Fingerprint warping using ridge curve correspondences
Fingerprint Warping Using
Ridge Curve Correspondences
Arun Ross,Member,IEEE,Sarat C.Dass,and Anil K.Jain,Fellow,IEEE Abstract—The performance of a fingerprint matching system is affected by the nonlinear deformation introduced in the fingerprint impression during image acquisition.This nonlinear deformation causes fingerprint features such as minutiae points and ridge curves to be distorted in a complex manner.A technique is presented to estimate the nonlinear distortion in fingerprint pairs based on ridge curve correspondences.The nonlinear distortion,represented using the thin-plate spline(TPS)function,aids in the estimation of an“average”
deformation model for a specific finger when several impressions of that finger are available.The estimated average deformation is then utilized to distort the template fingerprint prior to matching it with an input fingerprint.The proposed deformation model based on ridge curves leads to a better alignment of two fingerprint images compared to a deformation model based on minutiae patterns.An index of deformation is proposed for selecting the“optimal”deformation model arising from multiple impressions associated with a finger.Results based on experimental data consisting of1,600fingerprints corresponding to50different fingers collected over a period of two weeks show that incorporating the proposed deformation model results in an improvement in the matching performance.
Index Terms—Fingerprints,nonlinear deformation,ridge curves,thin plate spline,index of deformation,minutiae pattern,template selection.
?
1I NTRODUCTION
T HE uniqueness of a fingerprint is dictated by the topographic relief of its ridge structure and the presence of ridge anomalies,termed minutiae points.The problem of automatic fingerprint matching involves determining the degree of similarity between two fingerprint impressions by comparing their ridge structure and the spatial distribution of the minutiae points[2],[3],[4],[5].However,the image acquisition process introduces nonlinear distortions in the ridge structure due to the nonuniform finger pressure applied by the subject and the elastic nature of the skin.The effects of these nonlinear distortions must be addressed when matching two fingerprint images.Models based on affine transformations invariably lead to unsatisfactory matching results since the distortions are basically elastic (nonrigid)in nature(Fig.1).
To deal with the problem of nonlinear distortion in fingerprint images,four types of approaches have been discussed in the literature.The first approach accounts for distortion in the image acquisition stage by capturing the least distorted print from the user while rejecting the others.Ratha and Bolle[6]describe a system which does not accept a fingerprint image if the user applies excessive force on the sensor.The system operates by measuring the forces and torques applied on the sensor.Dorai et al.[7]observe a video sequence of the fingertip as it interacts with the sensor and measure the distortion across successive frames.When excessive distortion is observed,the system requests the user to provide another fingerprint.These systems require specialized hardware and the ability to perform extensive computations in real-time.As a result,they do not offer a practical solution to fingerprint deformation in real-time embedded fingerprint applications.
In the second approach,the distortion is estimated during the matching stage.Thebaud[8]uses a gradient descent technique to compute local warps when comparing two fingerprints.The fingerprint correlation score is used as the objective function.Besides being time consuming,this technique potentially results in a higher False Accept Rate (FAR)since it performs local warping to force a match between the two images.Kova′cs-Vajna[4]uses minutiae triplets to compare two minutiae sets.By not using the entire minutiae pattern at once,the cumulative effect of distortion is avoided.Bazen and Gerez[9]use a thin-plate spline(TPS)model to account for nonlinear distortions when comparing two minutiae sets.
In the third approach,the distortion is removed before the matching stage.Senior and Bolle[10]have developed a model which assumes that ridges in a fingerprint are constantly spaced and that deviations from this model indicate the presence of elastic distortions.They apply local warps in regions exhibiting such deviations so that local ridge distances nearly equal the average interridge spacing.Their experimental results show a significant improvement in genuine matching scores(i.e.,the matching score when comparing two impressions of the same finger),as indicated by the t-statistic.However,their assumption that interridge spacing in a fingerprint is a constant is not always valid. Watson et al.[11]construct distortion tolerant filters for each (template)fingerprint.These filters when applied to the image before matching are shown to result in improved system performance.
The fourth approach is more suited for introducing distortions in synthetic fingerprints.Cappelli et al.[12]have
. A.Ross is with the Lane Department of Computer Science and Electrical
Engineering,West Virginia University,PO Box6109,Morgantown,WV
26506.E-mail:arun.ross@https://www.wendangku.net/doc/ad11741597.html,.
.S.C.Dass is with the Department of Statistics,Michigan State University,
A-439Wells Hall,East Lansing,MI48824.E-mail:sdass@https://www.wendangku.net/doc/ad11741597.html,.
. A.K.Jain is with the Department of Computer Science and Engineering,
Michigan State University,3115Engineering Building,East Lansing,MI
48824.E-mail:jain@https://www.wendangku.net/doc/ad11741597.html,.
Manuscript received4Sept.2003;revised6Apr.2005;accepted7Apr.2005;
published online11Nov.2005.
Recommended for acceptance by K.Yamamoto.
For information on obtaining reprints of this article,please send e-mail to:
tpami@https://www.wendangku.net/doc/ad11741597.html,,and reference IEEECS Log Number TPAMI-0260-0903.
0162-8828/06/$20.00?2006IEEE Published by the IEEE Computer Society
attempted to model the distortions that could occur in a fingerprint image by considering three concentric regions in a fingerprint;the inner and outer regions are assumed to have no distortions although ridges in the outer region can be translated and rotated with respect to the ridges in the inner region;the region in between is assumed to undergo nonlinear distortions in order to accommodate the transi-tion of ridges from the inner to the outer region.The authors,however,do not use this model to perform fingerprint matching.Rather,they use it to synthesize multiple impressions of the same finger[13].
Most current techniques deal with the problem of non-linear distortion on a case by case basis,i.e.,for every pair of fingerprint impressions(or for every fingerprint impression), a distortion removal technique is applied and a matching score generated.No attempt has been made thus far to develop a finger-specific deformation model that can be computed offline and later used for matching.The advantage of such a scheme is that,once a finger-specific model has been computed and stored along with the template,recomputation of the model is not necessary during matching.In this paper, we propose a technique for computing the average deforma-tion model of a fingerprint impression by using the thin plate spline(TPS)warping model.It is assumed that multiple impressions of a user’s fingerprint are available during the training phase.The model is expected to capture the intraclass variability due to nonlinear deformations in a fingerprint impression.The relative distortion between two impressions is estimated based on their ridge curve corre-spondences.The average deformation model associated with an arbitrary fingerprint impression(called the baseline impression)is an indication of its average distortion with respect to other impressions of the same finger.For a single finger,an optimal baseline impression with the most consistent distortions(i.e.,distortions that deviate the least from the average)is selected based on its index of deforma-tion.We demonstrate that predistorting the baseline impres-sion using the average deformation model can improve matching performance.
Earlier work in modeling the nonlinear distortion in fingerprint images[14],[9],[15]used only the spatial distribution of the minutiae points(Fig.2).In this paper,the relative distortions are estimated based on ridge curve correspondence(Fig.3).Modeling the distortion using ridge curve correspondences offers several advantages over minutiae correspondences,resulting in improved matching performance.Unlike minutiae points,which can be sparsely distributed in certain regions of a fingerprint image,ridge curves are present all over the image domain,thereby permitting a more reliable estimate of the distortion.The spatial continuity of ridge curves enables sampling of a large number of points on the ridges for establishing correspon-dences,including points in the vicinity of undetected minutiae points.Also,in some poor quality images,minutiae information cannot be reliably extracted and,thus,cannot be used to construct a fingerprint distortion model.For these reasons,ridge curve-based warping techniques are expected to provide a robust and reliable estimate of the distortion in fingerprint impressions.
The rest of the paper is organized as follows:Section2lists a few warping(deformation)models commonly used in the literature and presents the warping model based on thin-plate splines(TPS)that has been used in this work,Section3 describes the average deformation model that we propose, Section3.1defines the index of deformation that is utilized to select the optimal deformation model from a given set of models,Section4describes the experiments conducted and the results obtained,and Section5summarizes the paper and presents future directions for research.
2T HE F INGERPRINT W ARPING M ODEL
Warping methods can be used to obtain global deformation models for image registration.Applications of warping techniques abound in the statistical,medical imaging,and computer vision literature.Examples include warping by elastic deformations[16],[17],optical or fluid flow[18],[19], [20],diffusion processes[21],Bayesian prior distributions [22],[23],and thin-plate splines(TPS)[24],[25],[26].Only recently have warping techniques based on deformation modelsbeenusedtodescribedistortions infingerprintimages for thepurposeofmatching[14],[9].BazenandGerez[9]show that the use of nonlinear deformation models,as opposed to
Fig.1.Alignment of two impressions of the same finger using affine transformation.(a)and(b)are the gray-scale images,(c)and(d)are the thinned(skeletonized)images of(a)and(b),respectively,and(e)shows the alignment based on the thinned images.Ridge lines do not align in(e).
simple rigid transformations,can result in significant im-provement in fingerprint matching performance.
In this work,a landmark-based registration scheme is employed.Specifically,we use a skeletonized version of a fingerprint image,known as the thinned image,to extract ridge curve information (see Fig.1)that is sampled at regular intervals in order to derive landmark points.Estimating the deformation model based on ridge curves offers several advantages over minutiae point patterns.First,ridge lines are distributed over the entire fingerprint image and,thus,a more reliable deformation model can be obtained.Second,the likelihood of incorrectly associating two ridge curves is much less than incorrectly associating two minutiae points due to the richer intrinsic information available in curves compared to points.Consequently,the deformation model based on ridge curves yields better matching performance compared to minutiae points.
When multiple impressions of a finger are available,the relative distortion between one pair can be significantly different from another pair (Fig.4).Furthermore,even in a
single impression,the deformation of the ridges can vary from region to region.Thus,we address the following two problems:1)Obtain a deformation model based on ridge curve correspondences that can be incorporated in the matching stage and 2)given multiple deformation models for a finger (each model corresponds to one impression of the finger),select the optimal model that has the most consistent distortion effects as measured from a baseline impression.Let I 0ex;y Tand I 1ex;y Tdenote two fingerprint impres-sions,where ex;y T2S for a domain S &R 2.Our conven-tion is to refer to I 0and I 1as the template and query (or input)images,respectively.A warping of I 0to I 1is defined as the function F :S !S such that
F eI 0T?I 1:
e1T
We register the two impressions I 0and I 1by matching corresponding ridge curves.Thus,in (1),the warping function,F :S !S ,registers two sets of ridge curves derived from I 0and I 1.Let u k u k et T?eu k 1et T;u k 2et TTT
ROSS ET
AL.:FINGERPRINT WARPING USING RIDGE CURVE CORRESPONDENCES 21
Fig.2.An
example of minutiae correspondences between two impressions of a finger.
Fig.3.An example of ridge curve correspondences between two impressions of a finger.
denote a discretized ridge curve in I0for k?1;2;...;n,and let v k v ketT?ev k1etT;v k2etTTT,k?1;2;...;n,denote the corresponding discretized ridge curves in I1.Here,t is the index of discrete points on a single ridge curve and n is the total number of corresponding curves.The two sets of ridge curves,one set in I0and the other in I1,with known correspondences is denoted by the paireU;VTwhere U?eu1;u2;...;u nTT and V?ev1;v2;...;v nTT.We assume that each correspondence pair is aligned as close as possible using rigid transformation prior to nonlinear warping.This can be achieved using the Procrustes analysis([27],[28]) after pairs of corresponding points are obtained using the methodology outlined in Section2.2.For n pairs of ridge curve correspondences,a warping function,F,that warps U to V,subject to perfect alignment,is given by the conditions
Feu kT?v ke2Tfor k?1;2;...;n.2.1Establishing Ridge Curve Correspondences Given a pair of gray-scale fingerprint images,I0and I1,we obtain their thinned versions,R0and R1,using the algorithm described in[29].A thinned image is a binary image(see Figs.1c and1d)with gray-scale values of0 (indicating ridges)and255(indicating valleys).Each thinned image can be thought of as a collection of ridge curves.In order to develop ridge curve correspondences, we proceed as follows:
1.Minutiae points are extracted from I0and I1using the
algorithm described in[29].Let M0?em0;1;m0;2;...;
m0;K
Tand M1?em1;1;m1;2;...;m1;K
1
Tdenote the two minutiae sets of cardinalities K0and K1,respectively.
Here,each minutiae point m i;j is characterized by its
location in the image,the orientation of the associated
ridge,and the gray-scale intensity of pixels in its
vicinity.
2.Minutiae correspondences between M0and M1is
obtained using the elastic string matching technique
described in[29].The output of the matcher is a
similarity score in the range[0,1000]and a set of
correspondences of the form C?fem0;a
j
;m1;b
j
T: j?1;2;...;K g,where K min f K0;K1g,and the a j s
(b j s)are all distinct.Fig.2shows an example of the
minutiae point pattern correspondence for two im-
pressions of a finger.
3.Once the correspondence between M0and M1is
established,the ridge curves associated with these
minutiae points are extracted from R0and R1using
a simple ridge tracing technique.A minutiae point
that is a ridge ending has one ridge curve associated
with it,while a ridge bifurcation has three associated
ridges.In the case of a ridge ending,the ridge curve
correspondence between the two images can be
easily established since each minutiae point has only
one associated ridge curve.However,in the case of a
ridge bifurcation,the problem of establishing ridge
curve correspondences is nontrivial due to the
presence of multiple ridge curves for each minutiae
point;each of the three component ridge curves of
one minutiae point can potentially match with any
component of the other impression.
To resolve this ambiguity,each ridge curve corre-sponding to the minutiae point in I0(I1)is represented
as a directional vector r j(s j),j?1;2;3,based on two
points on the ridge curve:the minutiae point and the
d th point(d?20)on th
e ridge from the minutiae(see
Fig.5).We define j;k( j;k)to be the angle that r j(s j)
makes with r k(s k),for k?j.We find the vector r j(s j)
for which theangles f j;k;k?j gef j;k;k?j gTareboth
obtuse.This establishes the first ridge curve corre-
spondence,say,r1$s1,without loss of generality.We
then compute the cross products c r?r2?r3and
c s?s2?s3.We assign the correspondence r2$s2
and r3$s3if c r and c s are of the same sign and r2$s3
and r3$s2,otherwise.Fig.3shows an example of
ridge curve correspondence for a pair of impressions
of a finger.
22IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL.28,NO.1,JANUARY
2006
Fig.4.Nonlinear deformations(with rotation and translation parameters removed)associated with two pairings involving the same template: (a)Template image.(b)and(c)Query images.(d)and(e)Nonlinear deformation of(a)into(b)and(c),respectively.
2.2Sampling Ridge Curves
Having determined the corresponding ridge curves,we next establish a correspondence between points on these curves by sampling every q th point (q ?20)on each of the ridge curves.For the correspondence pair eU;V T,we have u k u k et Tand v k v k et Tfor k ?1;2;...;n .The sampling of the k th corresponding ridge curves,say at points t 1;t 2;...;t g k ,yields g k pairings of the form eu k et j T;v k et j TTfor j ?1;2;...;g k .Thus,we have a total
of N ?P
n k ?1g k points in establishing the correspon-dence.We denote this set of corresponding points by
U ?eu ?1;u ?2;...;u ?N TT and V ?ev ?1;v ?2;...;v ?N TT
.We use TPS to estimate the nonlinear deformation F based on these points.TPS represents a natural parametric general-ization from rigid to mild nonrigid deformations.The deformation model for TPS is given in terms of the warping function F eu T,with
F eu T?c tA áu tW T s eu T;
e3T
where u 2S ,c is a 2?1translation vector,A is a 2?2affine matrix,W is a N ?2coefficient matrix,s eu T?e eu àu ?1T;
eu àu ?2T;...; eu àu ?
N TTT ,where
eu T?
jj u jj 2log ejj u jjT;jj u jj >0
0;jj u jj ?0:
e4TIn (3),there are 6and 2N parameters corresponding to the
rigid and nonrigid parts of the deformation model,respectively,resulting in a total of 2N t6parameters to be estimated.The restrictions
F eu ?j T?v ?
j ;
e5T
j ?1;2;...;N ,provide 2N constraints.For the parameters to be uniquely estimated,we further assume that W satisfies the
two conditions:1)1T N W ?0and 2)U T
s W ?0,where 1N is the vector of ones of length N .Thus,the parameters of the TPS model can be obtained from the matrix equation
H 1N U 1T N 00U
T
00266664377775W c T A T 2435?V 00
2435;e6Twhere H is the N ?N matrix with entries h ij ? eu ?i àu ?
j T.The matrix in (6)gives rise to a TPS model that minimizes the bending energy subject to the perfect alignment con-straints in (5).A more robust TPS model can be obtained by relaxing the constraints in (5)and instead determining the function F which minimizes the expression
X N j ?1
v ?j
à
F eu ?j T
T
ev ?j àF eu ?
j TTt J eF T;
e7T
where
J eF T?X
2j ?1
Z S @2F j ex;y T@x 2
2t2@2F j ex;y T
@x@y 2(t
@2F j ex;y T
@y 2
2)dx dy e8T
represents the bending energy associated with F ?eF 1;F 2TT ,
F j is the j th component of F ,and >0.The case ?0gives rise to the TPS model described by (6).For general >0,the parameters of the resulting TPS model can be obtained using (6)with H replaced by H t I N ,where I N is the N ?N Identity matrix.
3A VERAGE D EFORMATION M ODEL
Suppose we have L impressions of a finger,T 1;T 2;...;T L .
Each impression,T i ,can be paired with the remaining impressions,T j ;j ?i ,to create L à1pairs of the form eT i ;T j T.For the pair eT i ;T j T,we obtain a nonlinear transformation F ij by employing the technique described in Section 2.Note that F ij transforms every pixel in the template fingerprint,T i ,to a new location.Thus,we can compute the average deformation of each pixel u in T i as
"F
i eu T?1X
j ?i
F ij eu T:
e9T
There will be L average deformation models correspond-ing to the L impressions of the finger.The average deformation is the typical deformation that arises when we compare one fingerprint impression of a finger (the baseline impression)with other impressions of the same finger.Fig.6shows that changing the baseline impression for the finger will result in a different average deformation model for that finger (the èvalues are as discussed in Section 3.1).Fig.7shows the average deformation for three different fingers;it can be clearly seen that the average warping functions are different for the three fingers,indicating that the fingerprint deformation is finger-specific.
3.1The èIndex of Deformation
We consider the following two questions in this section:1.
Which of the L average deformation models can be considered to be the optimal model for this finger?
ROSS
ET AL.:FINGERPRINT WARPING USING RIDGE CURVE CORRESPONDENCES 23
Fig.5.Vector representation of ridge bifurcation used to establish correspondences between component ridge curves.O marks the bifurcation points in correspondence and X marks the points on the ridges at Euclidean distance d from O .
2.Will the optimal model,when incorporated in the
matching stage,result in improved performance
compared to the suboptimal models?
In order to address these questions,we first define the pixel-wise covariance matrix associated with the i th average deformation,"F i,as follows:
D"F
i euT?
1
Là1
X
j?i
eF ijeuTà"F ieuTTáeF ijeuTà"F ieuTTT;e10T
where F ij is the deformation function that warps T i to T j. The covariance matrix,defined at each pixel u,is a measure of the variability associated with the estimated deformation functions.Two choices of pixel-wise measures of variability
are given by1)the determinant, eD"F
i
euTT?j D"F
i
euTj,and
2)the trace, eD"F
i
euTT?treD"F
i
euTT.Pixels with large(small) values of indicate high(low)variability in the deforma-tions F ij.We propose using the values of to determine the optimal model for a given finger.We define the i th index of deformation,èi,as
èi?
1X j S j
u?1
eD"F
i
euT
T;e11T
24IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL.28,NO.1,JANUARY
2006 Fig.6a.The average deformation model(shown as deformations on a reference grid)corresponding to six templates of a finger sorted in increasing è-values.(a)is chosen to be the optimal template since it has the leastè-value.(a)è?15:54.(b)è?17:97.(c)è?48:79.
where eDT?treDTand j S j is the number of pixels in the domain S.Subsequently,we choose T i?as the template with the smallest variability in deformation if i??arg min ièi.In effect,we choose that template T i that minimizes the average variation across pixels measured in terms ofèi. Low(high)values of the index of deformation indicate that the warping functions are similar(dissimilar)to each other.
3.2Eliminating Erroneous Correspondences
For each baseline fingerprint impression,it is important to determine the set of minutiae points that are correctly paired to form a correspondence.The average deformation model is sensitive to the accuracy of the ridge curve correspondence,which,in turn,depends on the minutiae correspondence.It is, therefore,necessary to check the correctness of the minutiae correspondences prior to obtaining the ridge curve corre-spondences.Fig.8a presents an example of two incorrect minutiae correspondences which result in incorrect ridge curve correspondences(Fig.8b).These erroneous correspon-dences have to be eliminated prior to computing the average deformation model;failure to exclude such minutiae points results in a warping model exhibiting spurious distortions.
For the given baseline fingerprint impression,minu-tiae points that have a correspondence with at least‘(‘?5)of the remaining Là1impressions are extracted. We denote the set of all such minutiae points by
ROSS ET
AL.:FINGERPRINT WARPING USING RIDGE CURVE CORRESPONDENCES25 Fig.6b.(continued).(d)è?83:12.(e)è?94:34.(f)è?232:53.
M?f m i;i?1;2;...;K g.Each m i has a corresponding minutiae point in at least‘of the Là1impressions.We denote these pairings byem i;p1T;em i;p2T;...;em i;p‘
i
T, where‘i is the total number of pairings.We next develop a measure of reliability for minutiae point m i as follows:
1.Sampled ridge point correspondences are obtained
for eachem i;p jT,j?1;2;...;n i,based on which a TPS
deformation model,Fem
i ;p jT
is computed.The average
deformation model for the minutiae point m i is given by
"F
m i
euT?
1
‘i
X‘i
j?1
Fem
i
;p jT
euT:
Here,the average deformation model is obtained in
a10?10square region,say S m
i
,centered at m i. 2.Let
D"F
m i
euT?
1
‘i
X‘i
j?1
eFem
i
;p jT
euTà"F m
i
euTT
áeFem
i
;p jT
euTà"F m
i
euTTT
e12T
26IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL.28,NO.1,JANUARY
2006
Fig.7.The average deformation model(shown as deformations on a reference grid)of three different fingers.(a)è?46:68.(b)è?37:59.
(c)è?85:18.
denote the site-wise variability measure of the defor-
mations Fem
i ;p jT
around"F m
i
.The average variability is
measured by
R m
i ?
1
j S m
i
j
X j S m i j
u?1
traceeD"F
m i
euTT
with small values of R m
i
indicating better reliability.
Correspondences pertaining to those minutiae
points with R m
i
values lower than the p th percentile
(e.g.,p?60)are used to develop the average
deformation model for the template fingerprint.
For the incorrect minutiae correspondences in Fig.8,the value of R for the top minutiae point was93.2(the sixtieth percentile value of R was55:5for this template),while the lower minutiae point occurred in less than five correspond-ing pairs and was,therefore,eliminated.Fig.9a shows the average deformation model that results for this template when all correspondences are used(i.e.,p?100);Fig.9b gives the deformation model for p?60.
4E XPERIMENTAL R ESULTS
In order to apply the TPS model to reliably estimate fingerprint deformation,we need to have several impressions of the same https://www.wendangku.net/doc/ad11741597.html,rge number of impressions of a finger are not available in standard fingerprint databases(e.g.,FVC2002 [30]).Therefore,fingerprint images of50fingers(correspond-ing to five subjects)were acquired using the Identix sensor (256?255,380dpi)over a period of two weeks in our lab.The subjects did not deliberately distort their fingerprints while interacting with the sensor.There were32impressions corresponding to every finger,resulting in a total of1,600 impressions.One half of the impressions(L?16for each finger,resulting in800impressions)were used as templates to compute the average deformation model for each finger, while the remaining800impressions were used as query images for testing.For each template image,T,the minutiae set,M T,and the thinned image,R T,were extracted(Fig.10). The average deformation model of T,"F T,was obtained based on pairings with the remaining15impressions of the same finger((7)with ?0:1).The minutiae set M T was trans-formed to the deformed set,MD T "F TeM TTusing"F T.A total of800sets(50?16)of deformed minutiae points were thus obtained.In order to test the matching performance of the deformed minutiae sets and the utility of the index of deformation,è,the following two experiments were con-ducted.In both these experiments,the minutiae matcher described in[29]was used to generate the matching (similarity)score.
In the first experiment,the matching performance using the average deformation model was evaluated.Every template image,T,was compared with every query image, Q,and two types of matching scores were generated for each comparison:the matching score obtained by matching 1)M T with M Q and2)MD T with M Q.The Receiver Operating Characteristic(ROC)curve plotting the genuine accept rate(GAR)against the false accept rate(FAR)at various matching thresholds is presented in Fig.11.An
ROSS
ET AL.:FINGERPRINT WARPING USING RIDGE CURVE CORRESPONDENCES27 Fig.8.(a)Examples of incorrect minutiae correspondences.(b)These result in erroneous ridge curve correspondences.
overall improvement is observed when the average defor-mation model is used to distort M T prior to matching.
In the second experiment,the advantage of using the index of deformation is demonstrated.Theè-index of deformation(with eDT?treDT)of every template image is used to rank the templates according to their variability in the distortion.The template images can now be split into two sets:1)impressions with the leastèvalues for every finger(theè-optimal templates)and2)the remaining impressions for every finger(theè-suboptimal templates). We repeated the matching procedure outlined above using these two template sets.The resulting ROC curve is shown in Fig.12.From the figure,it is clear that usingè-optimal templates results in better performance compared to using è-suboptimal templates.Further,theè-suboptimal tem-plates still yield better performance compared to the nondistorted templates,thus demonstrating the importance of the average deformable model.
The registration between the query and the template minutiae sets is significantly improved when the average deformation model based on ridge curves is applied prior to the rigid transformation.The improved registration allows our minutiae matcher to specify stricter bounding boxes when the query and deformed template minutiae points are compared without compromising the genuine matching scores.The added advantage of specifying stricter bounding boxes is that the number of false accepts is highly reduced resulting in a better ROC curve.
5S UMMARY AND F UTURE W ORK
In this paper,we have developed a deformation model for estimating the distortion effects in fingerprint impressions based on ridge curve correspondence.The proposed model was observed to result in a better performance compared to a model based on minutiae pattern correspondence.Our warping model samples the ridge curve and uses thin-plate splines for estimating the nonlinear deformation.The average deformation model of a finger is a compact description of the intraclass variability due to nonlinear distortions.We have also proposed an index of deformation,è,for selecting the optimal average deformation model corresponding to a finger by minimizing distortion variability.It was shown that theè-optimal deformation models result in superior matching performance compared toè-suboptimal models.
The subjects did not consciously distort their fingerprints when interacting with the sensor and,hence,one cannot predict the nature of the distortions present in the acquired images before hand.The significance of the proposed technique is its ability to compare multiple impressions of
28IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL.28,NO.1,JANUARY
2006
Fig.9.Effect of eliminating unreliable minutiae correspondences on the average deformation model.(a)Template fingerprint,(b)average deformation model with p?100,è?102:16,and(c)average deformation model with p?60,è?67:78.
a finger with a baseline impression,thereby determining the nonlinear average distortion automatically.However,the accuracy of the technique is largely defined by the reliability of minutiae point correspondences generated by the algorithm.Therefore,excessive deformations may result in erroneous minutiae (and ridge)correspondences con-founding the average deformation model.Furthermore,the proposed model assumes that the elastic nature of the skin can be approximated using thin-plate splines.It may be instructive to use alternate models based on the Navier-Stokes equation in order to describe the deformation [31].Future work includes developing an incremental ap-proach to updating the average deformation model,i.e.,updating the current average deformation model of a finger by using information presented by newly acquired finger-print impressions.We have used a simple pixel-wise averaging measure to compute the average deformation
model in this paper.This measure is sensitive to extreme deformations born out by outliers;thus,we seek more robust measures of describing the finger specific average deforma-tion model.We are also working on developing ridge curve correspondences between pairs of fingerprint impressions by viewing the thinned images solely as a set of curves in R 2.
A CKNOWLEDGMENTS
A preliminary version of this work appeared in [1].
R EFERENCES
[1] A.Ross,S.C.Dass,and A.K.Jain,“Estimating Fingerprint Deformation,”Proc.Int’l Conf.Biometric Authentication,pp.249-255,July 2004.
[2]
A.M.Bazen,G.T.
B.Verwaaijen,S.H.Gerez,L.P.J.Veelenturf,and B.J.van der Zwaag,“A Correlation-Based Fingerprint Verification System,”Proc.ProRISC2000Workshop Circuits,Systems,and Signal Processing,Nov.2000.
ROSS
ET AL.:FINGERPRINT WARPING USING RIDGE CURVE CORRESPONDENCES 29
Fig.10.Improved alignment of template and query images using ridge curve correspondences (right panel).The alignment using minutiae correspondences is shown in the left panel.
Both sets of alignment use the TPS warping model.
Fig.11.Improvement in matching performance when ridge
curve correspondences is used to develop the average deformation model.
Fig.12.Matching performance when the èindex of deformation is used to select optimal templates.Both optimal and suboptimal templates using ridge curve correspondences result in superior matching performance compared to minutiae correspondences.
[3] D.Roberge, C.Soutar,and B.Vijaya Kumar,“High-Speed
Fingerprint Verification Using an Optical Correlator,”Proc.SPIE, vol.3386,pp.123-133,1998.
[4]Z.M.Kova′cs-Vajna,“A Fingerprint Verification System Based on
Triangular Matching and Dynamic Time Warping,”IEEE Trans.
Pattern Analysis and Machine Intelligence,vol.22,no.11,pp.1266-1276,Nov.2000.
[5] D.Maio and D.Maltoni,“Direct Gray-Scale Minutiae Detection in
Fingerprints,”IEEE Trans.Pattern Analysis and Machine Intelligence, vol.19,no.1,pp.27-40,Jan.1997.
[6]N.K.Ratha and R.M.Bolle,“Effect of Controlled Acquisition on
Fingerprint Matching,”Proc.Int’l Conf.Pattern Recognition,vol.2, pp.1659-1661,1998.
[7] C.Dorai,N.Ratha,and R.Bolle,“Detecting Dynamic Behavior in
Compressed Fingerprint Videos:Distortion,”https://www.wendangku.net/doc/ad11741597.html,puter Vision and Pattern Recognition,pp.320-326,June2000.
[8]L.R.Thebaud,“Systems and Methods with Identity Verification
by Comparison and Interpretation of Skin Patterns such as Fingerprints,”US Patent5,909,501,1999.
[9] A.M.Bazen and S.Gerez,“Fingerprint Matching by Thin-Plate
Spline Modeling of Elastic Deformations,”Pattern Recognition, vol.36,no.8,pp.1859-1867,Aug.2003.
[10] A.Senior and R.Bolle,“Improved Fingerprint Matching by
Distortion Removal,”IEICE https://www.wendangku.net/doc/ad11741597.html,rmation and Systems,vol.84, no.7,pp.825-831,july2001.
[11] C.Watson,P.Grother,and D.Cassasent,“Distortion-Tolerant
Filter for Elastic-Distorted Fingerprint Matching,”Proc.SPIE Optical Pattern Recognition,pp.166-174,2000.
[12]R.Cappelli, D.Maio,and D.Maltoni,“Modelling Plastic
Distortion in Fingerprint Images,”Proc.Second Int’l Conf.Advances in Pattern Recognition,Mar.2001.
[13]R.Cappelli,R.Erol, D.Maio,and D.Maltoni,“Synthetic
Fingerprint-Image Generation,”Proc.15th Int’l Conf.Pattern Recognition,vol.3,Sept.2000.
[14] A.Almansa and L.Cohen,“Fingerprint Image Matching by
Minimization of a Thin-Plate Energy Using a Two-Step Algorithm with Auxiliary Variables,”Proc.IEEE Workshop Application of Computer Vision,pp.35-40,Dec.2000.
[15] A.Ross,S.C.Dass,and A.K.Jain,“A Deformable Model for
Fingerprint Matching,”Pattern Recognition,vol.38,no.1,pp.95-103,Jan.2005.
[16] D.J.Burr,“A dynamic Model for Image Registration,”Computer
Graphics and Image Processing,vol.15,pp.102-112,1981.
[17]L.Younes,“Optimal Matching between Shapes via Elastic
Deformations,”Image and Vision Computing,vol.17,pp.381-389, 1999.
[18]J.L.Barron, D.J.Fleet,and S.S.Beauchemin,“Performance of
Optical Flow Techniques,”Int’l https://www.wendangku.net/doc/ad11741597.html,puter Vision,vol.12,pp.43-77,1994.
[19]G.E.Christensen,R.D.Rabbitt,and https://www.wendangku.net/doc/ad11741597.html,ler,“Deformable
Templates Using Large Deformation Kinetics,”IEEE Trans.Image Processing,vol.5,pp.1435-1447,1996.
[20]S.C.Joshi and https://www.wendangku.net/doc/ad11741597.html,ler,“Landmark Matching via Large
Deformation Diffeomorphisms,”IEEE Trans.Image Processing, vol.9,pp.1357-1370,2000.
[21]Y.Amit,U.Grenander,and M.Piccioni,“Structural Image
Restoration through Deformable Templates,”J.Am.Statistical Assoc.,vol.86,pp.376-387,1991.
[22]J.M.Cartensen,“An Active Lattice Model in a Bayesian Frame-
work,”Computer Vision and Image Understanding,vol.63,no.2, pp.380-387,1996.
[23] C.A.Glasbey and K.V.Mardia,“A Penalized Likelihood
Approach to Image Warping,”J.Royal Statistical Soc.,Series B, Statistical Methodology,vol.63,no.3,pp.465-514,2001.
[24] F.L.Bookstein,“Principal Warps:Thin-Plate Splines and the
Decomposition of Deformations,”IEEE Trans.Pattern Analysis and Machine Intelligence vol.11,pp.567-585,1989.
[25]K.V.Mardia and T.J.Hainsworth,“Image Warping and Bayesian
Reconstruction with Gray-Level Templates,”Statistics and Images: Volume1,K.V.Mardia and G.K.Kanji,eds.,Oxford:Carfax, pp.257-280,1993.
[26]K.V.Mardia,T.J.Hainsworth,and J.F.Haddon,“Deformable
Templates in Image Sequences,”Proc.Int’l Conf.Pattern Recogni-tion,pp.132-135,1992.
[27]I.L.Dryden and K.V.Mardia,Statistical Shape Analysis.John Wiley
and Sons,1998.[28] F.L.Bookstein,“Landmark Methods for Forms without Land-
marks,”Medical Image Analysis,vol.1,pp.225-243,1997.
[29] A.K.Jain,L.Hong,and R.Bolle,“Online Fingerprint Verification,”
IEEE Trans.Pattern Analysis and Machine Intelligence,vol.19,no.4, pp.302-314,Apr.1997.
[30] D.Maio,D.Maltoni,R.Cappelli,J.L.Wayman,and A.K.Jain,
“FVC2002:Fingerprint Verification Competition,”Proc.Int’l Conf.
Pattern Recognition,pp.744-747,Aug.2002.
[31]S.Novikov and https://www.wendangku.net/doc/ad11741597.html,hmaev,“Registration and Modeling of
Elastic Deformations of Fingerprints,”Proc.Int’l ECCV Workshop Biometric Authentication,pp.80-88,May2004.
Arun Ross received the BE(Hons.)degree in
computer science from the Birla Institute of
Technology and Science,Pilani,India,in1996.
He received the MS and PhD degrees in
computer science and engineering from Michi-
gan State University in1999and2003,respec-
tively.He is an assistant professor in the Lane
Department of Computer Science and Electrical
Engineering at West Virginia University.Be-
tween July1996and December1997,he worked with the Design and Development Group of Tata Elxsi(India) Ltd.in Bangalore.He also spent three summers(2000-2002)with the Imaging and Visualization Group at Siemens Corporate Research,Inc., Princeton,New Jersey working on fingerprint recognition algorithms.His research interests include multimodal biometrics,fingerprint/iris analy-sis,and statistical pattern recognition.He is a member of the IEEE.
Sarat C.Dass received the MSc degree and the
PhD degree in statistics from Purdue University
in1995and1998,respectively.He worked as a
visiting assistant professor(in the period1998-
2000)in the Department of Statistics,University
of Michigan.Starting in2000,he joined the
Department of Statistics and Probability at
Michigan State University.His research and
teaching interests include statistical image pro-
cessing and pattern recognition,shape analysis, spatial statistics,Bayesian computational methods,foundations of statistics,and nonparametric statistical methods.He has been actively collaborating with computer scientists in several pattern recognition and image processing problems.
Anil K.Jain received the BTech degree from
the Indian Institute of Technology,Kanpur,in
1969and the MS and PhD degrees from The
Ohio State University in1970and1973,
respectively.He is a university distinguished
professor in the Department of Computer
Science and Engineering at Michigan State
University.His research interests include statis-
tical pattern recognition,data clustering,texture
analysis,document image understanding,and biometric authentication.He has received a Fulbright Research Award, a Guggenheim fellowship,and the Alexander von Humboldt Research Award.He delivered the2002Pierre Devijver lecture sponsored by the International Association of Pattern Recognition(IAPR)and received the 2003IEEE Computer Society Technical Achievement Award.He holds six patents in the area of fingerprint matching and is the author of a number of books including Handbook of Face Recognition(Springer, 2005),Handbook of Fingerprint Recognition(Springer,2003)(received the PSP award from the Association of American Publishers),and BIOMETRICS:Personal Identification in Networked Society(Kluwer, 1999).He is a fellow of the AAAS,ACM,IAPR,and IEEE.
.For more information on this or any other computing topic, please visit our Digital Library at https://www.wendangku.net/doc/ad11741597.html,/publications/dlib.
30IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL.28,NO.1,JANUARY
2006
背词根记单词
利用词根线索记忆单词 A ac do 做，动 ag to do active 行动作用 actual 实际上的 exact 确切的 react 反应 interact 相互作用 agent 代理人，经办人 agency 代办处 agenda 议程 agile 敏捷的 audi hear 听 audible 听得见的 audio 音频的 andio and video 音频和视频 video 视频的 B bi bio life 生命 biology 生物学 biography 传记 biotic 生命的 antibiotic 抗生素 biology n.生物 biological 生物学的联想: ecological 生态学的 metabolic 新陈代谢的 metabolism 新陈代谢 brev short 短 abbreviate 缩写 breviary 短暂 brief 简短的 brief introduction 简要介绍C cap head 头 cap 帽子 captain 上尉 capital 首都；资本 caption 标题，加标题 capture 捕获，俘虏 cas 降fall 落 casual 偶然的 casualty 伤亡 occasion 场合，时机，机会 occasional 偶然 ced ceed
cess go, walk 行走 acces 接近 precede 先行 antecedence 居先的 recession 萧条衰退 retrocede 后退 intercede 调停 ceive ，cept take 拿，取 accept 接受 intercept 截取 incept 摄入 deceive 欺骗,行骗 deception 欺骗 except 除….外,都 receipt 收据 deceive 欺骗 except 除了….之外 receive 得到 reception 接待处 cid cis kill,cut, fall 切割，掉落 accident 事故 incident 事 incise 切割，雕刻 suicide 自杀 concise 简明的 precise 精确的 circ ring 环，圈 circus 马戏团 circle 圆 semicircle 半圆 circulate 循环 circumstances 环境，条件（复数）civ city urb 城市 civil 全民的，市民的 civil war 内战 civilize 使文明，使开化 civilization 文明 uncivilized 未开化的 claim clam cry, shout 哭喊，叫喊 acclaim 喝彩,欢呼 claim 提主张，要求，声称 declaim 演讲 exclaim 呼喊 proclaim 宣布声明 喝彩，称赞
盐类的水解市级公开课教案
第三章水溶液中的离子平衡 第三节盐类的水解 （第一课时教案） 【教学目标】 知识目标：⑴能正确分析强酸弱碱盐和强碱弱酸盐的水解原理和规律，正确判断盐溶液的酸碱性； ⑵能用化学平衡原理解释盐类水解的实质； ⑶初步了解盐类水解方程式和离子方程式的写法。 能力目标：⑴通过实验探究的方式探究不同类型的盐溶液呈不同的酸碱性，继而分析CH3COONa溶液呈碱性的原因，感悟科学探究的过程与方法； ⑵通过实验比较不同盐溶液酸碱性，培养学生实验观察能力和动手能力 情感目标：⑴体验科学探究方法，感受自主学习和合作学习的乐趣； ⑵在实验探究过程中，使学生体验到透过现象揭示事物本质的成功的喜悦，增强学习的信心。【重点、难点】 重点：盐类水解反应的概念、规律 难点：盐类水解的实质 【教学方法】 实验探究、小组合作、讨论学习 【教学过程】 【温故知新】 常见的强弱电解质： 1、常见的弱电解质 弱酸：CH3COOH、H2CO3、H2S、H2SO3、HF、HClO等弱碱：NH3.H2O等 盐:少数盐水 2、常见的强电解质 强酸：HCl、H2SO4、HNO3、HBr等强碱：NaOH 、KOH 、Ba(OH)2 、Ca(OH)2 等 盐：大多数盐 一、探究盐溶液的酸碱性 【创设情景，引入新课】： 新闻链接：被蜂蛰伤莫大意——大妈差点送了命！ 问题引入：酸溶液显酸性，碱溶液显碱性，盐溶液一定显中性吗？如何设计实验证明？ 探究盐溶液的酸碱性： （常温，中性溶液pH=7，判断溶液酸碱性最简易方案：用pH试纸测pH） 【自主学习，合作探究】 实验目的：测定物质的PH 实验仪器：PH试纸、点滴板、玻璃棒 实验药品：CH3COONa溶液、Na2CO3溶液、NH4Cl溶液、（NH4）2SO4溶液、NaCl溶液、Na2SO4溶液 实验并记录实验数据
看听学一册单词大全
//Lesson1 meet vt.见面,遇见and conj.和this pron.这is vi.是class n.班级her pron.她的teacher n.老师Mr 先生which pron.哪一个your pron.你的pen n.钢笔the art.(定冠词) red a.红色的sir n.先生here ad.这儿you pron.你;你们are vi.是thank vt.谢谢Lesson2 blue a.蓝色pencil n.铅笔green a.绿色的book n.书 brown a.咖啡色的 schoolbag n.书包 rubber n.橡皮 Lesson3 his pron.他的 Miss 小姐 whose pron.谁的 cap n.帽子 it pron.它 yes ad.是的,是 come vi.来 sit vi.坐 down prep.向下 sit down 坐下 please ad.请 Lesson4 ruler n.尺 Lesson5 kick vt.踢 ball n.球 all a.全部的 right a.对的 all right 行,可以 look vi.看,注意 oh int.嗬,哦 sorry a.对不起 It's all right.没关系 Lesson6 yellow a.黄色的 bicycle n.自行车 pencil-box n.铅笔盒 basket n.篮子 desk n.课桌 white a.白的 umbrella n.雨伞 car n.汽车 grey=gray a.灰的 table n.桌 子 black a.黑色的 chair n.椅子 Lesson7 who pron.谁 that pron.那一个 girl n.女孩 on prep.在...上面 in prep.在…里面 boy n.男孩 with 拿着 football n.足球 he pron.他 brother n.兄弟 Lesson8 woman n.妇女 man n.男人 sister n.姐妹 Lesson9 hullo int.喂 mum n.妈妈 tea n.茶,茶叶 ready a.准备好的 hungry a.饿的 am vi.是的 no ad.不 not ad.不 what pron.什么 now ad.现在 very ad.非常
英语词汇构词知识词根篇二十cur,curs,cour,ccurs
英语词汇构词知识词根篇二十cur，curs，cour，ccurs 20、cur，curs，cour，ccurs cur，curs，cour，ccurs=run跑 occur [oc-表示to ，或towards，cur跑；“跑来”→来临] 出现，发生 occurrence [见上，-ence名词后缀] 出现，发生，发生的事 occurrent [见上，-ent形容词后缀，…的] 偶然发生的，正在发生的 current [cur跑→行，-ent形容词后缀，…的] 流行的，通行的，流通的，现行的，当前的，现在的；[-ent 名词后缀] 流，水流，气流 undercurrent [under-底下，current流] 暗流，潜流currency [cur跑→行，-ency名词后缀] 流行，流通，流通货币，通货 excurse [ex-外，出，curs跑→行走；“跑出去”，“出行”] 远足，旅游，旅行 excursion [见上，-ion名词后缀] 远足，旅行，旅游 excursionist [见上，-ist表示人] 远足者，旅游者excursive [ex-外，出，curs跑→行走；“走出”→走
离→走离正题；-ive形容词后缀，…的] 离题的，扯开的excursus [见上，-us名词后缀；“离题”的话] 离题语，附注，附记 course [cours跑→行进] 行程，进程，路程，道路，课程 intercourse [inter-在…之间，cours跑→行走→来往；“彼此之间的来往”] 交往，交际，交流 concourse [con-共同，一起，cours跑；“跑到一起来”] 汇合，集合，合流 courser [cours跑，-er者；“善跑的人或动物”] 跑者，追猎者，猎犬，骏马 courier [cour跑，-ier 表示人，“跑路的人”] 送急件的人，信使 succour [suc-后，随后，cour跑；“随后赶到”] 救助，救援，援助 cursory [surs跑→急行，-ory形容词后缀，…的；“急行奔走的”] 耸促的，草率的，粗略的 cursorial [见上，-ial形容词后缀，…的] （动物）疾走的，善于奔跑的 cursive [curs跑→速行，-ive形容词后缀，…的，“疾行的”，“速走的”] （字体）草写的，行书，草书，草写体incursion [in-内，入内，curs跑→行走，-ion名词
公开课--盐类的水解-教学设计
教材先提出盐溶液的酸碱性的问题，然后通过实验得出盐溶液的酸碱性与其组成的关系，再通过微观分析得出其本质原因。针对这一内容组织形式、结合学生已经基本具备解决问题的弱电解质电离平衡知识和平衡移动原理知识，本节课适合于采用探究式教学模式。同时，本节课的核心任务是形成盐类水解的概念，根据概念的分类和学习的认知学心理，概念的形成要通过具体的例证进行概念感知、对例证进行分析、比较、辨别，在此基础上舍弃非本质特征、提取其本质特征、进行抽象概括，形成概念。综合上述两点，我设计了如下探究教学模式。 教学模式：提出问题――实验探究――得出结论――质疑――理论讲解――交流应用 教学环境、教学媒体选择 教学场地：高二（1）班 实验器材：试管、烧杯、玻璃棒等 本节课将用到 黑板，粉笔，计算机，投影仪，图片，课件PPT等。 板书设计 第三节盐类的水解 一、探究盐溶液的酸碱性 强酸强碱盐：中性 强酸弱碱盐：酸性 强碱弱酸盐：碱性 二、盐溶液呈现酸碱性的原因 1、理论分析：CH3OONa溶液（显碱性）： CH3OONa=CH3OO—+ Na+ + H2O H++ OH— CH3COOH 2、盐类水解的定义 3、实质 4、规律 5、条件 6、特征
三、水解方程式的书写规则 ①一般用，不用气体或沉淀符号； ②多元弱酸根分布写，以第一步为主； ③多元弱碱的阳离子一步到位。 教学目标 一、知识与技能 （1）使学生理解盐类水解的实质，能解释强碱弱酸盐和强酸弱碱盐的水解。 （2）能通过比较、分类、归纳、概括等方法得出盐类水解的规律，揭示盐类水解的本质。（3）能运用盐类水解的规律判断盐溶液的酸碱性，会书写盐类水解的离子方程式。 二、过程与方法 通过实验，培养学生的观察能力，加强基本操作训练，培养分析、综合的思维能力三、情感态度与价值观 （1）能在思考分析过程中倾听他人意见，相互启发，体会合作交流的重要与快乐。 （2）体验科学探究的乐趣，学会透过现象看本质。 （3）建立个性与共性、对立与统一的科学辩证观。 重点、难点分析 重点：盐类水解的过程、本质。 难点：盐类水解的本质。 课堂教学过程结构设计 教师活动学生活动设计意图 【引言】在酸溶液中， [H+]>[OH-],使溶液呈酸性；在碱溶液中，[H+]<[OH-],使溶液呈碱性；那么在由酸和碱发生中和反应产生的盐的水溶液中，是否一定是[H+]=[OH-],使溶液呈中性呢？ 【评价】同学们众说纷纭，观点各不相同。俗话说，实践出真知，我们还是用实验来检验思考回答： 部分学生：是中性 部分学生：不一定是中性， 也可能呈碱性或酸性， 设置问题情境，激发学习兴 趣，轻松进入学习状态。 ，
矩阵分解在优化方法中的应用
矩阵分解以及矩阵范数在数值计算中的应用 张先垒 （自动化与电气工程学院 控制科学与工程 2012210186） 【摘要】矩阵的分解是将一个矩阵分解为较为简单的或具有某种特性的若干矩阵的和或 者乘积，这是矩阵理论及其应用中比较常见的方法。由于矩阵的这些特殊的分解形式，一方面反映了矩阵的某些数值特性，如矩阵的秩、特征值、奇异值等；另一方面矩阵的分解方法与过程往往为某些有效的数值计算方法和理论分析提供了重要的依据，它是应用于解最优化问题、特征值问题、最小二乘方问题的主要数学工具。 关键词 ： 矩阵分解 对角化 逆矩阵 范数 条件数 1. 引言 矩阵分解在工程中的应用主要是在解线性方程组中,而这主要就是关系到储存和计算时间的问题上面,如何实现最小的储存和最少的计算时间是在工程计算中的头等问题。在这方年就牵涉到很多对矩阵进行怎样的分解，这篇文章介绍了基本的关于三角分解相关的内容以及关于界的稳定性的考虑。 2. 矩阵的三角分解求解线性方程组 数值求解线性方程组的方法中有一个主要是直接法，假设计算中没有舍入误差，经过有限次算术运算能够给出问题的精确解的数值方法。其中高斯消去法就是利用矩阵的分解实现的。矩阵论一种有效而且应用广泛的分解法就是三角分解法，将一个矩阵分解为一个酉矩阵（或正交矩阵）与一个三角矩阵的乘积或者三角矩阵与三角矩阵的乘积。（见课本P93例4.3）考虑一般的线性方程组，设其中的系数矩阵A 是可逆的， 1111 n m mn a a A a a ?? ? = ? ??? （1-1） 设矩阵A 的第一列中至少有一个是非零元素（否则A 就是奇异矩阵）不妨设为1i a 若一 般的记初等矩阵 [1] 如1-2式及矩阵论课本上的Givens 矩阵。
英文单词记忆法。词根法。
请大家想一想，英语是谁发明的？英国人呗！英国人认不认识汉语？不认识！那么英国人在学英语单词的时候需不需要记住单词的汉语意思？不需要，英国人的英语课本里根本就没有汉字，何谈记住单词的汉语意思？那么既然英国人学英语不需要记住(甚至根本就见不到)单词的汉语意思，那么中国人学英语为什么要去记住单词的汉语意思呢？这种做法大家不觉得奇怪吗？ 然而由于中国人学英语时都在背单词的汉语意思，因此大家反而觉不出“背汉字”有什么奇怪的了。其实仔细想一想，这个行为真的很奇怪，奇怪的根源不在于行为本身，而在于中国人普遍不会直接识别英语单词的意思，因而只好靠汉语符号来机械地帮助记忆英语单词的意思，这样去学英语不仅多此一举，而且必然会陷入苦海无边的符号记忆灾难中。 其实英语单词和汉字一样，存在着很多的“偏旁部首”，知道了偏旁部首你就可以根据它们直接来猜测单词的意思，虽不说百分之百猜准，但起码可以猜测个大概，至少在别人告诉过你单词的意思后你可以恍然大悟地领会它，这样就可以大大增强你对英语单词“见字识意”的能力，做到真正认识一个单词，而把它的汉语意思仅做为一般参考。 举几个例子来说吧： 比如单词representative，请别急着告诉我你认识这个单词，其实你不见得“认识”这个单词，你仅是凭着你的记忆力记住了这串英语字母和两个汉字符号“代表”之间的对应关系，这样去学英语你会多费劲？下面我来告诉你这个单词为什么是“代表”的意思。re在英语里是一个偏旁部首，它是“回来”的意思；pre也是一个偏旁部首，是“向前”的意思；sent也是一个偏旁部首，是“发出去、派出去”的意思；a仅是偏旁部首之间的一个“连接件”，没了它两个辅音字母t就要连在一起了，发音会分不开，会费劲，因此用一个元音字母a隔开一下；tive也是一个偏旁部首，是“人”的意思。那么这几个偏旁部首连在一起是什么意思呢？re-pre-sent-a-tive，就是“回来-向前-派出去-的人”，即“回来征求大家的意见后又被派出去替大家讲话的人”，这不就是“代表”的意思吗！这么去认识一个单词才是真正“认识”了这个单词，把它认识到了骨子里。 再举一个例子吧：psychology。 psy=sci，是一个偏旁部首，是“知道”的意思；cho是一个偏旁部首，是“心”的意思；lo是一个偏旁部首，是“说”的意思；gy是一个偏旁部首，是“学”的意思，logy合起来是“学说”的意思。因此psy-cho-logy连起来就是“知道心的学说”，因此就是“心理学”的意思。 依此类推，不多举例了，我要表达的观点已经清楚了，那就是，不要去死记硬背单词的汉语意思，而要用识别“偏旁部首”的方法去真正认识一个单词，真正认识了单词后，你会发现单词表里的汉语翻译原来其实很勉强，有时甚至根本翻译不出来，因为汉语和英语是两种不同的文字体系，两者在文字上本来就不是一一对应的，只背英语单词的汉字意思是不能真正认识这个单词的，会造成很多的后续学习困难，会造成你一辈子看英语单词如雾里看花，永远有退不掉的陌生感。 那么接下来的问题是，英语里有多少个“偏旁部首”，怎样知道和学会它们？ 回答这个问题时我才发现中国人对英语偏旁部首陌生的两个主要原因，一是这些重要内容不在学校的英语教材当中，大家在课堂上学不到(这是目前学校英语教材急需弥补的缺陷)；二是少数书店里销售的有关这方面内容的书过分复杂化，动辙几百上千页，内容苦涩庞大，影响了这些常识的普及，使得本来是常识的东西不常识。其实英语里偏旁部首的学名叫“字根”，常用的也就二百多个，它们就像26个字母一样普通而重要，就像汉语里的偏旁部首那样普通而重要，它们是学英语第一课里就应该学习的重要内容，学英语者应及早地掌握这些重要的常识，及早地摆脱死记硬背的蛮干状态，及早地进入科学、高效的识字状态。 【英语字根】 1,ag=do,act 做，动 2,agri=field 田地，农田(agri也做agro,agr) 3,ann=year年 4,audi=hear听 5,bell=war战争 6,brev=short短 7,ced,ceed,cess=go行走
优质课教案：盐类的水解
公开课教案 开课教师：林玉治时间：2008-12-5 星期五第三节班级：K二6班 第三单元盐类的水解 第一课时盐类的水解规律 教学目标： 1.使学生理解盐类水解的实质，能解释强碱弱酸盐和强酸弱碱盐的水解。 2.能通过比较、分类、归纳、概括等方法得出盐类水解的规律，揭示盐类水解的本质。 3.能运用盐类水解的规律判断盐溶液的酸碱性，会书写盐类水解的离子方程式。 教学重点：盐类水解的本质 教学难点：盐类水解方程式的书写和分析 教学过程： 【导课】生活中如果遇到火灾怎么办？ 打119，找水源，找泡沫灭火器····· 【切题】泡沫灭火器的使用：倒置，喷出大量泡沫隔绝空气灭火。 灭火器的成分是硫酸铝和碳酸氢钠溶液，为什么倒置可以产生二氧化碳和氢氧化铝？ 欲知详情如何，请听《盐类的水解》 ［问题引入］酸溶液显酸性，碱溶液显碱性，盐溶液一定显中性吗？如何设计实验证明？ 一、探究盐溶液的酸碱性：（常温，中性溶液PH=7，判断溶液酸碱性最简易方案：用PH试纸测PH）【活动与探究】用pH试纸检验下列（一）组溶液的酸碱性：【学生实验】 （一）NaCl、NH4Cl、CH3COONa（二）AlCl3、Na2CO3、Na2SO4、 溶液的酸碱性 盐的类型 （从生成该盐的酸和碱的强弱分） 【讨论】为什么不同的盐溶液的酸碱性不同？由上述实验结果分析，盐溶液的酸碱性与生成该盐的酸和 碱的强弱间有什么关系？ 盐的组成： 酸+ 碱= 盐+ 水 强酸强碱强酸强碱盐如：NaCl KNO3 强酸弱碱盐NH4Cl Al2(SO4)3 弱酸弱碱弱酸强碱盐CH3COONa Na2CO3 弱酸弱碱盐CH3COONH4 △正盐的组成与盐溶液酸碱性的关系：（谁强显谁性，都强显中性） ①强碱弱酸盐的水溶液显碱性c（H+）＜c（OH—） ②强酸弱碱盐的水溶液显酸性c（H+）＞c（OH—） ③强酸强碱盐的水溶液显中性c（H+）= c（OH—）
3L英语(看听学)第一册单词大全
3L英语（看听学）第一册单词大全（674个）meet 见面，遇见and 和 this 这 is 是 class 班级 her 她的 teacher 老师 Mr 先生 which 哪一个your 你的(所有格) pen 钢笔 the (定冠词) red红色的 sir先生 here这儿 you你；你们are是 thank谢谢 blue蓝色 pencil铅笔 green绿色的 book书 brown咖啡色的 schoolbag书包 rubber橡皮 his他的(所有格) Miss小姐 whose谁的 cap帽子 it它 yes是的，是 come来 sit坐 down向下 sit down坐下 please请 ruler尺 kick踢 ball球 all全部的 right对的 all right行，可以 look看，注意 oh嗬，哦 sorry对不起, 抱歉 It’s all right没关系 yellow黄色的 bicycle自行车 pencil-box铅笔盒 basket篮子 desk课桌 white白的 umbrella雨伞 car汽车 grey灰的 table桌子 black黑色的 chair椅子 who谁 that那一个
girl女孩 on在...上面 in在…里面 boy男孩 with在此作“拿着”解football足球 he他 brother兄弟woman妇女 man男人 sister姐妹 hullo喂 mum妈妈 tea茶，茶叶ready准备好的hungry饿的 am是的 no不 not不 what什么 now现在 very非常thirsty渴的 busy忙的 tired累的 hot热的 cold冷的 look at看... my我的(所有格) picture图画，图片 nice好的，美的 egg蛋 listen听 dad爸爸 eat吃 quickly快地 put放 cup杯子 eggcup放蛋的杯子 like像 evening晚上 good evening晚上 好 children孩子们 empty空的 apple苹果 orange桔子 ice-cream冰淇淋 school学校 put on穿上 hat有边的帽子 shoe鞋子 funny有趣的 too也 me我宾格 knock敲，打 door门 actor男演员 actress女演员 schoolboy男学生 schoolgirl女学生 policeman男警察 policewoman女警 察 postman邮递员 milkman送牛奶的人
基于矩阵分解的协同过滤算法
万方数据
万方数据
万方数据
万方数据
基于矩阵分解的协同过滤算法 作者：李改， 李磊， LI Gai， LI Lei 作者单位：李改,LI Gai(顺德职业技术学院,广东顺德528333;中山大学信息科学与技术学院,广州510006;中山大学软件研究所,广州510275)， 李磊,LI Lei(中山大学信息科学与技术学院,广州510006;中山大学软件研究 所,广州510275) 刊名： 计算机工程与应用 英文刊名：Computer Engineering and Applications 年，卷(期)：2011,47(30) 被引用次数：1次 参考文献(18条) 1.Wu J L Collaborative filtering on the Nefifix prize dataset 2.Ricci F.Rokach L.Shapira B Recommender system handbook 2011 3.Adomavicius G.Tuzhilin A Toward the next generation of recommender systems:a survey of the state-of-the-art and possible extenstions 2005(06) 4.Bell R.Koren Y.Volinsky C The bellkor 2008 solution to the Netflix prize 2007 5.Paterek A Improving regularized singular value decomposition for collaborative filtering 2007 6.Lee D D.Seung H S Leaming the parts of objects by non-negative matrix factorization[外文期刊] 7.徐翔.王煦法基于SVD的协同过滤算法的欺诈攻击行为分析[期刊论文]-计算机工程与应用 2009(20) 8.Pan R.Zhou Y.Cao B One-class collaborative filtering 2008 9.Pan R.Martin S Mind the Gaps:weighting the unknown in largescale one-class collaborative filtering 2009 https://www.wendangku.net/doc/ad11741597.html,flix Netflix prize 11.罗辛.欧阳元新.熊璋通过相似度支持度优化基于K近邻的协同过滤算法[期刊论文]-计算机学报 2010(08) 12.汪静.印鉴.郑利荣基于共同评分和相似性权重的协同过滤推荐算法[期刊论文]-计算机科学 2010(02) 13.Hadoop[E B/OL] 14.Apache MapReduce Architecture 15.Wbite T.周敏.曾大聃.周傲英Hadoop权威指南 2010 16.Herlocker J.Konstan J.Borchers A An algorithmic framework for performing collaborative filtering 1999 17.Linden G.Smith B.York J https://www.wendangku.net/doc/ad11741597.html, recommendations:Itemto-item collaborative filtering[外文期刊] 2003 18.Sarwar B.Karypis G.Konstan J ltem-based collaborative filtering recommendation algorithms 2001 引证文献(1条) 1.沈韦华.陈洪涛.沈锦丰基于最佳匹配算法的精密零件检测研究[期刊论文]-科技通报 2013(5) 本文链接：https://www.wendangku.net/doc/ad11741597.html,/Periodical_jsjgcyyy201130002.aspx
英语单词常见前后缀词根总结
英语单词常见前后缀词根总结 中国人在学习英语的过程中，感到最困难的就是记忆单词，这在很大程度上是因为我们对英语的构词规律缺乏熟悉和分析。假如我们对英语单词的结构像对汉语的偏旁、部首一样熟悉，记忆英语单词就不会那么枯燥、乏味了。在汉语中，当我们见到"财"、"贩"、"购"、"资"、"货"等字时，从部首"贝"就知道它们都与钱有关。英语单词同样是由词缀和词根构成的，词缀分为前缀和后缀 常见前缀 1. a，an: ①无，不，非astable不稳定的acentric无中心的 ②含有in,on,at,by,with,to等意义asleep在熟睡中ahead向前 2.ab: 脱离abnormal不正常的abaxial离开轴心的 3.anti: 反抗antiwar反战的anti?imperialist反帝的 4.auto:自动，自己autobiography自传autostable自动稳定 5.be: 使…加强belittle使缩小befriend友好相待 6.bi:二，双biweekly双周刊biligual两种语言的 7.co,col,com,con,cor 共同cooperation协作collaboration协作，勾结combine联合correlation相互关系 8.counter,contra: 反，对应counteraction反作用contrast对比,对照 9.di,dif,dis: 否定，相反diffident不自信的dislike不喜欢 10.en,em: 使… enlarge扩大enable使…能empower使…有权力 11.e,ex: 外，出external外部的erupt喷出 12.extra: 以外，超过extraordinary格外的extrasolar太阳系以外的 13.hyper: 在上，超hyperfrequency超高频hypersonic超声的 14.il,im,in,ir: 否定illogical不合逻辑的impossible不可能的invisible 不可见irrational不合理的 15.inter:互相interchange交换interlock连锁 16.intra,intro: 在内，内部intrapersonal个人内心的introspect内省 17.mal: 恶，不良maltreat虐待malfunction机能失常
看听学一册单词
Lesson1 meet vt.见面,遇见and conj.和this pron.这is vi.是class n.班级her pron.她的teacher n.老师Mr 先生which pron.哪一个your pron.你的pen n.钢笔the art.(定冠词) red a.红色的sir n.先生here ad.这儿you pron.你;你们are vi.是thank vt.谢谢Lesson2 blue a.蓝色pencil n.铅笔green a.绿色的book n.书 brown a.咖啡色的 schoolbag n.书包 rubber n.橡皮 Lesson3 his pron.他的 Miss 小姐 whose pron.谁的 cap n.帽子 it pron.它 yes ad.是的,是 come vi.来 sit vi.坐 down prep.向下 sit down 坐下 please ad.请 Lesson4 ruler n.尺 Lesson5 kick vt.踢 ball n.球 all a.全部的 right a.对的 all right 行,可以 look vi.看,注意 oh int.嗬,哦 sorry a.对不起 It's all right.没关系 Lesson6 yellow a.黄色的 bicycle n.自行车 pencil-box n.铅笔盒 basket n.篮子 desk n.课桌 white a.白的 umbrella n.雨伞 car n.汽车 grey=gray a.灰的 table n.桌 子 black a.黑色的 chair n.椅子 Lesson7 who pron.谁 that pron.那一个 girl n.女孩 on prep.在...上面 in prep.在…里面 boy n.男孩 with 拿着 football n.足球 he pron.他 brother n.兄弟 Lesson8 woman n.妇女 man n.男人 sister n.姐妹 Lesson9 hullo int.喂 mum n.妈妈 tea n.茶,茶叶 ready a.准备好的 hungry a.饿的 am vi.是的 no ad.不 not ad.不 what pron.什么 now ad.现在 very ad.非常
(完整word版)矩阵分解及其简单应用
对矩阵分解及其应用 矩阵分解是指将一个矩阵表示为结构简单或具有特殊性质若干矩阵之积或之和，大体分为三角分解、QR 分解、满秩分解和奇异值分解。矩阵的分解是很重要的一部分内容，在线性代数中时常用来解决各种复杂的问题，在各个不同的专业领域也有重要的作用。秩亏网平差是测量数据处理中的一个难点，不仅表现在原理方面，更表现在计算方面，而应用矩阵分解来得到未知数的估计数大大简化了求解过程和难度。 1. 矩阵的三角分解 如果方阵A可表示为一个下三角矩阵L和一个上三角矩阵U之积，即A=LU 则称A可作三角分解。矩阵三角分解是以Gauss消去法为根据导出的，因此矩阵可以进行三角分解的条件也与之相同，即矩阵A的前n-1个顺序主子式都不为0, 即?k工0.所以在对矩阵A进行三角分解的着手的第一步应该是判断是否满足这个前提条件，否则怎么分解都没有意义。矩阵的三角分解不是唯一的，但是在一定的前提下， A=LDU勺分解可以是唯一的，其中D是对角矩阵。矩阵还有其他不同的三角分解，比如Doolittle 分解和Crout 分解，它们用待定系数法来解求 A 的三角分解，当矩阵阶数较大的时候有其各自的优点，使算法更加简单方便。 矩阵的三角分解可以用来解线性方程组Ax=b。由于A=LU，所以Ax=b可以变换成LU x=b，即有如下方程组： Ly = b { {Ux = y 先由Ly = b依次递推求得y i, y2, ........ ，y n，再由方程Ux = y依次递推求得X n， x n-1 ， ... ，X1 . 必须指出的是，当可逆矩阵A不满足?k工0时，应该用置换矩阵P左乘A以便使PA 的n个顺序主子式全不为零，此时有： Ly = pb { { Ux = y 这样，应用矩阵的三角分解，线性方程组的解求就可以简单很多了。 2. 矩阵的QF分解 矩阵的QR分解是指，如果实非奇异矩阵A可以表示为A=QR其中Q为正交矩阵，R为实非奇异上三角矩阵。QR分解的实际算法各种各样，有Schmidt正交方
大学英语四级单词词根词缀
*******前缀********* a ，an ①无，不，非astable不稳定的acentric无中心的 ②含有in,on,at,by,with,to等意义asleep在熟睡中ahead向前 ab 脱离abnormal不正常的abaxial离开轴心的 anti 反抗antiwar反战的anti imperialist反帝的 auto 自动，自己autobiography自传autostable自动稳定 be 使…加强belittle使缩小befriend友好相待 bi 二，双biweekly双周刊biligual两种语言的 co,col,com,con,cor 共同cooperation协作collaboration协作，勾结combine联合correlation相互关系 counter ,contra 反，对应counteraction反作用contrast对比,对照 di ,dif ,dis 否定，相反diffident不自信的dislike不喜欢 en,em 使… enlarge扩大enable使…能empower使…有权力 e ,ex 外，出external外部的erupt喷出 extra 以外，超过extraordinary格外的extrasolar太阳系以外的 hyper 在上，超hyperfrequency超高频hypersonic超声的 il ,im ,in ,ir 否定illogical不合逻辑的impossible不可能的invisible 不可见irrational不合理的 inter 互相interchange交换interlock连锁 intra ,intro 在内，内部intrapersonal个人内心的introspect内省 mal 恶，不良maltreat虐待malfunction机能失常 micro 微microscrope显微镜microware微波 mid 中，中间mid air半空中midstream中流 mini 小minibus小公共汽车ministate小国 mis 错，坏mistake错误misspell拼错 multi 多multiparty多党的multilingual多种语言的 non 否定nonsmoker不抽烟的人nonexistent不存在的 out 超过，过度，外，出，除去 outgo走得比…远outgrow长得太大outdoor户外的outroot除根 over 上，过度overwork工作过度overbridge天桥 post 后postwar战后postnatal诞生后的 pre 前，领先prewar战前的prefix前缀 re 回，再return返回restart重新开始 semi 半semicircle半圆semiconductor半导体 sub ,s uc ,suf ,sup 次，亚，在下，低于 substandard低于标准规格的succeed继承suffix后缀supplement增补 trans 转换，变换，横过，越过transmit传送transatlantic横渡大西洋的 tri 三tricar三轮车triangle三角(形) un 否定unstable不稳定的unknown未知的
盐类的水解公开课教学设计
盐类的水解 （第1课时） 一、教材分析 盐类的水解以水的电离平衡为基础，在一个多个平衡共存的体系中，主要利用电离平衡的理论来揭示其实质，同时形成水解平衡的概念，所以本节内容安排在弱电解质的电离平衡以及水的电离和溶液的酸碱性学习之后，是对化学平衡、电离平衡以及水的离子积等知识的综合运用和补充，在新旧知识融合和迁移的过程中，可以很好地渗透对学生能力培养的要求 二、教学目标 知识与技能： 1、认识盐类水解的原理、盐类水解的规律； 2、掌握盐类水解离子方程式的书写 过程与方法： 1、通过实验探究并运用归纳法分析盐类溶液呈不同酸碱性的原因； 2、从微观角度探究盐类水解的本质原因，在学习过程中掌握从宏观到微观、从静态到 动态的探究方法。 情感、态度与价值观： 1、通过探究不同的盐溶液呈现不同酸碱性的本质原因，学会透过现象看本质； 2、领悟矛盾的“对立与统一”以及矛盾“特殊性与普遍性”的哲学观点 三、教学重、难点 重点：盐类水解的本质 难点：盐类水解方程式的分析书写 四、实验用品 PH试纸、标准比色卡、玻璃棒、表面皿 NaCl KNO3 NH4Cl (NH4)2SO4 CH3COONa Na2CO3 （均为溶液）五、设计思路 本节课主要分四个模块进行： 1、探究盐溶液的酸碱性：从学生比较熟悉的盐类物质入手，指导学生通过实验并运 用归纳法分析盐类的组成与盐溶液酸、碱性的对应关系，找出盐类溶液酸碱性的规律； 2、探究盐溶液呈酸碱性的原因：指导学生以水的电离平衡为基础，分析盐类电离出 的阴、阳离子与水电离出的H+和OH-结合成弱酸或弱碱的趋势，明确不同盐溶液呈现不同酸碱性的本质原因并分析盐类水解的条件，让学生在学习过程中掌握从宏观到微观、从静态到动态的探究方法，领悟矛盾的“对立与统一”以及矛盾“特殊性与普遍性”的哲学观点； 3、认识盐类水解的实质与规律：指导学生对探究的结果进行总结提炼，明确盐类水 解的实质与规律，完成新知识的构建； 4、教学效果强化与反馈：给出适量有针对性的练习，巩固加深学生对概念的理解，
矩阵分解及其简单应用
矩阵分解是指将一个矩阵表示为结构简单或具有特殊性质若干矩阵之积或之和，大体分为三角分解、分解、满秩分解和奇异值分解.矩阵地分解是很重要地一部分内容，在线性代数中时常用来解决各种复杂地问题，在各个不同地专业领域也有重要地作用.秩亏网平差是测量数据处理中地一个难点，不仅表现在原理方面，更表现在计算方面，而应用矩阵分解来得到未知数地估计数大大简化了求解过程和难度. 矩阵地三角分解 如果方阵可表示为一个下三角矩阵和一个上三角矩阵之积，即，则称可作三角分解.矩阵三角分解是以消去法为根据导出地，因此矩阵可以进行三角分解地条件也与之相同，即矩阵地前个顺序主子式都不为，即.所以在对矩阵进行三角分解地着手地第一步应该是判断是否满足这个前提条件，否则怎么分解都没有意义.矩阵地三角分解不是唯一地，但是在一定地前提下，地分解可以是唯一地，其中是对角矩阵.矩阵还有其他不同地三角分解，比如分解和分解，它们用待定系数法来解求地三角分解，当矩阵阶数较大地时候有其各自地优点，使算法更加简单方便.资料个人收集整理，勿做商业用途 矩阵地三角分解可以用来解线性方程组.由于，所以可以变换成，即有如下方程组：资料个人收集整理，勿做商业用途 先由依次递推求得,,……，，再由方程依次递推求得，，……，. 资料个人收集整理，勿做商业用途 必须指出地是，当可逆矩阵不满足时，应该用置换矩阵左乘以便使地个顺序主子式全不为零，此时有：资料个人收集整理，勿做商业用途 这样，应用矩阵地三角分解，线性方程组地解求就可以简单很多了. 矩阵地分解 矩阵地分解是指，如果实非奇异矩阵可以表示为，其中为正交矩阵，为实非奇异上三角矩阵.分解地实际算法各种各样，有正交方法、方法和方法，而且各有优点和不足.资料个人收集整理，勿做商业用途 ．正交方法地分解 正交方法解求分解原理很简单，容易理解.步骤主要有：）把写成个列向量（，，……，），并进行正交化得（，，……，）；) 单位化，并令（，，……，），（，，……，），其中；）. 这种方法来进行分解，过程相对较为复杂，尤其是计算量大，尤其是阶数逐渐变大时，就显得更加不方便.资料个人收集整理，勿做商业用途 ．方法地分解 方法求分解是利用旋转初等矩阵，即矩阵()来得到地，()是正交矩阵，并且(()).()地第行第列 和第行第列为，第行第列和第行第列分别为和，其他地都为.任何阶实非奇异矩阵可通过左连乘()矩阵（乘积为）化为上三角矩阵，另，就有.该方法最主要地是在把矩阵化为列向量地基础上找出和，然后由此把矩阵地一步步向上三角矩阵靠近.方法相对正交方法明显地原理要复杂得多，但是却计算量小得多，矩阵()固有地性质很特别可以使其在很多方面地应用更加灵活.资料个人收集整理，勿做商业用途 ．方法地分解 方法分解矩阵是利用反射矩阵，即矩阵，其中是单位列向量，是正交矩阵，.可以证明，两个矩阵地乘积就是矩阵，并且任何实非奇异矩阵可通过连乘矩阵（乘积为）化为上三角矩阵，则.这种方法首要地就是寻找合适地单位列向量去构成矩阵，
常见词根助记单词 ( 3)
常见词根助记单词（3） 1.cur，curs，cour，ccurs =run跑 occur [oc-表示to ，或towards，cur跑；“跑来”→来临] 出现，发生 occurrence [见上，-ence名词后缀] 出现，发生，发生的事 current [cur跑→行，-ent形容词后缀，…的] 流行的，流通的，现在的；[-ent名词后缀] 流，水流，气流undercurrent [under-底下，current流] 暗流，潜流 currency [cur跑→行，-ency名词后缀] 流行，流通，流通货币，通货 excurse [ex-外，出，curs跑→行走；“跑出去”，“出行”] 远足，旅游，旅行 excursion [见上，-ion名词后缀] 远足，旅行，旅游 excursionist [见上，-ist表示人] 远足者，旅游者 excursive [ex-外，出，curs跑→行走；“走出”→走离→走离正题ive形容词后缀，…的] 离题的，扯开的course [cours跑→行进] 行程，进程，路程，道路，课程 concourse [con-共同，一起，cours跑；“跑到一起来”] 汇合，集合，合流 courser [cours跑，-er者；“善跑的人或动物”] 跑者，追猎者，猎犬，骏马 cursory [surs跑→急行，-ory形容词后缀，…的；“急行奔走的”] 耸促的，草率的，粗略的 incursion [in-内，入内，curs跑→行走，-ion名词后缀；“走入”，“闯入”] 进入，侵入，入侵，侵犯precursor [pre-先，前，curs跑→行，-or者] 先行者，先驱者，前任，前辈 concur [con-共同，cur跑；“共同跑来”] 同时发生，同意 recur [re-回，复，cur跑；“跑回”→返回→重来] 再发生，（养病等）复发，（往事等）重新浮现recurrence [见上，-ence名词后缀] 复发，再发生，重新浮现 corridor [cor=cour跑→行走；“行走的地方”] 走廊 curriculum [cur=couse；a course of study] 课程 2、dict =say言，说（dict也作dic） contradict [contra相反，dict言；说出“相反之言”] 反驳，同…相矛盾，与…相抵触 contradiction [见上，-ion名词后缀] 矛盾，抵触，对立 dictate [dict言，说→吩咐，命令，，-ate动词后缀；“口授命令或指示”] 口述而令别人记录，使听写，命令dictation [见上，-ion名词后缀] 口授，命令，支配，口述，听写 dictator [见上，-or者；“口授命令者”→支配者，掌权者] 独裁者，专政者，口授者 predict [pre-前，先，预先，dict言] 预言，预告 prediction [见上，-ion名词后缀] 预言，预告 malediction [male-恶，坏，dict言，-ion名词后缀；“恶言”] 咒骂，诽谤 benediction [bene-好，dict言词，-ion名词后缀；“好的言词”] 祝福 indicate [in-表示at，dic言，说→说明，表明，表示，-ate动词后缀] 指示，指出，表明 indicator [见上，-or表示人或物] 指示者，指示器 dictionary [dict言，词，diction措词，-ary名词后缀，表示物；“关于措词的书”] 字典，词典 diction [dict言，词，-ion名词后缀] 措词，词令 dictum [dict言，词，-um名词后缀] 格言，名言 edict [c-外，出，dict言，词→指示；“统治者发出的话，“当局发出的指示”] 法令，布告 indict [in-表示on，upon，against，dict言，诉说] 控告，告发，对…起诉