Research Report2 (2)

8/13/2019 Research Report2 (2)

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An Investigation of Particle Image Velocimetry Techniques Applied to the Analysis ofWheel-Soil Interaction on Mars Terrain Simulant

Mobolaji Akinpelu

Dr. Karl Iagnemma

Department of Mechanical EngineeringMassachusetts Institute of Technology

Abstract

In 2009, the wheel of the Mars Rover got stuck because there was not enough traction.The aim of this project is to create or modify software that will track Martian soil particle

and show how the motion of the wheel affects the soil. The overall goal of the tasksdescribed in this report is to investigate available PIV software for the above purpose and

understand how to modify the parameters of the software, based on cross-correlationalgorithm, to give the most accurate information on the motion of the soil.

1. Introduction

After landing in January 2004 to probe the past geology and climate of Mars, in May2009, the Mars Rover Spirit got stuck in soft Martian sand [1]. Attempts to get it out onlydrove it deeper [2]. In early 2011, the Mars Rover went through a particularly harsh

Mars winter that sent it into hibernation while exposing the scientific and engineeringequipments on board to damage. NASA scientists held out hope that after the passing of

the winter, Spirit will get enough energy from the sun to recharge and resumecommunication with scientists and engineers on earth. But it did not. In May 2011,NASA abandoned efforts to resume communication with the Spirit Rover.

Consequently, studying the interaction between the wheel of the Mars Rover and Martian

soil has become an interesting and important problem, whose answer will help avoidfuture occurrences like the above. This project simulates the motion of a wheel of theMars Rover on a Mars soil simulant. The simulation is used to understand the forces the

wheel exerts on the soil and the movement and shearing pattern of the soil particles. Theinformation from these experiments is vital for understanding the mechanical properties

of Mars soil and the interaction between the soil and the wheel. The result of the study ofthese properties and interactions can be important for the design of future Mars roverwheels and motion mechanisms.

2. Problem Statement

To track the motion of the particles of the soil, we plan to use publicly available ParticleImage Velocimetry (PIV) software. However, a sampling of PIV software shows that

they are made for particular applications like the study of fluid flow in biological andgeological applications. Therefore, we had to conduct an analysis of the instrumentation

requirements (camera frame rate and p ixel resolution), software parameters ( interrogationwindow size, degree of overlap of interrogation windows) and physical conditions


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(lighting conditions and test rig container) and how to choose these variables so our PIVanalysis gives accurate and useful data about the flow patterns in the soil.

This analysis is important because it represents a preliminary study that will inform our

choice of instruments, software parameters and physical conditions for our experiments.

There have been attempts to conduct a more general analysis of the effects of choice ofparameters on the accuracy of PIV results [4]. However, our approach differs from that of

researchers like [4] because it is an investigation carried out for a specific applicationinstead of an analysis of the structure and results of the cross-correlation algorithm that is

the main feature in contemporary literature.

Figure 1: An artists rendering of the Mars Rover Spirit

Figure 2: The test-bed for our experiments


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3. Methods

In this section, we explain how PIV analysis works generally, how cross-correlation

works, and how we created a statistical test based on our understanding of our PIV and

cross-correlation work.

I.Particle Image Velocimetry

Particle Image Velocimetry (PIV) is a technique used in experimental fluid mechanics todetermine instantaneous velocity vector fields by measuring the displacements ofnumerous fine particles that accurately follow the motion of the fluid [3]. This velocity is

measured by recording images of the particles at more than one precise time anddeducing the displacement of the particles from the displacement of the image [3]. The

steps in a PIV analysis are typically as follows:

1.

A fluid is seeded with marker particles that refract, absorb or scatter light, havea high contrast with the rest of the fluid and do not interrupt the fluid flow.2. Then the particles in the fluid flow are illuminated by pulsed sheets of light at

exact time intervals and images of the illuminated particles are taken.3. Next, the resulting images are processed with software that is based on

algorithms like the cross-correlation algorithm.

The analysis of the recorded images to measure the particles displacement is an

important part of any fluid flow motion experiment. In particular, researchers have tomake a choice on the technique, algorithm and software that gives them the mostinformed and accurate understanding of the dynamics of the fluid flow. For example,

apart from PIV, there are other techniques for analyzing motion in a fluid like LaserSpeckle Velocimetry (Fomin 1998), Scalar Image Velocimtry (Dahm et al 1992), and

Image Correlation Velocimetry (Tokumaru and Dimotakis 1995) [3]. Compared withother velocity measurement techniques such as LaserDoppler AnemometryandHot-WireAnemometry, PIV offers many advantages for the study of fluid mechanics like

revealing the global structure of complicated and/or unsteady flow field quantitatively(Adrian, 1991) - so it has been studied intensely and developed rapidly in the past two

decades [4].

In our case, we started out applying Particle Tracking Velocimetry (PTV), a technique

quite similar to PIV, to our fluid flow. One difference between PIV and PTV is that thealgorithm that drives PTV attempts to track individual particles displacements to

determine velocities, whereas in PIV, regions of flow are tracked. This feature of PTVimplies that there has to be a low particle density in the regions of the flow that are beingcompared to determine the displacement to ensure that the software can recognize and

track the individual particle elements from image frame to image frame [2]. Thistheoretical knowledge, our understanding of the physical properties of the Martian soil

and a preliminary test of images of the soil with PTV software confirmed to us that PIVwas a better choice than PTV.


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Figure 3: Why we chose PIV over PTV

Figure 4: An outline of the PIV Steps

Figure 5: The Martian soil we are experimenting on


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II. Cross-Correlation

Cross-correlation is an example of an algorithm for processing images in a PIV analysis.

PIV images are processed by sub-dividing two consecutive images of the flow into a

regular grid of sub-areas that overlap and finding the velocity vector for each sub-area byan algorithm like cross-correlation. After obtaining the images for a PIV analysis as

explained above, a small sub-area of the first image, usually called an interrogation areaor interrogation window, is compared with a sub-area at the same location in the second

image using cross-correlation [piv8]. This processing produces a table of correlationvalues over a range of displacements, and the overall displacement of particles in thewindow is represented by a peak in this correlation table. [5]. In other words, the process

results in the most probable displacement vector for that particular particlepattern.(Adrian 1991; Willert and Gharib 1991; Stamhuis and Videler 1995) [6]. The

process is repeated for all interrogation areas of the pair of images to get a completevector diagram of the flow. Errors in an analysis using cross-correlation occur mainly

from insufficient data like a lack of imaged flow tracers or poor image quality, and/orfrom correlation abnormalities from unmatched tracer images in the correlated samplevolume [5]. The cross-correlation algorithm is based on the cross-correlation function:

K

Ki

L

Lj

yjxiIjiIyxR ),(),(),(II

The variablesI and I are the intensity values of the images where I is larger than the

template I. Essentially the template I is linearly shifted around in the sample I

without extending over edges ofI . For each choice of sample shift (x, y), the sum of the

products of all overlapping pixel intensities produces one cross-correlation value IIR (x,y). By applying this operation for a range of shifts (M x +M,N y +N), acorrelation plane the size of (2M + 1) (2N + 1) is formed. For shift values at which the

samples particle images align with each other, the sum of the products of pixel

intensities will be larger than elsewhere, resulting in a high cross-correlation value IIR

at this position. Essentially the cross-correlation function statistically measures the degreeof match between the two samples for a given shift. The highest value in the correlation

plane can then be used as a direct estimate of the particle image displacement [7]. One

can imagine this procedure as moving I over I until the best matching is found. Theexpression best matching is used because in practice there is never a 100% matchingdue to particles that have left or entered the imaged area in the second image compared

with the first [6].


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Figure 6: Example of the formation of the correlation plane by direct cross-correlation:here a 4 X 4 pixel template is correlated with a larger 8 X 8 pixel sample to produce a

5 X 5 correlation plane.

Figure 7: The cross-correlation function as computed from real data by correlating asmaller template I (3232 pixel) with a larger sample I (6464 pixel). The mean shift of

the particle images is approximately 12 pixels to the right.

Few systematical researches have been performed to evaluate the effectiveness and

accuracy of final PIV results obtained using cross correlation. Therefore, users of thecross correlation method have to spend a lot of time and cost to optimize variousparameters for PIV image acquiring and processing to get an accurate velocity field

[4].This absence of literature on the effectiveness and accuracy of PIV is inspiration forthis research project: to analyze, in an application-specific manner, the accuracy of

MATLAB-based PIV software we considered for our PIV analysis of motion in Martiansoil.


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Figure 8: Diagrams of steps in PIV analysis of successively recorded PIV patterns in aflow: two sub-images from the same location of two frames are compared in a cross-

correlation procedure resulting in a 2-D probability density distribution which shows apeak.

III. Rotated Images

To test the accuracy of the PIV software we considered using, we simulated circularmotion in our acquired PIV images used the PIV software in detecting this motion. First,

an image of the soil in the test bed was taken (see above) through the glass using a point-and-shoot camera. The image was taken through the glass to ensure that the image onwhich the analysis was conducted correctly simulated the conditions under which

eventual experiment will be conducted. Also, the acquired images was converted to

grayscale because PIV software works best with grayscale images since grayscale imagesensure that there is a higher contrast between the particles the software searches for andthe rest of the fluid. Then, MATLAB scripts were used to rotate this image about itscenter, for one revolution, in increments of 6 degrees. At the end of this process, there

was a stack of 60 images tilted 6 degrees from the previous image.

Figure 9: The image after being rotated 36 degrees.


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The MATLAB code that produced the series of images is in the Appendix. The

MATLAB option crop was chosen over the MATLAB option loose for the codebecause this ensures that the images that are produced by imrotate are all equal in size.

Although the crop option crops the images after they are rotated, a square region

inscribed in a circular region inscribed in the original image can be used for the analysisbecause it is never cropped out of the image. The square region is outlined in white in the

image below.

Figure 10: Sample image showing vectors used for analysis

Mathematically, the motion simulated by the process of rotating the images is circular

motion with a constant angular velocity. All the vectors shown in the diagram above haveknown theoretical velocity values based on the MATLAB code shown in the Appendix.

The analysis was conducted by inputting the series of images, 1 to 60, in pairs of 1-2, 2-3,

3-4 and so on, into three publicly available MATLAB-based PIV software (matpiv,pivlab, fluere) and setting up the parameters so that the software were measuring the

velocity at the same points as the theoretically derived ones. After this, the resultingvectors from each software were plotted on the same image as the theoretical vector toget a visual perception of the accuracy of the software results. It is important to note now

that each vector field like that below is the product of applying PIV to a pair of images.

The vectors in green are the theoretical vectors and those in red are the experimental onesfrom one of the software. The analysis carried out was percentage error for each pair ofvectors that lie in the white square in the field below, sum of percentage errors in thewhite square of each field of vectors (each field is the result of an analysis of a pair of

images by a PIV software), and the sum of all the sums derived for each vector fieldcreated by each software.


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Figure 11: Sample result from analysis

4. Results

I. MATPIV:

Matpiv is a toolbox for PIV created by Kristian Sveen of the University of Cambridge[matpiv manual]. A sample of a matpiv command for carrying out a PIV analysis is this:

[x,y,u,v] = matpiv (mpim1b.bmp,mpim1c.bmp, 64, 0.0012, 0,single);

The command above processes images mpim1b.bmp and mpim1c.bmpusing a 64 X 64

kernel with 0% overlap between each processed sub-area. 0.0012 refers to the timeseparation between the images and single is an option that specifies how many

iterations (one in this case) of cross-correlation should be carried out on the pair ofimages.

The result consists of four matrices x, y, u and v which are measured in pixels andpixels/second. x is a matrix of the x-coordinates where the vectors are drawn (in the

center of each sub-area). y is a matrix of the y-coordinates where the vectors are drawn

(in the center of each sub-area). u and v are the x-components and y-components of thevectors calculated in each sub-area. These results can be visualized with the MATLAB

command quiver(x,y,u,v).

For this statistical analysis, the matpivcommand used was:

[x,y,u,v] = matpiv(image1,image2,32,1,0.0,'single');


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Based on this analysis, the following results were recovered:

Sample experimental values of x-component of velocity

Sub-Area Coordinates 2 3 4 5 6 7

2 -8.21163 -8.66173 -8.73043 -8.482263681 -8.1661 -8.34833

3 -5.31049 -4.8456 -4.50717 -4.6753092 -5.28905 -5.06494

4 -1.83637 -1.62804 -1.75153 -1.801007877 -1.63236 -1.81915

5 1.616693 1.591556 1.464555 1.453186828 1.926127 1.849509

6 4.924812 5.141235 5.017183 5.251215307 5.112303 5.366289

7 8.199604 8.517288 8.517034 8.721054276 8.377348 8.159283

Sample experimental values of y-component of velocity

where the sub-area coordinates refer to the sub-areas that are in the white squarediscussed above.

Theoretical values of x-component of veloc ity


2 -8.37758 -8.37758 -8.37758 -8.37758 -8.37758 -8.37758

3 -5.02655 -5.02655 -5.02655 -5.02655 -5.02655 -5.02655

4 -1.67552 -1.67552 -1.67552 -1.67552 -1.67552 -1.67552

5 1.675516 1.675516 1.675516 1.675516 1.675516 1.675516

6 5.026548 5.026548 5.026548 5.026548 5.026548 5.026548

7 8.37758 8.37758 8.37758 8.37758 8.37758 8.37758


2 8.398358 5.077235 1.732083 -1.65306 -4.94357 -8.27928

3 8.274315 4.790196 1.318847 -1.61582 -5.2165 -8.28049

4 8.544504 4.796586 1.832139 -1.85545 -5.15183 -8.43121

5 8.34281 5.144725 1.584458 -1.84387 -4.99039 -8.25076

6 8.463819 5.087714 1.48174 -1.68683 -5.21205 -8.29548

7 8.17882 5.131278 1.746328 -1.74055 -5.24424 -8.43496


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Theoretical values of y-component of veloc ity


2 8.37758 5.026548 1.675516 -1.67552 -5.02655 -8.37758

3 8.37758 5.026548 1.675516 -1.67552 -5.02655 -8.37758

4 8.37758 5.026548 1.675516 -1.67552 -5.02655 -8.37758

5 8.37758 5.026548 1.675516 -1.67552 -5.02655 -8.37758

6 8.37758 5.026548 1.675516 -1.67552 -5.02655 -8.37758

7 8.37758 5.026548 1.675516 -1.67552 -5.02655 -8.37758

Percentage errors of x-component of velocities in field identified above


2 -0.0002 0.000339 0.000421 0.000125 -0.00025 -3.5E-05

3 0.000565 -0.00036 -0.00103 -0.0007 0.000522 7.64E-05

4 0.00096 -0.00028 0.000454 0.000749 -0.00026 0.000857

5 -0.00035 -0.0005 -0.00126 -0.00133 0.001496 0.001038

6 -0.0002 0.000228 -1.9E-05 0.000447 0.000171 0.000676

7 -0.00021 0.000167 0.000166 0.00041 -2.8E-07 -0.00026

Percentage errors of y-component of velocities in field identified above


2 2.48E-05 0.000101 0.000338 -0.00013 -0.00017 -0.00012

3 -0.00012 -0.00047 -0.00213 -0.00036 0.000378 -0.00012

4 0.000199 -0.00046 0.000935 0.001074 0.000249 6.4E-05

5 -4.2E-05 0.000235 -0.00054 0.001005 -7.2E-05 -0.00015

6 0.000103 0.000122 -0.00116 6.75E-05 0.000369 -9.8E-05

7 -0.00024 0.000208 0.000423 0.000388 0.000433 6.85E-05

After repeating the above process for the 59 vector fields produced by matpiv , the totalpercentage e rror for the x-components of velocities produced by matpivwas found to be

0.2277 and the total percentage error for the y-components of velocities produced bymatpiv was found to be 0.2328. This statistical analysis was also carried out for themagnitudes of the velocities and the angle (direction) of the velocities.


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Figure 12:MATLAB surf plot of percentage errors for a typical matpiv vector field

II. PIVLAB:

Pivlabis another MATLAB-based PIV software that we proposed using. It comes with a

GUI and was created by William Thielicke and Eize J. Stamhuis. It has options in itsinterface to carry out a similar kind of analysis as matpiv and output results in a .mat file.

The contents of the produced .mat file (x,y,u,v) was used to carry out the analysis inMATLAB in a similar way as above.



2 0 0 0 0 15.24937 -7.97261

3 4.82961 0 15.37748 -4.77448 0 -1.61923

4 8.261181 0 0 15.1449 -6.87253 0

5 0 7.477854 -4.97153 0 1.779358 0

6 0 13.0447 0 12.68634 0 -14.6783

7 0 0 11.07954 -2.25074 0 0

Sample experimental values of y-component of velocitySub-Area Coordinates 2 3 4 5 6 7

2 0 0 0 0 -11.5772 -9.75443

3 -8.0102 0 -3.5761 -1.76888 0 9.2950334 -3.9983 0 0 15.30452 14.97414 0

5 0 13.93374 12.95876 0 -4.20108 0

6 0 -14.5519 0 -3.66656 0 -5.13846

7 0 0 7.230583 7.163633 0 0


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The theoretical values are the same as identified under MATPIV. Also, the zerovalues above are the result of converting the NaN returned bypivlabto zero for the sake

of the error calculations.



2 -0.01 -0.01 -0.01 -0.01 -0.0282 -0.00048

3 -0.01961 -0.01 -0.04059 -0.0005 -0.01 -0.00678

4 -0.05931 -0.01 -0.01 -0.10039 0.031017 -0.01

5 -0.01 0.03463 -0.03967 -0.01 0.00062 -0.01

6 -0.01 0.015952 -0.01 0.015239 -0.01 -0.0392

7 -0.01 -0.01 0.003225 -0.01269 -0.01 -0.01



2 -0.01 -0.01 -0.01 -0.01 0.013032 0.001643

3 -0.01956 -0.01 -0.03134 0.000557 -0.01 -0.0211

4 -0.01477 -0.01 -0.01 -0.10134 -0.03979 -0.01

5 -0.01 0.01772 0.067342 -0.01 -0.00164 -0.01

6 -0.01 -0.03895 -0.01 0.011883 -0.01 -0.00387

7 -0.01 -0.01 0.033154 -0.05275 -0.01 -0.01

After repeating the above process for the 59 vector fields produced by pivlab, the total

percentage error for the x-components of velocities produced by pivlabwas found to be19.6012 and the total percentage error for the y-components of velocities produced by

pivlab was found to be 19.3999. This statistical analysis was also carried out for themagnitudes of the velocities and the angle (direction) of the velocities.

Figure 13:MATLAB surf plot of percentage errors for a typical pivlab vector field


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III. FLUERE:

Fluereis the third MATLAB-based PIV software that we proposed using. It comes with aGUI and was created by Kyle Lynch. It has options in its interface to carry out a similar

kind of analysis as matpiv and pivlab and output results in series of .dat files. The

contents of the produced .dat files (x,y,u,v) was used to carry out the analysis inMATLAB in a similar way as above.



2 -2.99462 -3.9254 -4.66279 -5.17047 -5.1741 -5.2004

3 -0.45405 -2.12058 -3.94811 -4.39776 -4.74465 -4.84281

4 0.63601 -0.22359 -1.52012 -1.81063 -1.72643 -2.93354

5 1.74266 1.49695 0.996216 1.07895 1.43909 0.005752

6 1.72815 1.68268 1.7567 2.22117 3.17362 2.4933

7 2.03651 1.78522 2.2365 2.82498 3.41779 3.7096

Sample experimental values of y-component of velocitySub-Area Coordinates 2 3 4 5 6 7

2 2.49051 1.94802 0.99023 0.264218 -2.60795 -4.70671

3 2.3307 2.23345 1.31027 -1.34655 -4.48242 -4.5719

4 2.44311 2.26915 1.51004 -1.79391 -4.78846 -4.12071

5 2.28715 2.12282 1.39524 -1.74566 -4.57704 -4.3442

6 1.96353 1.56194 0.768392 -1.44975 -4.29572 -4.63903

7 1.47992 0.97209 0.106774 -1.24525 -3.84474 -4.66745



2 -0.00643 -0.00531 -0.00443 -0.00383 -0.00382 -0.00379

3 -0.0091 -0.00578 -0.00215 -0.00125 -0.00056 -0.00037

4 -0.0138 -0.00867 -0.00093 0.000806 0.000304 0.007508

5 0.000401 -0.00107 -0.00405 -0.00356 -0.00141 -0.00997

6 -0.00656 -0.00665 -0.00651 -0.00558 -0.00369 -0.00504

7 -0.00757 -0.00787 -0.00733 -0.00663 -0.00592 -0.00557


Sub-Area Coordinate 2 3 4 5 6 7

2 -0.00703 -0.00612 -0.00409 -0.01158 -0.00481 -0.00438

3 -0.00722 -0.00556 -0.00218 -0.00196 -0.00108 -0.00454

4 -0.00708 -0.00549 -0.00099 0.000707 -0.00047 -0.00508

5 -0.00727 -0.00578 -0.00167 0.000419 -0.00089 -0.00481

6 -0.00766 -0.00689 -0.00541 -0.00135 -0.00145 -0.00446

7 -0.00823 -0.00807 -0.00936 -0.00257 -0.00235 -0.00443


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After repeating the above process for the 59 vector fields produced by fluere, the total

percentage error for the x-components of velocities p roduced by fluerewas found to be10.8576 and the total percentage error for the y-components of velocities produced by

fluere was found to be 10.7915. This analysis was also carried out for the magnitudes of

the velocities and the angle (direction) of the velocities.

Figure 14:MATLAB surf plot of percentage errors for a typical fluere vector field

5. Discussion

Based on these results, we chose matpiv for our analysis of the motion. Recently, we havealso begun to take a look at how the quality of our input images (image pre-processing)and the filtering tools available for each software (vector post-processing) may affect

these accuracy estimates. Also, there are default or basic settings that are not common toall of the three software. We took this into consideration in making decisions based on

these results. One limitation of this project is that we cannot tell how important otherchoices like kernel size will affect the accuracy results. Also we do not know if the factthat it is a simple circular motion affects the accuracy of the error values .

6. Appendix

I.

MATLAB code used to rotate images

functionrt = rotat(img1

E = 1;

fork = 1:6:360

figure(1);

A=imrotate(imread(img1),k,'crop');

imwrite(A,['rot''-'num2str(E) '.tif']);

E=E+1;


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end

end

II. MATLAB code for theoretica l value of circular velocity

xmax = 256;

dx = 32;tx = [32:dx:xmax];

tx = tx-xmax/2;

Nx = length(tx);

% y-dimension

ymax = 256;

dy = 32;

ty = [32:dy:ymax];

ty = ty-ymax/2;

Ny = length(ty);

% angular velocity

w = 6; %deg/sec

w = w*pi/180; %rad/sec

% Create velocity field matrices

vx = zeros(Ny,Nx);

vy = zeros(Ny,Nx);

% V = 1;

fori = 1:Nx

fork = 1:Ny

r = sqrt(tx(i).^2+ty(k).^2);%radius

V = w*r;

vx(k,i) = -V*ty(k)/r;

vy(k,i) = V*tx(i)/r;

end

end

[xx,yy] = meshgrid(tx+xmax/2,ty+ymax/2);

quiver(xx(1:1:end),yy(1:1:end),vx(1:1:end),vy(1:1:end));


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7. References

[1] Keane, Richard D., and Ronald J. Adrian. "Theory of Cross-Correlation Analysis of

PIV Images." Applied Scientific Research (1992): 1-25. Print.

[2] Muthanna, Chittiapaa. "Particle Image Velocimetry." (2006): 1-63. Web. July 2011.

[3] Adrian, R. J., and J. Westerweel. "Introduction." Particle Image Velocimetry. New

York: Cambridge UP, 2011. 1-36. Print.

[4] Hu H., T. Kobayashi, K. Okamoto, and N. Taniguchi. "Evaluation of the Cross

Correlation Method by Using PIV Standard Images." The Visualization Society of Japan

and Ohmsha: Journal of Visualization 1st ser. 1 (1998): 1-8. Print.

[5] Hart, Douglas P. "The Elimination of Correlation Errors in PIV Processing." 9th

International Symposium on Applications of Laser Techniques to Fluid Mechanics

(1998): 1-8. Print.

[6] Stamhuis, Eize J. "Basics and Principles of Particle Image Velocimetry (PIV) for

Mapping Biogenic and Biologically Relevant Flows."Aquatic Ecology(2006): 1-17.

Print.

[7] Raffel, Markus, Christian Willert, Jurgen Kompenhans, and Steve Wereley. "Image

Evaluation Methods for PIV." Particle Image Velocimetry: a Practical Guide. Heidelberg:

Springer, 2007. 123-76. Print.

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