Research Article Image Matching Based on Local Phase …downloads.hindawi.com/journals/mpe/2016/5182106.pdf · 2019. 7. 30. · Research Article Image Matching Based on Local Phase

Research ArticleImage Matching Based on Local Phase Quantization Applied forMeasuring the Tensile Properties of High Elongation Materials

Zhenjun Zhang12 Fengji Li1 Min Liu1 and Pawan Kumar Yadav1

1Department of Electric and Information Engineering Hunan University Yuelu District Changsha 410082 China2National Engineering Laboratory for Robot Visual Perception and Control Technology Yuelu District Changsha 410082 China

Correspondence should be addressed to Zhenjun Zhang zhenjunhnueducn

Received 10 May 2016 Revised 23 August 2016 Accepted 24 August 2016

Academic Editor Eric Florentin

Copyright copy 2016 Zhenjun Zhang 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

In recent years high elongation materials are widely used Therefore it is important to investigate the tensile properties of highelongationmaterials for engineering applications Video extensometer is equipment for measuring thematerialsrsquo tensile propertiesIt uses image processing technology to match data points and measures the actual deformation However when measuring highelongationmaterialsmotion blurwill appear on the collected images which can affect the accuracy of imagematching In this paperwe proposed an image matching method which is based on Local Phase Quantization (LPQ) features to reduce the interference ofthe motion blur and improve the accuracy of the image matching algorithms as well The experimental results on simulations showthat the proposed initialization method is sufficiently accurate to enable the correct convergence of the subsequent optimization inthe presence of motion blur The experiment of uniaxial tensile also verifies the accuracy and robustness of the method

1 Introduction

High elongation materials are an important class of mate-rials for structural applications such as transportation civilinfrastructures and biomedical applications In actual serviceconditions these materials are often subject to both mechan-ical and environmental loads These factors will change thematerial properties and thus have a great influence on theservice life and safety performance of these materials Inorder to study these factors the tensilemechanical test shouldbe carried out on these materials

At present the most commonly used equipment for thetensile mechanical test is the mechanical extensometer andvideo extensometer For the high elongation materials themechanical extensometer which is mounted directly ontothe material via blade causes many problems such as thefollowing (1) mutual friction will reduce the measurementaccuracy (2) the total deformation cannot be easilymeasuredin the uniaxial tensile test (3) the measuring range islimited Compared with mechanical extensometer the videoextensometer has the following advantages over mechanicalextensometer (1) it has no abrasion on the material (2)

it is applied for different types of specimen and materialproperties (3) its measuring range is not limited (4) it hashigh precision [1]

Compared with mechanical extensometer video exten-someter has obvious advantages in engineering applicationsHowever if higher accuracy is pursued some influencefactors cannot be ignored such as out-of-plane displacementself-heating of the camera lens distortion and image blurinduced by motion Reference [2] theoretically describes themeasurement errors caused by out-of-plane displacementand self-heating of the camera it further establishes a high-accuracy two-dimensional digital image correlation (2D-DIC) system using a bilateral telecentric lens to minimize theerrors Reference [3] investigates the systematic errors dueto lens distortion using the radial lens distortion model andin-plane translation tests it finds out that the displacementand strain errors at an interrogated image point not onlyare linear proportion to the distortion coefficient of thecamera lens used but also depend on the distance relativeto distortion center and its magnitude of the displacementthe paper also proposes a linear least-squares algorithm toestimate the distortion coefficients and then to eliminate

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5182106 10 pageshttpdxdoiorg10115520165182106

2 Mathematical Problems in Engineering

Clamp

Material

CCD camera

Light source

Computer

Figure 1 Measurement system overview

the errors Reference [4] proposes an off-axis digital imagecorrelation method for real-time noncontact and targetlessmeasurement of vertical deflection of bridges to achievesubpixel accuracy

Despite these advances few works about eliminat-ing extensometerrsquos measurement errors caused by motion-induced image blur to improve the accuracy have beenreported In this paper we will propose an image matchingmethod for video extensometer to measure the parametersby utilizing Local Phase Quantization (LPQ) feature Thismethod is robust and performs well on images with seriousmotion blur and deformation

The rest of the paper is organized as follows The basicprinciple of video extensometer is described in Section 2The generation mechanism of motion blur is introduced inSection 3The proposed matching method using Local PhaseQuantization is discussed in detail in Section 4 Experimentalresults on both simulated and real-world data as well ascomparative results with the existing method are presentedin Section 5 Finally Section 6 gives the conclusion

2 System Overview

A schematic diagram of the video extensometer system isshown in Figure 1 The main parts of the video extensometersystem are a light source tensile equipment CCD cameraand PC with digital image processing software

For measurement first make the spackle pattern on thematerial surface keep the CCD camerarsquos optical axis verticalto the test specimen surface choose the appropriate focallength and make sure the field of view covers the wholematerial In the tensile stress test the position of the datapoint will be changed before and after the deformationThrough the deformation images collected by CCD cameraat different times we can calculate the materialsrsquo tensileproperties

21 Digital Image Correlation The video extensometer usesDIC to analyze materials tensile properties [2ndash4] The prin-ciple of DIC is tracking the same point in undeformed anddeformed image and then yields the displacements of thedata point The information contained by the data pointcannot identify the same point in a series of deformed imagesSo we use a square subset centered on the data point toreplace the data point as shown in Figure 2(a) A smallsquare speckle subset of 2119873 times 2119873 pixel centered at the datapoint 119875 in the undeformed image is defined as the referencesubset At the same time a bigger subset of 2119872 times 2119872 pixelscentered at the corresponding data point 1198751015840 in the deformedimage is defined as the searching subset To calculate thecorrelation coefficient the reference subset is moved throughthe searching subset pixel by pixel The peak position of thecorrelation coefficient is the target subset as can be seen fromFigure 2(b)

As shown in Figure 2(b) the stress on the material willcause rigid body displacement scale distortion and rotationdeformation So the intensity of an arbitrary point in theundeformed subset region and the corresponding point in thedeformation subset region is related as follows

119892 (1199091015840119894 1199101015840119894 ) = 119865 (119891 (119909119894 119910119894)) (1)

For the shape function 119865 the most widely used functionis first-order shape function [5]

1199091015840 = 119909 + 119906 + 120597119906120597119909Δ119909 + 120597119906120597119910Δ1199101199101015840 = 119910 + V + 120597V120597119909Δ119909 + 120597V120597119910Δ119910

(2)

The corresponding matrix form is

[11988310158401198841015840] = [119909119910] + [119906 119906119909 119906119910V V119909 V119910

][[[1Δ119909Δ119910]]] (3)

Mathematical Problems in Engineering 3

xO

y

P(x0 y0)

Reference subset

Undeformed image

(a)

P998400(x9984000 y998400

0)

Target subset

Searching subset

xO

y

Deformed image

(b)

Figure 2 Procedures of DIC

where 120597119906120597119909 120597119906120597119910 120597V120597119909 and 120597V120597119910 are the displacementgradients of the subset Δ119909 and Δ119910 are the distances betweenpoint (119909 119910) and point (1199090 1199100) respectively

Images are obtained in discrete form and intensity valuesare recorded as ldquopixelsrdquo But the integer pixel locationsdo not satisfy the accuracy of extensometer In order toimprove the precision of video extensometer we use subpixeldisplacement measurement algorithm to get the subpixel dis-placement The correlation coefficient curve fitting methoditerative algorithm and gradient-based algorithm are thethree most common methods [6 7] In this paper we usedthe N-R iteration algorithm [8] as the subpixel displacementmeasurement algorithm

The displacement refined by optimizing correlation usingN-R iteration algorithm needs to calculate the correlationcoefficient which describes the similarity between the ref-erence subset and the target subset In the actual situationthe spackle pattern will be affected by the lighting conditionand the deformation Tong [9] used different correlationcoefficient to analyze the result of DIC and point out theZNSSD and ZNCC have better robustness and reliabilityTherelationship between two related functions is as follows

119862ZNSSD = 2 (1 minus 119862ZNCC) (4)

and it is easier to optimize ZNSSD which is expressed as thesum of squares of nonlinear functions

119862ZNSSD

= 119872sum119894=minus119872

119872sum119895=minus119872

[119891 (119909119894 119910119895) minus 119891119898]Δ119891 minus [119892 (1199091015840119894 1199101015840119895) minus 119892119898]Δ119892

2

(5)

where

119891119898 = 1(2119872 + 1)2119872sum119894=minus119872

119872sum119895=minus119872

119891 (119909119894 119910119895)

119892119898 = 1(2119872 + 1)2119872sum119894=minus119872

119872sum119895=minus119872

119892 (1199091015840119894 119910119895)

Δ119891 = radic 119872sum119894=minus119872

119872sum119895=minus119872

[119891 (119909119894 119910119895) minus 119891119898]2

Δ119892 = radic 119872sum119894=minus119872

119872sum119895=minus119872

[119892 (1199091015840119894 1199101015840119895) minus 119892119898]2(6)

Finally use the optimal parameter which minimizesZNSSD to calculate the data point displacement and get thematerials tensile properties

3 Motion Blur

When the CCD camera collects an image the image mayrepresent the scene over a period of time known as theexposure time But the relative motion between the cameraand material during exposure time may result in a blurringimage which is motion blur

The ideal lens imaging model is shown in Figure 3according to Gaussian lens law

1119906 + 1V = 1119891 (7)

where 119906 is the object distance V is image distance and 119891 isthe focal length Using (7) we can calculate image distance asfollows

V = 119891119906119906 minus 119891 (8)


Lens

Focal point

Imagesensor

Objectplane

u f

v

x

y

O

Figure 3 Ideal lens imaging model

(a) (b) (c)

Figure 4 Rubber tensile images

Based on the tension speed V0 we can get the movementdistance 119909 of material during exposure time and calculate theblur distance 119910 using the following formula

119910 = V0119905V119906 (9)

Using (9) we calculated the blur distance for differentfocal lengths and tensile rate list in Table 1

As we can see from Table 1 the blur distance can reach30 120583m and cause 5-pixel blur motions on images Figure 4shows the three images under different stretching conditionsthe collected image will blur as the material is stretchedimage details are lost and edges become blurredThe blurredimage significantly influences the accuracy of the videoextensometer

4 Integer Pixel Displacement Search Based onLocal Phase Quantization Features

In order to get more accurately measured result we use N-Riteration algorithm but the convergent range of N-R iterationalgorithm is only a few pixels [10] in order to make thealgorithm converge rapidly and accurately an accurate initialestimate of the deformation parameter must be provided

The conventional method for initializing the deformationparameter is using integer pixel displacement search It isused to find the peak position of the correlation coefficientin a deformed image pixel by pixel However this methoddepends on gray information it cannot handle large rotationand motion blur because it assumes the subset shape isunchanged [11ndash16] Recent works [17 18] use the optimizedparameter of the current point as the initial estimate for itsneighbors based on the assumption that the deformations ofneighboring points are very similar The calculation startedfrom a seed point manually selected in the undeformedimage The seed pointrsquos deformation parameter is initializedby using either manually specified correspondence or con-ventional search scheme After being optimized the param-eter is transferred to an adjacent point The whole procedurewas repeated until all the data points have been analyzedHowever it still has several drawbacks The series of imagesrequires that the shape of its local region remains unchangedin the entire deformation process This assumption is easilyviolated in large motion blur and deformation

It can be seen that image blur will change the imageinformation and make it difficult to identify the data pointthus affecting the video extensometermeasurement accuracyIn order to make the video extensometer more accurate weneed to eliminate the influence of motion blur on integer


Table 1 The blur distance for various types of lenses and tensile rate

Focal length(mm)

Imagedistance(mm)

Size of pixel(120583m)

Exposuretime (s)

Tensile rate(mmmin)

Objectdistance(mm)

Blur distance(120583m)

Blur pixel(pix)

20 2142 52 001 100 300 119 02350 60 52 001 100 300 333 06480 10909 52 001 100 300 606 11720 2142 52 001 200 300 238 04650 60 52 001 200 300 667 12880 10909 52 001 200 300 1212 23320 2142 52 001 500 300 595 11450 60 52 001 500 300 1666 32180 10909 52 001 500 300 3030 583

pixel displacement search For blurred images matchingconventional methods are based on a prior knowledge ofthe blurred image to restore a clear image [19ndash21] and thenmatching the data point in the clear image However imagerestoration is an ill-posed problem process Restoration effectof the blurred image depends on the prior knowledge aboutthe degradation process and the efficiency of the restorationalgorithm Actually we can directly extract invariant featuresfrom blurred images [22] to design matching algorithm [23]which not only consumes less computation resource but alsoimproves the accuracy and precision

Based on the above ideas this chapter proposes aninteger pixel searching algorithm based on LPQ feature [24]which is invariant to centrally symmetric blur and uniformillumination changes

41 Blur Invariance Using Fourier Transform Phase In digitalimage processing the discrete model for spatially shift-invariant blurring of an ideal image 119891(119909 119910) resulting in anobserved image 119892(119909 119910) can be expressed by a convolutiongiven by 119892 (119909 119910) = 119891 (119909 119910) lowast ℎ (119909 119910) + 119899 (119909 119910) (10)where ℎ(119909 119910) is the point spread function (PSF) of the system119899(119909 119910) is additive noise and lowast denotes 2D convolution

The LPQ featuremodel is based on the following assump-tions (1) the noise pollution can be ignored (2) the PSF is thecentrally symmetric Depending on the above assumption(10) can be written as119892 (119909 119910) = 119891 (119909 119910) lowast ℎ (119909 119910) (11)

In the Fourier domain this corresponds to119866 (119909 119910) = 119865 (119909 119910) lowast 119867 (119909 119910) (12)where 119866(119909 119910) 119865(119909 119910) and 119867(119909 119910) are the discrete Fouriertransforms (DFT) of the observed image 119892(119909 119910) the idealimage 119891(119909 119910) and the point spread function ℎ(119909 119910) Equa-tion (12) can be separated into themagnitude and phase partsresulting in 1003816100381610038161003816119866 (119909 119910)1003816100381610038161003816 = 1003816100381610038161003816119865 (119909 119910) times 119867 (119909 119910)1003816100381610038161003816 120593119892 (119906 V) = 120593119891 (119906 V) + 120593ℎ (119906 V) (13)

Because the PSF ℎ(119909 119910) is centrally symmetric namelyℎ(119909 119910) = ℎ(minus119909 minus119910) its Fourier transform is always real-valued and as a consequence its phase is only a two-valuedfunction given by

120593ℎ (119906 V) = 0 if 119867(119906 V) ge 0120587 if 119867(119906 V) lt 0 (14)

This means that

120593119892 (119906 V) = 120593119891 (119906 V) 119867 (119906 V) gt 0120593119892 (119906 V) + 120587 119867 (119906 V) lt 0 (15)

From the above equation when119867(119906 V) ge 0 the phase ofthe observed image 119866(119906 V) at the frequencies is invariant tocentrally symmetric blur

42 Extraction of LPQ Feature The LPQ feature is based onthe blur invariance property of the Fourier phase spectrumdescribed in Section 41 It uses the local phase informationextracted by the 2D DFT computed over a rectangularneighborhood 119873(119909119910) The 2D DFT on pixel (119909 119910) denotedas 119865(119909119910)(119906 V) is defined by

119865(119909119910) (119906 V)= sum(1199091015840 1199101015840)isin119873119909119910

119891 (119909 minus 1199091015840 119910 minus 1199101015840) 119890minus(11989521205871199061199091015840+11989521205871199061199101015840)= 119882119879119906 119891(119909119910)

(16)

where119882119906 is the basis vector of the 2D DFT at a frequency 119906and 119891(119909119910) is another vector containing all1198722 image samplesfrom119873(119909119910)

In LPQ only four frequency points are considered suchas 1199061 = [0 120572]119879 1199061 = [119886 0]119879 1199061 = [119886 120572]119879 and 1199061 = [119886 minus120572]119879where 119886 satisfies 119867(120572 120572) ge 0 For each pixel position thisresults in a vector119865(119909119910) = [119865 (1199061 (119909 119910)) 119865 (1199062 (119909 119910)) 119865 (1199063 (119909 119910))

119865 (1199064 (119909 119910))] (17)


807060504030201000 50 100 150 200 250

(a)

(b) (d)

(e)

807060504030201000 50 100 150 200 250

120100 80 60 40

200 0 20 40 60 80 100 120

yx

x = 41y = 41z = 323388

0

100

200

300

400

500

(c)

1205942

Figure 5 Flowchart for the algorithms routine (a) undeformed image (b) deformed image (c) LPQ descriptor of reference subset (d) LPQdescriptor of target subset and (e) Chi-Square value

The phase information in the Fourier coefficients isrecorded by examining the signs of the real and imaginaryparts of each component in 119865(119909119910) This is done by using asimple scalar quantization

119890119894 = 1 if 119892119894 ge 00 if 119892119894 lt 0 (18)

where 119892119894 is the 119894th component of the vector 119866(119909119910) = [Re119865119909Im119865119909] The resulting eight binary coefficients 119890119894 are repre-sented as integer values within 0ndash255 using binary coding

119891LPQ (119909 119910) = 8sum119894=1

119890119894 (119909 119910) 2119894minus1 (19)

Finally a histogram is formed by all the positions in therectangular region and used as a 256-dimensional featurevector in the match

43 Matching Based on the LPQ Feature In this sectionwe introduce the key procedure of the matching algorithmFirst input a series of images select the first one as theun-deformed image and then select a data point 119875 on itSecondly a rectangular119872times119872neighborhood at the data point119875 is selected as the reference subset as shown in Figure 5(a)The LPQ descriptor of the reference subset is computedaccording to the method of the previous section as shown inFigure 5(c) Thirdly a region bigger than the reference subsetarea centered at the location of corresponding1198751015840 is defined asthe searching subset on the deformed imagesThen we dividesearching subset into subimages which have the same sizeas the reference subset and calculate the LPQ descriptor foreach subimage as shown in Figure 5(d) Finally calculate the

similarity between the reference subset and the rectangularregions of searching subset Since the LPQ descriptor is ahistogram we can use the histogram distance calculationmethod tomeasure image similarityThe common histogramdistance calculation methods are histogram intersectionmethod Bhattacharyya distance method and Chi-Squarestatistics method In this paper we use Chi-Square statisticsas the similarity metric It is defined as follows

1205942 (119878 119862) = sum119895

120596119895 (119878119895 minus 119862119895)2

119878119895 + 119862119895 (20)

where 119878 and119862 are the histograms of reference subset and rect-angular regions respectively 119878119895 are 119862119895 are the 119895th subregionof the histogram of the reference subset and the rectangularregions 120596119895 is the weight values of the 119895th subregion

According to Chi-Square statistics the location of theminimum value of 1205942(119878 119862) as shown in Figure 5(e) is thecorresponding 1198751015840 on the deformed image The location 1198751015840 isan initial estimate of the deformation parameter for the N-Riteration

5 Experimental Results

In order to assess the performance of the initialization usingLPQ matching we performed uniaxial tensile experimentsIn the DIC calculation deformed images are generated bythe bicubic spline interpolation ZNSSD is selected as theobjective function The optimization is terminated if thechange of ZNSSD is less than 10minus8 or a maximum numberof 20 iterations is reached Unless specified first-order shapefunction is used to model the deformation in the subset


(a) (b)

Figure 6 Simulate spackle pattern (a) undeformed image (b) the data point in the undeformed image

51 Generation of Simulated Images The simulated imagesare assumed to be the sum of individual Gaussian speckles

119868 (119909 119910) = 119873sum119870=1

119868119896 exp((119909 minus 119909119896)2 + (119910 minus 119910119896)21198772 ) (21)

where119873 is the total number of speckles and 119877 is the specklesize119909119896 and119910119896 are the positions of each specklewith a randomdistribution and 119868119896 is the peak intensity of each speckle Weuse a speckle pattern with dimensions of 700 times 700 pixels119873 = 1200 and 119877 = 4 as the undeformed image as shown inFigure 6(a)

52 Accuracy and Preassigned Displacement In this experi-ment a series of deformed images are generated using thebicubic spline interpolation method with a range of 01ndash09pixels by step of 01 pixels between successive images Weuse the first image as the undeformed image as shown inFigure 6(a) Next 50 data points are uniformly sampled in arectangular region by a grid step of 5 pixels to measure thedisplacement of the other deformation images as shown inFigure 6(b)The reference subset of each data point is 41times41The measured mean error maximum error and standarddeviation of the 50 data points selected in the 9 deformedimages are calculated The result of the conventional method[13] is listed in Table 2 and the result of proposed method islisted in Table 3 depending on the focal length a pixel on theimage is equivalent to 30 120583m on the material

53 Simulated Deformation Combined with Large MotionBlur To further illustrate the advantage of using LPQ featurein the initialization of DIC we add the motion blur on thespackle pattern range from 0 to 10 pixels and step by 2 pixelsConsidering when the materials fracture the extension ratewill increase rapidly and add the maximum fuzzy scale to10 pixels as well The proposed method is compared withthe conventional method in the blurred spackle pattern Themean error and the standard deviation of the two differentmethods on different motion blur are shown in Figures 7 and8 respectively

Table 2 Results of the conventional method

Exact value Mean error Maximum error Standard deviation03 0005421 0007142 000694206 0004254 0016541 000598709 0004852 0023214 000853212 0004024 0019654 000897415 0006104 0012100 001274618 0005845 0008451 001092421 0005112 0013542 001057224 0007521 0016587 001134827 0007310 0018547 0012679Unit 120583m

Table 3 Results of proposed method


It can be analyzed that with the increase of the motionblur the mean error and the standard deviation of theconventional method increase however the accuracy of theproposed method is steady because of LPQ feature butthe conventional method cannot search the integral pixeldisplacement accurately whenmotion blur changes the imagegray information


1 2 3 4 5 6 7 8 9 100Fuzzy scale (pixel)

115

225

335

445

5

Mea

n er

ror

times10minus3

Proposed methodConventional method

Figure 7 Comparison of mean error between the two searchmethods


25

3

35

4

45

5

55

Stan

dard

dev

iatio

n


Figure 8 Comparison of standard deviation between the two searchmethods

54 Simulated Deformations Combined with Different Size ofSubset Region The size of reference subset is an importantparameter in DIC When the size of subset region is smallercomputational efficiency is higher However less informationcontained in the smaller subset may reduce the matchaccuracy If we increase the subset size it can reduce thenoise impact and get more accurate results However thecalculation will increase proportionally with the subset sizeincrease

We choose reference subset size from 11 times 11ndash71 times 71 pixelsand then increase it by 20 times 20 pixels The mean error fordifferent reference subset size is shown in Figure 9 We cansee that with the reference subset increase in the size from11 times 11 to 41 times 41 pixels the precision increases whereas itstarts to increase slowly as the size exceeds 41times41 pixel Herewe chose the 41 times 41 pixels for the size of subset region55 Displacement Measurement of Uniaxial Tensile Experi-ment In the presented video extensometer system imagesare capturedwithBasler avA1000-120KMCamera Link analogmonochrome cameras and a Matrox Meteor-II analog framegrabber cardThe cameras are equippedwithCCTVampVIDEO

Size of subset (pixel)

02468

1012

Mea

n er

ror (

pixe

l)

0 11 lowast 11 21 lowast 21 31 lowast 31 41 lowast 41 51 lowast 51 61 lowast 61 71 lowast 71

Figure 9 Comparison of standard deviation between differentsubset regions


Number Average displacement 119862znssd

1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940

LENS AVENIR ZOOMLENS ST16160 lenses with focal lengthof 16ndash160mm

The shape of the high elongation specimen typically usedfor uniaxial tensile tests follows the GBT 528-2009 standardThe width of the specimen is 50 plusmn 10mm thickness of thespecimen is 20 plusmn 02m

During the experiment we use the first image as theundeformed image and other 5 images at different times asthe deformed images On the undeformed image we choose50 data points on the same line and calculate its displacementThe average displacement and average ZNSSD values forthese data points at different deformed images are listedin Table 4 Meanwhile the measurements of conventionalmethod are listed in Table 5

In Table 4 it is shown that the average displacement issimilar in the first and second deformed images but afterthe third picture images appear with large deformation andmotion blur the ZNSSD of conventional method increasesignificantly up to 0094 at the last deformed images Theresults of the proposed method are still accurate and theZNSSD does not have obvious change Also the displacementof conventional method is larger than that of the proposedmethod on the y-axis which is stretching direction


6 Discussion and Conclusion

In this paper we propose a novel integer pixel searchingalgorithm based on LPQ feature utilized in the video exten-someter system that can match the data points during theuniaxial tensile testing

Experimental results on simulated speckle images verifythat this algorithm is better than conventional methods themaximum mean error is only 0003995 120583m and the relativeerror is 133 which satisfies the accuracy of extensometerThen we addmotion blur on the speckle image experimentalresults showed that accuracy of the proposed method willnot be affected by the motion blur The impact on accuracythat the size of subset region has is also studied we findthat considering both accuracy and efficiency the size of41 times 41 a pixels was the optimal option Finally the proposedmethod is tested in the deformation measurement of a highelongation specimen The experiment results show that theproposed method has excellent performance in the actualapplication

In conclusion the algorithm can bring a more accurateand more intelligent measurement technique for measuringfull-field displacements of high elongation materials

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The authors would like to thank Xiaofang Yuan for hisconstructive suggestions This work is supported by theNational Natural Science Foundation of China (Grant no61501181) by the National Science and Technology MajorProject of China (Grant no 07-Y30B10-9001-1416) by thePriority Academic Program Development of Jiangsu HigherEducation Institutions and by Jiangsu Collaborative Inno-vation Center on Atmospheric Environment and EquipmentTechnology

References

[1] C Yueting ldquoPrimary discussion on measuring technology invideo extensometerrdquo Engineering amp Test vol 9 pp 50ndash53 2009

[2] B Pan L Yu and D F Wu ldquoHigh-accuracy 2D digital imagecorrelationmeasurements with bilateral telecentric lenses erroranalysis and experimental verificationrdquo Experimental Mechan-ics vol 53 no 9 pp 1719ndash1733 2013

[3] B Pan L Yu D Wu and L Tang ldquoSystematic errors in two-dimensional digital image correlation due to lens distortionrdquoOptics and Lasers in Engineering vol 51 no 2 pp 140ndash147 2013

[4] B Pan L Tian and X L Song ldquoReal-time non-contact andtargetless measurement of vertical deflection of bridges usingoff-axis digital image correlationrdquo NDT amp E International vol79 pp 73ndash80 2016

[5] B Pan K Qian H Xie and A Asundi ldquoTwo-dimensional digi-tal image correlation for in-plane displacement and strain mea-surement a reviewrdquo Measurement Science and Technology vol20 no 6 Article ID 062001 2009

[6] B Pan H Xie Z Guo and T Hua ldquoFull-field strain measure-ment using a two-dimensional Savitzky-Golay digital differen-tiator in digital image correlationrdquo Optical Engineering vol 46no 3 Article ID 033601 2007

[7] B Gu V S Sheng Z Wang D Ho S Osman and S Li ldquoIncre-mental learning for ]-Support Vector Regressionrdquo Neural Net-works vol 67 pp 140ndash150 2015

[8] B Pan H Xie Y Xia and Q Wang ldquoLarge-deformation meas-urement based on reliable initial guess in digital image corre-lation methodrdquo Acta Optica Sinica vol 29 no 2 pp 400ndash4062009

[9] W Tong ldquoAn evaluation of digital image correlation criteria forstrain mapping applicationsrdquo Strain vol 41 no 4 pp 167ndash1752005

[10] M A Sutton C Mingqi W H Peters Y Chao and S McNeillldquoApplication of an optimized digital correlation method toplanar deformation analysisrdquo Image amp Vision Computing vol4 no 3 pp 143ndash150 1986

[11] B Gu V S Sheng K Y TayW Romano and S Li ldquoIncrementalsupport vector learning for ordinal regressionrdquo IEEE Transac-tions on Neural Networks amp Learning Systems vol 26 no 7 pp1403ndash1416 2015

[12] F Hild B Raka M Baudequin S Roux and F CantelaubeldquoMultiscale displacement field measurements of compressedmineral-wool samples by digital image correlationrdquo AppliedOptics vol 41 no 32 pp 6815ndash6828 2002

[13] Z-F Zhang Y-L Kang H-W Wang Q-H Qin Y Qiu andX-Q Li ldquoA novel coarse-fine search scheme for digital imagecorrelation methodrdquo Measurement vol 39 no 8 pp 710ndash7182006

[14] B Pan FW Da and X Yong ldquoIncremental calculation for largedeformation measurement using reliability-guided digitalimage correlationrdquo Optics and Lasers in Engineering vol 50no 4 pp 586ndash592 2012

[15] J Li X Li B Yang and X Sun ldquoSegmentation-based imagecopy-move forgery detection schemerdquo IEEE Transactions onInformation Forensics and Security vol 10 no 3 pp 507ndash5182015

[16] B Pan ldquoReliability-guided digital image correlation for imagedeformation measurementrdquo Applied Optics vol 48 no 8 pp1535ndash1542 2009

[17] B Pan Z Wang and Z Lu ldquoGenuine full-field deformationmeasurement of an object with complex shape using reliability-guided digital image correlationrdquo Optics Express vol 18 no 2pp 1011ndash1023 2010

[18] H Cheong E Chae E Lee G Jo and J Paik ldquoFast imagerestoration for spatially varying defocus blur of imaging sensorrdquoSensors vol 15 no 1 pp 880ndash898 2015

[19] B Chen H Shu G Coatrieux G Chen X Sun and J L Coa-trieux ldquoColor image analysis by quaternion-type momentsrdquoJournal of Mathematical Imaging and Vision vol 51 no 1 pp124ndash144 2015

[20] M R Banham and A K Katsaggelos ldquoDigital image restora-tionrdquo IEEE Signal Processing Magazine vol 14 no 2 pp 24ndash411997

[21] Y Peng T Wu S Wang N Kwok and Z Peng ldquoMotion-blurred particle image restoration for on-line wearmonitoringrdquoSensors vol 15 no 4 pp 8173ndash8191 2015

[22] Z Xia X Wang X Sun and B Wang ldquoSteganalysis of leastsignificant bit matching using multi-order differencesrdquo Securityand Communication Networks vol 7 no 8 pp 1283ndash1291 2014


[23] X Wen L Shao Y Xue and W Fang ldquoA rapid learning algo-rithm for vehicle classificationrdquo Information Sciences vol 295no 1 pp 395ndash406 2015

[24] V Ojansivu E Rahtu and J Heikkila ldquoRotation invariant localphase quantization for blurinsensitive texture analysisrdquo inProceedings of the 19th International Conference on PatternRecognition (ICPR rsquo08) pp 3330ndash4005 Tampa Fla USA 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Clamp

Material

CCD camera

Light source

Computer

Figure 1 Measurement system overview

the errors Reference [4] proposes an off-axis digital imagecorrelation method for real-time noncontact and targetlessmeasurement of vertical deflection of bridges to achievesubpixel accuracy

Despite these advances few works about eliminat-ing extensometerrsquos measurement errors caused by motion-induced image blur to improve the accuracy have beenreported In this paper we will propose an image matchingmethod for video extensometer to measure the parametersby utilizing Local Phase Quantization (LPQ) feature Thismethod is robust and performs well on images with seriousmotion blur and deformation

The rest of the paper is organized as follows The basicprinciple of video extensometer is described in Section 2The generation mechanism of motion blur is introduced inSection 3The proposed matching method using Local PhaseQuantization is discussed in detail in Section 4 Experimentalresults on both simulated and real-world data as well ascomparative results with the existing method are presentedin Section 5 Finally Section 6 gives the conclusion

2 System Overview

A schematic diagram of the video extensometer system isshown in Figure 1 The main parts of the video extensometersystem are a light source tensile equipment CCD cameraand PC with digital image processing software

For measurement first make the spackle pattern on thematerial surface keep the CCD camerarsquos optical axis verticalto the test specimen surface choose the appropriate focallength and make sure the field of view covers the wholematerial In the tensile stress test the position of the datapoint will be changed before and after the deformationThrough the deformation images collected by CCD cameraat different times we can calculate the materialsrsquo tensileproperties

21 Digital Image Correlation The video extensometer usesDIC to analyze materials tensile properties [2ndash4] The prin-ciple of DIC is tracking the same point in undeformed anddeformed image and then yields the displacements of thedata point The information contained by the data pointcannot identify the same point in a series of deformed imagesSo we use a square subset centered on the data point toreplace the data point as shown in Figure 2(a) A smallsquare speckle subset of 2119873 times 2119873 pixel centered at the datapoint 119875 in the undeformed image is defined as the referencesubset At the same time a bigger subset of 2119872 times 2119872 pixelscentered at the corresponding data point 1198751015840 in the deformedimage is defined as the searching subset To calculate thecorrelation coefficient the reference subset is moved throughthe searching subset pixel by pixel The peak position of thecorrelation coefficient is the target subset as can be seen fromFigure 2(b)

As shown in Figure 2(b) the stress on the material willcause rigid body displacement scale distortion and rotationdeformation So the intensity of an arbitrary point in theundeformed subset region and the corresponding point in thedeformation subset region is related as follows

119892 (1199091015840119894 1199101015840119894 ) = 119865 (119891 (119909119894 119910119894)) (1)

For the shape function 119865 the most widely used functionis first-order shape function [5]

1199091015840 = 119909 + 119906 + 120597119906120597119909Δ119909 + 120597119906120597119910Δ1199101199101015840 = 119910 + V + 120597V120597119909Δ119909 + 120597V120597119910Δ119910

(2)

The corresponding matrix form is

[11988310158401198841015840] = [119909119910] + [119906 119906119909 119906119910V V119909 V119910

][[[1Δ119909Δ119910]]] (3)


xO

y

P(x0 y0)

Reference subset

Undeformed image

(a)

P998400(x9984000 y998400

0)

Target subset

Searching subset

xO

y

Deformed image

(b)





119862ZNSSD = 2 (1 minus 119862ZNCC) (4)


119862ZNSSD

= 119872sum119894=minus119872

119872sum119895=minus119872


2

(5)

where

119891119898 = 1(2119872 + 1)2119872sum119894=minus119872

119872sum119895=minus119872

119891 (119909119894 119910119895)

119892119898 = 1(2119872 + 1)2119872sum119894=minus119872

119872sum119895=minus119872

119892 (1199091015840119894 119910119895)

Δ119891 = radic 119872sum119894=minus119872

119872sum119895=minus119872

[119891 (119909119894 119910119895) minus 119891119898]2

Δ119892 = radic 119872sum119894=minus119872

119872sum119895=minus119872

[119892 (1199091015840119894 1199101015840119895) minus 119892119898]2(6)


3 Motion Blur



1119906 + 1V = 1119891 (7)


V = 119891119906119906 minus 119891 (8)


Lens

Focal point

Imagesensor

Objectplane

u f

v

x

y

O


(a) (b) (c)



119910 = V0119905V119906 (9)









Focal length(mm)

Imagedistance(mm)


Exposuretime (s)

Tensile rate(mmmin)

Objectdistance(mm)


Blur pixel(pix)

20 2142 52 001 100 300 119 02350 60 52 001 100 300 333 06480 10909 52 001 100 300 606 11720 2142 52 001 200 300 238 04650 60 52 001 200 300 667 12880 10909 52 001 200 300 1212 23320 2142 52 001 500 300 595 11450 60 52 001 500 300 1666 32180 10909 52 001 500 300 3030 583







120593ℎ (119906 V) = 0 if 119867(119906 V) ge 0120587 if 119867(119906 V) lt 0 (14)

This means that

120593119892 (119906 V) = 120593119891 (119906 V) 119867 (119906 V) gt 0120593119892 (119906 V) + 120587 119867 (119906 V) lt 0 (15)



119865(119909119910) (119906 V)= sum(1199091015840 1199101015840)isin119873119909119910


(16)



119865 (1199064 (119909 119910))] (17)


807060504030201000 50 100 150 200 250

(a)

(b) (d)

(e)

807060504030201000 50 100 150 200 250

120100 80 60 40

200 0 20 40 60 80 100 120

yx

x = 41y = 41z = 323388

0

100

200

300

400

500

(c)

1205942



119890119894 = 1 if 119892119894 ge 00 if 119892119894 lt 0 (18)


119891LPQ (119909 119910) = 8sum119894=1

119890119894 (119909 119910) 2119894minus1 (19)




1205942 (119878 119862) = sum119895

120596119895 (119878119895 minus 119862119895)2

119878119895 + 119862119895 (20)






(a) (b)



119868 (119909 119910) = 119873sum119870=1

119868119896 exp((119909 minus 119909119896)2 + (119910 minus 119910119896)21198772 ) (21)











115

225

335

445

5

Mea

n er

ror

times10minus3




25

3

35

4

45

5

55

Stan

dard

dev

iatio

n






02468

1012

Mea

n er

ror (

pixe

l)





1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940










Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










xO

y

P(x0 y0)

Reference subset

Undeformed image

(a)

P998400(x9984000 y998400

0)

Target subset

Searching subset

xO

y

Deformed image

(b)





119862ZNSSD = 2 (1 minus 119862ZNCC) (4)


119862ZNSSD

= 119872sum119894=minus119872

119872sum119895=minus119872


2

(5)

where

119891119898 = 1(2119872 + 1)2119872sum119894=minus119872

119872sum119895=minus119872

119891 (119909119894 119910119895)

119892119898 = 1(2119872 + 1)2119872sum119894=minus119872

119872sum119895=minus119872

119892 (1199091015840119894 119910119895)

Δ119891 = radic 119872sum119894=minus119872

119872sum119895=minus119872

[119891 (119909119894 119910119895) minus 119891119898]2

Δ119892 = radic 119872sum119894=minus119872

119872sum119895=minus119872

[119892 (1199091015840119894 1199101015840119895) minus 119892119898]2(6)


3 Motion Blur



1119906 + 1V = 1119891 (7)


V = 119891119906119906 minus 119891 (8)


Lens

Focal point

Imagesensor

Objectplane

u f

v

x

y

O


(a) (b) (c)



119910 = V0119905V119906 (9)









Focal length(mm)

Imagedistance(mm)


Exposuretime (s)

Tensile rate(mmmin)

Objectdistance(mm)


Blur pixel(pix)

20 2142 52 001 100 300 119 02350 60 52 001 100 300 333 06480 10909 52 001 100 300 606 11720 2142 52 001 200 300 238 04650 60 52 001 200 300 667 12880 10909 52 001 200 300 1212 23320 2142 52 001 500 300 595 11450 60 52 001 500 300 1666 32180 10909 52 001 500 300 3030 583







120593ℎ (119906 V) = 0 if 119867(119906 V) ge 0120587 if 119867(119906 V) lt 0 (14)

This means that

120593119892 (119906 V) = 120593119891 (119906 V) 119867 (119906 V) gt 0120593119892 (119906 V) + 120587 119867 (119906 V) lt 0 (15)



119865(119909119910) (119906 V)= sum(1199091015840 1199101015840)isin119873119909119910


(16)



119865 (1199064 (119909 119910))] (17)


807060504030201000 50 100 150 200 250

(a)

(b) (d)

(e)

807060504030201000 50 100 150 200 250

120100 80 60 40

200 0 20 40 60 80 100 120

yx

x = 41y = 41z = 323388

0

100

200

300

400

500

(c)

1205942



119890119894 = 1 if 119892119894 ge 00 if 119892119894 lt 0 (18)


119891LPQ (119909 119910) = 8sum119894=1

119890119894 (119909 119910) 2119894minus1 (19)




1205942 (119878 119862) = sum119895

120596119895 (119878119895 minus 119862119895)2

119878119895 + 119862119895 (20)






(a) (b)



119868 (119909 119910) = 119873sum119870=1

119868119896 exp((119909 minus 119909119896)2 + (119910 minus 119910119896)21198772 ) (21)











115

225

335

445

5

Mea

n er

ror

times10minus3




25

3

35

4

45

5

55

Stan

dard

dev

iatio

n






02468

1012

Mea

n er

ror (

pixe

l)





1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940










Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Lens

Focal point

Imagesensor

Objectplane

u f

v

x

y

O


(a) (b) (c)



119910 = V0119905V119906 (9)









Focal length(mm)

Imagedistance(mm)


Exposuretime (s)

Tensile rate(mmmin)

Objectdistance(mm)


Blur pixel(pix)

20 2142 52 001 100 300 119 02350 60 52 001 100 300 333 06480 10909 52 001 100 300 606 11720 2142 52 001 200 300 238 04650 60 52 001 200 300 667 12880 10909 52 001 200 300 1212 23320 2142 52 001 500 300 595 11450 60 52 001 500 300 1666 32180 10909 52 001 500 300 3030 583







120593ℎ (119906 V) = 0 if 119867(119906 V) ge 0120587 if 119867(119906 V) lt 0 (14)

This means that

120593119892 (119906 V) = 120593119891 (119906 V) 119867 (119906 V) gt 0120593119892 (119906 V) + 120587 119867 (119906 V) lt 0 (15)



119865(119909119910) (119906 V)= sum(1199091015840 1199101015840)isin119873119909119910


(16)



119865 (1199064 (119909 119910))] (17)


807060504030201000 50 100 150 200 250

(a)

(b) (d)

(e)

807060504030201000 50 100 150 200 250

120100 80 60 40

200 0 20 40 60 80 100 120

yx

x = 41y = 41z = 323388

0

100

200

300

400

500

(c)

1205942



119890119894 = 1 if 119892119894 ge 00 if 119892119894 lt 0 (18)


119891LPQ (119909 119910) = 8sum119894=1

119890119894 (119909 119910) 2119894minus1 (19)




1205942 (119878 119862) = sum119895

120596119895 (119878119895 minus 119862119895)2

119878119895 + 119862119895 (20)






(a) (b)



119868 (119909 119910) = 119873sum119870=1

119868119896 exp((119909 minus 119909119896)2 + (119910 minus 119910119896)21198772 ) (21)











115

225

335

445

5

Mea

n er

ror

times10minus3




25

3

35

4

45

5

55

Stan

dard

dev

iatio

n






02468

1012

Mea

n er

ror (

pixe

l)





1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940










Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Focal length(mm)

Imagedistance(mm)


Exposuretime (s)

Tensile rate(mmmin)

Objectdistance(mm)


Blur pixel(pix)

20 2142 52 001 100 300 119 02350 60 52 001 100 300 333 06480 10909 52 001 100 300 606 11720 2142 52 001 200 300 238 04650 60 52 001 200 300 667 12880 10909 52 001 200 300 1212 23320 2142 52 001 500 300 595 11450 60 52 001 500 300 1666 32180 10909 52 001 500 300 3030 583







120593ℎ (119906 V) = 0 if 119867(119906 V) ge 0120587 if 119867(119906 V) lt 0 (14)

This means that

120593119892 (119906 V) = 120593119891 (119906 V) 119867 (119906 V) gt 0120593119892 (119906 V) + 120587 119867 (119906 V) lt 0 (15)



119865(119909119910) (119906 V)= sum(1199091015840 1199101015840)isin119873119909119910


(16)



119865 (1199064 (119909 119910))] (17)


807060504030201000 50 100 150 200 250

(a)

(b) (d)

(e)

807060504030201000 50 100 150 200 250

120100 80 60 40

200 0 20 40 60 80 100 120

yx

x = 41y = 41z = 323388

0

100

200

300

400

500

(c)

1205942



119890119894 = 1 if 119892119894 ge 00 if 119892119894 lt 0 (18)


119891LPQ (119909 119910) = 8sum119894=1

119890119894 (119909 119910) 2119894minus1 (19)




1205942 (119878 119862) = sum119895

120596119895 (119878119895 minus 119862119895)2

119878119895 + 119862119895 (20)






(a) (b)



119868 (119909 119910) = 119873sum119870=1

119868119896 exp((119909 minus 119909119896)2 + (119910 minus 119910119896)21198772 ) (21)











115

225

335

445

5

Mea

n er

ror

times10minus3




25

3

35

4

45

5

55

Stan

dard

dev

iatio

n






02468

1012

Mea

n er

ror (

pixe

l)





1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940










Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










807060504030201000 50 100 150 200 250

(a)

(b) (d)

(e)

807060504030201000 50 100 150 200 250

120100 80 60 40

200 0 20 40 60 80 100 120

yx

x = 41y = 41z = 323388

0

100

200

300

400

500

(c)

1205942



119890119894 = 1 if 119892119894 ge 00 if 119892119894 lt 0 (18)


119891LPQ (119909 119910) = 8sum119894=1

119890119894 (119909 119910) 2119894minus1 (19)




1205942 (119878 119862) = sum119895

120596119895 (119878119895 minus 119862119895)2

119878119895 + 119862119895 (20)






(a) (b)



119868 (119909 119910) = 119873sum119870=1

119868119896 exp((119909 minus 119909119896)2 + (119910 minus 119910119896)21198772 ) (21)











115

225

335

445

5

Mea

n er

ror

times10minus3




25

3

35

4

45

5

55

Stan

dard

dev

iatio

n






02468

1012

Mea

n er

ror (

pixe

l)





1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940










Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










(a) (b)



119868 (119909 119910) = 119873sum119870=1

119868119896 exp((119909 minus 119909119896)2 + (119910 minus 119910119896)21198772 ) (21)











115

225

335

445

5

Mea

n er

ror

times10minus3




25

3

35

4

45

5

55

Stan

dard

dev

iatio

n






02468

1012

Mea

n er

ror (

pixe

l)





1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940










Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











115

225

335

445

5

Mea

n er

ror

times10minus3




25

3

35

4

45

5

55

Stan

dard

dev

iatio

n






02468

1012

Mea

n er

ror (

pixe

l)





1 22500 9026 000922 22500 8922 002183 22512 8536 000814 22556 7151 001165 22548 1955 00209



1 22500 9086 001602 22500 8904 003203 22520 8452 003604 22596 7098 006805 22558 1834 00940










Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Competing Interests


Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Image Matching Based on Local Phase …downloads.hindawi.com/journals/mpe/2016/5182106.pdf · 2019. 7. 30. · Research Article Image Matching Based on Local Phase