CN111008977B - Image segmentation method of high-order MRF model based on multi-node topological overlap measure - Google Patents
Image segmentation method of high-order MRF model based on multi-node topological overlap measure Download PDFInfo
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Abstract
The invention discloses an image segmentation method of a high-order MRF model based on multi-node topological overlap measurement, which comprises the steps of firstly inputting a natural image to be segmented; then initializing parameters; reconstructing a high-order MRF priori energy item based on MTOM; establishing a high-order MRF image segmentation energy model based on multi-node topological overlap measure according to the WGMM likelihood energy of local region consistency and a partial second-order Potts prior model based on the region; and optimizing the high-order MRF image segmentation energy model by using a Gibbs sampling algorithm, and determining an image segmentation result. The method can not only effectively resist strong noise and abrupt texture disturbance of the image and has better robustness, but also has more accurate image segmentation edges.
Description
Technical Field
The invention belongs to the technical field of image processing, and particularly relates to an image segmentation method of a high-order MRF model based on multi-node topological overlap measurement.
Background
Image segmentation is one of the core problems of computer vision field research, and is the basis for higher-layer analysis and understanding of images. In recent years, image segmentation methods based on a Markov Random Field (MRF) model have received a great deal of attention, and have become a hot spot for research in the field of image segmentation. Under the probability framework, MRF utilizes Gibbs distribution of image pixel labels to describe image local space priori knowledge, and combines the image space priori knowledge with likelihood features based on Bayesian theorem, so that the MRF has been successfully applied to the field of image segmentation.
Because the low-order MRF model only expresses simple priori knowledge such as neighborhood smoothing, the simple priori knowledge often leads to over-smoothing of segmentation results, thereby preventing further application of MRF in the field of image segmentation. The high-order MRF introduces more neighborhood information and can express more complex priori knowledge and statistical information, but the conventional high-order MRF model for constrained region consistency has limited expression capability on the priori of the image local region, and particularly has difficult effective expression on some high-dimensional features in the image local region, such as high-order topological structure features in the local region and the like.
To improve the ability of the MRF model to describe structural features of local regions of an image, local spatial correlation of the image is often described using a distance metric method of contiguous pixels. The local space prior information of the image is introduced based on the similarity measurement of the point to the pixel, and adjacent pixels with the closer prior information constraint distance tend to take the same label, but the similarity of the local pixels cannot be effectively described by the conventional similarity measurement based on the Euclidean distance due to the high dimensionality of complex image features.
The spatial prior information expression method based on the distance measurement effectively captures the image local prior knowledge, but the method describes the structural characteristics of the local area only through the correlation between the low-order point pair adjacent pixels in the local area, so that the expression method simply utilizing the point pair pixel correlation cannot effectively capture the high-order structural correlation characteristics of the complex image, and therefore, the method cannot effectively resist the strong noise and the complex texture mutation interference of the image, and the image is segmented more accurately.
Disclosure of Invention
The technical problem to be solved by the invention is to provide an image segmentation method of a high-order MRF model based on multi-node topological overlap measure aiming at the defects in the prior art, so as to solve the problem that the image segmentation model of the high-order MRF with conventional constraint area consistency cannot effectively describe the high-order topological structure characteristics of a complex image and effectively improve the image segmentation effect.
The invention adopts the following technical scheme:
an image segmentation method of a high-order MRF model based on multi-node topological overlap measurement firstly inputs a natural image to be segmented; then initializing parameters; reconstructing a high-order MRF priori energy item based on MTOM; establishing a high-order MRF image segmentation energy model based on multi-node topological overlap measure according to the WGMM likelihood energy of local region consistency and a partial second-order Potts prior model based on the region; and optimizing the high-order MRF image segmentation energy model by using a Gibbs sampling algorithm, and determining an image segmentation result.
Specifically, the natural image X to be segmented is specifically:
X＝{x s |x s ∈Ω,s∈S}
where Ω= {0,1, …,255} represents the observed pixel x in the image s S represents a finite set of lattice points;
defining the label field of the segmented image as:
Y＝{y s |y s ∈Λ,s∈S}
where Λ= {0,1, …, L }, L represents the total number of image split tags.
Specifically, the parameter initialization is specifically:
local area w s Mean and variance of class label number L, WGMM
Randomly initializing; a priori parameter beta; normalized parameter ρ, power adjacency parameter γ; gibbs sampling algorithm initial temperature T (0) 。
Specifically, constructing a high-order MRF prior energy term based on MTOM specifically comprises the following steps:
s301, for=1 to S, and according to the input image x= { X s |x s e.OMEGA, s.e.S }, calculating the MTOM of the adjacent pixels;
s302, establishing a local area w s Is a priori of the higher order topology space of (2)
And S303, if s=S, obtaining the prior energy of the high-order topological space of all areas of the image X.
Specifically, establishing the WGMM likelihood energy of the local region consistency is specifically as follows:
s401, measuring the center pixel x by adopting Hamming distance s And its neighborhood pixel x r Is introduced into the adjacent pixel x s And x r Similarity between them, defining a weight function w (y r )；
S402, introducing the constructed weight into a Gaussian mixture model (Gaussian Mixture Model, GMM) to obtain a WGMM likelihood model of local region consistency;
s403, adopting an EM algorithm to iteratively estimate GMM parameters
S404, establishing the WGMM likelihood energy P (X|Y, theta) of the local area consistency.
Further, in step S404, the partial area consistency WGMM likelihood energy P (x|y, θ) is specifically:
wherein P (x) s |y s θ) is a GMM, parameter
Mean and variance of the first GMM distribution, Λ= {0,1, …, L }, respectively, L representing the total number of image segmentation labels, +.>
Is a normalization function.
Specifically, the partial second-order Potts prior model based on the region specifically comprises the following steps:
wherein Z (beta) is a normalized constant; beta epsilon [0.1,5 ]]The prior parameters of the Potts model are obtained; delta (y) s ,y r ) As a delta function.
Further, in the local area w s In, assume a local area w s Is of (1)
Is an MRF, let y s ,y r For the corresponding adjacent pixel x s ,x r Assigned label, delta (y s ,y r ) For delta function, build local region w s Is->
Specifically, the high-order MRF image segmentation energy model based on the multi-node topological overlap measure is specifically:
E G (Y|X,Θ)＝E d (X|Y,θ)+E s (Y|β)+E h (X|Υ)
wherein Θ= (θ, β, y); e (E) d (X|Y, θ) represents the partial region consistency WGMM likelihood energy, θ is the likelihood energy model parameters, E s (Y|beta) represents part of second-order Potts prior, beta is Potts type prior energy model parameter, E h (X|y) represents a high order MRF prior term based on MTOM, y being a high order prior energy model parameter.
Specifically, the optimization of the high-order MRF image segmentation energy model by using the Gibbs sampling algorithm is specifically as follows:
s701, setting the maximum iteration number t of Gibbs sampling algorithm max ；
S702, pre-dividing the image according to the MAP criterion:
s703, for=1 to S, each pixel x is calculated s Local area w s Distributing energy of the tag;
s704, in local area w s Center pixel y s According to probability P (y s =η) receives a new label configuration η;
s705 calculating Global energy
If->
Epsilon=10e-6, the segmentation result Y is output * ＝Y (n) The method comprises the steps of carrying out a first treatment on the surface of the Otherwise the temperature t=0.95T is reduced (n) The process returns to step S703 to continue the iteration.
Compared with the prior art, the invention has at least the following beneficial effects:
the invention provides an image segmentation method of a high-order MRF model based on multi-node topological overlap measurement. The model is put forward, a high-order MRF priori model of the image is established by utilizing multi-node topological overlap measure, topological space structure information among multiple pixels in a local area of the image is effectively described, and priori knowledge expression capability of the MRF priori model on complex natural images is improved; meanwhile, a partial second-order Potts prior model based on a local area is established, more label node information is introduced by utilizing a larger local area, and local label consistency constraint based on local area inconsistency punishment is established; then, the image local spatial correlation is introduced by using the Hamming distance of the adjacent pixels, and WGMM with local region consistency is proposed, effectively describing the complex likelihood feature distribution between the observed image field and the tag field. Finally, under the MRF framework, a high-order MRF segmentation model based on multi-node topological overlap measurement is provided, and model optimization is realized by utilizing a Gibbs sampling algorithm.
Further, a local area w s The larger the local information is contained, the more complex the model is calculated, and the experiment verifies that the given local area w s When the model is=3, the model is balanced in the aspects of segmentation precision and optimization time; by adopting a trial-and-error method, when the normalized parameter rho=0.2, the power adjacent parameter gamma=4 and the Gibbs sampling algorithm initial temperature T 0 =4.0, the proposed model has the optimal segmentation result.
Furthermore, the high-order spatial topological relation of multiple pixels is built based on the high-order MRF priori of the MTOM, an MRF priori knowledge expression model with high-order spatial correlation is built, high-order priori knowledge such as complex topological spatial structural features and the like contained in the local area of the image is described more effectively, and false correlation caused by 'strong noise' is reduced.
Furthermore, the local area consistency WGMM likelihood model introduces image local spatial correlation by utilizing the Hamming distance of adjacent pixels in the local area, and effectively improves the description capability of likelihood feature distribution of an observation image field and a tag field thereof.
Furthermore, a part of the second-order Potts prior model introduces a larger local area, contains more label node information, and improves the noise resistance of the segmentation model.
Furthermore, under the MRF framework, the WGMM likelihood feature and partial second-order Potts prior knowledge based on MTOM (methyl thiazolyl tetrazolium) and local region consistency are fused, a high-order MRF image segmentation model based on multi-node topological overlap measurement is established, the expression capability of the MRF image segmentation model on complex high-order features of a natural image is effectively improved, and the method has robustness and effectiveness on image strong noise and texture mutation interference.
Further, the Gibbs sampling algorithm is a markov continuous monte carlo (Markov Chain Monte Carlo, MCMC) algorithm, where each random variable is iteratively resampled from a conditional distribution of given residual variables, and the resulting samples obey a distribution of true samples, which is a simple and often efficient way to perform posterior reasoning in the MRF model.
In conclusion, the method provided by the invention can not only effectively resist strong noise and abrupt texture disturbance of the image and has better robustness, but also has more accurate image segmentation edges.
The technical scheme of the invention is further described in detail through the drawings and the embodiments.
Drawings
FIG. 1 is a flow chart of the present invention;
FIG. 2 is a graph showing experimental results of an embodiment of the present invention; wherein, (a) is 196073, 62096, 167062 and 253036 original pictures; (b) Segmentation results based on the Pairwise MRF segmentation model are 196073, 62096, 167062, 253036; (c) Based on Robust for 196073, 62096, 167062, 253036
Dividing results of the MRF model; (d) The invention proposes the segmentation result of the model for 196073, 62096, 167062 and 253036.
Detailed Description
Referring to fig. 1, the image segmentation method of the high-order MRF (MTOM-HMRF) model based on multi-node topological overlap measurement of the present invention comprises the following steps:
s1, inputting a natural image X= { X to be segmented s |x s e.OMEGA.s.e.S., where Ω.= {0,1, …,255} represents the observed pixel x in the image s S represents a finite set of lattice points; tag field y= { Y defining divided image s |y s E Λ, S e S }, Λ= {0,1, …, L }, L representing the total number of image segmentation tags;
s2, initializing parameters;
given a local area w s A classification tag number L; mean and variance of WGMM
Randomly initializing; a priori parameter beta; normalized parameter ρ, power adjacency parameter γ; gibbs sampling algorithm initial temperature T (0) ；
The method comprises the steps of manually determining a segmentation class number L according to an image to be segmented; given local area w in the present example s =3; 2-order prior Potts model parameter beta E [0.1,5 ]]The method comprises the steps of carrying out a first treatment on the surface of the Normalized parameter ρ=0.2, power adjacency parameter γ=4; gibbs sampling algorithm initial temperature T 0 ＝4.0。
S3, constructing a high-order MRF priori energy item based on MTOM (message transmission optimization mechanism);
s301, for=1 to S, and according to the input image x= { X s |x s E, omega, S E S, calculating a message transmission optimization mechanism MTOM of adjacent pixels;
s3011, calculate the local area w s Center pixel x of (2) s And its neighborhood pixel x r Is a normalized euclidean distance of:
wherein,, ||x s -x r || 2 Representing the euclidean distance of adjacent pixel pairs; ρ=0.2 is a normalization parameter.
Establishing a local area w s Euclidean distance metric vector of (c):
wherein d (x s ,x r ) Representing the center pixel x s And its neighborhood pixel x r And satisfies 0.ltoreq.d sr ≤1；|w s |×|w s And I is the size of the local area of the image.
S3012, utilizing power adjacency function to make local area w s Similarity metric vector of (2)
Conversion to a contiguous intensity metric vector +.>
The power adjacency function is defined as follows:
a sr (x s ,x r )＝|d(x s ,x r )| γ
wherein a is sr (. Cndot.) represents the adjacent pixel pair x s ,x r Adjacent matrix weights of (a); γ=4 is a power exponent;
established image local area w s Is (are) adjacency strength metric vector
The expression is as follows:
s3013, calculate the adjacent pixel pair x s ,x r Topological overlap measure t sr (x s ,x r )；
Wherein s is not equal to r, a sr Representing adjacent pixel pairs x s ,x r The connection strength between the two is defined as:
wherein,, ||x s -x r || 2 Representing the euclidean distance of adjacent pixel pairs; ρ=0.2 is a normalization parameter; γ=4 is the similarity penalty factor of the adjacent pixels to the power adjacency function.
S302, establishing a local area w s Is a priori of the higher order topology space of (2)
The method comprises the following steps:
wherein t is up (·),t down (·),t left (·),t right (. Cndot.) each represents a center pixel x s And a topological overlap measure with pixels in the upper, lower, left and right neighborhoods.
I.e. local area w s Is a priori of the higher order topology space of (2)
Defined as center pixel x s And its neighborhood pixel x up ,x down ,x left ,x right Is a sum of topological overlap measures of (a).
S303, if s=S, obtaining the prior energy of the high-order topological space of all areas of the image X;
local area w s Is a priori energy of higher order topology of (2)
The method comprises the following steps:
center pixel x s Adjacent to it is a pixel x r Connection strength a between sr The method comprises the following steps:
wherein y= { ρ, γ } is the similarity penalty factor of the normalized function parameter and the power adjacency function, respectively.
S4, establishing partial region consistency WGMM likelihood energy;
s401, measuring the center pixel x by adopting Hamming distance s And its neighborhood pixel x r Is introduced into the adjacent pixel x s And x r Similarity between them, defining a weight function w (y r ) The method comprises the following steps:
wherein, the similarity of adjacent pixels is x s -x r The magnitude of the weights is controlled.
S402, introducing the constructed weight into a Gaussian mixture model (Gaussian Mixture Model, GMM) to obtain a WGMM likelihood model with local region consistency, wherein the WGMM likelihood model is as follows:
wherein P (x) s |y s θ) is a mixed gaussian distribution (GMM), GMM parameters
Mean and variance of the ith GMM distribution, Λ= {0,1, …, L }, L representing the total number of image segmentation labels, respectively;
Is a normalization function; w (y) r ) Is a weight function containing spatial information, the magnitude of which uses the center pixel x s And its neighborhood pixel x r Is determined by similarity.
S403, adopting an EM algorithm to iteratively estimate GMM parameters
S404, establishing a local area consistency WGMM likelihood energy of the model provided by the invention, wherein the WGMM likelihood energy is specifically as follows:
wherein P (x) s |y s θ) is a GMM, parameter
Mean and variance of the ith GMM distribution, respectively, where Λ= {0,1,..l }, L represents the total number of image segmentation labels.
Is a normalization function.
S5, establishing partial second-order Potts priori knowledge based on the region;
s501, in local area w s In, assume a local area w s Is of the tag field Y of (2) ws Is an MRF, let y s ,y r For the corresponding adjacent pixel x s ,x r Assigned label, delta (y s ,y r ) Is a delta function:
establishing a local area w s Inconsistent penalty of (1) is
S502, establishing a partial second-order Potts prior model based on the region is described as follows:
wherein Z (beta) is a normalized constant; beta epsilon [0.1,5 ]]The prior parameters of the Potts model are used for controlling the smoothness of the segmentation result of the local area; delta (y) s ,y r ) As a delta function.
S6, establishing a high-order MRF (Higher-order MRF model with multi-node topological overlap measure, MTOM-HMRF) image segmentation energy model based on multi-node topological overlap measure:
E G (Y|X,Θ)＝E d (X|Y,θ)+E s (Y|β)+E h (XΥ)
wherein Θ= (θ, β, y); e (E) d (X|Y, θ) represents the local region consistency WGMM likelihood energy, θ is the likelihood energy model parameters. E (E) s (Y|beta) represents part of the second order Potts prior, and beta is the Potts type prior energy model parameter. E (E) h (X|gamma) represents a high-order MRF prior term based on MTOM, gamma is a high-order prior energy model parameter.
S7, optimizing the high-order MRF image segmentation energy model by using Gibbs sampling algorithm, and outputting an image segmentation result Y * ：
S701, setting the maximum iteration number t of Gibbs sampling algorithm max Initial temperature T 0 =4.0, annealing speed 0.95;
s702, pre-dividing the image according to MAP criteria:
s703, for=1 to S, for each pixel x s Calculate the local area w s Dispensing energy from the tag:
s704, gibbs sampling: in the local area w s Center pixel y s A new label configuration η is accepted according to the following probability:
s705 calculating Global energy
If->
The segmentation result Y is output * ＝Y (n) The method comprises the steps of carrying out a first treatment on the surface of the Otherwise the temperature t=0.95T is reduced (n) The process returns to step S703 to continue the iteration.
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more apparent, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments of the present invention. The components of the embodiments of the present invention generally described and illustrated in the figures herein may be arranged and designed in a wide variety of different configurations. Thus, the following detailed description of the embodiments of the invention, as presented in the figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of selected embodiments of the invention. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Referring to fig. 2, fig. 2 (a) shows a natural image to be segmented from top to bottomThe lower four graphs are respectively divided into class 2, class 3 and class 3; FIG. 2 (b) shows the segmentation result based on the Pairwise MRF segmentation model; FIG. 2 (c) is based on Robust
Dividing results of the MRF model; (d) the segmentation result of the invention.
As can be found from the comparison graph of the segmentation results, the segmentation results based on the pariwise MRF model are poor, and because the 4-neighborhood point structure of the pariwise MRF model is difficult to describe more complex image space correlation features, the interference robustness to image noise and texture features is weak, such as the abundant 'sand grain' features in the graph '196073', the texture features of 'leaves' in the graph '167062', and the like, the segmentation results are greatly interfered, and more 'speckle' mistaken segmentation areas appear. Based on Robust compared with the Pairwise MRF model
The MRF model has improved segmentation result to a certain extent, such as the segmentation result of figure 196073, the speckle erroneous segmentation region caused by the sand texture interference is suppressed to a certain extent, but due to Robust->
The MRF model assumes that the prior weights are the same at all locations in the local region, thus resulting in inaccurate segmentation results of the image edges or detail structure portions, such as the "cross hand grip" of the "sail" in the graph "62096" not being better extracted, the boundary line between the "forest" and the "sloping field" in the graph "167062" being "edge banding" or the like. The segmentation method of the model introduces richer high-order spatial correlation characteristics by utilizing a high-order topological structure, has stronger robustness to the interference of strong noise and texture mutation of images, and can obtain more accurate segmentation, such as boundary lines between a tree forest and a sloping field in a graph ' 167062 ', a horizontal hand handle of a sail in a graph ' 62096 ', and the like, an airplane in a graph ' 3096 ', sand grains in a graph ' 196073 ', and a graph ' 62096The "water wave", the "snow" in the graph "167062", the "grassy" in the graph "253036", etc., gives smoother segmentation results. Therefore, the model provided by the invention has stronger robustness to the interference of strong noise and abrupt texture change of the complex natural image, obviously improves the image segmentation precision, and particularly has higher segmentation precision at the image segmentation edge.
The above is only for illustrating the technical idea of the present invention, and the protection scope of the present invention is not limited by this, and any modification made on the basis of the technical scheme according to the technical idea of the present invention falls within the protection scope of the claims of the present invention.
Claims (6)
1. The image segmentation method of the high-order MRF model based on the multi-node topological overlapping measure is characterized by comprising the steps of firstly inputting a natural image to be segmented; then initializing parameters; reconstructing a high-order MRF priori energy item based on MTOM; establishing a high-order MRF image segmentation energy model based on multi-node topological overlap measure according to the WGMM likelihood energy of local region consistency and a partial second-order Potts prior model based on the region; optimizing the high-order MRF image segmentation energy model by using a Gibbs sampling algorithm, and determining an image segmentation result;
the establishment of the local area consistency WGMM likelihood energy is specifically as follows:
s401, measuring the center pixel x by adopting Hamming distance s And its neighborhood pixel x r Is introduced into the adjacent pixel x s And x r Similarity between them, defining a weight function w (y r )；
S402, introducing the constructed weight into a Gaussian mixture model (Gaussian Mixture Model, GMM) to obtain a WGMM likelihood model of local region consistency;
s403, adopting an EM algorithm to iteratively estimate GMM parameters
S404, establishing partial area consistency WGMM likelihood energy P (X|Y, theta), wherein the partial area consistency WGMM likelihood energy P (X|Y, theta) is specifically as follows:
wherein P (x) s |y s θ) is a GMM, parameter
Mean and variance of the first GMM distribution, Λ= {0, 1..once, L }, L represents the total number of image segmentation labels, { a }>
Is a normalization function;
the partial second-order Potts prior model based on the region is specifically as follows:
wherein Z (beta) is a normalized constant; beta epsilon [0.1,5 ]]The prior parameters of the Potts model are obtained; delta (y) s ,y r ) Delta function;
the high-order MRF image segmentation energy model based on the multi-node topological overlap measure is specifically as follows:
E G (Y|X,Θ)＝E d (X|Y,θ)+E s (Y|β)+E h (X|Υ)
wherein Θ= (θ, β, y); e (E) d (X|Y, θ) represents the partial region consistency WGMM likelihood energy, θ is the likelihood energy model parameters, E s (Y|beta) represents part of second-order Potts prior, beta is Potts type prior energy model parameter, E h (X|y) represents a high order MRF prior term based on MTOM, y being a high order prior energy model parameter.
2. The image segmentation method of a high-order MRF model based on multi-node topological overlap measure according to claim 1, wherein the natural image X to be segmented is specifically:
X＝{x s |x s ∈Ω,s∈S}
where Ω= {0,1,..255 } represents the observed pixel x in the image s S represents a finite set of lattice points;
defining the label field of the segmented image as:
Y＝{y s |y s ∈Λ,s∈S}
where Λ= {0,1,..l }, L represents the total number of image segmentation tags.
3. The image segmentation method of a higher order MRF model based on multi-node topological overlap measure according to claim 1, wherein the parameter initialization is specifically:
local area w s Mean and variance of class label number L, WGMM
Randomly initializing; a priori parameter beta; normalized parameter ρ, power adjacency parameter γ; gibbs sampling algorithm initial temperature T (0) 。
4. The image segmentation method of a high-order MRF model based on multi-node topological overlap measure according to claim 1, wherein constructing the MTOM-based high-order MRF prior energy term specifically comprises:
s301, for=1 to S, and according to the input image x= { X s |x s e.OMEGA, s.e.S }, calculating the MTOM of the adjacent pixels;
s302, establishing a local area w s Is a priori of the higher order topology space of (2)
And S303, if s=S, obtaining the prior energy of the high-order topological space of all areas of the image X.
5. The multi-node topology overlay measure based on claim 1An image segmentation method of a higher-order MRF model of (2), characterized in that, in a local area w s In, assume a local area w s Is of (1)
Is an MRF, let y s ,y r For the corresponding adjacent pixel x s ,x r Assigned label, delta (y s ,y r ) For delta function, build local region w s Inconsistent penalty of (1) is
6. The image segmentation method of the high-order MRF model based on the multi-node topological overlap measure according to claim 1, wherein the optimization of the high-order MRF image segmentation energy model by using the Gibbs sampling algorithm is specifically as follows:
s701, setting the maximum iteration number t of Gibbs sampling algorithm max ；
S702, pre-dividing the image according to the MAP criterion:
s703, for=1 to S, each pixel x is calculated s Local area w s Distributing energy of the tag;
s704, in local area w s Center pixel y s According to probability P (y s =η) receives a new label configuration η;
s705 calculating Global energy
If->
ε＝10e -6 Outputting the division result Y * ＝Y (n) The method comprises the steps of carrying out a first treatment on the surface of the Otherwise the temperature t=0.95T is reduced (n) The process returns to step S703 to continue the iteration. />
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