Assessing the accuracy of applying photogrammetry to take geometric measurements on building products

The main objective of the research Assessing the accuracy of applying photogrammetry to take geometric measurements on building products is to characterize the errors of the photogrammetryderived geometric measurements on building products in a systematic, practical, and statistically significant way. In this research, we intend to use the offtheshelf, portable digital cameras, instead of highend, expensive cameras specially manufactured... Đề tài Hoàn thiện công tác quản trị nhân sự tại Công ty TNHH Mộc Khải Tuyên được nghiên cứu nhằm giúp công ty TNHH Mộc Khải Tuyên làm rõ được thực trạng công tác quản trị nhân sự trong công ty như thế nào từ đó đề ra các giải pháp giúp công ty hoàn thiện công tác quản trị nhân sự tốt hơn trong thời gian tới.

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Assessing the Accuracy of Applying Photogrammetry to Take Geometric Measurements on Building Products

Fei Dai 1 and Ming Lu, M.ASCE 2

Abstract: The present research describes the fundamental working mechanism of photogrammetry and characterizes the errors of the

photogrammetry-derived geometric measurements on building products in a systematic, practical, and statistically significant way A site engineer simply takes snapshots of a building product with a digital camera from different angles Back in office, the engineer derives as-built measurements through postprocessing those photos by use of photogrammetry software The twelve objects sampled in our experiments were building products and building facilities found on the campus of Hong Kong Polytechnic University, yielding 79 paired geometric measurements 共length, width, and height兲 by photogrammetry and by measurement tape, respectively The biases and limita-tions of analyzing the agreement between two sets of measurements by regression and correlation coefficient techniques were first revealed Then, the “95% limits of agreement” method was applied on the sample data and the confidence intervals were established for the limits of agreement derived, so as to ensure validity and statistical significance of the results In short, the main contribution of this research lies in formalizing a statistically significant, quantitatively reliable technique to assess the accuracy of applying photogrammetry

in particular applications of construction engineering Through weighing the accuracy level achievable by photogrammetry against the accuracy level desirable in a particular application, the engineer makes the final decision on the applicability of the photogrammetry-based approach.

DOI: 10.1061/ 共ASCE兲CO.1943-7862.0000114

CE Database subject headings: Photogrammetry; Measurement; Errors; Photography; Geometry; Construction management

Author keywords: Photogrammetry; Dimension measurement; Errors; Photography; 95% limits of agreement; Confidence interval

Introduction

The surveying technique of photogrammetry extracts input data from two-dimensional 共2D兲 photo images and maps them onto a three-dimensional 共3D兲 space In general, photogrammetry can be used to acquire the profiles of an object, quantify its geometric dimensions, and track its status change 共e.g., tracking the object’s orientation and its spatial relationships with other objects 兲 In construction engineering, photogrammetry has been applied in building components modeling 共Proctor and Atkinson 1972; Dai and Lu 2008 兲, project progress control 共Quiñones-Rozo et al.

2008; Kim and Kano 2008; Memon et al 2005 兲, and keeping evidence of the impact of tunnel construction on historical build-ings 共Luhmann and Tecklenburg 2001兲.

The accuracy of photogrammetry is dependent on the preci-sion of the camera used and the quality of the photos taken, and the functionality of the photoprocessing software applied Al-though photogrammetry holds great potential to provide an

alter-native for quantity surveying in construction management, a formal method for assessing the accuracy of geometric measure-ments taken by photogrammetry is yet to be developed The main objective of the present research is to characterize the errors of the photogrammetry-derived geometric measurements on building products in a systematic, practical, and statistically significant way.

In this research, we intend to use the off-the-shelf, portable digital cameras, instead of high-end, expensive cameras specially manufactured for photogrammetry applications Our research falls into the category of close-range photogrammetry measurement, which usually applies to those situations where the target object is away from the camera at a distance ranging from 1 to 300 m 共Luhmann et al 2006兲 We further narrow the shooting range to 关1

m, 6 m兴 so to be aligned with practical application needs for quantity surveying in building construction The application set-ting is given as follows:

A site engineer is responsible for taking geometric measure-ments on building products that have been just placed or partially completed Those measurements represent the as-built informa-tion and are used 共1兲 to ascertain the actual quantity of work completed; 共2兲 to check the quality of finished products against the building design and technical specifications; and 共3兲 to certify payment requests filed by the contractor In the conventional way, the engineer would apply a measurement tape to determine the length of each dimension of a building product and record the data in a form and on the spot.

As an alternative, the engineer simply takes snapshots of the building product with a digital camera from different angles Back

in office, the engineer derives as-built measurements through post processing those photos by use of photogrammetry software In

1 Ph.D Candidate, Dept of Civil and Structural Engineering, Hong Kong Polytechnic Univ., Hung Hom, Hong Kong E-mail: dai.f@polyu.

edu.hk 2 Associate Professor of Construction Engineering and Management, Dept of Civil and Structural Engineering, Hong Kong Polytechnic Univ., Hung Hom, Hong Kong 共corresponding author兲 E-mail: cemlu@polyu.

edu.hk Note This manuscript was submitted on January 6, 2009; approved

on June 22, 2009; published online on January 15, 2010 Discussion period open until July 1, 2010; separate discussions must be submitted for

individual papers This paper is part of the Journal of Construction Engineering and Management, Vol 136, No 2, February 1, 2010.

©ASCE, ISSN 0733-9364/2010/2-242–250/$25.00.

242 / JOURNAL OF CONSTRUCTION ENGINEERING AND MANAGEMENT © ASCE / FEBRUARY 2010

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addition, applying the photogrammetry method at a building site would produce two “by-product” benefits: First, measurements can be taken effortlessly on those building elements situated in hazardous areas that are unsafe to access Second, the alignment

of a building product can be continuously monitored by taking site pictures at different times.

The remainder of this paper is organized as follows: The fun-damental mechanism of photogrammetry is described first We then explain major reasons that account for errors in the geomet-ric measurements obtained by photogrammetry; they are 共1兲 the system error due to distortion of the camera lens and 共2兲 the random error due to human factors Next, we describe the steps of method application, experiment design, and the sample data ac-quired After revealing biases and limitations of applying regres-sion and correlation coefficient methods for error analysis, we resort to the 95% limits of agreement method to assess the accu-racy of photogrammetry based on the sample data We further establish the confidence intervals for the limits of agreement in order to ensure validity and statistical significance of the results.

The practical implication and applicability of the photogrammetry-based approach to construction engineering ap-plications is discussed before drawing conclusions.

Fundamentals of Photogrammetry

The basic mathematical equations underlying photogrammetry

are called Collinearity Equations 共given in Appendix I兲, which unify the image coordinates system in the camera with the object coordinates system in the global space 共Wong 1980; Wolf 1983;

McGlone 1989兲 Thus, the coordinates 共x,y兲 of an image point in the image plane can be analytically transformed into its coordi-nates 共X, Y, and Z兲 in the global space The very basic technique

of photogrammetry is effective and computationally simple.

It is notable that the photogrammetry algorithm is based on definitions of the interior orientation and the exterior orientation

of a photographic system Fig 1 gives the pinhole camera model

to illustrate how a camera forms the image of an object The

interior orientation is described by the principle point and the principle distance of a camera in the image coordinates system.

The principle point refers to the projected position of the perspec-tive center 共O in Fig 1兲 on the image plane 共x o , y o in Fig 1 兲 while

the principle distance 共c in Fig 1兲 is the distance between the

perspective center and the image plane The exterior orientation is defined by six parameters of the camera in the global space,

namely, the location coordinates (X o , Y o , and Z o ) and the Euler

orientation angles 共␻, ␾, and ␬兲 of the camera’s perspective cen-ter If a camera’s internal parameters are known, any spatial point can be fixed by intersecting two rays of light that are projected from two different camera stations 共Fig 2兲 Thus, with two

pic-tures taken from different angles, it is possible to determine the coordinates of a point in the image coordinates system inside the camera as well as in the object coordinates system in the global space Eventually, a collection of the points fixed by photogram-metry computing suffice to produce a skeleton model of the ob-ject Fig 3 gives an example of the sample photos of a façade and the different perspectives of its 3D model resulting from photo-grammetry Next, we discuss two major factors that induce the measurement errors of photogrammetry.

System Error due to Lens Distortion

Measurement errors due to camera lens distortion can be treated

as the system error with a consistent effect 共Viswanathan 2005兲 It causes an image point on the image plane to shift from its true position 共x n ⬘ , y n ⬘ 兲 to a perturbed position 共x n , y n 兲 Thus, the true coordinates of any image point can be compensated by Eq 共1兲

Fig 1 Pinhole camera model

Fig 2 Fixing a spatial point by intersecting two rays of light

Fig 3 Sample photos of a façade and different perspectives of the

3D model resulting from photogrammetry

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x n ⬘ = x n + dx

The camera lens distortion 共i.e., dx and dy兲 can be taken as the

aggregate of the radial distortion and the decentering distortion 共Beyer et al 1995; Fraser 1996兲 As the lens of a camera is actu-ally composed of a combination of lenses, the centers of those lens elements are not strictly collinear, giving rise to decentering distortion In contrast, the radial distortion occurs in each single optical lens and the distortion effect is magnified along the radial direction of the lens: the further a point is away from the center of the lens, the larger error is produced for its projected image point.

Therefore, dx, dy can be decomposed by Eq. 共2兲

dx = dx r + dx d

Assume the optical axis of the lens is perpendicular to the image plane, then

dx r = K 1 共x n ⬘ − x p 兲r 2 + K 2 共x n ⬘ − x p 兲r 4

dy r = K 1 共y n ⬘ − y p 兲r 2 + K 2 共y n ⬘ − y p 兲r 4

dx d = P 1 关r 2 + 2共x n ⬘ − x p 兲 2 兴 + 2P 2 共x n ⬘ − x p 兲共y n ⬘ − y p 兲

dy d = P 2 关r 2 + 2共y n ⬘ − y p 兲 2 兴 + 2P 1 共x n ⬘ − x p 兲共y n ⬘ − y p 兲

r 2 = 共x n ⬘ − x p 兲 2 + 共y n ⬘ − y p 兲 2 共3兲

Here, x p and y p = coordinates of the principal point; K 1 and K 2

= radial distortion parameters; and P 1 and P 2 = decentering distor-tion parameters When the lens distordistor-tion is small, the system

error due to the lens distortion can be ignored, namely, x n ⬘ ⬇x n and

y n ⬘ ⬇y n ; otherwise, the system error should be corrected.

Those system parameters (K 1 , K 2 , P 1 , and P 2 ) need to be first

determined by following analytical procedures to calibrate the camera 共Tsai 1987; Rüther 1989兲 In our research, we applied the software of PhotoModeler 共Eos System Inc 2007兲 to calibrate a Canon EOS 400 D camera with its focal length fixed at 18 mm

共K 1 = 5.167e-004; K 2 = −1.120e-006; P 1 = 3.924e-005; and P 2

= 3.684e-005 兲 The calibration results indicate the lens distortion

of the camera is relatively small.

Random Error due to Human Factors

Theoretically, one point captured in two different photos is suffi-cient to fix its 3D coordinates To complete this, this step requires identifying and marking the point in the two photos Any human error in point marking gives rise to another form of error—the random error 共Viswanathan 2005兲.

As shown in Fig 4, we assume that the point of P ⬘ 共x ⬘ , y ⬘ 兲 is

the true position of a target point, whereas the point of P 1 共x 1 , y 1 兲

is fixed by photogrammetry computing The discrepancy between the two points is attributed to imprecise point marking To reduce this error, it is advisable to include the target point in three or more photos At the expense of redundancy, the random error on any of the photos can be compensated by the others For example,

if the target point is covered in three photos, then any two can be used to derive the point by photogrammetry, resulting in a total of

three points (P 1 , P 2 , and P 3 兲共Fig 4兲 As the true position of

P ⬘ 共x ⬘ , y ⬘ 兲 actually is unknown, the most likely coordinates of the target point can be determined by least-squares adjustment

共Luh-mann et al 2006 兲, which minimizes the sum of the squares of the residuals as in Eq 共4兲

兺 i=1 n 共 v i 兲 2 = 共 v 1 兲 2 + 共 v 2 兲 2 + ¯ + 共 v n 兲 2 = min 共4兲 where v 1 , v 2 , , v n = residuals on the n measurements Given our

three-point example, we have Eq 共4a兲 for least-squares

adjust-ment

兺 i=1 3 共 v xi 兲 2 = 共x − x 1 兲 2 + 共x − x 2 兲 2 + 共x − x 3 兲 2

兺

i=1

3

共 v yi 兲 2 = 共y − y 1 兲 2 + 共y − y 2 兲 2 + 共y − y 3 兲 2 共4a兲

Taking derivatives with respect to each unknown and equating them to zero, we have Eq 共4b兲

d 兺 i=1 3 共 v xi 兲 2

dx = 2共x − x 1 兲 + 2共x − x 2 兲 + 2共x − x 3 兲 = 0

d 兺 i=1 3 共 v yi 兲 2

dy = 2 共y − y 1 兲 + 2共y − y 2 兲 + 2共y − y 3 兲 = 0 共4b兲

Therefore, an approximation of the target point coordinates is as

Eq 共4c兲:

x

¯ = x = x 1 + x 2 + x 3

3

y

¯ = y = y 1 + y 2 + y 3

The resulting 共x¯,y¯兲, from a statistical perspective, is more reliable

than any single point measured Note, unlike the system error, the random error due to human factors cannot be analytically re-moved In fact, our present research is mainly concerned with assessing the random error of photogrammetry in taking geomet-ric measurements on building products.

Experiment Design and Sample Data

Our experiment designed for assessing the measurement error of photogrammetry includes the following six steps: 共1兲 identifying

a set of target objects, taking measurement of geometric

dimen-( , )

1 ( , 1 1 )

P x y

3 ( , 3 3 )

P x y

2 ( , 2 2 )

P x y

Fig 4 Simple illustration of random error: various points’

coordi-nates derived for one identical point

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sions by tape for each object, and recording measurement data;

共2兲 taking sufficient photos of the same set of target objects by using a digital camera with fixed focal length; 共3兲 processing photos into 3D representations of the target objects by using pho-togrammetry software; 共4兲 fixing the scale of each object model

by identifying a reference line; 共5兲 taking geometric measure-ments on each object based on its 3D model; and 共6兲 conducting accuracy analysis by comparing the two sets of measurements.

A photo-based 3D model resulting from the above step 3 only represents the relative scale of each edge on the object To convert the relative scales into the absolute measurements requires deter-mination of the length of a reference line in the absolute unit of measure This reference line can be one edge on the object that can be easily measured by tape In case that the target object is not accessible, the reference line can be taken by one edge of an adjacent object which can be spatially related with the current object For instance, a concrete block sits on top of a tall plat-form; one edge on the platform is parallel to one edge of the concrete block We can take the edge on the platform as the ref-erence line, and include the platform in the photos of the concrete block In this way, the absolute measures on all the edges of the concrete block can be fixed by photogrammetry.

The twelve objects sampled in our experiment were the build-ing products and buildbuild-ing facilities found on the campus of Hong Kong Polytechnic University We simply took one edge on each object as the reference line for scaling purpose Table 1 lists the sample data consisting of 79 paired dimension measurements by tape and by photogrammetry, respectively Note, in Table 1, as the first measurement on each subject is used as the reference line for scaling, it is excluded from ensuing error analysis Thus, the sample data available for error analysis consists of 67 pairs of geometric measurements It is noted that the sample size is statis-tically significant to the following measurement error analysis in consideration of the expected accuracy level being in the order of

1 cm and the relatively small variation on the measurement errors 共the sample standard deviation of measurement error being 6.81

mm 兲

Analysis of Photogrammetry Accuracy

Visual Assessment of Agreement

That two measurement methods agree with each other means they yield comparable, interchangeable results when applied on the same object Fig 5 contrasts photo-based measurements against tape readings based on our sample data All sample points would lie on the line of equality 共the diagonal line in Fig 5兲, indicating the two sets of measurements agree with each other However, when the range of variation on the measurements is large com-pared with the difference between the two sets of measurements, this plot may become obscure and inadequate to substantiate the agreement between the two sets of measurements 共Bland and Alt-man 1999 兲 Our case serves an example: the geometric measure-ments in the sample data vary in meters 共ranging from 0.75 to 2.8

m兲 while the differences between the two sets of measurements only differ in millimeters 共ranging from ⫺10 to 20 mm兲.

A better way to visualize the agreement of data are to plot the difference between the two sets of measurements against zero, as given in Fig 6 We can observe a good agreement between the two sets of measurements: the differences are enveloped within

10 to 20 mm, except for three outliers.

Analytical Assessment of Agreement

Applying regression or correlation coefficient techniques to evaluate the agreement between two sets of measurements taken

on the same objects possibly produces biased results 共Altman and Bland 1983兲 To shed light on the biases, we generate two sets of

pseudomeasurement data, X and Y, as plotted in Fig 7 Both the

regression line 共slope=1.02, intercept=0.83兲 and the correlation coefficient 共r=0.93兲 imply the two sets of measurements are well

associated; but it can be seen that nearly all the points lie to the left of the line of equality, thus suggesting a lack of agreement between the two sets of data.

In addition, the regression and correlation coefficient tech-niques share one limitation: their results may vary as different data ranges are considered, while the true indicator of agreement should remain stable irrespective of data ranges 共Bland and Alt-man 2003 兲 To illuminate this problem, we segregate the data of the pseudomeasurements at an arbitrary cut point of 5 Fig 8 shows that the resulting regression lines and correlation coeffi-cients much depend on the subrange of measurements For samples whose values are less than 5, the regression line has a slope of 0.74 and an intercept of 1.36, while the correlation coef-ficient is 0.73 关Fig 8共a兲兴; for samples whose values are greater than or equal to 5, the slope is 0.67 and the intercept is 3.42 as of the regression line, while the correlation coefficient is 0.75 关Fig.

8 共b兲兴 Note in each case, the value of correlation coefficient 共0.75 and 0.73 兲 has considerably decreased compared with the original value of 0.93 derived without dividing the data.

Ninety-Five Percent Limits of Agreement

Applying the “95% limits of agreement” method to assess the agreement of two measurement methods was originally proposed

in the medical research 共Bland and Altman 1986兲 This technique has been applied in a wide range of research disciplines 共as evi-denced by more than 10,000 citations of the original research publication 兲 In the medical discipline, one classical example of applying the 95% limits of agreement was to evaluate the inter-changeability of blood pressure measurements between a new type of electronic instrument and the commonplace sphygmoma-nometer 共mercury bars兲 The new instrument did not pass the test

as the 95% limits of agreement for the differences between the two sets of measurements were found to be 关⫺54.7, 22.1兴 mmHg, far exceeding the generally accepted error of margin in medicine 共i.e., within ⫾10 mmHg兲共Bland and Altman 1999兲 In the present research, we intend to assess the discrepancy between geometric measurements taken on the same building products by photo-grammetry and by tape The nature of our problem is analogous to the blood pressure measurement problem in medicine, lending it well to applying the 95% limits of agreement.

The “95% limits of agreement” method is based on two as-sumptions on the sample data: 共1兲 the mean and the standard deviation of the differences between the two sets remain constant along the entire range of measurements; and 共2兲 the differences between the two sets roughly follow a normal distribution 共Bland and Altman 1995兲 Fig 9 presents the two plots used to validate the above assumptions for our present problem, namely: 共1兲 the scatter plot of the difference against the average values of the two sets of measurements; and 共2兲 the histogram of the differences In Fig 9 共a兲, all the points scatter around the horizontal axis along the range of measurements, without displaying particular diver-gence or converdiver-gence patterns This indicates the mean and

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Trang 5

Table 1 Seventy-Nine Paired Dimension Measurements Taken from Twelve Subjects by Tape and by Photogrammetry, Respectively

Subject name

Tape measurement 共mm兲

Photo-based measurement 共mm兲 Subject name

Photo-based measurement 共mm兲

“Air conditioner” 5,720 5,720 “Power-transmission equipment” 610 610 “Jockey-hall signboard” 2,830 2,830

“Windows of classroom” 2,120 2,120 “Door of classroom” 2,520 2,520 “Pavilion” 690 690

“Road signboard” 830 830 “Ventilation equipment” 870 870 “Medals podium” 350 350

Trang 6

dard deviation of the differences remain constant Fig 9共b兲 shows that the differences between the two sets appear to follow a nor-mal distribution.

Note as the magnitude of measurement increases, any diver-gence or converdiver-gence trend identified in regard to the differences between the two sets of measurements implies a relationship be-tween the error and the magnitude of measurement In such cases,

to determine the limits of agreement first entails transforming all the measurements by taking logarithm 共Bland and Altman 1986兲

or using a ratio of the differences over the averaged measure-ments 共Linnet and Bruunshuus 1991兲.

Given the sample mean x ¯ and the sample standard deviation s

of the differences between the two sets of measurements, we have the lower and upper limits of agreement determined by Eq 共5兲

Lower limit = x ¯ − 1.96s Upper limit = x ¯ + 1.96s 共5兲 Note that 1.96 in Eq 共5兲 is the 95% two-tailed cut value on the standard normal distribution Then, we would expect with 95%

likelihood, the differences between the two sets of measurements fall between the two limits 共Bland and Altman 2003兲 As for our sample data, the mean difference of the photo-based measurement subtracting the tape measurement is 1.96 mm 共i.e., x¯兲, and the

standard deviation of the difference is 6.81 mm 共i.e., s兲 Hence, by

Eq 共5兲, the lower limit and upper limit are determined to be minus 15.30 and 11.39 mm, respectively We can state that with 95% likelihood, any geometric measurement of a building prod-uct taken by photogrammetry would differ from the correspond-ing tape measurement by no less than 15.30 mm and no more than 11.39 mm.

Confidence Intervals on Limits of Agreement

Analogous to the sample mean and the sample standard deviation, the derived limits of agreement are only estimates based on lim-ited sample data and are subject to change as different samples are taken To complete the statistical analysis, it is necessary to establish confidence intervals around the estimated values of the limits of agreement so as to infer their true values with respect to the whole population.

First, we establish the 95% confidence intervals for the mean difference between the two sets of measurements by employing

the statistic of the t-distribution with n − 1 degrees of freedom.

For 95% level of confidence, the interval is represented in Eq 共6兲

关x¯ − t n−1,0.025 s / 冑 n, ¯ + t x n−1,0.025 s / 冑 n 兴共6兲

In the case of our sample data, the sample mean difference x ¯ is

⫺1.96 mm, the sample size is 67, and t 66,0.025 is 1.998 The 95%

confidence interval for the mean difference is determined as 关⫺3.62 mm, ⫺0.29 mm兴.

Next, we establish the 95% confidence intervals for the limits

of agreement by Eq 共7兲

-30 -20 -10 0 10 20 30

Number of Dimension

Fig 6 Difference between photo-based measurements and tape

mea-surements

regression line

y = 1.02 x + 0.83

0 2 4 6 8 10 12

Measurement A

line of equality

Fig 7 Regression analysis on two sets of pseudomeasurement data

共dimensionless兲

y = 0.74 x + 1.36 0

2 4 6 8 10 12

0 2 4 6 8 10 12 Measurement A

(a)

y = 0.67 x + 3.42

0 2 4 6 8 10 12

0 2 4 6 8 10 12 Measurement A

(b)

Fig 8 Regression analysis on the two pseudomeasurement data at

the cut point of 5 共dimensionless兲

0 500 1000 1500 2000 2500 3000

Tape Measurement (mm)

Fig 5 Contrasting photo-based measurements against tape

measure-ments

Trang 7

关LL − t n−1,0.025 1.71s / 冑 n, LL + t n−1,0.025 1.71s / 冑 n 兴

关UL − t n−1,0.025 1.71s / 冑 n, UL + t n−1,0.025 1.71s / 冑 n 兴共7兲

in which LL = lower 95% limit of agreement; UL = upper 95%

limit of agreement; and 1.71s / 冑 n = standard error of the 95%

lim-its of agreement Note the mathematical deduction of this stan-dard error is not commonly found in the literature and hence is given in Appendix II.

In our case, 1.71s / 冑 n equals 1.42 mm Hence, the 95%

confi-dence interval for the lower limit of agreement is 关−15.30

− 1.998 ⫻1.42兴 to 关−15.30+1.998⫻1.42兴, namely, ⫺18.14 mm to

⫺12.46 mm Similarly, the 95% confidence interval for the upper limit of agreement is 关11.39−1.998⫻1.42兴 to 关11.39+1.998

⫻1.42兴, namely, 8.55 mm to 14.23 mm Fig 10 depicts the 95%

confidence intervals for the sample mean difference and the lower and upper limits of agreement in dashed lines.

The relatively narrow intervals suggest that the 95% limits of agreement derived from the sample data 共i.e., 关⫺15.30, 11.39 mm兴兲 can be taken to represent such statistical descriptors for the population Given particular accuracy requirements in a given construction application, this finding provides the quantitative basis to make decisions on whether to accept or reject photogram-metry as an alternative to conventional tape measurements, as discussed in the next section.

Applicability of Photogrammetry-Based Approach

Photogrammetry provides a potential alternative to the conven-tional approach to measuring geometric dimensions of building products by tape Nonetheless, the resulting accuracy of photo-grammetry is largely dependent on three factors, namely, 共1兲 the quality of the camera used 共such as the optical precision of the lens and the quantity of pixels in forming a digital image 兲, 共2兲 the quality of the photos taken 共such as the clarity, the lighting, and the contrast of the picture; the shooting distance between the ob-ject and the camera兲 and 共3兲 the functionality of the photoprocess-ing software applied 共e.g., the calibration of a camera, resulting in the determination of the camera’s internal parameters for photo-grammetry computing.兲

The photogrammetry-based approach can lend itself well to a particular application setting of construction engineering, such as checking the geometric dimensions of “as-built” building prod-ucts or monitoring the settling displacements of control points on

an existing building Nonetheless, it should be ensured that the achievable accuracy level of the photogrammetry-based approach matches up to the desired accuracy level for a particular applica-tion before implementing the approach on site For instance, dur-ing the course of the present research, experienced consultant engineers in Hong Kong were interviewed, revealing that the commonly acceptable error tolerance for building settlement monitoring should fall in the order of ⫾25 mm of the actual vertical dimension measurement In fact, the photogrammetry-based measurement approach being evaluated throughout the present research has produced the accuracy level sufficient to building settlement monitoring, namely, 关⫺15.30 mm, 11.39 mm兴

in terms of the 95% limits of agreement as benchmarked against the tape measurements.

In short, the main contribution of the research presented is formalizing a statistically significant, quantitatively reliable method to assess the accuracy of applying photogrammetry in particular applications of construction engineering Through weighing the accuracy level achievable by photogrammetry against the accuracy level desirable in a particular application, the engineer makes the final decision on the applicability of the photogrammetry-based approach.

Conclusions

The surveying technique of photogrammetry extracts input data from 2D photo images and maps them onto a 3D space In

gen 30 -20 -10 0 10 20 30

0 500 1000 1500 2000 2500 3000

Dimension Measurements (mm)

(a)

0 2 4 6 8 10 12 14 16 18 20

Difference (Photo-based - Tape) (mm)

-15 -12 -9 -6 -3 0 3 6 9 12 15

(b)

Fig 9 Plots of: 共a兲 the scattered measurement difference against the average of photogrammetry and tape measurements; 共b兲 the histo-gram of the differences

-25 -20 -15 -10 -5 0 5 10 15 20 25

0 500 1000 1500 2000 2500 3000 3500 4000

Dimension Measurements (mm)

Limit of agreement

Mean

14.23 8.55 -0.29 -3.62 -12.46 -18.14

Fig 10 Ninety-five percent confidence intervals for sample mean

difference and 95% limits of agreement

Trang 8

eral, photogrammetry provides a potential alternative to the con-ventional approach to measuring geometric dimensions of building products by tape The photogrammetry-based approach can lend itself well to a particular application setting of construc-tion engineering; examples are checking the geometric dimen-sions of as-built building products or monitoring the settling displacements of control points on an existing building By sim-ply taking snapshots of the building product with a digital camera with different angles, a site engineer is able to derive as-built measurements through post processing those photos by use of photogrammetry software.

It is reemphasized that the achievable accuracy level for the photogrammetry-based approach should match up to the desired accuracy level for a particular application prior to implementing the approach on site The resulting accuracy of photogrammetry is largely dependent on 共1兲 the quality of the camera used; 共2兲 the quality of the photos taken; and 共3兲 the functionality of the pho-toprocessing software applied The main contribution of the re-search presented is formalizing a statistically significant, quantitatively reliable technique to assess the accuracy of apply-ing photogrammetry for geometric dimension measurements in particular applications of construction engineering By weighing the accuracy level achievable by the methodology against the accuracy level desirable according to particular application re-quirements, the engineer makes the final decision on the applica-bility of the photogrammetry-based approach.

In summary, the very basic technique of photogrammetry is effective and computationally simple As photogrammetry has been digitized, its application cost has been much reduced while its accuracy keeps improving with technological advances in digi-tal cameras and computer software The systematic approach we have proposed for assessing the accuracy of photogrammetry is conducive to finding new applications of photogrammetry in con-struction engineering and management.

Acknowledgments

The research presented in this paper was substantially funded by Hong Kong Research Grants Council 共Grant No PolyU 5245/

08E兲.

Appendix I Collinearity Equation of Photogrammetry

冤 x n − x o

y n − y o

− c 冥 = ␭M 冤 X n − X o

Y n − Y o

where M = rotation matrix; ␭=scale factor; 共X o , Y o and Z o 兲

= location of the perspective center in the object space; and p n

= 共x n , y n 兲 T and P n =共X n , Y n , Z n 兲 T = coordinates of the nth target in

the image plane and object space, respectively Algebraic manipu-lation of Eq 共8兲 yields the well-known Collinearity Equations relating the nth target location in object space to the

correspond-ing point on image plane

x n − x o = − c m 11 共X n − X o 兲 + m 12 共Y n − Y o 兲 + m 13 共Z n − Z o 兲

m 31 共X n − X o 兲 + m 32 共Y n − Y o 兲 + m 33 共Z n − Z o 兲

y n − y o = − c m 21 共X n − X o 兲 + m 22 共Y n − Y o 兲 + m 23 共Z n − Z o 兲

m 31 共X n − X o 兲 + m 32 共Y n − Y o 兲 + m 33 共Z n − Z o 兲共9兲

where m ij 共i, j=1,2,3兲=elements of the rotation matrix M that

are functions of the Euler orientation angles 共␻, ␾, ␬兲

m 11 = cos ␾ cos ␬

m 12 = sin ␻ sin ␾ cos ␬ + cos ␻ sin ␬

m 13 = − cos ␻ sin ␾ cos ␬ + sin ␻ sin ␬

m 21 = − cos ␾ sin ␬

m 22 = − sin ␻ sin ␾ sin ␬ + cos ␻ cos ␬

m 23 = cos ␻ sin ␾ sin ␬ + sin ␻ cos ␬

m 31 = sin ␾

m 32 = − sin ␻ cos ␾

The orientation angles (␻, ␾, and ␬ ) are essentially the pitch,

yaw, and roll angles of the camera in the object space.

Appendix II Standard Error of 95% Limits of Agreement

The standard error of the 95% limits of agreement can be denoted by: 冑 Var 共X¯⫾1.96S兲, where X¯=random variable of the sample mean; and S = random variable of the sample standard deviation.

As X ¯ and S are independent, Var共X¯⫾1.96S兲, which is the

vari-ance of the 95% limits of agreement, can be written as

Var共X¯ ⫾ 1.96S兲 = Var共X¯兲 + 1.96 2 Var共S兲共11兲 The Var共X¯兲 is ␴ 2 /n, and approximated to s 2 /n To determine

Var共S兲, we first derive the expected value and variance of S 2 , i.e.,

E 关S 2 兴 and Var共S 2 兲, by calculating the expected value and variance

of ␴ 2 ␹ n−1 2 /共n−1兲 on the grounds that S 2 is distributed as the sta-tistic ␴ 2 ␹ n−1 2 /共n−1兲共␹ n−1 2 is the Chi-square distribution with n

− 1 degrees of freedom兲 According to Rohatgi 共1976兲, the ex-pected value and variance of ␹ n−1 2 are denoted by

E 关␹ n−1 2 兴 = n − 1

Var 共␹ n−1

Thus

E 关S 2 兴 = E关␴ 2 ␹ n−1 2 /共n − 1兲兴 = ␴ 2

Var共S 2 兲 = Var关␴ 2 ␹ n−1

2 /共n − 1兲兴 = 2␴ 4 /共n − 1兲 共13兲 Then we employ the delta method 共Oehlert 1992兲 to derive the

variance of S This method is to take second-order Taylor

expan-sions to approximate the variance of a function of one or more

random variables Given X be a random variable with E 关X兴=␮ x

and Var共X兲=␴ x 2 , then the approximate variance of a function of X

is given by

Trang 9

Var 关f共X兲兴 ⬇ 冋 d

dX f 共X兲兩 ␮ x 册 2

⫻ ␴ x

2 共14兲

provided that f is twice differentiable and that the mean and vari-ance of X are finite Let f 共X兲= 冑 X, Eq. 共14兲 becomes

Var共冑 X 兲 ⬇ 冋 d

dX 冑 X 兩 ␮ x 册 2

⫻ ␴ x 2 = 冋冏 1

2 冑 X 冏 ␮ x 册 2

⫻ ␴ x 2 = ␴ x

2

4 ␮ x

共15兲

Let X = S 2 , and denote ␮ x , ␴ x

2 in Eq 共15兲 by Eq 共13兲, we have Var共S兲 = Var共冑 S 2 兲 = ␴ 2

To use s 2 to represent ␴ 2 , we finally approximate Var共S兲 by

s 2 /2共n−1兲.

Now put the formulae of Var 共X¯兲 and Var共S兲 back into Eq 共11兲

Var 共X¯ ⫾ 1.96S兲 = Var共X¯兲 + 1.96 2 Var 共S兲 = s

2

n + 1.96

2 s 2

2 共n − 1兲

= 冉 1

n +

1.96 2

When n = large, let n − 1 ⬇n, this equation can be approximated into 2.92s 2 /n Hence, standard errors of X¯−1.96S and X¯+1.96S are approximated as 1.71s / 冑 n.

Thus, the standard error for the 95% limits of agreement can

be estimated in Eq 共7兲.

Appendix III Statistical Symbols Used

• ¯ = 1 x /n 兺

i=1

n

x i , the sample mean of a sample x 1 , x 2 , , x n

• s 2 = 1 /共n−1兲兺

i=1

n

共x i − x ¯ 兲 2 = 1 /共n−1兲关兺

i=1

n

x i 2 − nx ¯ 2 兴, the sample vari-ance of a sample.

• s = 冑 s 2 , the sample standard deviation.

• ␮, the mean of the whole population.

• ␴ 2 , the variance of the whole population.

• X ¯ , the random variable of the sample mean.

• S, the random variable of the sample standard deviation.

• E 关X兴, the expected value/mean of the random variable X.

• Var共X兲, the variance of the random variable X.

• ␹ n−1

2 , the statistic of the t-distribution with n − 1 degrees of

freedom.

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Tiêu đề	Assessing the Accuracy of Applying Photogrammetry to Take Geometric Measurements on Building Products
Tác giả	Fei Dai, Ming Lu, M.ASCE
Trường học	Hong Kong Polytechnic University
Chuyên ngành	Construction Engineering
Thể loại	thesis
Thành phố	Hong Kong

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