioned shortcoming of Snake model.In this model, initial profile can be arranged on any position of target area, the practicality of model is further strengthened, but, for ultrasonoscopy, because it is subject to the severe jamming of speckle noise in imaging process, therefore,, at tumor boundaries near zone, speckle noise tends to form the bulk of some catastrophe points or sudden change, thereby, caused Chan-Vese model to be easy to that speckle noise is considered as to edge in the time extracting lesion boundary and extracted.In addition, Chan-Vese model, to multiobject Image, easily occurs that location, edge is inaccurate.

The content of innovation and creation

The present invention seeks to the defect existing for above-mentioned cutting techniques, a kind of Ultrasonic Breast Tumor Image extraction method is provided.

Order of the present invention can be realized the tumor of breast edge extracting that prosthetic is intervened, can not only extract rapidly tumor focus region, and greatly improve accuracy, thereby for good technical foundation is established in differentiation and the computer-aided diagnosis of tumor of breast, be conducive to advance the application of Medical Image Processing in tumor of breast clinical diagnosis, for numerous doctors provide the useful information of tumor region.

The present invention is achieved by the following technical solutions, specifically comprises the following steps and carry out:

1. choose target image: user utilizes mouse to draw and gets a square frame, within tumour is included in to this square frame, its object one is in order to cut out irrelevant information, reduces and disturbs; The 2nd, in order to improve the real-time of system.

2. tumour ultrasonoscopy edge extracts automatically: adopt SRAD anisotropy broadcast algorithm, carry out squelch to cutting out the image obtaining; The outstanding advantages of this algorithm is embodied in: can effectively remove speckle noise preserving edge information, can strengthen step-like edge simultaneously; Therefore, the interference of spot (Speckle) noise that adopts this algorithm to reduce as much as possible to be caused by ultrasonic imaging mechanism, for subsequent treatment provides good basis; The iterative equation of SRAD algorithm is as follows:

$\left\{\begin{array}{c}\∂I(x,y;t)/\∂t=\mathrm{div}[c\left(q\right)\▿I(x,y;t)]\\ I(x,y;0)=I_0(x,y),(\∂I(x,y;t)/\∂\stackrel{\→}{n})|\∂\mathrm{\Ω}=0\end{array}\right.$

Wherein, I ₀(x, y) is the image after cutting out; C (q) is coefficient of diffusion; The supporting domain of Ω presentation video, for the border of Ω, and for outer normal vector;

I (x, y; T): Ω × [0 ,+∞) → R is the output image after iteration each time;

3. obtain the image I 1 after squelch by step (2), utilize improved geometric active contour model to carry out auto Segmentation to tumor image:

3a image pre-service: the image I 1 of the dilation and erosion method of utilizing geometric shape after to squelch carried out pre-service, automatically generates initial active outline line C ₀, i.e. zero level collection, and generate symbolic distance function SDF, i.e. level set function φ ₀(x, y), the computing formula that dilation and erosion method adopts is as follows:

$(I1\⊕S)(x,y)=\max \{I1(x-i,y-j)+S(i,j)\},i,j\∈S$

$(I1\⊕S)(x,y)=\max \{I1(x+i,y+j)-S(i,j)\},i,j\∈S$

Wherein S (x, y) represents to have definite shape and big or small structural element; I1 is the image after squelch;

3b model optimization: the huge problem of calculated amount developing in order to improve original geometry active contour, system has adopted arrowband algorithm, i.e. and the near zone of method considering zero level set only only upgrades the SDF in this narrowband region at every turn, thereby improves the work efficiency of algorithm;

3c Edge extraction: poor and multiple goal is extracted to the problem such as inaccurate to the noise immunity of ultrasonoscopy in order to improve traditional Chan-Vese model, the gradient information of system combining image and area grayscale information, propose a new energy term based on gradient:

$\begin{array}{c}E\left(\mathrm{\φ}\right)=E_{\mathrm{ext}}+E_{\mathrm{int}}\\ =\mathrm{\α}\underset{\mathrm{inside}\left(C\right)}{\∫}{|I(x,y)-c_a|}^2\mathrm{dxdy}+\mathrm{\β}\underset{\mathrm{outside}\left(C\right)}{\∫}{|I(x,y)-c_b|}^2\mathrm{dxdy}+\\ \mathrm{\μ}\underset{\mathrm{inside}\left(C\right)}{\∫}\stackrel{\‾}{{|A_I(x,y)-D_a|}^2}\mathrm{dxdy}+\mathrm{\η}\underset{\mathrm{outside}\left(C\right)}{\∫}\stackrel{\‾}{{|A_I(x,y)-D_b|}^2}\mathrm{dxdy}+\\ \mathrm{\γ}\underset{\mathrm{\Ω}}{\∫}(1/2){\left(\right|\▿\mathrm{\φ}|-1)}^2\mathrm{dxdy}\end{array}$

Wherein, φ is the level set function after reinitializing; c _aand D _abe respectively pixel average and the gradient mean value of inside, target area; c _band D _bbe respectively pixel average and the gradient mean value of outside, target area; α, β, μ, η and γ are adjustable parameter; And have:

$\begin{array}{c}\stackrel{\‾}{{|A_I(x,y)-D_a|}^2}=0.1*{|A_I(x,y+1)-D_a|}^2+0.1*{|A_I(x,y-1)-D_a|}^2+\\ 0.1*{|A_I(x+1,y)-D_a|}^2+0.1*{|A_I(x-1,y)-D_a|}^2+\\ 0.5*{|A_I(x,y)-D_a|}^2\end{array}$

And $A_I(x,y)=\sqrt{{(\∂I(x,y)/\∂x)}^2+{(\∂I(x,y)/\∂y)}^2}$

3d model modification: in the time that zero level collection curve is close or touch border, arrowband, according to evolution formula, automatically upgrades level set function, and recalculate new narrowband region;

The convergence of 3e model and criterion: check when whether iteration restrains, if convergence does not forward step 3b to, otherwise, calculating and stop, zero level collection curve stops developing, and the Rule of judgment of getting iteration convergence is:

$\mathrm{De}=\frac{\underset{\left|{\mathrm{\&phi;}}^n\right|<s}{\mathrm{\&Sigma;}}|{\mathrm{\&phi;}}^{n+1}-{\mathrm{\&phi;}}^n|}{\mathrm{Sum}}\&le;s^2\mathrm{\&Delta;t}$

Wherein, φ ⁿbe the level set function that the n time iteration obtains, Δ t is time step, and Sum represents to meet | φ ⁿ| the grid sum of≤s.

The advantage of the invention and effect

The advantage of the invention and effect

The present invention compared with prior art has the following advantages:

1, the present invention has adopted advanced geometric shape method auto-initiation level set function, can not only effectively avoid artificial participation, has improved the automatization level of this system.

2, the present invention utilizes arrowband method to improve original geometric active contour model, has greatly reduced calculated amount, has improved the real-time performance of system.

3, the present invention proposes a new energy function master pattern is improved, make this model be more suitable for extracting in the focus of medical ultrasonic image tumour, further improved accuracy and the practicality of algorithm.

Accompanying drawing and explanation thereof

Fig. 1 is process flow diagram of the present invention;

Fig. 2 is the present invention and the contrast and experiment figure of original geometry movable contour model algorithm on the noisy image of multiple goal;

Wherein, 2-a is former figure; 2-b is the experimental result of Li; 2-c is experimental result of the present invention;

Fig. 3 is the comparative test result figure that the present invention and original geometry movable contour model algorithm extract for Ultrasonic Breast Tumor Image;

Wherein, 3-a is former figure; 3-b is the experimental result of Li; 3-c is experimental result of the present invention;

Fig. 4-a is the ultrasonoscopy that contains stringiness tumor of breast that diagnostic ultrasonic equipment collects, and white rectangle square frame is that user draws the general region of tumour of getting; Fig. 4-b draws the subimage of getting rear formation; Fig. 4-c is the image obtaining after SRAD squelch, and the zero level collection curve automatically being generated by this image; Fig. 4-d is that the improvement Chan-Vese model algorithm that adopts the present invention to propose carries out in automatic focus extraction, the automatic evolutionary process of zero level collection curve; Fig. 4-e is that zero level collection curve evolvement finally stops, and the contour curve obtaining extracts and obtained tumor of breast edge; Fig. 4-f is the final extraction result demonstration figure that this diagnostic ultrasonic equipment uses the present invention to obtain.

The present invention brings forward a new energy function, has realized the effective improvement to original Chan-Vese model, makes Chan-Vese model be more suitable for extracting in the focus of medical ultrasonic image tumour.Compare through simulation study and experiment, the present invention is all obviously better than Chan-Vese model on extraction performance and time loss, concrete visible lab diagram 2 and Fig. 3.Wherein, Fig. 2 is the present invention and the contrast and experiment figure of Chan-Vese model algorithm on the noisy image of multiple goal; Fig. 3 is the comparative test result figure that the present invention and Chan-Vese model algorithm extract for Ultrasonic Breast Tumor Image.In addition, the present invention is in the leaching process of tumor focus, and without manual intervention, automatization level is higher.Fig. 4 is that the present invention is loaded on diagnostic ultrasonic equipment, carries out the instance graph of clinical assistant diagnosis.In the selection of this instance parameter, q=0.5, α=η=0.2, μ=β=0.4, Δ t=0.5; SRAD iterations is set to 5 times, and the iteration stopping number of times of algorithm is 50 times.

Concrete embodiment:

Mainly for the extraction of Ultrasonic Breast Tumor Image, specialized designs is implemented in the present invention.On the basis of feature of fully having studied ultrasonic imaging mechanism and ultrasonoscopy, the present invention has applied spot noise reduction anisotropy diffusion (SRAD) algorithm ultrasonoscopy has been carried out to necessity ground noise remove, and has effectively retained the information at tumor region edge and strengthened edge contour; Following closely, adopt improved geometric active contour model (Chan-Vese model) to process the image after denoising, extract the focus region of tumor of breast.Obviously, whole processing procedure is without manual intervention, and the present invention can automatically extract and obtain final result.

With reference to Fig. 1, the medical ultrasonic image tumour extraction that the present invention is based on improvement movable contour model comprises:

Step 1: cut out, obtain area-of-interest.

The present invention only needs user probably to confirm the position of tumour, can utilize mouse on the medical image showing, to draw the approximate range of getting tumour, and system just can automatically propose the important information such as concrete shape and size of tumour.

Step 2: squelch, improve ultrasonoscopy readability, be convenient to subsequent treatment.

The speckle noise of ultrasonoscopy has reduced ultrasonic image quality widely, has had a strong impact on the subsequent treatment of ultrasonoscopy, especially the extraction to tumor focus and identification.The present invention introduces SRAD anisotropy broadcast algorithm, can effectively remove speckle noise preserving edge information, can strengthen step-like edge simultaneously.Its iterative equation is as follows:

$\left\{\begin{array}{c}\&PartialD;I(x,y;t)/\&PartialD;t=\mathrm{div}\left[g\right|\&dtri;I(x,y;t)\left|\right]\&dtri;I(x,y;t)\\ I(X,Y;0)=I_0(x,y),(\&PartialD;I(x,y;t)/\&PartialD;\stackrel{\&RightArrow;}{n})|\&PartialD;\mathrm{\&Omega;}=0\end{array}\right.$

Wherein coefficient of diffusion c (q) is:

$c\left(q\right)=\frac{1}{1+[q^2-q_0^2\left(t\right)]/\left[q_0^2\left(t\right)(1+q_0^2\left(t\right))\right]}$

In formula, q is the instantaneous coefficient operator being calculated by local variance; Q0 (t) is spot scale coefficient, for controlling level and smooth degree.Experiment shows, q0 (t) gets [0,1], and effect is good.Here get q0 (t)=0.4.Instantaneous coefficient operator q is defined as:

$q(x,y,t)=\sqrt{\frac{{(\&dtri;I/I)}^2/2-{({\&dtri;}^{2}I/I)}^2/16}{{[1+({\&dtri;}^{2}I/I)/4]}^2}}$

This operator inclusion gradient operator and Laplace operator, for detection of the edge in spot image.Obtain higher value at edge and high-contrast profile place, and obtain smaller value at homogeneous area.

Step 3: system extracts borderline tumor automatically, without man-machine interactively.

If the image obtaining after noise suppression is I1, next, utilize improved geometric active contour model, the edge of realizing tumor of breast extracts automatically:

1) image I after utilizing dilation and erosion in geometric shape to denoising is processed,

Automatically generate initial active outline line C ₀, and then, can calculate symbolic distance function,

Namely level set function φ ₀(x, y).Dilation and corrosion computing is defined as follows:

$(I1\&CirclePlus;S)(x,y)=\max \{I1(x-i,y-j)+S(i,j)\},i,j\&Element;S$

$(I1\&CirclePlus;S)(x,y)=\max \{I1(x+i,y+j)-S(i,j)\},i,j\&Element;S$

Wherein S (x, y) represents to have definite shape and big or small structural element.

2) for improving real-time, system has adopted arrowband algorithm to be optimized model, only considers that all pixels are that 4 pixels are with interior region to zero level collection curve distance; In iterative computation after this, only upgrade the SDF in this narrowband region at every turn, thereby improve the work efficiency of algorithm.

3) native system makes full use of visual gradient information and area grayscale information, proposes a more efficiently energy function:

$\begin{array}{c}E\left(\mathrm{\&phi;}\right)=E_{\mathrm{ext}}+E_{\mathrm{int}}\\ =\mathrm{\&alpha;}\underset{\mathrm{inside}\left(C\right)}{\&Integral;}{|\mathrm{Ku}(x,y)-T_i|}^2+{|u(x,y)-c_i|}^2\mathrm{dxdy}\\ +\mathrm{\&beta;}\underset{\mathrm{outside}\left(C\right)}{\&Integral;}{|\mathrm{Ku}(x,y)-T_o|}^2+{|u(x,y)-c_o|}^2\mathrm{dxdy}\\ +\mathrm{\&gamma;}\underset{\mathrm{\&Omega;}}{\&Integral;}(1/2){\left(\right|\&dtri;\mathrm{\&phi;}|-1)}^2\mathrm{dxdy}\end{array}$

Wherein, c _iand T _ibe respectively pixel average and the gradient mean value of inside, target area; c _oand T _obe respectively pixel average and the gradient mean value of outside, target area.Ku (x, y) is illustrated in the Grad that point (x, y) is located, and φ is the level set function after reinitializing.

Utilize the variational method to this total energy function minimization, obtain level set function φ evolutionary process and be:

$\begin{array}{c}\frac{\&PartialD;\mathrm{\&phi;}}{\&PartialD;t}=\mathrm{\&beta;}{[\mathrm{Ku}(x,y)-T_o]}^2+\mathrm{\&beta;}{[u(x,y)-c_o]}^2-\mathrm{\&alpha;}{[\mathrm{Ku}(x,y)-T_i]}^2\\ -\mathrm{\&alpha;}{[u(x,y)-c_i]}^2+\mathrm{\&gamma;}[\mathrm{\&Delta;\&phi;}-\mathrm{div}\left(\frac{\&dtri;\mathrm{\&phi;}}{|\&dtri;\mathrm{\&phi;}|}\right)]\end{array}$

Wherein, symbol represent gradient, symbol Δ is expressed as Laplace operator.

For the numerical solution of level set function φ EVOLUTION EQUATION, adopt implicit iterative solution.Can prove the time-based level set function φ EVOLUTION EQUATION being obtained by variation minimization for above-mentioned, its implied format iterative solution method is unconditional stability, and therefore, time step Δ t can suitably strengthen, with the evolution of acceleration curve, the present invention gets Δ t=0.5.

4), according to evolution formula, in the time that zero level collection curve is close or touch border, arrowband, system will automatically be upgraded level set function and recalculate new narrowband region.

5) check when whether iteration restrains, if convergence does not forward step 2 to), otherwise, calculating and stop, zero level collection curve stops developing, and now, this curve will drop on the edge of tumour exactly, thereby obtains the extraction result of final tumor's profiles.For fear of unnecessary iterative computation, the Rule of judgment of getting iteration convergence is:

$\mathrm{De}=\frac{\underset{\left|{\mathrm{\&phi;}}^n\right|<s}{\mathrm{\&Sigma;}}|{\mathrm{\&phi;}}^{n+1}-{\mathrm{\&phi;}}^n|}{\mathrm{Sum}}\&le;s^2\mathrm{\&Delta;t}$

Wherein, φ ⁿbe the level set function that the n time iteration obtains, and Sum represents to meet | φ ⁿ| the grid sum of≤s.

Verify below validity and the practicality of the inventive method by emulation experiment and concrete clinical practice.The method being compared, is the method that the people such as Li propose, concrete list of references " T.F.Chan, L.A.Vese, Active contours without edges, IEEE Trans.Image Processing, 2001, vol.10, no.2, pp.266-277. "

Fig. 2 and Fig. 3 are respectively noisy multiple goal composograph and medical ultrasonic tumor of breast image, and the comparative result of experiment.As can be seen from the figure, the present invention can be correctly by Target Segmentation out, compares the scheduling algorithm with Li, has higher accuracy rate, and have good real-time, as shown in table 1.

Two kinds of methods of table 1 are cut apart the comparison of required time

	Fig. 2	Fig. 3
			Li method	11.54	12.33
The present invention	0.97	1.25

Fig. 4 is that the present invention is loaded on diagnostic ultrasonic equipment, carries out the instance graph of clinical assistant diagnosis.The result of clinical testing shows, the present invention has very high practicality, and the important information of tumour can be provided for clinician practically.

Claims (1)

1. a Ultrasound Image of Breast Tumor extraction method, carries out according to the following steps:

(1) choose target image: user utilizes mouse to cut out to include the block scheme picture of pending tumor of breast;

(2) tumour ultrasonoscopy edge extracts automatically: adopt SRAD anisotropy broadcast algorithm, carry out squelch to cutting out the image obtaining, the iterative equation of SRAD algorithm is as follows:

$\left\{\begin{array}{c}\&PartialD;I(x,y;t)/\&PartialD;t=\mathrm{div}[c\left(q\right)\&dtri;I(x,y;t)]\\ I(x,y;0)=I_0(x,y),(\&PartialD;I(x,y;t)/\&PartialD;\stackrel{\&RightArrow;}{n})|\&PartialD;\mathrm{\&Omega;}=0\end{array}\right.$

Wherein, I ₀(x, y) is the image after cutting out; C (q) is coefficient of diffusion; The supporting domain of Ω presentation video, for the border of Ω, and for outer normal vector;

I (x, y; T): Ω × [0 ,+∞) → R is the output image after iteration each time;

(3) obtain the image I 1 after squelch by step (2), utilize improved geometric active contour model to carry out auto Segmentation to tumor image:

3a image pre-service: the image I 1 of the dilation and erosion method of utilizing geometric shape after to squelch carried out pre-service, automatically generates initial active outline line C ₀, i.e. zero level collection, and generate symbolic distance function SDF, i.e. level set function φ ₀(x, y), the computing formula that dilation and erosion method adopts is as follows:

$(I1\&CirclePlus;S)(x,y)=\max \{I1(x-i,y-j)+S(i,j)\},i,j\&Element;S$

$(I1\&CirclePlus;S)(x,y)=\max \{I1(x+i,y+j)-S(i,j)\},i,j\&Element;S$

Wherein S (x, y) represents to have definite shape and big or small structural element; I1 is the image after squelch;

3b model optimization: adopted arrowband algorithm, i.e. the near zone of method considering zero level set only, upgrades the SDF in this narrowband region again, thus the work efficiency of algorithm improved;

3c Edge extraction: the gradient information of combining image and area grayscale information, adopt minimization of energy function E (φ) to calculate, zero level collection curve is developed along borderline tumor direction:

$\begin{array}{c}E\left(\mathrm{\&phi;}\right)=E_{\mathrm{ext}}+E_{\mathrm{int}}\\ =\mathrm{\&alpha;}\underset{\mathrm{inside}\left(C\right)}{\&Integral;}{|I(x,y)-c_a|}^2\mathrm{dxdy}+\mathrm{\&beta;}\underset{\mathrm{outside}\left(C\right)}{\&Integral;}{|I(x,y)-c_b|}^2\mathrm{dxdy}+\\ \mathrm{\&mu;}\underset{\mathrm{inside}\left(C\right)}{\&Integral;}\stackrel{\&OverBar;}{{|A_I(x,y)-D_a|}^2}\mathrm{dxdy}+\mathrm{\&eta;}\underset{\mathrm{outside}\left(C\right)}{\&Integral;}\stackrel{\&OverBar;}{{|A_I(x,y)-D_b|}^2}\mathrm{dxdy}+\\ \mathrm{\&gamma;}\underset{\mathrm{\&Omega;}}{\&Integral;}(1/2){\left(\right|\&dtri;\mathrm{\&phi;}|-1)}^2\mathrm{dxdy}\end{array}$

Wherein, φ is the level set function after reinitializing; c _aand D _abe respectively pixel average and the gradient mean value of inside, target area; c _band D _bbe respectively pixel average and the gradient mean value of outside, target area, α, β, μ, η and γ are adjustable parameter; And have:

$\begin{array}{c}\stackrel{\&OverBar;}{{|A_I(x,y)-D_a|}^2}=0.1*{|A_I(x,y+1)-D_a|}^2+0.1*{|A_I(x,y-1)-D_a|}^2+\\ 0.1*{|A_I(x+1,y)-D_a|}^2+0.1*{|A_I(x-1,y)-D_a|}^2+\\ 0.5*{|A_I(x,y)-D_a|}^2\end{array}$

And $A_I(x,y)=\sqrt{{(\&PartialD;I(x,y)/\&PartialD;x)}^2+{(\&PartialD;I(x,y)/\&PartialD;y)}^2}$

3d model modification: in the time that zero level collection curve is close or touch border, arrowband, according to evolution formula, automatically upgrades level set function, and recalculate new narrowband region;

The convergence of 3e model and criterion: check when whether iteration restrains, if convergence does not forward step 3b to, otherwise, calculating and stop, zero level collection curve stops developing, and the Rule of judgment of getting iteration convergence is:

$\mathrm{De}=\frac{\underset{\left|{\mathrm{\&phi;}}^n\right|<s}{\mathrm{\&Sigma;}}|{\mathrm{\&phi;}}^{n+1}-{\mathrm{\&phi;}}^n|}{\mathrm{Sum}}\&le;s^2\mathrm{\&Delta;t}$

Wherein, φ ⁿbe the level set function that the n time iteration obtains, Δ t is time step, and Sum represents to meet | φ ⁿ| the grid sum of≤s.