CN101639948A - Method for calculating characteristic value and characteristic vector of flexible body point distribution model based on interpolation algorithm - Google Patents

Abstract

A kind of characteristic value and feature vector calculation method of the flexible body points distribution models based on interpolation algorithm, comprising the following steps: 1) be loaded into flexible body the distributed model Ma and Mb of two adjacent moments Ta and Tb, in which:

; 2) by flexible body the distributed model Ma and Mb, average shape is calculated separately With

, eigenmatrix Φ a and Φ b and eigenvalue λ a and λ b, setting flexible body is in consecutive variations from Ta to Tb; 2.1) calculating of the characteristic value of mid-module; 2.2) estimation of the feature vector of mid-module. It is good that the present invention is capable of forming the middle transition model of real-time distribution, model accuracy height, practicability.

Description

Eigenwert and proper vector computing method based on the flexible body point distributed model of interpolation algorithm

Technical field

The present invention relates to flexible body point distributed model analytical technology, in conjunction with the calculation method of parameters of graphics, matrix theory, linear algebra, medical image analysis, computer image processing technology and flexible body model.

Background technology

Flexible body is with respect to rigid body, in computing machine, carry out modeling and analyze more complicated, and the flexible in time situation about changing of object is very general in actual life, as document 1: Wang Min qin Korea S is coated with the swimming autumn by force. based on the medical image segmentation summary [J] of deformation model. and the health care equipment, 2009,30 (2) :-37-39; Document 2 " Munsell, B.C.; Dalal, P.; Song Wang, " Evaluating Shape Correspondence forStatistical Shape Analysis:A Benchmark Study; " IEEE Transactions on PatternAnalysis and Machine Intelligence, vol.30, no.11, pp.2023-2039, Nov.2008, be Marcel, B.C; Wear and draw, P.; Wang Song, " add up the form fit estimation in the shape analysis: a kind of calibration is studied ", IEEE pattern analysis and machine intelligence proceedings, vol.30, no.11,2008 (11): 2023-2039; And document 3: Gong Yong Yiluo laughs at southern Jia Weijia Huang and improves people's living condition. based on the medical figure registration of improved spring proton model. and Chinese journal of computers, 2008 (7): 1224-1233.For many biomedical tissues, as human heart, both hands, bone, kidney, brain, cell or the like, these people can describe its outer shape feature well but be difficult to provide rigid model accurately, and as document 4: sky, Chen Yu village dagger-axe king closes. and histogram is estimated the application of mutual information in non-rigid image registration. Chinese journal of computers, 2000 (4): 444-448; Document 5: the clear bear of simple river great waves Feng Huan advances. the active profile and the application in brain MR image segmentation thereof of the constraint of some distributed model. and Chinese biological engineering in medicine journal, 2006,25 (5): 513-517; And document 6: Zhou Shoujun, Liang Bin, Chen Wufan. heart sequence image estimation new method: based on the inflection curves estimation and the tracking of generalized fuzzy gradient vector flow. Chinese journal of computers .2003.26 (11).Point distributed model (PDM) is a kind of technology that emerges recently, and it can well be described research object, and the decomposition that can successfully be applied to non-rigid model is handled with synthetic.In recent years, the statistics modeling of flexible body and expedited the emergence of some advanced applied software with combining of image interpretation technology, these softwares are not only at biomedicine, also relate to target following, identification, so that fields such as video display, as document 7: Tang fruit Zhao Xiao east Wang Yuan U.S.. 3 d medical images is cut apart and visual research. Chinese journal of computers, 1998 (3): 204-209; Document 8: the outstanding horse of the section people of the same generation is finished cutting edge of a knife or a sword and opens skill Bao Houkaibaoshanglian. energy conduction model and the application [J] in medical image segmentation. and software journal, 2009 (5) :-1106-1115; Document 9: Hou Lihua. improve the Snake model at medical ultrasonic image Application in Segmentation [J]. Computer Simulation, 2008,25 (8) :-183-185,322; And document 10: the bright Dai Peishan of the little flat Wang Bo in Min yellow dawn of sun. based on cut apart [J] of the medical image sequence of active snape changing mode. system emulation journal, 2007,19 (22) :-5331-5335.

PDM is a kind of statistical model of analyzing based on object samples.From people such as Cootes, this body idea about modeling based on gauge point on the training sample is implemented always, as document 11:Timothy F.Cootes, Christopher J.Taylor, " Combining point distribution models with shape models basedon finite element analysis ", Image Vision Comput.13 (5): 403-409 (1995), it is the ancient thatch of base of a fruit nurse F., Christoffer Taylor J., " finite element analysis mid point distributed model combines with shape ", computer picture and vision, 1995,13 (5): 403-409.Most typical idea is that gauge point is alignd automatically to determine the position and the deformation type of figure.Can be easily after the registration image change in location of gauge point in different illustrations more on the same group, the coordinate that only needs to calculate them is just passable.Thereby the PDM model is finished by the point vector for the description of shape, and it is made up of intermediate shape and change of shape pattern.By to this distribution situation modeling, we can obtain the new shape similar to original training set.

At three dimensions, gauge point is described if each figure in the training set image is n by one group of quantity all, so its vector representation (if two-dimensional object is the 2n n dimensional vector n) that can tie up with a 3n.Markers work in the training set is extremely important, is accomplished manually usually.In addition, we will collect an example of hundreds of term and general principle flexible bodies to obtain reasonable statistics, as document 12:Guoyan Zheng under study for action; XiaoDong; Rajamani, K.T.; Xuan Zhang; Styner, M.; Thoranaghatte, R.U.; Nolte, L.-P.; Ballester, M.A.G., " Accurate and Robust Reconstruction of a Surface Model of theProximal Femur From Sparse-Point Data and a Dense-Point Distribution Model forSurgical Navigation; " IEEE Transactions on Biomedical Engineering, vol.54, no.12, pp.2109-2122, Dec.2007, be Zhang Guoyan, Dong Xiao, A Ya Manny, K, T; Zhang Xuan, Si Dina, M; Suo Lana breathes out enlightening, R U.; Nuo Di, L, P.; Ba Yesiteer, M.A.G., " rebuilding based on the accurate robust of the near end of thighbone of sparse point data and point of density distributed model in the operation guiding system ", IEEE biomedical engineering proceedings, vol.54, no.12,2007 (12): 2109-2122; Document 13:Koikkalainen, J.; Tolli, T.; Lauerma, K.; Antila, K.; Mattila, E.; Lilja, M.; Lotjonen, J., " Methods of Artificial Enlargementof the Training Set for Statistical Shape Models; " IEEE Transactions on MedicalImaging, vol.27, no.11, pp.1643-1654, Nov.2008, i.e. cock Richard Crenna, J.; The Tuoli, T.; Raul's agate, K.; By lifting K.; Ma Dila, E.; Li Ya, M.; Luo Tina, J., " the statistical shape model training set manually enlarges method ", IEEE medical image proceedings, vol.27,2008 (11): 1643-1654.The probability model dimension that training set forms may be very big (in a cardiac module, can reach dimensions up to ten thousand usually), the process of PDM is very uninteresting to waste time and energy again so set up.Yet in some applications, an independent PDM model or limited PDM model all are not enough to good description object to be changed.In fact, no matter be that continuous P DM model or model insertion method all are in order to satisfy Properties of Objects, i.e. variation on object time and the space.For example when the heart modeling, need the model of different time sections could describe of the variation of ventricle shape well in the heartbeat different phase.But, when making up the continuous model in whole heart movement cycle, we can only be obtained the model in specific several moment by known model sequence, and the model of other any times is just unknown, although occur having about ten years in image segmentation field PDM modeling, also do not carry out for the further investigation of model interpolation and related content based on model.

Summary of the invention

The deficiency of, poor practicability poor for the model that can not form real-time distribution, the model accuracy that overcome existing flexible body point distributed model the invention provides a kind of middle transition model, model accuracy height, practicality that can form real-time distribution good eigenwert and proper vector computing method based on the flexible body point distributed model of interpolation algorithm.

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

A kind of eigenwert of the flexible body point distributed model based on interpolation algorithm and the computing method of proper vector, described computing method may further comprise the steps:

1), is written into two adjacent moment T _aAnd T _bFlexible body distributed model M _aAnd M _b, its calculating formula is:

M _a＝<y _a，Φ _a，λ _a>，M _b＝<y _b，Φ _b，λ _b>；

2), by described flexible body distributed model M _aAnd M _b, calculate average shape y respectively _aAnd y _b, eigenmatrix Φ a and Φ b and eigenvalue _aAnd λ _b, set flexible body from T _aTo T _bBe continuous variation,

2.1) calculating of the eigenwert of mid-module:

λ _t≈a ²λ _a+b ²λ _b， (25)

Here λ a and λ b are respectively the feature value vector of former and later two adjacent PDM models, promptly

λ _a＝(λ _a1，λ _a2，λ _a3，...)， (26)

λ _b＝(λ _b1，λ _b2，λ _b3，...). (27)

2.2), the estimation of the proper vector of mid-module:

The covariance matrix that inserts mid-module is approximately:

S _t≈a ²S _a+b ²S _b (28)

Wherein, covariance matrix Sa and Sb obtain by eigenmatrix Φ a and Φ b are approximate, and Φ a and Φ b obtain the known some distributed model before interpolation; Earlier training set data is carried out PCA before the setting mid-module is set up and analyze, and choose the deformation that t kind strain mode is described whole object with the minimizing number of parameters; Because Φ is a top t eigenvector in the covariance matrix, then has:

$S_a\&ap;{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T---\left(29\right)$

$S_b\&ap;{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T---\left(30\right)$

Wherein, Λ a and Λ b are the diagonal matrix that is combined to form by eigenvector λ a and λ b; With the method for svd from formula $S_t=a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Obtain inserting the eigenvector of mid-module:

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T=\mathrm{svd}({\mathrm{<figure-callout\; id="2"\; label="a"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">a</figure-callout>}}^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)---\left(31\right)$

Determine orthogonal matrix Ut by two known adjacent PDM models and correlation parameter a, b, Λ a, Λ b, Φ a and Φ b, the preceding t of described orthogonal matrix Ut is capable to be exactly eigenmatrix Φ t.

Further, also comprise described step 2):

2.3), the covariance matrix of mid-module derived by following formula and draws:

$S_t=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t){(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yti}}{\mathrm{\&Delta;}}_{\mathrm{yti}}^T$

$=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}}){(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}})}^T$

$=\frac{a^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T+\frac{b^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T---\left(15\right)$

Order

$S_{\mathrm{ab}}=\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(16\right)$

Because

$S_a=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T---\left(17\right)$

$S_b=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(18\right)$

Obviously have

$S_{\mathrm{ab}}=S_{\mathrm{ba}}^T---\left(19\right);$

According to covariance matrix, matrix S _tBy S _a, S _bAnd S _AbCalculate, by following formula:

$S_{\mathrm{ab}}=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Phi;}}_a{b}_{\mathrm{ai}}{b}_{\mathrm{bi}}^T{\mathrm{\&Phi;}}_b^T---\left(32\right)$

Use singular value decomposition method, can make:

$S_{\mathrm{ab}}={\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T---\left(33\right)$

Common Φ _Ab≠ Φ _Ba, because matrix S _AbAnd unsymmetrical matrix, Φ _AbAnd Φ _BaAll comprise t vector;

When interpolation, some distributed model keeping characteristics value and proper vector, the covariance matrix of mid-module calculates by following formula:

$S_t={\mathrm{<figure-callout\; id="2"\; label="a"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">a</figure-callout>}}^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ba}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ab}}^T---\left(34\right).$

Further again, described step 2) also comprise in:

2.3), the calculating of proper vector group

When through type (31) comes the calculated characteristics vector, (31) are rewritten as:

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T={\mathrm{\&Phi;}}_a(a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)={\mathrm{\&Phi;}}_{a}L---\left(35\right)$

Wherein $L=a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Be the matrix of a r * N dimension, wherein r is the major component number, and N is the sample dimension, i.e. the product of the quantity of calibration point and gauge point dimension;

If U _iBe matrix G=L Φ _aThe principal character vector, the character pair value is λ _i, (Φ then _au _i) and λ _iBe respectively matrix S=Φ _aThe proper vector of L and eigenwert.

By Gu _i=λ _iu _i,:

S(Φ _au _i)＝Φ _aL(Φ _au _i)＝Φ _aGu _i＝Φ _a(λ _iu _i)＝λ _i(Φ _au _i) (36)。

Technical conceive of the present invention is: the PDM method is a kind of priori modeling method, and it is based upon on the known basis of one group of sample (being training set) of destination object.This group sample set can provide the shape of object and the statistics of deformation to describe.In actual applications, the shape description of this group objects sample typically uses (as the coordinate figure of one group of pixel on the picture) that profile carries out.Further, can on profile, choose some gauge points and describe the object shape, and these gauge points often are associated with certain feature of object.

Correspondence markings point on the different samples is being followed the principle of " feature locations is roughly the same " when choosing, but often this location is accurate inadequately, has certain deviation.This deviation will be owing to interindividual natural deformation.We expect that this deviation is very small for the yardstick of whole body.The PDM method allows us to imitate this small difference, and points out the small really of which deviation, and which is bigger comparatively speaking.

Because the body in the training set is to obtain from different pictures, and therefore the coordinate system of different pictures and disunity are necessary earlier all training set bodies to be carried out coordinate normalization, are about to their alignment.In fact the process of alignment is exactly each sample pattern to be sought the process of suitable similarity conversion, comprise translation, convergent-divergent and rotation, so that it is approaching as much as possible between the different samples, on the other hand, Xuan Ding conversion also should make each sample pattern approaching as far as possible with the averaging model that is obtained by all samples.Statement supposes that our sample has only two for convenience, and normalization is reduced to these two samples of alignment.The shape of each sample is all represented by the vector that a gauge point coordinate is formed:

x＝(x ₁，y ₁，z ₁，...，x _n，y _n，z _n) ^T， (1)

(x wherein _i, y _i, z _i) be the volume coordinate of certain gauge point in 3-D view.If two-dimensional medical images, then gauge point (x _i, y _i) expression.

Gauge point has obtained multiple mode, mark or adopt automatically that labeling algorithm all is fine manually on the picture.For example, a kind of MRI image tagged method that people such as Izard propose, use unified algorithmic rule in different people mind map sheet, to carry out the institutional framework mark, as document 14:C.Izard, B.Jedynak, and C.Stark, Automatic Landmarking of Magnetic Resonance brain Images, Proc.of theSPIE International Symposium on Medical Imaging, 12-17 February 2005, San Diego, California, USA, be the C. Izard, B. pricks nanogram, the C. Stark, " the automatic mark of cerebral nucleus magnetic resonance image (MRI) ", SPIE medical image webinar, 12-17 day in February, 2005, the U.S., California, Santiago.Obtain one group through behind the training sample of mark, we need snap to them under the unified coordinate system.Can use broad sense alignment algorithm (GPA) with the training sample alignment, and make all sample shape arrive the square distance and the minimum of averaging model, in fact just seek conversion T _i(translation, rotation, convergent-divergent) minimizes following formula

D _m＝∑|m-T _i(x _i)| ²， (2)

Wherein

$m=\frac{1}{n}\mathrm{\&Sigma;}{T}_i\left(x_i\right),\left|m\right|=1,---\left(3\right)$

And x _iBe a shape sample in the training set, it is the vector representation (having in three dimensions under the situation of n gauge point) of a 3n dimension.

Training set after the alignment has been formed the some cloud on the three dimensions, can be regarded as a probability density function.In order to reduce computing cost and EMS memory occupation, we adopt the pivot in principal component analysis (PCA) (PCA) the searching point cloud, and set up model with the minority component of prostatitis maximum.These components that are selected can the main deformation of description object.Model with left ventricle in cardiac cycle is an example, and we can enough describe the shape of ventricle through 60 proper vectors of usefulness after the principal component analysis (PCA) and change.At last, the PDM model can be expressed as:

x＝x+p ₁b ₁+…+p _tb _t＝x+Φc (4)

Wherein, x is the average shape vector after the alignment, and Φ is the matrix of a 3n * t, and the vector of unit length on the major component direction is shown in each tabulation of Φ, the t dimensional vector that c is made up of form parameter.As seen, the dimension through PCA analysis back shape has dropped to t by 3n.

We have obtained a statistical models that is similar to a distributed model through above step, and this model can be applicable to the object modeling of flexible body, and in new image location and cut apart new example.We can obtain the variation range of form parameter b in the learning process of training set, as long as change in this scope, the value of b can generate rational new samples (emulation body) arbitrarily.Common b _iBe changed to λ _j(the big eigenwert of corresponding i among the matrix Φ) or

After from sample set, training a statistics appearance model, just can explain a new image example with PDM.For with model and images match, adopt a kind of method of active iteration usually.This method makes model constantly be out of shape the profile that adapts to destination object in the new images.The variation of mould shapes is subjected to the constraint of some statisticss, also promptly can only change in the scope that training sample concentrates gained to go out.For shape, we also require image outline roughly be positioned at model points near.Can be expressed as the variation of crossing the normal direction of this point on the gradient of certain point and the profile.Make the profile of statistical model constantly move along image outline, the mahalanobis distance that calculates two profiles simultaneously just minimizes and can finally obtain real boundary position.Therefore, the shape that obtain the target flexibility object in particular instance needs to carry out repeatedly following two steps up to convergence: (1) is along the corresponding point that are complementary on the normal searching image outline of model each point the mahalanobis distance minimum of model points and corresponding diagram picture point (make); (2) changing modal position and form parameter makes each gauge point of model more near the corresponding point that find on the image outline.That is to say that in fact the fit procedure of model is exactly the process of searching the most similar position of model points and constantly revising whole geometry conversion T and form parameter c on image, makes it to minimize.Can be expressed as on the mathematics:

f＝|X-T(x+Pc)| ²＝f(b，X _c，Y _c，Z _c，s，α，β，γ)

(5)

＝|X-T(x+Pc；X _c，Y _c，Z _c，s，α，β，γ)| ²

Wherein X is the temporary pattern that obtains when current iteration.

This is a problem of seeking minimum value, can find the solution by the alternative manner of some nonlinear optimizations.Finally we can determine integral transformation parameter T, and the form parameter b of individual instances:

c＝P ^T(T ^-1(X)-x). (6)

In recent years, people have just carried out deep research based on many relevant issues of the flexible body modeling method of PDM, as body alignment, automatic mark, average body generation, deformation modeling, model analysis, image segmentation or the like problem, as document 15: Jiang Xiaoyue, Zhao Rongchun, a kind of improved active contour image Segmentation Technology, Chinese image graphics journal: A collects, 2004,9 (9): 1019-1024; Document 16: Ma Feng, Tang Zesheng, Xia Shaowei. multiple dimensioned geometric active curve and MR image boundary are extracted. Chinese journal of computers .2000.23 (8); Document 17:Gilliam, A.D.; Epstein, F.H.; Acton, S.T., " Cardiac Motion Recovery via ActiveTrajectory Field Models, " IEEE Transactions on Information Technology inBiomedicine, vol.13, no.2, pp.226-235, March 2009, i.e. Terry Gilliam, A.D.; Epstein, F.H.; The Exxon, S.T., " based on the recovery aroused in interest of track field model ", IEEE biomedical information technology proceedings, vol.13,2009 (3): 226-235.In addition, be widely used in this thought on the basis of non-rigid body object modeling, and can extend further to such as movable body model (ASM) and active performance modeling (AAM) etc. to be applied to more deep graphical analysis.At present, the expression the most successful to the PDM model obtains by the PCA method, as document 18:Tobon-Gomez, C.; Butakoff, C.; Aguade, S.; Sukno, F.; Moragas, G.; Frangi, A.F., " Automatic Construction of 3D-ASM Intensity Models by Simulating ImageAcquisition:Application to Myocardial Gated SPECT Studies; " IEEE Transactions onMedical Imaging, vol.27, no.11, pp.1655-1667, Nov.2008, i.e. Tuo Dun-Ge Maici, C.; Crust tower koff, C.; A Guaide, S.; Sa Kenuo, F.; Mo Lagaisi, G.; The Forlan base, A.F., " the three-dimensional ASM density model based on the analog image collection is rebuild automatically: be applied to heart door lock SPECT image studies ", IEEE medical image proceedings, vol.27,2008 (11): 1655-1667.The PCA method is easily understood very much, as document 19:Engelen, S., Hubert, M., Vanden Branden, K., A comparison of three procedures forrobust PCA in high dimensions, Workshop on Robustness for High-dimensional Data (RobHD2004), 2004, it is the English Glenn, S., Herbert, M., the Fan Dengbu Landon, K., " three process contrasts of high-dimensional PCA Robust Analysis ", the academic discussion of high dimensional data robustness, 2004, at first a body is projected to through study PCA space (orthogonal intersection space that is formed by the linear independence direction), from training sample, extract the parameter (be called principal ingredient) relevant on a small quantity then, control the distortion of body again with these parameters with strain mode.Obviously, this deformation is carried out along the determined largest deformation direction of sample set.Owing to only study deformation on several main directions, be computation complexity or the requirement of storage space all greatly reduced.From the mathematics angle, at first be to obtain average shape y and body covariance matrix S from the training collective estimation:

$\stackrel{\&OverBar;}{y}=\frac{1}{s}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{y}_i,$ $S=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_i-\stackrel{\&OverBar;}{y}){(y_i-\stackrel{\&OverBar;}{y})}^T---\left(7\right)$

Like this, a rational shape can be expressed as

y＝y+Φc (8)

Vectorial c is made up of the deformation parameter corresponding to each proper vector in the formula.Matrix Φ has then comprised one group of proper vector that is obtained by covariance matrix S decomposition, but this group vector linear combination becomes new shape y arbitrarily.According to the PCA theory, each parameter among the c is all controlled the deformation of body on a certain direction, and all directions are all independently of one another, parameter according to body in counterparty's descending series arrangement of deformation that makes progress.The present invention considers only to comprise average shape, eigenwert and proper vector and do not comprise under the data conditions of original training set and carry out interpolation arithmetic at a PDM model.That is:

M＝<y，Φ，λ> (9)

As mentioned above, single PDM model can be obtained by the body of one group of flexible object, for example, and the heart picture of the patient in hospital that we study (some flexible bodies that are observed example).Yet we will observe these flexible bodies situation over time sometimes, can obtain a series of shape information like this, for example the 3 D deformation of heart in whole cycle of activity.And hospital's existing equipment can only obtain the 3-D view in some moment, promptly corresponding to some Ti model M i constantly.

To a specific dynamic changing object, suppose we discrete time coordinate (T1, T2 ...) and on set up one group of PDM model (M ₁, M ₂...).We can be problem description: provide two known PDM models, T _aModel M constantly _aAnd T _bModel M constantly _b(model is known to mean that we have known average shape y _aAnd y _b, eigenmatrix Φ a and Φ b and eigenvalue _aAnd λ _b), how to obtain (T at t _a≤ t≤T _b) constantly mid-module M _tCorrelation parameter.These parameters have average shape, eigenwert and proper vector, i.e. M _t=＜yt, Φ t, λ t〉(Fig. 1).Here we suppose that at Ta a body is from Y in the time period of Tb _AiVary continuously to Y _Bi, the two all is the n dimension, as Y _Ai=(x _Ai1, x _Ai2, x _Ai3..., x _Ain) ^THere also defined two constants:

$a=\frac{t-T_a}{T_b-T_a}---\left(10\right)$

b＝1-a (11)

A and b have reflected that mid-module is with respect to M _aAnd M _bDistance.

(1) average shape of mid-module

According to above definition and hypothesis, obtain Y with linear interpolation (can satisfy most practical application) _AiAnd Y _BiBetween mid-module be shaped as:

y _ti＝ay _ai+by _bi (12)

A and b are by formula (10) and (11) definition, i.e. a+b=1.

The average shape of inserting is:

${\stackrel{\&OverBar;}{y}}_t=\frac{1}{s}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ti}}=\frac{1}{s}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}({\mathrm{ay}}_{\mathrm{ai}}+b{y}_{\mathrm{bi}})=\frac{a}{s}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{y}_{\mathrm{ai}}+\frac{b}{s}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{y}_{\mathrm{bi}}=a{\stackrel{\&OverBar;}{y}}_a+b{\stackrel{\&OverBar;}{y}}_b---\left(13\right)$

So we obtain to draw a conclusion:

During theorem 1 linear interpolation, the PDM mid-module average shape of insertion is the linear combination of adjacent model average shape, and its parameter is determined by formula (13).

(2) deformation of mid-module

Similar, the deformation of middle PDM model can be calculated like this:

Δy _ti＝y _ti-y _t＝(ay _ai+by _bi)-(ay _a+by _b)

＝a(y _ai-y _a)+b(y _bi-y _b) (14)

＝aΔ _yai+bΔ _ybi

Can obtain to draw a conclusion from following formula:

During theorem 2 linear interpolations, the change of shape speed of the PDM mid-module of insertion is the linear combination of adjacent model speed of deformation, and its parameter is determined by formula (14).

(3) covariance matrix of mid-module

The covariance matrix of mid-module can be derived by following formula and be drawn

$S_t=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t){(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yti}}{\mathrm{\&Delta;}}_{\mathrm{yti}}^T$

$=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}}){(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}})}^T$

$=\frac{a^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T+\frac{b^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T---\left(15\right)$

Order

$S_{\mathrm{ab}}=\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(16\right)$

Because

$S_a=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T---\left(17\right)$

$S_b=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(18\right)$

Obviously have

$S_{\mathrm{ab}}=S_{\mathrm{ba}}^T---\left(19\right)$

Finally, obtain following inference.

During inference 1 linear interpolation, the covariance matrix of mid-module can be obtained by following formula

$S_t=a^2{S}_a+b^2{S}_b+\mathrm{ab}{S}_{\mathrm{ab}}+\mathrm{ab}{S}_{\mathrm{ab}}^T---\left(20\right)$

Here S _AbRepresentation model M _aAnd M _bThe alternation matrix.We know that covariance matrix is to draw by the sampled data of Different Individual in the same time period is carried out statistical study, and the alternation matrix has then been described the deformation process of two object models that obtain in different sampling stages.

The estimation of mid-module:

(1) alternation matrix

For making things convenient for our discussion, hypothesized model is the n dimension, as

Δy _ai＝[d _a1i?d _a2i?...?d _ani] ^T (21)

Then, the alternation matrix is

$S_{\mathrm{ab}}=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}\left[\begin{array}{cccc}{d}_{a1i}{d}_{b1i}& d_{a1i}{d}_{b2i}& ...& d_{a1i}{d}_{\mathrm{bni}}\\ d_{a2i}{d}_{b1i}& d_{a2i}{d}_{b2i}& ...& d_{a2i}{d}_{\mathrm{bni}}\\ ...& ...& ...& ...\\ d_{\mathrm{ani}}{d}_{b1i}& d_{\mathrm{ani}}{d}_{b2i}& ...& d_{\mathrm{ani}}{d}_{\mathrm{bni}}\end{array}\right]---\left(22\right)$

Generally, sample is normal state or is evenly distributed that each element in the alternation matrix goes to zero, that is:

$S_{\mathrm{ab}}(m,n)=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{d}_{\mathrm{ami}}{d}_{\mathrm{ani}}\&RightArrow;0,i\&RightArrow;+\&infin;---\left(23\right)$

So alternation diagonal of a matrix element is much smaller than matrix S _aAnd S _bIn corresponding element, that is:

$\left\{\begin{array}{c}{S}_{\mathrm{ab}}(m,n)<<S_a(m,n)\\ S_{\mathrm{ab}}(m,n)<<S_b(m,n)\end{array}\right.,m=n---\left(24\right)$

(2) estimation of eigenwert

As can be seen, alternation matrix S _AbAnd S _BaCalculating influence for the mid-module eigenwert is very little, so we can estimate:

λ _t≈a ²λ _a+b ²λ _b， (25)

Here λ a and λ b are respectively the feature value vector of former and later two adjacent PDM models, promptly

λ _a＝(λ _a1，λ _a2，λ _a3，...)， (26)

λ _b＝(λ _b1，λ _b2，λ _b3，...). (27)

The precision of element estimation among the λ t is depended primarily on the quantity and the distribution situation of sample set.Find that through computer simulation when sample set is 30 when evenly distributing at random, error rate can maintain below 5 percent.

(3) method of estimation of proper vector

When sample set was very big, the covariance matrix that inserts mid-module can be approximated to be:

S _t≈a ²S _a+b ²S _b (28)

So covariance matrix Sa and Sb can obtain by eigenmatrix Φ a and Φ b are approximate, and Φ a and Φ b obtain can be before the interpolation known some distributed model.Suppose earlier training set data to be carried out the PCA analysis with the minimizing number of parameters before the PDM modelling, and choose the deformation that t kind strain mode is described whole object.Because Φ is a top t proper vector in the covariance matrix, we know:

$S_a\&ap;{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T---\left(29\right)$

$S_b\&ap;{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T---\left(30\right)$

Wherein Λ a and Λ b are the diagonal matrix that is combined to form by proper vector λ a and λ b.Then, we can use the method for svd (SVD) from formula $S_t=a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Obtain inserting the proper vector of mid-module:

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T=\mathrm{svd}(a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)---\left(31\right)$

Now, by two known adjacent PDM models and correlation parameter a, b, Λ a, Λ b, Φ a and Φ b, determined orthogonal matrix Ut, and the preceding t of Ut capable be exactly eigenmatrix Φ t.

(4) Method for Accurate Calculation of proper vector

By top formula, we can be obtained inserting each parameter (as average shape, eigenwert, proper vector etc.) of mid-module by two known PDM Model Calculation.Yet because we have ignored the effect of alternation matrix, therefore there is certain error in this method, and is especially very little or its this error that distributes when relatively more unusual is just more obvious when the sample set capacity.In fact we have a kind of accurate method, are exactly the alternation matrix that directly generates all adjacent two PDM models in the training set.According to formula (20), matrix S _tCan be by S _a, S _bAnd S _AbCalculate, this means matrix S _AbM need made up _aAnd M _bIn time, just obtain.In fact, can pass through following formula:

$S_{\mathrm{ab}}=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Phi;}}_a{b}_{\mathrm{ai}}{b}_{\mathrm{bi}}^T{\mathrm{\&Phi;}}_b^T---\left(32\right)$

Use singular value decomposition method, can make:

$S_{\mathrm{ab}}={\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T---\left(33\right)$

Common Φ _Ab≠ Φ _Ba, this is because matrix S _AbIt is not a symmetric matrix.Φ _AbAnd Φ _BaAll comprise t vector.

When interpolation, owing to put the information that distributed model has only kept eigenwert and proper vector, the covariance matrix of mid-module should calculate by following formula:

$S_t=a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ba}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ab}}^T---\left(34\right)$

Similar, the proper vector of mid-module can be obtained by formula (31).Therefore this method is a kind of solution more accurately owing to do not train the influence of sample distribution.

(5) calculating of proper vector group

When direct through type (31) came the calculated characteristics vector, we can find that computation process all has very high requirement to storage space and operation efficiency because matrix is very big usually.For example, in our experiment three-dimensional cardiac model is arranged, comprise 2848 gauge points, by having selected 90 kinds of strain modes after the principal component analysis (PCA), this model covariance matrix will occupy the memory headroom of 584MB so, the simplest add reducing may be also will be with several minutes clock times.Therefore, in order to simplify computation complexity, we are rewritten as (31):

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T={\mathrm{\&Phi;}}_a(a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)={\mathrm{\&Phi;}}_{a}L---\left(35\right)$

Wherein $L=a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Be the matrix of a r * N dimension, wherein r is the major component number, and N is the sample dimension, i.e. the product of the quantity of calibration point and gauge point dimension.Below we describe computing method after the simplification.

Theorem 3: if U _iBe matrix G=L Φ _aThe principal character vector, the character pair value is λ _i, (Φ then _au _i) and λ _iBe respectively matrix S=Φ _aThe proper vector of L and eigenwert.

Its proof is quite simple, by Gu _i=λ _iu _i,:

S(Φ _au _i)＝Φ _aL(Φ _au _i)＝Φ _aGu _i＝Φ _a(λ _iu _i)＝λ _i(Φ _au _i) (36)

Beneficial effect of the present invention mainly shows: middle transition model, model accuracy height, the practicality that can form real-time distribution are good.

Description of drawings

Fig. 1 be from time series close on model be interpolated into the synoptic diagram of instantaneous PDM model.

Fig. 2 is the synoptic diagram that typical cardiac cycle is divided into six stages.

Fig. 3 is the synoptic diagram of six PDM models of heart different phase feature.

Fig. 4 is that interpolation generates new PDM model to carry out the synoptic diagram that flexible body is analyzed.

Fig. 5 is the synoptic diagram by the time smoothing variation of the PDM cardiac module of interpolation generation.

Embodiment

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

With reference to Fig. 1～Fig. 5, a kind of eigenwert of the flexible body point distributed model based on interpolation algorithm and the computing method of proper vector, it is characterized in that: described computing method may further comprise the steps:

1), is written into two adjacent moment T _aAnd T _bFlexible body distributed model M _aAnd M _b, its calculating formula is:

M _a＝<y _a，Φ _a，λ _a>，M _b＝<y _b，Φ _b，λ _b>；

2), by described flexible body distributed model M _aAnd M _b, calculate average shape y respectively _aAnd y _b, eigenmatrix Φ a and Φ b and eigenvalue _aAnd λ _b, set flexible body from T _aTo T _bBe continuous variation,

2.1) calculating of the eigenwert of mid-module:

λ _t≈a ²λ _a+b ²λ _b， (25)

Here λ a and λ b are respectively the feature value vector of former and later two adjacent PDM models, promptly

λ _a＝(λ _a1，λ _a2，λ _a3，...)， (26)

λ _b＝(λ _b1，λ _b2，λ _b3，...). (27)

2.2), the estimation of the proper vector of mid-module:

The covariance matrix that inserts mid-module is approximately:

S _t≈a ²S _a+b ²S _b (28)

Wherein, covariance matrix Sa and Sb obtain by eigenmatrix Φ a and Φ b are approximate, and Φ a and Φ b obtain the known some distributed model before interpolation; Earlier training set data is carried out PCA before the setting mid-module is set up and analyze, and choose the deformation that t kind strain mode is described whole object with the minimizing number of parameters; Because Φ is a top t eigenvector in the covariance matrix, then has:

$S_a\&ap;{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T---\left(29\right)$

$S_b\&ap;{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T---\left(30\right)$

Wherein, Λ a and Λ b are the diagonal matrix that is combined to form by eigenvector λ a and λ b; With the method for svd from formula $S_t=a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Obtain inserting the eigenvector of mid-module:

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T=\mathrm{svd}(a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)---\left(31\right)$

Determine orthogonal matrix Ut by two known adjacent PDM models and correlation parameter a, b, Λ a, Λ b, Φ a and Φ b, the preceding t of described orthogonal matrix Ut is capable to be exactly eigenmatrix Φ t;

2.3), the covariance matrix of mid-module derived by following formula and draws:

$S_t=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t){(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yti}}{\mathrm{\&Delta;}}_{\mathrm{yti}}^T$

$=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}}){(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}})}^T$

$=\frac{a^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T+\frac{b^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T---\left(15\right)$

Order

$S_{\mathrm{ab}}=\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(16\right)$

Because

$S_a=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T---\left(17\right)$

$S_b=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(18\right)$

Obviously have

$S_{\mathrm{ab}}=S_{\mathrm{ba}}^T---\left(19\right);$

According to covariance matrix, matrix S _tBy S _a, S _bAnd S _AbCalculate, by following formula:

$S_{\mathrm{ab}}=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Phi;}}_a{b}_{\mathrm{ai}}{b}_{\mathrm{bi}}^T{\mathrm{\&Phi;}}_b^T---\left(32\right)$

Use singular value decomposition method, can make:

$S_{\mathrm{ab}}={\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T---\left(33\right)$

Common Φ _Ab≠ Φ _Ba, because matrix S _AbAnd unsymmetrical matrix, Φ _AbAnd Φ _BaAll comprise t vector;

When interpolation, some distributed model keeping characteristics value and proper vector, the covariance matrix of mid-module calculates by following formula:

$S_t=a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ba}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ab}}^T---\left(34\right).$

2.3), the calculating of proper vector group

When through type (31) comes the calculated characteristics vector, (31) are rewritten as:

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T={\mathrm{\&Phi;}}_a(a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)={\mathrm{\&Phi;}}_{a}L---\left(35\right)$

Wherein $L=a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="A2009101021870025H3.png,A2009101021870026H1.png"\; state="\{\{state\}\}">b</figure-callout>}}^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Be the matrix of a r * N dimension, wherein r is the major component number, and N is the sample dimension, i.e. the product of the quantity of calibration point and gauge point dimension;

If U _iBe matrix G=L Φ _aThe principal character vector, the character pair value is λ _i, (Φ then _au _i) and λ _iBe respectively matrix S=Φ _aThe proper vector of L and eigenwert.

By Gu _i=λ _iu _i,:

S(Φ _au _i)＝Φ _aL(Φ _au _i)＝Φ _aGu _i＝Φ _a(λ _iu _i)＝λ _i(Φ _au _i) (36)。

In our research project about cardiac image analysis, can be according to defining a normalized time coordinate axle cardiac cycle.For example, a heart beat cycle can be divided into six physiological stages: atrial contraction, ventricular contraction, Ve, ventricular diastole, ventricle are filled fast and diastasis period is had a rest (Fig. 2).Also there is the people further the process of Ve to be divided into PEP and to penetrate the blood later stage.The left side heart is similar but incomplete same to the motion process of the right heart.Main difference is, the systolic pressure of left ventricle and artery is than the big three to four-fold of the right heart, and also there is remarkable difference the time aspect of left and right sides heart pump blood process, as document 20: willow, Wang Tianfu, Lin Jiangli etc., the axis coupling and the image interpolation of rotation sweep ultrasound cardiac images, biomedical engineering magazine, 2004,21 (1): 28-33,41; Document 21: Chen Qiang, Zhou Zeming, Wang Pingan, Xia Deshen, the cutting apart automatically of Tagged Left Ventricle MR Images, Chinese image graphics journal: A collects, and 2004,9 (6): 666-673; Document 22: Luo Yulan, Zheng Changqiong etc. are cut apart and algorithm research the Sichuan Teachers University journal based on the cavity of the B ultrasonic cardiac image of Snake model: natural science edition, 2002,25 (1): 66-69.Because the spatial shape of whole heart altered a great deal in a cardiac cycle, and single model is difficult to the dynamic heart movement process of accurate description, so we need set up the cardiac shape that a group model is described different phase aroused in interest.

In the image studies of gated-SPECT, can carry out the time standardization with reference to ECG signal.We wish by can obtaining six whole models after heart shape is added up, and these models can be used for describing exactly the particular state of heart, as heart diastasis and heart contraction latter stage.Fig. 3 has marked the position that each stage model occurs in standard cardioelectric figure curve.We have passed through correction to the force value amplitude among the figure, make that the force value at crest place is normal human's a systolic pressure (120mm mercury column), so also represent a typical left ventricular pressure tracing among the figure.

If in normal cardiac cycle 6 PDM models are arranged, then the Ti (representing the moment at a certain phase place place) in the formula (6) can be defined as:

T is the time value that obtains the gated-SPECT image in the formula, H _cBe heart rate inverse,

Expression is not more than the maximum integer of certain decimal.

Yet in real example, regular meeting produces the phase place (as there being 8 among Fig. 4) of different numbers, and this is because the sampling of cardiac image has fixing frequency usually, as every 100ms (Fig. 4).So just be necessary to obtain the PDM model of sampling instant, in the hope of obtaining cutting apart of better three-dimensional cardiac shape by interpolation.

Flexible body point distributed model interpolation algorithm adopts Computer Simulation and true heart data to carry out a large amount of experiments respectively in author's research process.In emulation, we can suppose the model of one group of nine dimension (or higher), yi=[x _1i, x _2i..., x _9i] ^TThe data set of two PDM model M 1 and M2 all produces at random.Each is organized data and all contains 100 samples, as the training set that generates the some distributed model, for example

${\stackrel{\&OverBar;}{y}}_1=\frac{1}{100}\underset{i=1}{\overset{100}{\mathrm{\&Sigma;}}}{y}_{1i},$ ${\mathrm{<figure-callout\; id="1"\; label="S"\; filenames="A2009101021870026H2.png"\; state="\{\{state\}\}">S</figure-callout>}}_1=\frac{1}{99}\underset{i=1}{\overset{100}{\mathrm{\&Sigma;}}}(y_{1i}-{\stackrel{\&OverBar;}{y}}_1){(y_{1i}-{\stackrel{\&OverBar;}{y}}_1)}^T$

Insert model for generating one, we are from S ₁Calculate eigenwert and the proper vector of M1, to S ₂Too.Suppose to insert model about distance be a=0.3, b=0.7.In order to compare, we also calculate their alternation matrix S _AbWith corresponding proper vector group.Experimental result shows that the method described in this can well calculate the various parameters of inserting model.The eigenwert error that is calculated by formula (31) is less than 1.83%, and the eigenwert error that is calculated by formula (35) is less than 0.05%.The C++ program of this emulation experiment is finished by the place development in laboratory.

Test with real cardiac data.In the great amount of images data that obtain by perfusion SPECT imaging method, we have set up left ventricle in advance at difference five PDM models constantly.Each model obtains (image is mainly provided by Barcelona hospital) by the training of the true picture of 246 groups of cases, comprises that the VTK of an average shape expresses a stack features value vector and a proper vector.The insertion computing of the processing procedure of all medical images and model is also by all C++ program realizations (seminar has developed about 600MB program code altogether).During experiment, we insert four mid-modules equably between per two adjacent PDM, and it is very satisfied that the experimental result that obtains is made us.Fig. 5 shows that the PDM model that generates by interpolation can change the flexible model of left ventricle in time smoothly.

The point distributed model is as a kind of statistical models of describing flexible object, and it is very effective being proved to be on a lot the application, especially in the medical image analysis field.In order to analyze the dynamic change situation of flexible body, the present invention initiates the interpolation algorithm of having studied the some distributed model.Result of study shows, can be generated the mid-module of any time by the parameters such as average shape, eigenwert and proper vector of PDM.In fact, the average shape and the deformation of insertion model are exactly the linear combination of its adjacent front and back model respective items, i.e. y _t=ay _a+ by _b, Δ y _Ti=a Δ _Yai+ b Δ _YbiIts eigenwert then can adopt formula λ _t≈ a ²λ _a+ b ²λ _bCalculate.At last, for obtaining covariance matrix and the proper vector of inserting model, the invention discloses alternation matrix form and corresponding calculation method thereof.

Claims (3)

1, a kind of eigenwert of the flexible body point distributed model based on interpolation algorithm and the computing method of proper vector, it is characterized in that: described computing method may further comprise the steps:

1), is written into two adjacent moment T _aAnd T _bFlexible body distributed model M _aAnd M _b, its calculating formula is:

M _a＝<y _a，Φ _a，λ _a>，M _b＝<y _b，Φ _b，λ _b>；

2), described flexible body distributed model M _aAnd M _b, comprise and calculate average shape y _aAnd y _b, eigenmatrix Φ _aAnd Φ _b, and eigenvalue _aAnd λ _b, set flexible body from T _aTo T _bBe continuous variation, then

2.1) calculating of the eigenwert of mid-module:

λ _t≈a ²λ _a+b ²λ _b， (25)

Here λ _aAnd λ _bBe respectively the feature value vector of former and later two adjacent PDM models, promptly

λ _a＝(λ _a1，λ _a2，λ _a3，...)， (26)

λ _b＝(λ _b1，λ _b2，λ _b3，...). (27)

2.2), the estimation of the proper vector of mid-module:

The covariance matrix that inserts mid-module is approximately:

S _t≈a ²S _a+b ²S _b (28)

Wherein, covariance matrix Sa and Sb obtain by eigenmatrix Φ a and Φ b are approximate, and Φ a and Φ b obtain the known some distributed model before interpolation; Earlier training set data is carried out PCA before the setting mid-module is set up and analyze, and choose the deformation that t kind strain mode is described whole object with the minimizing number of parameters; Because Φ is a top t eigenvector in the covariance matrix, then has:

$S_a\&ap;{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T---\left(29\right)$

$S_b\&ap;{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T---\left(30\right)$

Wherein, Λ _aAnd Λ _bBe by eigenvector λ _aAnd λ _bThe diagonal matrix that is combined to form; With the method for svd from formula $S_t=a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+b^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Obtain inserting the eigenvector of mid-module:

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T=\mathrm{svd}(a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+b^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)---\left(31\right)$

By two known adjacent PDM models and correlation parameter a, b, Λ _a, Λ _b, Φ a and Φ b determine orthogonal matrix Ut, the preceding t of described orthogonal matrix Ut is capable to be exactly eigenmatrix Φ t.

2, as claimed in claim 1 based on the time become the eigenwert of flexible body point distributed model of interpolation algorithm and the computing method of proper vector, it is characterized in that: also comprise described step 2):

2.3), the covariance matrix of mid-module derived by following formula and draws:

$S_t=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t){(y_{\mathrm{ti}}-{\stackrel{\&OverBar;}{y}}_t)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yti}}{\mathrm{\&Delta;}}_{\mathrm{yti}}^T$

$=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}}){(a{\mathrm{\&Delta;}}_{\mathrm{yai}}+b{\mathrm{\&Delta;}}_{\mathrm{ybi}})}^T---\left(15\right)$

$=\frac{a^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T+\frac{b^2}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T+\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T$

Order

$S_{\mathrm{ab}}=\frac{\mathrm{ab}}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(16\right)$

Because

$S_a=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{yai}}{\mathrm{\&Delta;}}_{\mathrm{yai}}^T---\left(17\right)$

$S_b=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}{\mathrm{\&Delta;}}_{\mathrm{ybi}}^T---\left(18\right)$

Obviously have

$S_{\mathrm{ab}}=S_{\mathrm{ba}}^T---\left(19\right);$

According to covariance matrix, matrix S _tBy S _a, S _bAnd S _AbCalculate, by following formula:

$S_{\mathrm{ab}}=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}(y_{\mathrm{ai}}-{\stackrel{\&OverBar;}{y}}_a){(y_{\mathrm{bi}}-{\stackrel{\&OverBar;}{y}}_b)}^T=\frac{1}{s-1}\underset{i=1}{\overset{s}{\mathrm{\&Sigma;}}}{\mathrm{\&Phi;}}_a{b}_{\mathrm{ai}}{b}_{\mathrm{bi}}^T{\mathrm{\&Phi;}}_b^T---\left(32\right)$

Use singular value decomposition method, can make:

$S_{\mathrm{ab}}={\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T---\left(33\right)$

Common Φ _Ab≠ Φ _Ba, because matrix S _AbAnd unsymmetrical matrix, Φ _AbAnd Φ _BaAll comprise t vector;

When interpolation calculation, some distributed model keeping characteristics value and proper vector, the covariance matrix of mid-module calculates by following formula:

$S_t=a^2{\mathrm{\&Phi;}}_a{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+b^2{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ab}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ba}}^T+\mathrm{ab}{\mathrm{\&Phi;}}_{\mathrm{ba}}{\mathrm{\&Lambda;}}_{\mathrm{ab}}{\mathrm{\&Phi;}}_{\mathrm{ab}}^T---\left(34\right).$

3, the eigenwert of the flexible body point distributed model based on interpolation algorithm as claimed in claim 1 or 2 and the computing method of proper vector is characterized in that: also comprise described step 2):

2.3), the calculating of proper vector group

When through type (31) comes the calculated characteristics vector, (31) are rewritten as:

$U_t{\mathrm{\&Lambda;}}_t{U}_t^T={\mathrm{\&Phi;}}_a(a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+b^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T)={\mathrm{\&Phi;}}_{a}L---\left(35\right)$

Wherein $L=a^2{\mathrm{\&Lambda;}}_a{\mathrm{\&Phi;}}_a^T+b^2{\mathrm{\&Phi;}}_a^T{\mathrm{\&Phi;}}_b{\mathrm{\&Lambda;}}_b{\mathrm{\&Phi;}}_b^T$ Be the matrix of a r * N dimension, wherein r is the major component number, and N is the sample dimension, i.e. the product of the quantity of calibration point and gauge point dimension;

If U _iBe matrix G=L Φ _aThe principal character vector, the character pair value is λ _i, (Φ then _au _i) and λ _iBe respectively matrix S=Φ _aThe proper vector of L and eigenwert.

By Gu _i=λ _iu _i,:

S(Φ _au _i)＝Φ _aL(Φ _au _i)＝Φ _aGu _i＝Φ _a(λ _iu _i)＝λ _i(Φ _au _i) (36)。

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