Int. J. Metrol. Qual. Eng. 5, 201 (2014)c© EDP Sciences 2014DOI: 10.1051/ijmqe/2014007

Quantification of the uncertainties of high-speed camerameasurementsC. Robbe1,2,�, N. Nsiampa1,2, A. Oukara1,2,3, and A. Papy1

1 Royal Military Academy, Department of Weapons Systems and Ballistics, 30 avenue de la Renaissance,1000 Brussels, Belgium

2 University of Liege (ULg), Aerospace & Mechanical Engineering Department (LTAS), 1, Chemin des Chevreuils,4000 Liege, Belgium

3 Polytechnic Military School of Algiers (EMP), Algeria, 5 Bordj El Bahri, Algiers, Algeria

Received: 25 February 2014 / Accepted: 9 June 2014

Abstract. This article proposes a combined theoretical and experimental approach to assess and quantifythe global uncertainty of a high-speed camera velocity measurement. The study is divided in five sections:firstly, different sources of measurement uncertainties performed by a high-speed camera are identifiedand quantified. They consist of geometrical uncertainties, pixel discretisation uncertainties or optical un-certainties. Secondly, a global uncertainty factor, taking into account the previously identified sources ofuncertainties, is computed. Thirdly, a sensibility study of the camera set-up parameters is performed, al-lowing the experimenter to optimize these parameters in order to minimize the final uncertainties. Fourthly,the theoretical computed uncertainty is compared with experimental measurements. Good concordance hasbeen found. Finally, the velocity measurement uncertainty study is extended to continuous displacementmeasurements as a function of time. The purpose of this article is to propose all the mathematical toolsnecessary to quantify the individual and global uncertainties, to highlight the important aspects of theexperimental set-up, and to give recommendations on how to improve a specific set-up in order to minimizethe global uncertainty. Taking all these into account, it has been shown that highly dynamic phenomenasuch as a ballistic phenomenon can be measured using a high-speed camera with a global uncertainty ofless than 2%.

Keywords: High-speed camera; uncertainties; velocity measurement; displacement measurement; ballistics

1 Introduction

During the past fifteen years, the high-speed camera hasbecome an essential measurement tool. Recent technolo-gies can achieve a frame rate up to one million imagesper second, making it very convenient to assess highly dy-namic phenomena such as ballistics. The results consistof measurement of specific displacements as a function oftime, or velocity measurements. Thanks to the technolog-ical evolution, the high-speed camera is more and moreprecise, relatively easy to setup, and almost always guar-antee no modification of the phenomenon to measure.

However, behind this apparent ease of use and greatprecision, many optical and geometrical phenomena areimplicitly involved when proceeding to an effective mea-surement. Therefore, measurements are often affected byerrors that can be quantified up to a certain range,hence the necessity to use the concept of uncertainty.The measurement uncertainty is defined as the possibleerror on the value of a quantity to measure, provided

� Correspondence: cyril.robbe@rma.ac.be

by the result of the measure. The uncertainty quanti-fies then the maximum expected error when proceedingto measurements [1, 2].

The topic of this article concerns the measurement ofthe average velocity of a ballistic projectile travelling aspecified distance. The presented results remain applica-ble to any dynamic phenomenon. To clarify the compu-tations, the projectile’s movement is supposed to be per-fectly straight, always in the same 2D plane, and alwayshorizontal. The employed camera is a Photron SA-5 high-speed camera [3].

The different sections presented in this article are thefollowing:

– determining the useful theoretical concepts in order toproceed to the uncertainty computations;

– identifying and quantifying all the sources of errors anduncertainties involved in the measurement;

– studying their influence on the velocity measurement;– gathering the results to get the value of the total ve-

locity measurement uncertainty;– studying the influence of the parameters of the camera

set-up on the uncertainty;

Article published by EDP Sciences

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Fig. 1. Illustration of the motion blur when filming a projectileflying at different velocities.

– comparing the results with reference experimentalmeasurements;

– extending the results on the continuous displacementof the projectile measurement as a function of time.

2 Theoretical background

2.1 Definitions

A few theoretical concepts are necessary to carry out theuncertainty study:

The frame rate FR [Hz] is defined as the number of im-ages that a camera can take in one second. A typical cam-era is characterised by a frame rate of 24 frames per second(FPS), a high-speed camera can go up to 1 000 000 FPS.

The geometrical resolution resg [pix] is defined as thenumber of pixels per unit of surface. Many high-speedcamera parameters are the result of a compromise betweenthe frame rate and the resolution, meaning that the high-est possible frame rates can only be achieved at the priceof a reduced geometrical resolution, and vice versa.

The angle of view Ω [◦] describes the angular extentof a given scene that is imaged by the camera.

The shutter time ST [s] is the time interval duringwhich the sensor of the camera is exposed at each imagecapture. The maximal shutter time is equals to the inverseof the frame rate. For filming highly dynamic phenomenasuch as ballistics, the shutter time is generally reduced.Indeed, when the shutter time is too high in comparisonwith the velocity of the phenomenon, the movement ofthe object induces blur around its representation on theimage, leading to motion blur [2,4], illustrated in Figure 1.

Mathematically, the motion blur MB [pix] is quantifiedas shown in equation (1):

MB [pix] = vpix ST (1)

where vpix [pix/s] is the velocity of the studied object.

2.2 The process of a velocity measurement

The setup to perform a velocity measurement is illustratedon Figure 2. The process to perform the measurement isdescribed as follows.

Fig. 2. Illustration of the experimental set-up.

The camera is placed at a certain distance L [m] of theprojectile’s trajectory. The axis of the camera should beorthogonal to this trajectory (as discussed in the sectiongamma angle uncertainty). A reference object, with onespecific dimension dc [mm], is placed in a vertical plane.Ideally, the distance dh [mm] between this vertical cali-bration plane and the vertical projectile plane is null (asdiscussed in the section calibration uncertainties). Aftertaking a picture, the distance dc [pix], which correspondsto number of pixels characterising the distance dc [mm],is measured on the image. This measurement allows com-puting the calibration factor calib [pix/mm], which is theratio between distances in pixel on the image and real dis-tances. The factor calib can also be determined thanks tothe L, resg and Ω parameters:

calib [pix/mm] =dc [pix]dc [mm]

=resg

500 tan(

Ω2

)L

. (2)

In the present case, calib is measured thanks to the ref-erence object, which is easier to perform and gives betterprecision.

Afterwards, the flight motion of the projectile isrecorded and dp [pix], corresponding to displacement ofthe projectile during a certain time interval Δt, is mea-sured. This measurement is achieved by locating the pro-jectile’s position on two different images of the video. Thedistance dp [pix] is converted to the real distance dp [mm]thanks to the calibration factor calib. The time interval Δtis measured by the chronometer of the camera. The veloc-ity v [m/s] can then ultimately be computed (Eq. (3)).

v[m

s

]=

dp [pix]1000 calib [pix/mm]Δt [s]

=dp [pix] dc [mm]

1000 dc [pix] Δt [s]. (3)

All the parameters influencing the velocity measurementuncertainty are identified and quantified for the presentedset-up in Table 1.

C. Robbe et al.: Quantification of the uncertainties of high-speed camera measurements 201-p3

Table 1. The experimental set-up parameters.

Parameter Value

L [m] 0.7Ω [◦] 22.86

FR [Hz] 50 000Resg [pix] 1024

ST [s] 10−5

dc [mm] 120dp [mm] 100v [m/s] 100

All these steps are performed thanks to the use ofa homemade dedicated tracking software [2, 5]. An extraobjective on the present study is to update this software,taking the presented results into consideration.

2.3 The uncertainty computation

The total velocity measurement uncertainty due to dif-ferent sources of independent linear uncertainties is ex-pressed by equation (4) [1, 4, 5]:

Uv =

√√√√ n∑1

(∂f

∂xi

)2

u2(xi) (4)

where Uv is the velocity measurement uncertainty, f is thecomputed velocity (Eq. (3)) and u(xi), is the individualuncertainty i driven by the parameter xi. This equationcan also be expressed as given by equation (5) [1]:

Uv =

√√√√ n∑1

U2i (5)

where Ui is the individual contribution of a specific uncer-tainty i on the velocity measurement uncertainty. In thefollowing sections, individual uncertainties are identified,and their individual influence on the velocity measure-ment uncertainty Ui is determined. Uv can be computedafterwards.

3 Identification and quantificationof the sources of uncertainties

3.1 Calibration uncertainties

During the calibration procedure, which consists in find-ing the calibration factor calib [pix/mm], three sources ofuncertainties are identified:

A first source of uncertainty is linked to the precisionon the measurement of the reference distance dc [mm].This uncertainty is called reference length uncertainty.

Fig. 3. Effect of the three calibration uncertainties on therelative uncertainty of the velocity measurement.

A second source of uncertainty is linked to the dis-tance dc [pix]. Indeed, depending on the quality of theimage, finding the exact pixels corresponding to thereference distance dc [mm] may only be possible upto a certain precision, linked to an uncertainty calledreference pixel uncertainty.Finally, a third source of uncertainty concerns the dis-tance dh [mm]. Ideally, this value is null. Otherwise,an error is committed in the computation of calib[pix/mm]. The related uncertainty is called verticalplane uncertainty.

These three uncertainties can be quantified as a functionof the error on dc [mm], dc [pix] and dh [mm], respectively.The two first cases are solved by comparing the results ofequation (3) with and without taking into account theuncertainty. Solving the third case requires to resolve thefollowing Thales system:

L

L + dh/1000=

dc

dc, corrected(6)

where dc, corrected [mm] is the corrected value of dc [mm] inthe vertical projectile plane. The results are obtained bycomparing the results of equation (3), using dc [mm] anddc, corrected [mm]. The results are presented on Figure 3.All the uncertainties evolve linearly as a function of theconsidered distances. The values of dc [mm], dc [pix] anddh [mm] are estimated at 0.2 mm, 1 pixel, and 20 mm,respectively, for the presented experimental set-up. Theinduced relative uncertainties of the velocity measurementUi are equal to 0.11%, 0.26% and 0.29%, respectively.

3.2 Gamma angle uncertainty

So far, the vertical projectile plane was always orthogonalto the axis of the camera. This section studies the likelycase of a non-null angle γ, as schematised on Figure 4.The related uncertainty, called gamma angle uncertainty,consists in measuring the apparent distance dp [mm] in-stead of the real distance y [mm]. The resolution of thisgeometry leads to equation (7):

y =dp

cos γ + tan β sin γ. (7)

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Fig. 4. Illustration of the geometry involved in the computa-tion of the gamma angle uncertainty.

Fig. 5. Effect of the gamma angle uncertainty on the relativevelocity measurement uncertainty of the velocity measurement.

In the present case, an extra assumption is made:

[AD] = [BD] =dp

2(8)

and thus,

β = tan−1

(dp

2L

). (9)

The relation between the distance dp and y is presentedon Figure 5. This relation is not linear and not symmetri-cal around the vertical axis, due to the asymmetry of thegeometry around the axis of the camera.

For the present set-up, the maximum uncertainty onthe gamma angle is estimated at 3◦.

Both the distances dp [mm] and dc [mm] are influencedby the gamma angle. The biggest error is committed whendp [mm] is measured with a gamma equals to 3◦ and whendc [mm] is measured with a gamma equals to –3◦. Finally,the induced uncertainty is equals to 1.03%.

3.3 Uncertainty related to the measurement of dp [pix]

An error can be committed while determining dp [pix] bychoosing the pixels corresponding to the position of theprojectile on the two images used for the velocity compu-tation. As the projectile is moving, the motion blur MB

Fig. 6. Illustration of the geometry involved in the computa-tion of the parallax error.

[pix] influences this error and the related uncertainty. Em-pirically, equation (10) has been determined for the pre-sented set-up:

UMB =MBkMB

=vpixST

kMB(10)

where UMB [pix] is the considered uncertainty, called mo-tion blur uncertainty, and kMB [–] is an empirical coef-ficient that reduces the effect of the motion blur on thevelocity measurement uncertainty. Among other things,its value depends on the quality of the image, the camerafocus adjustment, the lighting and the contrast betweenthe projectile and the background of the image. For thepresented setup, its value has been empirically assessedat 2. This value has to be refined for each new exper-imental set-up. The relative uncertainty of the velocitymeasurement due to the motion blur is performed thanksto equation (3), computed with a nominal value dp [pix]and the corrected value (dp +UMB) [pix]. The motion bluruncertainty is equal to 0.62%.

3.4 The parallax

The parallax is the disparity between views of an objectdue to a difference in view points [6]. Consequently, whenthe camera is fixed, an object in motion like the consideredprojectile is viewed differently depending of its position.The concept of error instead of uncertainty is used here, asthis error is fully defined by the geometry of the scene. Theparallax is illustrated on Figure 6 and explained as follows.Proceeding to the measurement dp [pix], one has to locatea specific position of the projectile on two images, whichinduces two parallaxes. If the experimenter looks at thefront of the projectile, he locates on image 1 the projectionof the point G on the plane of the projectile’s trajectory,which corresponds to the point B, instead of the point A.Similarly, he locates the point D instead of the point C onimage 2. The parallax error consists in measuring dp [pix]and thus dp [mm] instead of y = dp − d1 + d2 [mm].

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As presented on Figure 6, the distances d1 [mm] andd2 [mm] partially compensated each other. Therefore, theworst-case scenario occurs when one of these two distancesis equal to zero, and the other is at its maximal value,which corresponds to the situation where α = 0 and β =Ω2 . The measurement dp [pix] is then performed only onhalf of the image.

The result is:

tan β =dp + d2

L

sin (90− β) =(

r

r + d2

). (11)

For the present example, with a spherical projectile of a ra-dius r = 40 mm, the parallax error influence on the veloc-ity measurement remains under 0.0002%. An uncertaintyon the radius r leads to an even smaller uncertainty onthe velocity and is therefore not considered here. In con-clusion, to ensure a good measurement, one has to verifythat the parallax remains low enough in comparison withthe other uncertainties. Otherwise, it has to be taken intoaccount.

3.5 The perspective

The perspective is used by a {camera-lens} system to rep-resent three-dimensional objects and depth relationshipson a two-dimensional surface. This representation goeswith optical distortions that bring perspective errors. Thequantification and correction of these errors is the topicof this section.

Mathematically, the perspective can be modelled asa system of transformations that changes the 3D real-unit coordinates describing the filmed situation into 2Dpixels coordinates describing the image. These transfor-mations vary from one {camera-lens} system to another,and can be determined thanks to a geometrical calibrationprocedure, that quantifies the parameters of the chosenmodel. A well-known model in the literature is the pin-hole model [7–11]. Its parameters can be classified in twotypes: the internal camera geometric and optical charac-teristics (intrinsic parameters) and the 3-D position andorientation of the camera frame relative to a certain coor-dinate system (extrinsic parameters).

For the presented set-up, due to the orthogonality ofthe projectile plane with the axis of the camera, the extrin-sic parameters are fully characterized by the parametersL, Ω, and Resg, leading to a single calibration factor calib.The intrinsic parameters characterise the remaining dis-tortions produced by the {camera-lens} system (Fig. 7).

For example, a camera objective with a wide anglebrings more distortions and induces more measurementerrors than one with a normal angle of view. The intrin-sic parameters of the {camera-lens} system can be de-termined by performing a calibration procedure based ona pin-hole model [7, 12]. These parameters are then usedto correct the distortion on the images generated by thecamera. The difference between the measurements with

Fig. 7. Effect of lens distortion on an image. (a) Distortedimage, (b) ideally projected image, (c) overly corrected image.

and without the correction of the distortion is consideredas a fully characterised error called perspective error.

The calibration process is performed using an opensource calibration toolbox for Matlab [7,12]. The principleis to take pictures of a calibration pattern and compare thedimensions on the image to the corresponding real dimen-sions. The parameters of the model are optimised to fit theimage dimensions with the real dimensions. The employeddistortion model is described on equation (12) [7, 12].

r2d = x2

d + y2d

rcor = 1 + k1r2d + k2r

4d

xcor =fcxxd

rcor+ p0x

ycor =fcyyd

rcor+ p0y (12)

where

– xd [pix] and yd [pix] are the x-coordinate and y-coordinate of a considered point in the distorded im-age; rd [pix] is the distance between this point and thecentre of the image;

– xcor [pix] and ycor [pix] are the corrected x-coordinateand y-coordinate of the considered point; rcor [pix] isthe distance between this point and the centre of theimage;

– fcx [pix] and fcy [pix] are the x-coordinate focal lengthand y-coordinate focal length of the camera [7, 12];

– p0x [pix] and p0y [pix] are the x-coordinate prin-cipal point and y-coordinate principal point of thecamera [7, 12];

– k1 [pix] and k2 [pix] are the radial distortionparameters [7, 12].

More complex pin-hole models exist, which take into ac-count the tangential distortions and higher degree radialdistortions [7,12]. These distortions are not significant forthe presented {camera-lens} system.

The results obtained with the calibration toolbox arepresented in Table 2. The induced distortion model is il-lustrated on Figure 8.

As one can see in Figure 8, the distortion is null at thecentre of the image, and increases from the centre to theborders.

Besides, when dc [pix] and dp [pix] are measured inthe same zone on the image (for example on the top leftof the image), the distortions related to the two measure-ments are identical. Consequently, dc [pix] and dp [pix] aremultiplied by the same distortion factor in equation (3).The induced error on the velocity due the distortion is

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Table 2. Values of the camera intrinsic parameters obtainedthanks to the calibration procedure [12].

Parameter Value [pix] Uncertainty (3σ) [pix]fcx 2685.88 10.24fcy 2682.32 10.19p0x 526.26 5.68p0y 527.51 5.66k1 –0.13961 0.0241k2 –0.42466 0.4496

Fig. 8. Illustration of the distortion correction [12].

Fig. 9. Influence of the distances between the centre of theimage and the measurements on the induced perspective error.

then null. If the distance between the two measurementsincreases, the error also increases.

All in all, the perspective error depends both on the lo-cation where the measurement on the image is performedand on the distance between dc and dp (Fig. 9).

For the present example, the quantification of the per-spective error is assessed when dc [pix] is centred whiledp [pix] is measured on the middle-left zone of the image(Fig. 8). The relative velocity error due to the perspectiveerror is then equals to 0.77%. Once the distortion is cor-rected, the only uncertainties that remain are based onthe uncertainties on the parameters presented in Table 2.Supposing that all these uncertainties are independent,the relative distortion uncertainty Udist on the velocitymeasurement is computed using equation (13):

Udist =

√√√√ n∑1

U ,2i,persp (13)

where Ui,persp are the relative uncertainties on the velocitymeasurement due to the uncertainties i on the parameterspresented in Table 2. The computed uncertainty is equalto 0.16%.

The computed error and uncertainty value remain rel-atively low, due to the excellent quality of the camera.Nevertheless, they remain significant in comparison withthe other identified uncertainties.

3.6 Uncertainty on the chronometer of the camera

This uncertainty called chrono uncertainty is linked to theprecision of the chronometer Uc [ns] of the camera. Thisvalue can be found in the datasheet of the camera. In thepresent case, this value is equal to 0.001% [3].

3.7 Discretisation errors

In every step of the velocity measurement process thatinvolves a transformation of real distances into pixel dis-tances, discretisation errors appear. Their effect is takeninto account by rounding up the different computeduncertainties.

4 Results and discussions

All the results of the previous sections are gathered inTable 3.

Assuming that all these individual uncertainties areindependent, the total relative uncertainty on the veloc-ity measurement is obtained with equation (5), with andwithout correcting the perspective. These results lead tothe following conclusions:

– For the proposed set-up, the most important sourceof uncertainty in terms of contribution to the totalvelocity uncertainty is the gamma uncertainty. Havinga precise system to measure and reduce this angle istherefore essential.

– The second most important parameter is the perspec-tive error. Even with an excellent camera, this er-ror remains significant. The correction of the distor-tion is essential to reach a certain level of precision.

C. Robbe et al.: Quantification of the uncertainties of high-speed camera measurements 201-p7

Table 3. Global results of the uncertainty study.

Parameter that Value Value of the induced

Type of uncertainty/error drives this of the relative velocity

uncertainty/error parameter uncertainty/error

Gamma angle γ [◦] (Fig. 4) 3 1.01%

error (before [AC] [pix](–512, 0) 0.77%

Perspectivecorrection) (Fig. 1)

uncertaintyUi,persp Table 2 0.16%

(after correction)

Motion blur kMB [–] 2 0.62%

Vertical plane dh [mm] 20 0.29%

Reference pixel dc [pix] 1 0.26%

Reference length dc [mm] 0.2 0.11%

Chronometer Uc [ns] 10 0.001%

ParallaxRadius r [mm]

40 0.0002%(Fig. 5)

TOTAL (with perspective correction) 1.26%

TOTAL (without perspective correction) 1.47%

Even with the correction, the perspective uncertaintyremains significant.

– The motion blur uncertainty comes in the third place.Besides the empirical kMB [–] parameter, this uncer-tainty is directly influenced by the ST and the velocityof the projectile (Eq. (10)). The high-speed cameraused in the proposed set-up is good enough to keepthis value at an acceptable level.

– The three calibration uncertainties (vertical plane, ref-erence pixel and reference length uncertainties) follow.In particular, the value of the vertical plane uncertaintycan be reduced by designing a system that allow mea-suring precisely and reducing dh.

– The parallax and the chrono uncertainties are insignif-icant for the proposed set-up.

Another experimental set-up, with other camera param-eters, and different values of the parameters driving theindividual uncertainties can lead to different conclusions.Therefore, for every set-up, all the presented mathematicaltools should be used to identify the critical uncertaintiesand to improve the precision of the measurements. Inte-grating these computations in the tracking software allowssystematically quantifying the uncertainty related to anycamera measurement [2].

5 Sensibility of the input parameters

The sensibility studies allow the experimenter to refine thesetup parameters presented in Table 1 in order to minimizethe velocity measurement uncertainty.

In this section, sensibility studies of two input parame-ters presented in Table 1 are performed. These parametersare the distance of the camera L and the travelling dis-tance of the projectile dp. The sensibility studies of theother parameters have been previously presented [2]. Themeasurements of dc [pix] and dp [pix] are performed on

Fig. 10. Influence of the input parameter L on the relativeuncertainty of the velocity measurement.

the centre of the image in order to nullify the perspectiveand parallax errors.

5.1 Sensibility of the L parameter

The results are presented on Figure 10. A low value ofL leads to a significant velocity measurement uncertainty,due to the individual contributions of the gamma angleuncertainty, and the vertical plane uncertainty. Besides,the contribution of the motion blur uncertainty, and thereference pixel uncertainty increases with L. As each pixelcorresponds to a bigger real distance as L increase, themeasurement error of one pixel has more repercussion onthe velocity measurement. For the same reason, oscilla-tions on the curves characterizing pixels measurements(motion blur uncertainty, reference pixel uncertainty, andtotal uncertainty) also increase. These oscillations are dueto discretisation errors, which increase as the distance rep-resented by one pixel increases.

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Fig. 11. Influence of the input parameter dp on the relativeuncertainty of the velocity measurement.

The minimum velocity measurement uncertainty is ob-served for a distance L included between 0.6 m and 1.3 m,which corresponds to the value proposed in Table 1.

5.2 Sensibility of the dp parameter

The results are presented on Figure 11. When dp [mm]increases, dp [pix] also increases, and the impact of anerror of one pixel due to the motion blur uncertainty isreduced. At the same time, the gamma angle uncertaintyconstantly increases.

When the distance dp becomes higher than 100 mm,the contribution of gamma angle uncertainty angle contin-ues to increase, and is not compensate by the reduction ofthe motion blur uncertainty, leading to an increase of thetotal uncertainty.

6 Experimental validation

To validate the computation, velocity measurements us-ing the high-speed camera are compared with a referencemeasurement. Statistical indicators are compared with thecomputed uncertainty results. The set-up consists in mea-suring the velocity of a projectile flying at around 55 m/s.The velocity measurements are performed with the cam-era, as presented on Figure 1, and with an optical lightscreen barrier Drello LS9iN, frequently used in ballistic.A laser angle measurement system allow reducing the un-certainty on the gamma angle up to a value of 1◦ instead ofthe aforementioned 3◦. Eighteen projectiles are shot. Theresults are summarised on Figure 12. One can see the in-dividual results in terms of relative difference between thelight screen barrier measurements and the camera mea-surements. The camera velocity measurement uncertaintyis estimated at 0.766%, and the light screen barrier ischaracterized by an uncertainty on the velocity measure-ment of 1% [2, 13]. The total theoretical uncertainty ofthe relative difference between measurements, computedwith equation (5), is equal to 1.26%. The standard devi-ation of the relative difference between the measurements

Fig. 12. Comparison of the uncertainty computed with theproposed approach and experimental measurements realisedwith a reference measurement.

is estimated thanks to the experimental samples and thestatistical law [2, 14, 15]:

σ2diff =

√σ2ech

χ2−1(0, 975; n− 1)(14)

where σdiff is the standard deviation of the relative differ-ence of measurements, σech is the standard deviation ofthe presented sample, and χ2−1 is the Khi2 reverse law.The total theoretical uncertainty is compared with theconfidence interval determined by the average plus or mi-nus three times the standard deviation computed by equa-tion (14). The theoretical uncertainty is very close to theconfidence interval characterizing the measurements. Theuncertainty computation approach is therefore reliable.

7 Applying this study to continuousdisplacement

All these uncertainty computation steps can be appliedto the continuous measurement of the displacement of theprojectile as a function of time (Fig. 13). This exampleshows the displacement measurement of a non-lethal de-formable projectile impacting a rigid target.

The measurement is performed with the same experi-mental set-up, and is post-processed by the tracking soft-ware [2, 16]. The obtained curve rises up to a maximumvalue, and then declines, characterizing the rebound of theprojectile. The perspective and the parallax have not beenconsidered. The uncertainty computations allow defining acorridor around the nominal curve in which the real mea-surement should figure. For relatively low displacements,the critical uncertainty is the motion blur uncertainty. Forthe maximal displacement, the critical uncertainty is thegamma uncertainty. These results are in perfect agreementwith Figure 11. In conclusion, the precision of the resultsis fully characterised and is globally acceptable for thepresented application.

C. Robbe et al.: Quantification of the uncertainties of high-speed camera measurements 201-p9

Fig. 13. Application of the uncertainty study on the con-tinuous displacement of the projectile as a function of timemeasurement.

8 Conclusions

A complete theoretical uncertainty study on the veloc-ity measurement of a flying projectile, using a high-speedcamera, has been presented. All these results are applica-ble to other measurements performed by a camera. Manysources of uncertainties have been identified and quanti-fied. On the one hand, the velocity measurement uncer-tainties have been quantified. On the other hand, it showswhich aspects of the set-up should be improved to reducethe uncertainty. In particular, the authors draw the atten-tion on the importance of mastering the geometry of theset-up, and on systematically correcting the distortionsinduced by the “camera-lens” system.

A sensibility study of two input parameters has beenexposed, allowing the experimenter to optimize these pa-rameters as to reduce the total velocity measurementuncertainty.

The results have been validated experimentally thanksto a reference velocity measurement. Good correspondenceis observed.

Finally, all these results have been used to generatea confidence corridor around a continuous displacementmeasurement, as a function of time, using the high-speedcamera. The results show how precise such measurementscan be.

Thanks to the exposed study, the velocity mea-surements can achieve a very high degree of precision,and the uncertainty related to the performed measure-ment can be systematically quantified, which makes these

measurements very robust and reliable. The methods canalso help the buyer to dimension the requested materialto obtain a desired uncertainty on the measurements.

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