Research Article Robust Adaptive Principal Component...

Research ArticleRobust Adaptive Principal Component Analysis Based onIntergraph Matrix for Medical Image Registration

Chengcai Leng12 Jinjun Xiao3 Min Li4 and Haipeng Zhang1

1Key Laboratory of Nondestructive Testing Ministry of Education School of Mathematics and Information ScienceNanchang Hangkong University Nanchang 330063 China2State Key Laboratory of Management and Control for Complex Systems Institute of Automation Chinese Academy of SciencesBeijing 100190 China3Information Center Jiangxi School of Electronics and Information Engineering Nanchang 330096 China4School of Material Science and Engineering Nanchang Hangkong University Nanchang 330063 China

Correspondence should be addressed to Chengcai Leng chengcailenggmailcom

Received 22 October 2014 Revised 19 March 2015 Accepted 26 March 2015

Academic Editor Francesco Camastra

Copyright copy 2015 Chengcai Leng et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper proposes a novel robust adaptive principal component analysis (RAPCA) method based on intergraph matrix for imageregistration in order to improve robustness and real-time performanceThe contributions can be divided into three parts Firstly anovel RAPCAmethod is developed to capture the common structure patterns based on intergraph matrix of the objects Secondlythe robust similarity measure is proposed based on adaptive principal component Finally the robust registration algorithm isderived based on the RAPCAThe experimental results show that the proposed method is very effective in capturing the commonstructure patterns for image registration on real-world images

1 Introduction

Image registration is a fundamental task in medical imageprocessing and has been widely used in multimodal imagefusion and tumor detection Generally speaking image reg-istration methods can be classified into two categories area-based and feature-based methods [1] Area-based methodsdeal with the images without detecting salient features andadopt optimization algorithms Mutual information (MI) isone of the most frequently used techniques in area-basedmethods because of its ability to measure the similarityof the pixels between the reference image and the sensedimage [2 3] Rivaz et al [4] proposed an efficient regis-tration method by using Contextual Conditioned MutualInformation (CoCoMI) as the similarity measure in a reg-ularized cost function with a B-spline deformation fieldand efficiently optimized the cost function by a stochasticgradient descent method but CoCoMI does not significantlychange the registration time These methods also have someintrinsic limitations due to the intensity distribution varying

illumination and geometric deformations and are caused forinstance by noise Feature-basedmethods directly use salientfeatures extracted from two images which ismore suitable forilluminated change and complicated geometric deformationTherefore comparing with area-based methods feature-based methods have also been widely used in remote sensingregistration [5ndash7]

Feature-based image registration methods consist of foursteps [8] Among them themost difficult part of a registrationprocess is the determination of the correspondence betweentwo given feature point sets of the images to be registeredIf some correspondences are incorrect they will produce anincorrect transformation function which could yield totallywrong resultsTherefore the correct feature correspondencesare a key problem for accurate image registration Graphspectral theory is a powerful tool which characterizes theglobal structural properties of graphs using the eigenvaluesand eigenvectors of either the adjacency matrix or theclosely related Laplacianmatrix [9]Therefore graph spectralmethods have been widely used in computer vision fields for

Hindawi Publishing CorporationComputational Intelligence and NeuroscienceVolume 2015 Article ID 829528 7 pageshttpdxdoiorg1011552015829528

2 Computational Intelligence and Neuroscience

feature matching such as graph matching [10ndash14] Scott andLonguet-Higgins [10] first introduced a Gaussian weightedfunction to build an interimage proximitymatrix to get corre-spondences from the strength matrix based on singular valuedecomposition which is sensitive to the degree of rotationTo resolve this problem Shapiro and Brady [11] constructedan intraimage proximity matrix for the individual point setsbeing matched which aims to capture relational image struc-tures but a sign correction stage is necessary Carcassoni andHancock [12] have shown how the modal structure of pointsets can be embedded into the framework of the expectationmaximization (EM) algorithm and improved the accuracy ofcorrespondences

Graph spectral method is effective in characterizing theglobal structure of image and kernel principal componentanalysis (KPCA) [7 15 16] has a close relationship withgraph spectral method which also has the similar meritsand is effective for pattern recognition regression analysisand nonparametric estimation But in order to improvethe real-time performance the dimensionality reductionbecomes a necessity One of the most popular dimensionalityreduction algorithms may be principal component analy-sis (PCA) [15] which performs dimensionality reductionby projecting the original 119899-dimensional data into the 119903-dimensional linear subspace by the leading eigenvectorsof the datarsquos covariance matrix Caelli and Kosinov [17]have extended the Shapiro and Brady method of seekingcorrespondences by searching for matching that maximizethe inner product of the truncated and renormalized eigen-vectors Xu andKing [18] exploited the PCAalgorithmwhichcan be used to quickly calculate the approximate matchingerror of two attributed graphs and reduce the executioncomplexity

Although many researchers have applied the PCA intocomputer vision and pattern recognition how to chooseadaptively the principle component based on the theo-rem is still not answered in detail In this paper we willgive an error analysis theorem and show how to chooseadaptively the principal component based on error analysistheorem which can further extend and improve the the-ory of PCA Therefore we exploit error analysis theoremand propose a novel robust adaptive principal componentanalysis (RAPCA) method based on intergraph matrix ofthe objects for image registration Firstly the RAPCA isextracted to capture the common structure pattern based onintergraph matrix Secondly the robust similarity measureis proposed based on adaptive principal component byprojecting both the reference image and the sensed imageinto the same lower dimensional feature space Thirdlythe robust registration algorithm is derived based on theRAPCA The experimental results show that the proposedmethod is very effective for image registration on real-worldimages

This paper is organized as follows Section 2 gives theerror analysis theorem and a novel image registration algo-rithm based on RAPCA is proposed Section 3 reportsthe experimental results and we draw the conclusion inSection 4

2 Robust Adaptive Principal ComponentAnalysis Based on Error Analysis Theorem

21 Error Analysis Theorem An adaptive principal compo-nent analysis based on error analysis theorem is exploited toextract principal feature component to describe the originaldata and abandon the interference of small eigenvaluescorresponding to eigenvectors We propose error analysistheorem according to Karhunen-Loeve Transform (KLT) [1920] and give a quantitative and qualitative analysis to showhow to choose adaptively the principal component to avoidchoosing the principal component 119903 by trial and error

Theorem 1 (error analysis theorem) Let image (matrix) 119883

represent an119873times119872matrix and119883 is transformed intoΦ119883 byorthogonal transform Φ that is 119884 = Φ119883 then mean squareerror can be expressed as follows

120576 (119903) =

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894=

119873

sum

119894=119903+1

120582119894 (1)

where Φ = (12060111206012 sdot sdot sdot 120601

119873)119879 is the orthogonal matrix 119862

119883=

119864[(119883 minus 119883)(119883 minus 119883)119879] is the covariance matrix and 120601

119894is

the eigenvector corresponding to the 119894th eigenvalue 120582119894of the

covariance matrix of 119862119883

Proof Let 119884 = Φ119883 be the transformed image (matrix) givingan orthogonal transform 119883 The original image (matrix) canbe rewritten as

119883 = Φ119879119884 = (120601

11206012sdot sdot sdot 120601119873)(

1199101

1199102

119910119873

) =

119873

sum

119894=1

120601119894119910119894 (2)

The estimate of119883 can be written as follows

119883(119903) =

119903

sum

119894=1

119910119894120601119894+

119873

sum

119894=119903+1

120572119894120601119894 (3)

where 120572119894are undetermined coefficients

The mean square error is defined as

120576 (119903) = 119864 [119883 minus 119883 (119903)]

119879

[119883 minus 119883 (119903)]

=

119873

sum

119894119895=119903+1

119864 [120601119879

119894(119910119894minus 120572119894)119879(119910119895minus 120572119895) 120601119895]

(4)

We have the following formulation according to orthogonal-ity condition120601

119879

119894120601119895= 1 if 119894 = 119895 otherwise120601119879

119894120601119895= 0 Consider

120576 (119903) =

119873

sum

119894=119903+1

119864 [(119910119894minus 120572119894)2] (5)

The gradient of the function 120576(119903) with respect to 120572119894is given

and let 120597120576(119903)120597120572119894= 0 Thus

120572119894= 119864 [119910

119894] = 119864 [120601

119879

119894119883] = 120601

119879

119894119883 (6)

Computational Intelligence and Neuroscience 3

Substituting (6) into (5) we have

120576 (119903) =

119873

sum

119894=119903+1

119864 [(119910119894minus 120572119894) (119910119894minus 120572119894)119879]

=

119873

sum

119894=119903+1

120601119879

119894119864 [(119883 minus 119883) (119883 minus 119883)

119879

] 120601119894

=

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894

(7)

Consequently we will prove the following formula

120576 (119903) =

119873

sum

119894=119872+1

120582119894 (8)

Firstly we construct an auxiliary function

119871 (119903) = 120576 (119903) minus

119873

sum

119894=119903+1

120582119894(120601119879

119894120601119894minus 1)

=

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894minus 120582119894(120601119879

119894120601119894minus 1)

(9)


and let 120597119871(119903)120597120601119894= 0 Thus

[119862119883] 120601119894= 120582119894120601119894 (10)

From (10) we can see that 120582119894is the eigenvalue of covariance

matrix119862119883 and120601

119894is the corresponding eigenvectorTherefore

120576 (119903) =

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894=

119873

sum

119894=119903+1

120582119894 (11)

So we can choose adaptively principal component accordingto mean square error (1) in order to avoid choosing theprincipal component 119903 by trial and error

In order to better exploit error analysis theorem wewill give an error rate (ER) as measure criterion to chooseadaptively principal component and the error rate is definedas follows

ER =

sum119873

119894=119903+1120582119894

sum119873

119894=1120582119894

(12)

Let the size of the matrix 119883 be a 119873 times 119872 dimensionalitythe matrix 119883 can be written as 119883 = (119909

1 1199092 119909

119872) We

construct the covariance matrix 119862119883

= (1119872)sum119872

119894=1(119909119894minus

120583119883)(119909119894minus 120583119883)119879 where 120583

119883= (1119872)sum

119872

119894=1119909119894and we also

obtain adaptively the 119903 most principal component 119880119883

=

[1199061 1199062 119906

119903] according to error rate (12) based on singular

value decomposition

22 Registration Algorithm Based on RAPCA First we con-struct an intergraphmatrix of the two sets of features and giveadaptive principal component based on intergraph matrix

to capture the common structure pattern Then we canproject the intranormalized Laplacian graph matrix into thesame lower dimensional feature space to reveal its structurepatterns respectively The detailed registration algorithmbased on RAPCA is described as follows

Step 1 Construct an intergraph matrix119883 isin 119877119872times119872

of the twopoint sets of 119866 and 119867 and the 119883 is defined as 119883

119894119895= 119890minus1199032

11989411989521205902

where 119903

2

119894119895= 119866119894minus 1198671198952 is the Euclidean distance between the

two point sets and the scale parameter 120590 controls the degreeof interaction between the two sets of features

Step 2 Construct the covariance matrix 119862119883

=

(1119872)sum119872

119894=1(119909119894minus120583119883)(119909119894minus120583119883)119879 based on an intergraphmatrix

119883 and then compute its eigenvalues and eigenvectors

Step 3 Choose adaptively the principal component119880119883

= [1199061

1199062 119906

119903] according to error rate (12)

Step 4 Construct intranormalized Laplacian graph matrices119871119866and 119871

119867of point sets and the normalized Laplacian graph

matrix 119871119866is defined as

119871119866(V119894 V119895) =

minus

10038171003817100381710038171003817V119894minus V119895

10038171003817100381710038171003817

2

radic119889V119894

119889V119895

119894 = 119895

1 119894 = 119895

(13)

and 119889V119894

= sum119872

119895=1V119894minus V1198952 The normalized Laplacian matrix

119871119867is similarly defined Please see [9] for details

Step 5 Project119871119866V119894

and119871119867V119895

into the same lower dimensionalfeature space to reveal the structure patterns which is definedas 119875119866V119894

= 119880119879

119883119871119866V119894

and 119875119867V119895

= 119880119879

119883119871119867V119895

Step 6 Compute the similarity matrix 119878119894119895

= 119875119866V119894

minus 119875119867V119895

2

119865

where 119878119894119895reflects the similarity between two point sets 119866 and

119867 The points 119866V119894

and 119867V119895

having a 1 1 correspondence withone another are given based on the elements of 119878

119894119895being

both the smallest element in its row and the smallest elementin its column Otherwise it is many-to-one or one-to-many correspondence Finally the transformation functionis obtained by the matching relationship

3 Experimental Results and Discussion

In this section we provide some experimental evaluations ofregistration algorithm based on RAPCA for image registra-tion We can choose adaptively the principal component 119903

according to the error rate when it is less than 10minus1 and all theexperiments have been done on a personal computer usingMATLAB R2010a with Intel(R) Core (TM) CPU 253GHzand 400GB RAM There are analyses on real-world imageswhich aim to demonstrate that the proposed method is effi-cient and feasible comparing with Caellirsquos method (clusteringmatching method) [17]


(a) (b) (c)

(d) (e) (f)

Figure 1 Matching results for different feature points Top row matching results based on Caellirsquos method Bottom row matching resultsbased on our method

To test RAPCA algorithm we applied it to medicalimages Figure 1 shows the comparison of matching resultsusing our method and Caellirsquos method [17] to test on T1and T2 of the 24th slice of a magnetic resonance imaging(MRI) sequence The 18 feature points 25 feature pointsand 29 feature points are extracted by the Harris CornerDetector [21] from Figures 1(a) and 1(d) 1(b) and 1(e) and1(c) and 1(f) respectively From Figures 1(a) and 1(d) wecan see that the feature points matching are one-to-onecorrespondence with Caellirsquos method and our method Withincrease in the number of feature points Caellirsquos methodproduces more many-to-one correspondence as shown inFigures 1(b) and 1(c) However our method still achievesa one-to-one correspondence as shown in Figures 1(e) and1(f) Our RAPCA algorithm has high matching ability byprojecting intranormalized Laplacian graph matrix into thesame lower dimensional feature space based on intergraphmatrix which can reveal the internal geometrical structureinformation of two point sets Caellirsquos method produces somemany-to-one correspondence because the distance betweensome points is very close which are considered to be in thesame class In addition Caellirsquos method is not also stableand can produce different matching results with differentfeature points extracted These MRI images are examples toillustrate that the features matching of our method is betterthan Caellirsquos method

Figure 2 shows the matching performance of Caellirsquosmethod and our method From the experimental results wecan see that the feature points correspondences are shownin Figures 2(a) 2(b) and 2(c) which reflect the correctcorrespondence relationship of Figures 1(a) and 1(d) 1(b) and1(e) and 1(c) and 1(f) respectively The results indicate thatour method is robust and better than Caellirsquos method

To further test our algorithm we applied the proposedmethod to the 217 times 181 medical images from the samepatient of different modality from the brain datasets [22]Figure 3 (top row and bottom row) gives matching resultsfor different modality images with Caellirsquos method and ourmethod respectivelyThefirst and second columngive T1 andPD matching results and the third and fourth column givePD and T2 matching results respectively Figures 3(a) and3(b) give T1 and PD matching results with different featurepoints which produce the different results and the matchingresults are bad as shown in Figure 3(b) Figures 3(c) and 3(d)give PD and T2matching results with different feature pointswhich also produce the different results and the matchingresults are also bad as shown in Figure 3(c)Therefore Caellirsquosmethod is not stable with different feature points from the toprowof Figure 3However ourmethod can find correct featurecorrespondences which show that the proposed method iseffective and feasible for different modality images The rea-son is thatwe incorporate the intergraphmatrix to capture thecommon structure pattern and obtain the adaptive principalcomponent based on error analysis theorem Meanwhilethe robust similarity measure is proposed based on robustprincipal component by projecting both the reference imageand the sensed image into the same lower dimensional featurespace to reduce computational complexity Table 1 also showsthe comparison of computation time which indicates that thecomputation time of our method is less than Caellirsquos method

4 Conclusion

In this paper we present a novel RAPCA method basedon intergraph matrix for image registration Firstly wegive an error analysis theorem and an adaptive principal


0 2 4 6 8 10 12 14 16 180

2

4

6

8

10

12

14

16

18

The number of feature points

Feat

ure c

orre

spon

denc

e

(a)

0 5 10 15 20 250

5

10

15

20

25


Feat

ure c

orre

spon

denc

e

(b)

0 5 10 15 20 25 300

5

10

15

20

25

30


Feat

ure c

orre

spon

denc

e

Caellis methodOur method

(c)

Figure 2 Performance comparison of features correspondence on the MRI images of Caellirsquos method and our method (a) Results of Figures1(a) and 1(d) (b) results of Figures 1(b) and 1(e) and (c) results of Figures 1(c) and 1(f)

Table 1 Comparison of the computation time of Figures 1 and 3

Figure and computation time Caellirsquos and our method (seconds)

Figure 1 (a) and (d) (b) and (e) (c) and (f)0560803639 0626803834 0712204057

Figure 3 (a) and (e) (b) and (f) (c) and (g) (d) and (h)0507102931 0803703573 0403603030 0525903670

component is extracted based on error analysis theoremby incorporating intergraph matrix to capture the com-mon structure pattern of the objects Secondly the robustsimilarity measure is proposed based on robust principalcomponent by projecting both the reference image andthe sensed image into the same lower dimensional featurespace Thirdly the robust registration algorithm is givenbased on the RAPCA The experimental results indicatethat the proposed method is effective and feasible for imageregistration

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported in part by the National NaturalScience Foundation of China under Grant no 61363049 theproject funded by China Postdoctoral Science Foundationunder Grant no 2014M550881 the Open Project Program


(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 3 Matching results for different modality images Top row matching results based on Caellirsquos method Bottom row matching resultsbased on our method

of the State Key Laboratory of Management and Control forComplex Systems under Grant no 20140101 the ScientificResearch Fund of Jiangxi Provincial Education Departmentunder Grant no GJJ14541 the Open Project Program ofthe Key Laboratory of Nondestructive Testing Ministry ofEducation under Grant no ZD201429007 and the DoctorScientific Research Starting Foundation under Grant noEA201307044

References

[1] J B A Maintz andM A Viergever ldquoA survey of medical imageregistrationrdquoMedical Image Analysis vol 2 no 1 pp 1ndash36 1998

[2] A Myronenko and X B Song ldquoIntensity-based image registra-tion by minimizing residual complexityrdquo IEEE Transactions onMedical Imaging vol 29 no 11 pp 1882ndash1891 2010

[3] K Ikeda F Ino and K Hagihara ldquoEfficient accelerationof mutual information computation for nonrigid registrationusing CUDArdquo IEEE Journal of Biomedical and Health Informat-ics vol 18 no 3 pp 956ndash968 2014

[4] H Rivaz Z Karimaghaloo V S Fonov and D L CollinsldquoNonrigid registration of ultrasound andMRI using contextualconditioned mutual informationrdquo IEEE Transactions on Medi-cal Imaging vol 33 no 3 pp 708ndash725 2014

[5] Q L Li G Y Wang J G Liu and S B Chen ldquoRobustscale-invariant feature matching for remote sensing imageregistrationrdquo IEEE Geoscience and Remote Sensing Letters vol6 no 2 pp 287ndash291 2009

[6] Z Xiong and Y Zhang ldquoA novel interest-point-matching algo-rithm for high-resolution satellite imagesrdquo IEEETransactions onGeoscience and Remote Sensing vol 47 no 12 pp 4189ndash42002009

[7] X Duan Z Tian M Ding and W Zhao ldquoRegistration ofremote-sensing images using robust weighted kernel principal

component analysisrdquo AEU-International Journal of Electronicsand Communications vol 67 no 1 pp 20ndash28 2013

[8] B Zitova and J Flusser ldquoImage registration methods a surveyrdquoImage and Vision Computing vol 21 no 11 pp 977ndash1000 2003

[9] F R K Chung Spectral GraphTheory AmericanMathematicalSociety Providence RI USA 1997

[10] G L Scott and H C Longuet-Higgins ldquoAn algorithm forassociating the features of two imagesrdquo Proceedings of the RoyalSociety B Biological Sciences vol 244 no 1309 pp 21ndash26 1991

[11] L S Shapiro and J M Brady ldquoFeature-based correspondencean eigenvector approachrdquo Image and Vision Computing vol 10no 5 pp 283ndash288 1992

[12] M Carcassoni and E R Hancock ldquoSpectral correspondence forpoint pattern matchingrdquo Pattern Recognition vol 36 no 1 pp193ndash204 2003

[13] A Egozi Y Keller and H Guterman ldquoImproving shaperetrieval by spectral matching andmeta similarityrdquo IEEE Trans-actions on Image Processing vol 19 no 5 pp 1319ndash1327 2010

[14] J Ma J Zhao J Tian A L Yuille and Z Tu ldquoRobust pointmatching via vector field consensusrdquo IEEE Transactions onImage Processing vol 23 no 4 pp 1706ndash1721 2014

[15] B Scholkopf A Smola and K-R Muller ldquoNonlinear compo-nent analysis as a Kernel eigenvalue problemrdquo Neural Compu-tation vol 10 no 5 pp 1299ndash1319 1998

[16] H Sahbi ldquoKernel PCA for similarity invariant shape recogni-tionrdquo Neurocomputing vol 70 no 16ndash18 pp 3034ndash3045 2007

[17] T Caelli and S Kosinov ldquoAn eigenspace projection clusteringmethod for inexact graph matchingrdquo IEEE Transactions onPattern Analysis andMachine Intelligence vol 26 no 4 pp 515ndash519 2004

[18] L Xu and I King ldquoA PCA approach for fast retrieval ofstructural patterns in attributed graphsrdquo IEEE Transactions onSystems Man and Cybernetics Part B Cybernetics vol 31 no5 pp 812ndash817 2001


[19] R C Zhao Introduction to Digital Image Processing North-western Polytechnical University Press Xirsquoan China 2000(Chinese)

[20] C M Bishop Pattern Recognition and Machine LearningSpringer New York NY USA 2006

[21] C Harris and M Stephens ldquoA combined corner and edgedetectorrdquo in Proceedings of the Alvey Vision Conference pp 147ndash151 1988

[22] httpbiomedicdocicacukbrain-developmentindexphpn=MainDatasets

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feature matching such as graph matching [10ndash14] Scott andLonguet-Higgins [10] first introduced a Gaussian weightedfunction to build an interimage proximitymatrix to get corre-spondences from the strength matrix based on singular valuedecomposition which is sensitive to the degree of rotationTo resolve this problem Shapiro and Brady [11] constructedan intraimage proximity matrix for the individual point setsbeing matched which aims to capture relational image struc-tures but a sign correction stage is necessary Carcassoni andHancock [12] have shown how the modal structure of pointsets can be embedded into the framework of the expectationmaximization (EM) algorithm and improved the accuracy ofcorrespondences

Graph spectral method is effective in characterizing theglobal structure of image and kernel principal componentanalysis (KPCA) [7 15 16] has a close relationship withgraph spectral method which also has the similar meritsand is effective for pattern recognition regression analysisand nonparametric estimation But in order to improvethe real-time performance the dimensionality reductionbecomes a necessity One of the most popular dimensionalityreduction algorithms may be principal component analy-sis (PCA) [15] which performs dimensionality reductionby projecting the original 119899-dimensional data into the 119903-dimensional linear subspace by the leading eigenvectorsof the datarsquos covariance matrix Caelli and Kosinov [17]have extended the Shapiro and Brady method of seekingcorrespondences by searching for matching that maximizethe inner product of the truncated and renormalized eigen-vectors Xu andKing [18] exploited the PCAalgorithmwhichcan be used to quickly calculate the approximate matchingerror of two attributed graphs and reduce the executioncomplexity

Although many researchers have applied the PCA intocomputer vision and pattern recognition how to chooseadaptively the principle component based on the theo-rem is still not answered in detail In this paper we willgive an error analysis theorem and show how to chooseadaptively the principal component based on error analysistheorem which can further extend and improve the the-ory of PCA Therefore we exploit error analysis theoremand propose a novel robust adaptive principal componentanalysis (RAPCA) method based on intergraph matrix ofthe objects for image registration Firstly the RAPCA isextracted to capture the common structure pattern based onintergraph matrix Secondly the robust similarity measureis proposed based on adaptive principal component byprojecting both the reference image and the sensed imageinto the same lower dimensional feature space Thirdlythe robust registration algorithm is derived based on theRAPCA The experimental results show that the proposedmethod is very effective for image registration on real-worldimages

This paper is organized as follows Section 2 gives theerror analysis theorem and a novel image registration algo-rithm based on RAPCA is proposed Section 3 reportsthe experimental results and we draw the conclusion inSection 4

2 Robust Adaptive Principal ComponentAnalysis Based on Error Analysis Theorem

21 Error Analysis Theorem An adaptive principal compo-nent analysis based on error analysis theorem is exploited toextract principal feature component to describe the originaldata and abandon the interference of small eigenvaluescorresponding to eigenvectors We propose error analysistheorem according to Karhunen-Loeve Transform (KLT) [1920] and give a quantitative and qualitative analysis to showhow to choose adaptively the principal component to avoidchoosing the principal component 119903 by trial and error

Theorem 1 (error analysis theorem) Let image (matrix) 119883

represent an119873times119872matrix and119883 is transformed intoΦ119883 byorthogonal transform Φ that is 119884 = Φ119883 then mean squareerror can be expressed as follows

120576 (119903) =

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894=

119873

sum

119894=119903+1

120582119894 (1)

where Φ = (12060111206012 sdot sdot sdot 120601

119873)119879 is the orthogonal matrix 119862

119883=

119864[(119883 minus 119883)(119883 minus 119883)119879] is the covariance matrix and 120601

119894is

the eigenvector corresponding to the 119894th eigenvalue 120582119894of the

covariance matrix of 119862119883

Proof Let 119884 = Φ119883 be the transformed image (matrix) givingan orthogonal transform 119883 The original image (matrix) canbe rewritten as

119883 = Φ119879119884 = (120601

11206012sdot sdot sdot 120601119873)(

1199101

1199102

119910119873

) =

119873

sum

119894=1

120601119894119910119894 (2)

The estimate of119883 can be written as follows

119883(119903) =

119903

sum

119894=1

119910119894120601119894+

119873

sum

119894=119903+1

120572119894120601119894 (3)

where 120572119894are undetermined coefficients

The mean square error is defined as

120576 (119903) = 119864 [119883 minus 119883 (119903)]

119879

[119883 minus 119883 (119903)]

=

119873

sum

119894119895=119903+1

119864 [120601119879

119894(119910119894minus 120572119894)119879(119910119895minus 120572119895) 120601119895]

(4)

We have the following formulation according to orthogonal-ity condition120601

119879

119894120601119895= 1 if 119894 = 119895 otherwise120601119879

119894120601119895= 0 Consider

120576 (119903) =

119873

sum

119894=119903+1

119864 [(119910119894minus 120572119894)2] (5)


and let 120597120576(119903)120597120572119894= 0 Thus

120572119894= 119864 [119910

119894] = 119864 [120601

119879

119894119883] = 120601

119879

119894119883 (6)



120576 (119903) =

119873

sum

119894=119903+1

119864 [(119910119894minus 120572119894) (119910119894minus 120572119894)119879]

=

119873

sum

119894=119903+1

120601119879

119894119864 [(119883 minus 119883) (119883 minus 119883)

119879

] 120601119894

=

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894

(7)


120576 (119903) =

119873

sum

119894=119872+1

120582119894 (8)


119871 (119903) = 120576 (119903) minus

119873

sum

119894=119903+1

120582119894(120601119879

119894120601119894minus 1)

=

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894minus 120582119894(120601119879

119894120601119894minus 1)

(9)


and let 120597119871(119903)120597120601119894= 0 Thus

[119862119883] 120601119894= 120582119894120601119894 (10)


matrix119862119883 and120601


120576 (119903) =

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894=

119873

sum

119894=119903+1

120582119894 (11)



ER =

sum119873

119894=119903+1120582119894

sum119873

119894=1120582119894

(12)


1 1199092 119909

119872) We


= (1119872)sum119872

119894=1(119909119894minus

120583119883)(119909119894minus 120583119883)119879 where 120583

119883= (1119872)sum

119872

119894=1119909119894and we also


=

[1199061 1199062 119906


value decomposition





119894119895= 119890minus1199032

11989411989521205902

where 119903

2




=

(1119872)sum119872




= [1199061

1199062 119906





119871119866(V119894 V119895) =

minus

10038171003817100381710038171003817V119894minus V119895

10038171003817100381710038171003817

2

radic119889V119894

119889V119895

119894 = 119895

1 119894 = 119895

(13)

and 119889V119894

= sum119872



Step 5 Project119871119866V119894

and119871119867V119895


= 119880119879

119883119871119866V119894

and 119875119867V119895

= 119880119879

119883119871119867V119895


= 119875119866V119894

minus 119875119867V119895

2

119865


119867 The points 119866V119894

and 119867V119895


119894119895being






(a) (b) (c)

(d) (e) (f)





4 Conclusion



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(c)









Acknowledgments



(a) (b) (c) (d)

(e) (f) (g) (h)



References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in





120576 (119903) =

119873

sum

119894=119903+1

119864 [(119910119894minus 120572119894) (119910119894minus 120572119894)119879]

=

119873

sum

119894=119903+1

120601119879

119894119864 [(119883 minus 119883) (119883 minus 119883)

119879

] 120601119894

=

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894

(7)


120576 (119903) =

119873

sum

119894=119872+1

120582119894 (8)


119871 (119903) = 120576 (119903) minus

119873

sum

119894=119903+1

120582119894(120601119879

119894120601119894minus 1)

=

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894minus 120582119894(120601119879

119894120601119894minus 1)

(9)


and let 120597119871(119903)120597120601119894= 0 Thus

[119862119883] 120601119894= 120582119894120601119894 (10)


matrix119862119883 and120601


120576 (119903) =

119873

sum

119894=119903+1

120601119879

119894[119862119883] 120601119894=

119873

sum

119894=119903+1

120582119894 (11)



ER =

sum119873

119894=119903+1120582119894

sum119873

119894=1120582119894

(12)


1 1199092 119909

119872) We


= (1119872)sum119872

119894=1(119909119894minus

120583119883)(119909119894minus 120583119883)119879 where 120583

119883= (1119872)sum

119872

119894=1119909119894and we also


=

[1199061 1199062 119906


value decomposition





119894119895= 119890minus1199032

11989411989521205902

where 119903

2




=

(1119872)sum119872




= [1199061

1199062 119906





119871119866(V119894 V119895) =

minus

10038171003817100381710038171003817V119894minus V119895

10038171003817100381710038171003817

2

radic119889V119894

119889V119895

119894 = 119895

1 119894 = 119895

(13)

and 119889V119894

= sum119872



Step 5 Project119871119866V119894

and119871119867V119895


= 119880119879

119883119871119866V119894

and 119875119867V119895

= 119880119879

119883119871119867V119895


= 119875119866V119894

minus 119875119867V119895

2

119865


119867 The points 119866V119894

and 119867V119895


119894119895being






(a) (b) (c)

(d) (e) (f)





4 Conclusion



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(b)

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(c)









Acknowledgments



(a) (b) (c) (d)

(e) (f) (g) (h)



References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




(a) (b) (c)

(d) (e) (f)





4 Conclusion



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(b)

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30


Feat

ure c

orre

spon

denc

e


(c)









Acknowledgments



(a) (b) (c) (d)

(e) (f) (g) (h)



References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




0 2 4 6 8 10 12 14 16 180

2

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Feat

ure c

orre

spon

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(a)

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Feat

ure c

orre

spon

denc

e

(b)

0 5 10 15 20 25 300

5

10

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20

25

30


Feat

ure c

orre

spon

denc

e


(c)









Acknowledgments



(a) (b) (c) (d)

(e) (f) (g) (h)



References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in




(a) (b) (c) (d)

(e) (f) (g) (h)



References
































Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in















Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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