CN105184266A - Finger vein image recognition method - Google Patents

Abstract

The invention provides a finger vein image recognition method, which comprises the steps of: dividing finger vein ROI images into training and testing samples; normalizing image sizes to 46*102 pixels; utilizing a Gabor filter for enhancement; subjecting each pixel point to comparison and selection to obtain an ROI enhanced image; adopting a PCA for reducing dimensionality of the ROI enhanced image; subjecting the ROI enhanced image after dimensionality reduction to hyperspherical granulation to obtain a new granular set; and calculating distance between each large hypersphere in each finger vein ROI image of the testing samples and the new granular set for carrying out steps such as recognition. The finger vein image recognition method adopts the PCA and hyperspherical granulation combined method for granulating and fusing all finger vein ROI images belonging to the same individual into a large hypersphere, and better describes the generality of samples of the same individual acquired at different time. During recognition, the samples to be tested are processed into a hypersphere, and then subjected to distance comparison with the fused large hypersphere.

Description

A kind of finger venous image recognition methods

Technical field

The invention belongs to biometrics identification technology field, particularly relate to a kind of finger venous image recognition methods.

Background technology

At present, because the security of traditional biometrics identification technology is lower, therefore can not meet the demand of people to high precision identification.Along with the development of biometrics identification technology, there is a lot of advantage as a kind of novel biometrics identification technology in finger vena.First, finger vena is identified as vivo identification, and different people's blood vessel network structures is almost unchangeable after growing up; Secondly, finger vena is internal feature, there is not the cognitive disorders that any extraneous factor (as: wearing and tearing etc.) is brought; The acquisition system of finger vena is untouchable, does not exist and easily steals problem.

It is a lot of that finger vena knows method for distinguishing, but most of classic method is not all considered in database from the general character between the sample of same individuality (classification), all need samples all in sample to be identified and database indistinguishably to contrast one by one, therefore recognition efficiency is low.

Summary of the invention

In order to solve the problem, the object of the present invention is to provide a kind of finger venous image recognition methods that can improve recognition efficiency.

In order to achieve the above object, finger venous image recognition methods provided by the invention comprises the following step carried out in order:

1) all finger vena ROI images to be detected are divided into two parts, a part is as training sample, and a part is as test sample book, then above-mentioned all images are classified, and given class label, if some image comes from same finger, then class label is identical;

2) size of above-mentioned all finger vena ROI images is normalized to 46*102 (4692) pixel, obtains ROI normalization vein image;

3) utilize Gabor filter to carry out Gabor enhancing to above-mentioned all ROI normalization vein images, obtain that is 0 °, 8 directions respectively, 22.5 °, 45 °, 67.5 °, 90 °, 112.5 °, the ROI characteristic image of 135 ° and 157.5 °;

4) the pixel gray-scale value of same position in the ROI characteristic image in each 8 direction above-mentioned is compared one by one, the direction of ROI characteristic image corresponding for maximum gradation value is strengthened the direction character of image slices vegetarian refreshments as ROI, after so successively each pixel being compared selection, obtain ROI and strengthen image, and each ROI enhancing image table is shown as ROI enhancing image array A _46*102;

5) adopt PCA to strengthen image to all ROI and carry out dimensionality reduction, and obtain major component feature, by the contribution rate regulating major component feature to account for whole feature space, ROI is strengthened image and drop to different dimensions;

6) sampling feature vectors after the ROI after each dimensionality reduction being strengthened image dimensionality reduction characterizes, strengthen the image hypersphere grain of in higher dimensional space by the ROI after each dimensionality reduction to represent, the centre of sphere of hypersphere grain is the sampling feature vectors after dimensionality reduction, radius is set to 0, ROI after each like this dimensionality reduction strengthens image and is conceptualized as in higher dimensional space a former grain with the centre of sphere and radius, then hypersphere granulation carried out to it and obtain the large hypersphere grain of x, obtaining a new grain collection thus;

7) each finger vena ROI image G in Euclidean distance formulae discovery test sample book is utilized _tconcentrate the distance of each large hypersphere grain with above-mentioned grain newly, select with it apart from that minimum large hypersphere grain G _m, so large hypersphere grain grain G _mclassification be this finger vena ROI image G in test sample book _tclassification.

In step 3) in, the expression formula of described Gabor filter is such as formula shown in (1):

$G(x,y)=\frac{1}{2{\mathrm{\π\σ}}^2}\exp \left\{-\frac{1}{2}\left(\frac{x_{{\mathrm{\θ}}_k}^2}{{\mathrm{\σ}}_x^2}+\frac{y_{{\mathrm{\θ}}_k}^2}{{\mathrm{\σ}}_y^2}\right)\right\}\·\exp \left(\hat{j}2{\mathrm{\πf}}_k{x}_{{\mathrm{\θ}}_k}\right),k=0,1,\mathrm{...},7---\left(1\right)$

Wherein, σ represents the yardstick of Gabor filter, σ=4, and 5,6; θ _krepresent the angle value in a kth direction.

In step 5) in, described PCA dimensionality reduction detailed process is as follows:

(1) above-mentioned ROI is strengthened image array A _46*102each row as one dimension, the data of every one dimension are all deducted the average of this dimension, make its eigencenter and obtain matrix B;

(2) the covariance matrix C of compute matrix B;

(3) eigenwert and the proper vector of covariance matrix C is calculated;

(4) arrange descending for eigenwert, if front n eigenwert sum has exceeded 97% of all eigenwert sums, then get front n eigenwert characteristic of correspondence vector, obtain a new data set.

In step 6) in, described image strengthened to the ROI after dimensionality reduction to carry out the concrete steps of hypersphere granulation as follows:

(1) image is strengthened as operand using the ROI after dimensionality reduction in all training samples;

(2) certain former grain G in the Euclidean distance formulae discovery training sample below shown in formula (3) is utilized _jwith all former grain G _ithe distance d of (i ∈ [1, n]) _ji, and record, preserve in jth row in a matrix, until calculated last former grain G by order from small to large _n, obtain the matrix D of a n × n thus; Euclidean distance formula is as follows:

d(G _i,Gj)＝||C _i-C _j|| ₂-r _i-r _j ⁽³⁾

Wherein, G _i=(C _i, r _i), G _j=(C _j, r _j), C _i, C _jbe respectively G _i, G _jthe centre of sphere, r _i, r _jbe respectively G _i, G _jradius;

(3) maximal value capable for y+1 in matrix D is set to threshold value ρ, y is the number of training of same individuality, assuming that certain G _jwith other former grain G _ithe distance of (i ∈ [1, n], i ≠ j) is d _ji, these distances are compared with threshold value successively, compare with threshold value by all distance values in jth row, if meet d≤ρ, illustrate that the feature of these two former grains is comparatively similar, be likely same class grain, then by former grain G _ipick out and be kept in a jth cellular, finally like this can obtain n cellular, comprise in each cellular several may with former grain G _jfor of a sort former grain; Then the class label of the class label of all former grain preserved in a jth cellular and a jth former grain is compared, if consistent, then illustrate that it and a jth former grain are same class, to continue to be retained in a jth cellular, if inconsistent, then illustrate that with a jth former grain be not a class, then reject in its jth cellular; Finally just obtain new cellular collection, n cellular is contained in the inside;

(4) cellular of contained identical former grain is come out, only retain one, all the other are deleted, finally from n cellular, select x cellular, x is the classification number finally marked off, if classification is without any mistake, then x is true classification number, if there is mistake, then x is close to true classification number, so just training sample set be divide into x class;

(5) adopt formula (4) to merge the former grain in each class in above-mentioned x class respectively, obtain x large hypersphere grain, obtain a new grain collection thus; Fusion formula is:

Wherein, P=C _i-r _i(C _ij/ || C _ij||), Q=C _j+ r _j(C _ij/ || C _ij||), C _ijby C _ipoint to C _jvector, C _ij=C _j-C _i, C is the center of circle of large hypersphere grain after merging, and R is the radius of large hypersphere grain after merging.

Finger venous image recognition methods provided by the invention is granulated all finger vena ROI images belonging to same individuality is fused into a large hypersphere grain by PCA and hypersphere being granulated the method that combines, better describes the general character of the same sample body coming from different time collection.During identification, only sample to be tested also need be treated to hypersphere grain, carry out distance comparison with all large hypersphere grain after merging, do not need to mate one by one with each sample coming from same individuality, thus improve the low problem of recognition efficiency.

Accompanying drawing explanation

Fig. 1 is ROI normalization vein image in the present invention.

Fig. 2 is the ROI characteristic image in 8 directions in the present invention.

Fig. 3 is that in the present invention, ROI strengthens image.

Fig. 4 is the large hypersphere grain schematic diagram after merging in the present invention.

Embodiment

Below in conjunction with the drawings and specific embodiments, finger venous image recognition methods provided by the invention is described in detail.

Finger venous image recognition methods provided by the invention comprises the following step carried out in order:

1) all finger vena ROI (area-of-interest) images to be detected are divided into two parts, a part is as training sample, a part is as test sample book, then above-mentioned all images are classified, and given class label, if some image comes from same finger, then class label is identical.

Elect any 7 in 10 finger vena ROI images of the same finger (such as right hand forefinger) from same person as training sample in the present invention, other 3 as test sample book, and these 10 finger vena ROI images are because belong to a finger together, so classification is identical, then given identical class label.

2) size of above-mentioned all finger vena ROI images is normalized to 46*102 (4692) pixel, obtains ROI normalization vein image as shown in Figure 1;

3) because the vein texture of above-mentioned ROI normalization vein image and feature are not very clear, therefore the present invention utilizes Gabor filter to carry out Gabor enhancing to above-mentioned all ROI normalization vein images, and the expression formula of Gabor filter is such as formula shown in (1).

$G(x,y)=\frac{1}{2{\mathrm{\π\σ}}^2}\exp \left\{-\frac{1}{2}\left(\frac{x_{{\mathrm{\θ}}_k}^2}{{\mathrm{\σ}}_x^2}+\frac{y_{{\mathrm{\θ}}_k}^2}{{\mathrm{\σ}}_y^2}\right)\right\}\·\exp \left(\hat{j}2{\mathrm{\πf}}_k{x}_{{\mathrm{\θ}}_k}\right),k=0,1,\mathrm{...},7---\left(1\right)$

Wherein, σ represents the yardstick of Gabor filter, σ=4, and 5,6; θ _krepresent the angle value in a kth direction, by calculating, the ROI characteristic image of acquisition 8 directions (0 °, 22.5 °, 45 °, 67.5 °, 90 °, 112.5 °, 135 ° and 157.5 °) respectively, as shown in Figure 2.

4) the pixel gray-scale value of same position in the ROI characteristic image in each 8 direction above-mentioned is compared one by one, the direction of ROI characteristic image corresponding for maximum gradation value is strengthened the direction character of image slices vegetarian refreshments as ROI, after so successively each pixel being compared selection, the ROI obtained as shown in Figure 3 strengthens image, and each ROI enhancing image table is shown as ROI enhancing image array A _46*102.

5) method for expressing of information in higher dimensional space is a major issue of Granule Computing always.If but directly regard each ROI enhancing image as a hypersphere grain, feature space dimension can very high (4692 dimension), this becomes difficulty with regard to making hypersphere grain in the expression of higher dimensional space, and calculation of complex, recognition efficiency is very low, so this step adopts PCA (principal component analysis (PCA)) to strengthen image to all ROI carry out dimensionality reduction, and obtains major component feature, by the contribution rate regulating major component feature to account for whole feature space, ROI is strengthened image and drop to different dimensions.

PCA dimensionality reduction detailed process is as follows:

(1) above-mentioned ROI is strengthened image array A _46*102each row as one dimension, the data of every one dimension are all deducted the average of this dimension, make its eigencenter and obtain matrix B;

(2) the covariance matrix C of compute matrix B;

(3) eigenwert and the proper vector of covariance matrix C is calculated;

(4) arrange descending for eigenwert, if front n eigenwert sum has exceeded 97% of all eigenwert sums, then get front n eigenwert characteristic of correspondence vector, obtain a new data set.

Such as, if the first two eigenwert sum has exceeded 97% of all eigenwert sums, meet the demands, then got the first two eigenwert characteristic of correspondence vector, obtain the matrix M of a 102*2.

A′ _46*2＝A _46*102×M _102*2(2)

Just the data set A of 46*102 has been mapped to the data set A ' of 46*2 according to formula (2), feature has reduced to 2 by 102.

In the present invention, PCA dimensionality reduction result is as shown in table 1.The class label often opening the data that ROI enhancing image obtains after dimensionality reduction is still consistent with former finger vena ROI image.

Characteristics of image dimension after table 1PCA dimensionality reduction

6) without before PCA dimensionality reduction, if each ROI is strengthened image R ' direct representation to become hypersphere grain, then all features of sample have 4692, and namely the row vector of feature composition has 4692 dimensions, and dimension is too high.Therefore have employed PCA dimensionality reduction, as table 1 set contribution rate as 80% time, the dimension fallen is 62 dimensions, then the row vector of feature composition just reduces to 62 dimensions, is dropped to 62 dimensions can be greatly reduced operand by 4692 dimensions.So far the sampling feature vectors after the ROI after each dimensionality reduction being strengthened image dimensionality reduction in this step characterizes; strengthen the image hypersphere grain of in higher dimensional space by the ROI after each dimensionality reduction to represent; the centre of sphere of hypersphere grain is the sampling feature vectors after dimensionality reduction; radius is set to 0; ROI after each like this dimensionality reduction strengthens image and is conceptualized as in higher dimensional space a former grain with the centre of sphere and radius; then hypersphere granulation carried out to it and obtain the large hypersphere grain of x, obtaining a new grain collection thus.Concrete steps are as follows:

(1) image is strengthened as operand using the ROI after dimensionality reduction in all training samples;

(2) certain former grain G in the Euclidean distance formulae discovery training sample below shown in formula (3) is utilized _jwith all former grain G _ithe distance d of (i ∈ [1, n]) _ji, and record, preserve in jth row in a matrix by order from small to large.Such as first calculate first former grain G ₁with the distance of n former grain, then by this n distance value by the first row being kept at matrix from small to large successively, then calculating second former grain G ₂distance with n former grain, is kept at secondary series, by that analogy, until calculated last former grain G _n, obtain the matrix D of a n × n thus.Euclidean distance formula is as follows:

d(G _i,G _j)＝||C _i-C _j|| ₂-r _i-r _j(3)

Wherein, G _i=(C _i, r _i), G _j=(C _j, r _j), C _i, C _jbe respectively G _i, G _jthe centre of sphere, r _i, r _jbe respectively G _i, G _jradius.

(3) maximal value capable for y+1 in matrix D is set to threshold value ρ, y is the number of training of same individuality, as y=7 in the present invention.Assuming that certain G _jwith other former grain G _i(i ∈ [1, n], i ≠ j ⁾distance be d _ji, these distances are compared with threshold value successively, compare with threshold value by all distance values in jth row, if meet d≤ρ, illustrate that the feature of these two former grains is comparatively similar, be likely same class grain, then by former grain G _ipick out and be kept in a jth cellular.Finally like this can obtain n cellular, comprise in each cellular several may with former grain G _jfor of a sort former grain.Then the class label of the class label of all former grain preserved in a jth cellular and a jth former grain is compared, if consistent, then illustrate that it and a jth former grain are same class, to continue to be retained in a jth cellular, if inconsistent, then illustrate that with a jth former grain be not a class, then reject in its jth cellular.Finally just obtain new cellular collection, n cellular is contained in the inside.

(4) in this n cellular, contained by some cellular certain, former grain is identical, such as supposes G ₁~ G ₄four former grains belong to other sample of same class, and when training precision is 100%, the former grain deposited in 1st ~ 4 cellulars should be all G ₁, G ₂, G ₃, G ₄, so only retain first cellular, because this cellular characterizes G ₁, G ₂, G ₃, G ₄for same class, other three cellulars repeat, and can delete.Come out by the cellular of contained identical former grain, only retain one, all the other are deleted, and finally from n cellular, select x cellular, and in this x cellular, contained former grain is all not identical.X is the classification number finally marked off, if classification is without any mistake, then x is true classification number, and if there is mistake, then x is close to true classification number.So just training sample set be divide into x class.

(5) adopt formula (4) to merge the former grain in each class in above-mentioned x class respectively, obtain x large hypersphere grain, obtain a new grain collection thus.Fusion formula is:

Wherein, P=C _i-r _i(C _ij/ || C _ij||), Q=C _j+ r _j(C _ij/ || C _ij||), C _ijby C _ipoint to C _jvector, C _ij=C _j-C _i, C is the center of circle of large hypersphere grain after merging, and R is the radius of large hypersphere grain after merging.Than the former grain if any two two-dimensional spaces, be G respectively ₁=(2,6,2), G ₂=(5,7,3), then calculate by formula (4), the large hypersphere grain after fusion is G=(3.97,6.66,4.08), as shown in Figure 4.

7) each finger vena ROI image G in the Euclidean distance formulae discovery test sample book shown in formula (3) is utilized _tconcentrate the distance of each large hypersphere grain with above-mentioned grain newly, select with it apart from that minimum large hypersphere grain G _m, so large hypersphere grain grain G _mclassification be this finger vena ROI image G in test sample book _tclassification.

By the method for this distance measure, each finger vena ROI image in all test sample books is identified, can (the present invention be because will verify recognition correct rate when not knowing test sample book classification, so prior given class label, be used as to contrast with recognition result), by contrasting with training new grain collection out, thus judge its classification.If the classification identified is consistent with former given class label, then identify correct, otherwise, identification error.

In order to verify effect of the present invention, present inventor has performed following experiment:

Experiment sample finger venous image database in the present invention is obtained by self-control system acquisition.Database comprises the finger of 185 Different Individual, and each individuality comprises 10 width and refers to vein ROI image, altogether 1850 width finger vena ROI images.All finger vein images are all normalized to 46*102 (4692), and experimental situation is PC, MatlabR2010a.

Here mainly finger venous image recognition methods provided by the invention is discussed at measuring accuracy Ts (%), test duration Ts (s), training precision Tr (%), recognition performance in training time Tr (s) four.Specific experiment result is as table 2.

Table 2 recognition performance

As can be seen from Table 2, when contribution rate is 75% (when dimension reduces to 49 dimension), when ensureing that measuring accuracy Ts (%) is the highest, the performance of other three aspects also reaches optimum.And when 99%, be that measuring accuracy or test duration are all relatively poor, reason may be that higher characteristic number causes overmatching problem.In the scope of 95% ~ 70%, Ts (%) is more or less the same, and this illustrates that the dimensionality reduction mode PCA that the present invention selects in very large range shows very stable, so be conveniently when arranging contribution rate number percent.It is because have several fixing images correctly to identify in database that Tr (%) remains unchanged always.Relevant as Ts (s), Tr (s) dimension all with fallen, so when ensureing that measuring accuracy is the highest, dimension is more low better.

Result as can be seen from table, the finger venous image recognition methods provided in the present invention is practicable.

Meanwhile, the present invention is fused into a large hypersphere grain all finger vena ROI image granulations belonging to same individuality, better describes the general character coming from the same sample body that different time gathers.During identification, only sample to be tested also need be treated to hypersphere grain, carry out distance comparison with all large hypersphere grain after merging, do not need to mate one by one with each sample coming from same individuality, thus substantially increase recognition efficiency.

Claims (4)

1. a finger venous image recognition methods, is characterized in that: described finger venous image recognition methods comprises the following step carried out in order:

1) all finger vena ROI images to be detected are divided into two parts, a part is as training sample, and a part is as test sample book, then above-mentioned all images are classified, and given class label, if some image comes from same finger, then class label is identical;

2) size of above-mentioned all finger vena ROI images is normalized to 46*102 (4692) pixel, obtains ROI normalization vein image;

3) utilize Gabor filter to carry out Gabor enhancing to above-mentioned all ROI normalization vein images, obtain that is 0 °, 8 directions respectively, 22.5 °, 45 °, 67.5 °, 90 °, 112.5 °, the ROI characteristic image of 135 ° and 157.5 °;

4) the pixel gray-scale value of same position in the ROI characteristic image in each 8 direction above-mentioned is compared one by one, the direction of ROI characteristic image corresponding for maximum gradation value is strengthened the direction character of image slices vegetarian refreshments as ROI, after so successively each pixel being compared selection, obtain ROI and strengthen image, and each ROI enhancing image table is shown as ROI enhancing image array A _46*102;

5) adopt PCA to strengthen image to all ROI and carry out dimensionality reduction, and obtain major component feature, by the contribution rate regulating major component feature to account for whole feature space, ROI is strengthened image and drop to different dimensions;

6) sampling feature vectors after the ROI after each dimensionality reduction being strengthened image dimensionality reduction characterizes, strengthen the image hypersphere grain of in higher dimensional space by the ROI after each dimensionality reduction to represent, the centre of sphere of hypersphere grain is the sampling feature vectors after dimensionality reduction, radius is set to 0, ROI after each like this dimensionality reduction strengthens image and is conceptualized as in higher dimensional space a former grain with the centre of sphere and radius, then hypersphere granulation carried out to it and obtain the large hypersphere grain of x, obtaining a new grain collection thus;

7) each finger vena ROI image G in Euclidean distance formulae discovery test sample book is utilized _tconcentrate the distance of each large hypersphere grain with above-mentioned grain newly, select with it apart from that minimum large hypersphere grain G _m, so large hypersphere grain grain G _mclassification be this finger vena ROI image G in test sample book _tclassification.

2. finger venous image recognition methods according to claim 1, is characterized in that: in step 3) in, the expression formula of described Gabor filter is such as formula shown in (1):

$G\left(x,y\right)=\frac{1}{2{\mathrm{\π\σ}}^2}\exp \left\{-\frac{1}{2}\left(\frac{x_{{\mathrm{\θ}}_k}^2}{{\mathrm{\σ}}_x^2}+\frac{y_{{\mathrm{\θ}}_k}^2}{{\mathrm{\σ}}_y^2}\right)\right\}\·\exp \left(\hat{j}2{\mathrm{\πf}}_k{x}_{{\mathrm{\θ}}_k}\right),k=0,1,...,7---\left(1\right)$

Wherein, σ represents the yardstick of Gabor filter, σ=4, and 5,6; θ _krepresent the angle value in a kth direction.

3. finger venous image recognition methods according to claim 1, is characterized in that: in step 5) in, described PCA dimensionality reduction detailed process is as follows:

(1) above-mentioned ROI is strengthened image array A _46*102each row as one dimension, the data of every one dimension are all deducted the average of this dimension, make its eigencenter and obtain matrix B;

(2) the covariance matrix C of compute matrix B;

(3) eigenwert and the proper vector of covariance matrix C is calculated;

(4) arrange descending for eigenwert, if front n eigenwert sum has exceeded 97% of all eigenwert sums, then get front n eigenwert characteristic of correspondence vector, obtain a new data set.

4. finger venous image recognition methods according to claim 1, is characterized in that: in step 6) in, described image strengthened to the ROI after dimensionality reduction to carry out the concrete steps of hypersphere granulation as follows:

(1) image is strengthened as operand using the ROI after dimensionality reduction in all training samples;

(2) certain former grain G in the Euclidean distance formulae discovery training sample below shown in formula (3) is utilized _jwith all former grain G _ithe distance d of (i ∈ [1, n]) _ji, and record, preserve in jth row in a matrix, until calculated last former grain G by order from small to large _n, obtain the matrix D of a n × n thus; Euclidean distance formula is as follows:

d(G _i,G _j)＝||C _i-C _j|| ₂-r _i-r _j(3)

Wherein, G _i=(C _i, r _i), G _j=(C _j, r _j), C _i, C _jbe respectively G _i, G _jthe centre of sphere, r _i, r _jbe respectively G _i, G _jradius;

(3) maximal value capable for y+1 in matrix D is set to threshold value ρ, y is the number of training of same individuality, assuming that certain G _jwith other former grain G _ithe distance of (i ∈ [1, n], i ≠ j) is d _ji, these distances are compared with threshold value successively, compare with threshold value by all distance values in jth row, if meet d≤ρ, illustrate that the feature of these two former grains is comparatively similar, be likely same class grain, then by former grain G _ipick out and be kept in a jth cellular, finally like this can obtain n cellular, comprise in each cellular several may with former grain G _jfor of a sort former grain; Then the class label of the class label of all former grain preserved in a jth cellular and a jth former grain is compared, if consistent, then illustrate that it and a jth former grain are same class, to continue to be retained in a jth cellular, if inconsistent, then illustrate that with a jth former grain be not a class, then reject in its jth cellular; Finally just obtain new cellular collection, n cellular is contained in the inside;

(4) cellular of contained identical former grain is come out, only retain one, all the other are deleted, finally from n cellular, select x cellular, x is the classification number finally marked off, if classification is without any mistake, then x is true classification number, if there is mistake, then x is close to true classification number, so just training sample set be divide into x class;

(5) adopt formula (4) to merge the former grain in each class in above-mentioned x class respectively, obtain x large hypersphere grain, obtain a new grain collection thus; Fusion formula is:

Wherein, P=C _i-r _i(C _ij/ || C _ij||), Q=C _j+ r _j(C _ij/ || C _ij||), C _ijby C _ipoint to C _jvector, C _ij=C _j-C _i, C is the center of circle of large hypersphere grain after merging, and R is the radius of large hypersphere grain after merging.