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Manufacturing Engineering Society International Conference 2017, MESIC 2017, 28-30 June Manufacturing Engineering Society Conference 2017, International Vigo (Pontevedra), Spain2017, MESIC 2017, 28-30 June 2017, Vigo (Pontevedra), Spain

Bayesian model for subpixel uncertainty determination in optical

Bayesian model for Society subpixel uncertainty determination in 28-30 optical Manufacturing Engineering International Conference 2017, MESIC 2017, June 2017, measurements Vigo (Pontevedra), Spain measurements M. Berzala,*, E. Gómeza, J. de Vicenteb, J. Cajaa, C. Barajasa Costing models for*,capacity inCaja Industry 4.0: Trade-off M. Berzal E. Gómezoptimization , J. de Vicente , J. , C. Barajas Dpto. Ingeniería Mecánica, Química y Diseño Industrial, ETS de Ingeniería y Diseño Industrial, Universidad Politécnica de Madrid, between used capacity and operational Dpto. Ingeniería Mecánica, Química y Diseño Industrial, ETS de Ingeniería y Diseño Industrial, efficiency Universidad Politécnica de Madrid, Madrid 28012, Spain a,

a

b

a

a

a a

b b

MadridIndustriales, 28012, Spain Laboratorio de Metrología y Metrotecnia, ETS de Ingenieros Universidad Politécnica de Madrid. Madrid 28006, Spai. Laboratorio de Metrología y Metrotecnia, ETS Universidad Madrid 28006, Spai. a de Ingenieros Industriales, a,* b Politécnica de Madrid. b

A. Santana , P. Afonso , A. Zanin , R. Wernke a

University of Minho, 4800-058 Guimarães, Portugal

Abstract b Unochapecó, 89809-000 Chapecó, SC, Brazil Abstract Uncertainty determination can be obtained by two procedures: GUM and the Monte Carlo Method. This work presents a model Uncertainty can be obtained by two procedures: GUM the measuring Monte Carlo Method.when This using work presents a model that helps todetermination evaluate the uncertainty in measurements collected by and optical machines the Monte Carlo that helpsInitially, to evaluate the uncertainty measurements collectedprobability, by optical measuring machines using thecamera Monte into Carloa method. the model converts in intensity, using Bayesian from the pixel imagewhen derived from Abstract method. thethree model intensity, Bayesianare probability, from thesquares pixel image derived into a polygonalInitially, area with to converts five vertexes. The using outer vertexes fitted using least procedures to from obtaincamera a measurand polygonal with three to five vertexes. The outer vertexes fitted leastverified squares procedures to obtain a measurand shape approximation inofa "Industry subpixel range. have beenareprogrammed into Matlab using synthetic images Under thearea concept 4.0",Algorithms production processes willusing be and pushed to be increasingly interconnected, shape approximation in a subpixel range. Algorithms have been programmed and verified into Matlab using synthetic images with differentbased triangles. a detailed analysis, the usefulness of a new tool, the willcapacity be demonstrated as an information on aThrough real time basis and, necessarily, much more efficient. In parameter, this context, optimization with different triangles. Through uncertainty a detailed analysis, the usefulness ofimages. a new tool, the parameter, will be demonstrated as an alternative method for estimating of measurements of pixel goes beyond the traditional aim of capacity maximization, contributing also for organization’s profitability and value. alternative method forPublished estimatingbyuncertainty of measurements of pixel images. © 2017 The Authors. B.V. Indeed, lean management andElsevier continuous improvement approaches suggest capacity optimization instead of © 2017 The Authors. Published by B.V. committee of the Manufacturing Engineering Society International Conference Peer-review under responsibility of Elsevier the scientific © 2017 The Authors. Published by Elsevier B.V. maximization. The study of capacity optimization andofcosting models is Engineering an important research topic that deserves Peer-review under responsibility of the scientific committee the Manufacturing Society International Conference 2017. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. contributions from both the practical and theoretical perspectives. This paper presents and discusses a mathematical 2017.

model for capacity management based on different costing models (ABCMonte and Carlo TDABC). Keywords:subpixel approximation, bayesian probability, optical measurement, uncertainty, method.A generic model has been developed and itapproximation, was used to bayesian analyzeprobability, idle capacity and to design uncertainty, strategies Monte towards themethod. maximization of organization’s Keywords:subpixel optical measurement, Carlo value. The trade-off capacity maximization vs operational efficiency is highlighted and it is shown that capacity optimization might hide operational inefficiency. 1. Introduction 1. 2017 Introduction © The Authors. Published by Elsevier B.V. Peer-review under responsibility of are the ascientific committeetoofachieve the Manufacturing SocietyinInternational Conference Nowadays, vision machines real alternative measurandEngineering characterization a submillimetre range. 2017. Nowadays, vision machines are a real alternative their to achieve characterization in aautomation submillimetre range. These machines’s speed in obtaining measurements, lack ofmeasurand need of physical contact and has earned These measurements, lack of need of physicalscientific contact and automation hastowards earned them amachines’s prominentspeed placeininobtaining current metrology, thustheir generating a considerable interest oriented Keywords: Cost Models;place ABC; TDABC; Capacity Management; Capacity; Operational Efficiencyscientific interest oriented towards them a prominent in current metrology, thusIdle generating a considerable 1. Introduction * Corresponding author. Tel.: E-mail address:[email protected] cost of idle capacity is a fundamental information for companies and their management of extreme importance * The Corresponding author. Tel.: E-mail address:[email protected]

in modern production systems. In general, it is defined as unused capacity or production potential and can be measured 2351-9789© 2017 The Authors. Published by Elsevier B.V. 2351-9789© 2017 The Authors. Published by Elsevier B.V.hours Peer-review responsibility of the scientific committee of the Manufacturing Engineering Conference in several under ways: tons of production, available of manufacturing, etc.Society The International management of the 2017. idle capacity Peer-review under Tel.: responsibility the761; scientific committee the Manufacturing Engineering Society International Conference 2017. * Paulo Afonso. +351 253of 510 fax: +351 253 604of741 E-mail address: [email protected]

2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 2351-9789 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the Manufacturing Engineering Society International Conference 2017. 10.1016/j.promfg.2017.09.042

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optimizing their operation, especially regarding estimates in uncertainties of measurement. Scientific research in the field of optical metrology has grown in last decade. [1-4]. Nowadays, modern optical equipment can provide a threecoordinate measuring machine, CMM, which already has a standard model developed for measurement. This paper proposes stripping off the Z-axis and using parameter in order to simplify the problem of uncertainties of measurement by performing them in two dimensions. Since the digital optical equipment can produce other sources of uncertainty a specific study will be required which will be accounted for in this paper as well. An optical instrument of measurement can be divided into two parts: as a machine and as an optical system. The machine is a monobloc structure that is supported by the table and allows for measuring by the displacement of the shafts. The optical system is made up of a complex set of lens and devices, such a CCD sensor, which allows the acquisition and transfer of images to the computer. Therefore, CCD camera replaces the "system of contact of the probe" used in a conventional CMM. This division of machine parts allows for the separate analysis of both systems to obtain a simplified model as a basis for the evaluation of the uncertainties of measurement [5-7]. In order to carry out the study of a complex optical model, all of the inner physical properties of lenses and devices should be known. Although due to the complication of this task, this type of study has been disregarded as a possibility in this paper but could be taken into account for future studies. Additionally, the pinhole camera [8, 9] is an alternative model which can be taken into account the amplification and radial and tangential distortions [5, 6]. The new model is based in former procedures, supposing that the optical system is well designed; and the image that CCD receives is a close representation of measurand. The first hypothesis assumes that these geometric variations come from information losses of reflected light in measurand and subsequently guided into optical device, when it is converted in CCD sensor. Therefore, a possible way to represent previous process is modelled such as the transformation from a real number to a natural one, i.e, areas are transformed into intensities. In this paper this conversion is defined as the direct method, and the inverse process from image pixel intensities to areas and edge coordinates, the reverse method. This method and partial area effect share main idea to acquire an accurate edge location [12]. The direct method is mainly used in this paper to generate synthetic images to uses as pattern in verifications tests. Subpixel algorithms give an extra accuracy in measurements although they use to need an extra computing time. The above algorithms output, from the image segmentation, the measuring edges obtained in subpixel range as a set of coordinates in a plane. If the measurement uncertainty is estimated by the Monte Carlo Method, supplement 1 of the GUM [13], some probability density function, PDF, will have to be defined. This task will be done by the proposed 3x3 pixels geometric model that takes into account the behaviour of central pixel and their first neighbours. Afterwards using the Bayesian probability as data filter, all PDFs will be saved in a database to speed up the algorithms that later could be used in the Monte Carlo method. The validation of the proposed method will be carrying out using synthetic triangle images generated by the direct method and fitted by least squares [14, 15]. 2. Code and execution time All code described in this paper are compiled in Matlab. Some models, mainly the 3x3 pixels model, need an extra computing time about 60 hours, so this code is evaluated using the CeSViMaMagerit Supercomputer. The Magerit supercomputer is a general purpose cluster with dual architecture, Intel and POWER that covers most computing needs. The POWER configuration is able to provide sustained power of more than 72 TFLOPS over a theoretical peak of almost 103.5 TFLOPS. The Intel partition provides a sustained power of more than 14.8 TFLOPS over a theoretical peak of 15.9 TFLOPS. After data base is generated all algorithms can be run in a i7 processor laptop with 12 Gb of RAM under Linux. The proposed procedure and others which were compared need almost the same run time amount. 3. The direct method One of first CCD model description due to Janesick and Blouke [16] uses a "rain to bucket" physical simile. Using this idea is supposed that the camera CCD outputs are proportional to the light reflected from measurand and guided through optical devices. Under other point of view, if the measurand image is formed in focal plane of optical devices, just in front the CCD sensor, the output image pixel intensities will be proportional to the area that is

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projected in focal plane, partial area effect hypothesis. Same concept is described by Bayley [17] and by Trujillo Pinoet al.[12]. The design of CCD model is founded in the approximation that there are three different coordinate systems locate at common plane between the focal plane and the CCD surface, and which the sensor effective pixel number is roughly the image pixel number. The first coordinate system is defined at focal plane where an almost real image of measurand is projected. Due to be direct related with the physic measurand, it uses length units. The second and third coordinate systems are linked to CCD sensor, one to measures the intensity and another to locate a pixel in the image, both systems have dimensionless scales that use only natural numbers. At the focal plane, an area is limited by a set of coordinates (x,y), Fig. 1 (left). This area is the projection of the reflected light from real measurand, and then the CCD converts this light into digital information. The CCD pixel coordinate system (u,v) has its origin at top left point of working plane, while Cartesian origin of focal plane (x,y) is at bottom left, both are related by a pure translation, Eq.(2), where px ,pyccdyare, respectively the size of the pixel in x directions, the size of the pixel in y direction, and the size of the CCD in y direction, all variables are in [mm]. As example for a 1/2'' CCD sensor with dimensions 6.4 [mm] × 4.8[mm], the size of pixel is of 8.33 [µm] × 8.33 [µm], px=py=8.33 [µm]. u  0     v    1 / px 1  0   

1 / py 0 0

ccdy  x    0  y   1   1   

(1)

U = TCCD  X  

(2)

The Fig. 1 shows schematically how the direct model works: from a polygonal area at focal plane to an image of 20x20 pixels in the CCD. Thus, the measurand limits go from being a line to a set of pixels. If a sensor pixel is covered by measurand area at focal plane, the CCD will indicate I A (dark blue) intensity; on the other hand if the pixel receives all the intensity of the light source, the CCD will display IB (white), intensity of the background.

a)

b)

Fig.1. (a) Theoretical shape at focal plane;(b) The proposed transformation where area intensity I A (dark blue), background intensity IB (white) and the edge is displayed are displayed.

In these ideal conditions, a transformation equation can be defined as:

I  IA  IB ∙A  IB  

(3)

The Eq. 3.represents the direct method, since it transforms an area “A” from focal plane into pixel intensity “I”in final image, so it is used to generate synthetic images and verify algorithms with different zooms, Fig. 2. The

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calculated areas can not be used in this form, because just the outer vertexes belong to real edge. These points are located as intersections between areas and pixel grid.

a)

b)

Fig. 2. (a) Synthetic triangle images for zoom equals to 2.5; (b) for zoom equals to 4.5.

4. The reverse method In this section is described the inverse procedure which is described in the former section. After many image observations and taking into account just a linear approximation, the edge pixels are classified as in Fig. 3. Consequently, when an ideal image is captured at edge only three pixels type will be consider: T290, T3 and T4. This classification is possible by the information that gets in from first neighbours. Hence to verify it, a simple model of 3x3 pixels is designed where the central pixel is the study pixel, as can see at left of the Fig. 4. The model needs many parameters to establish a statistical study; in this case all covered pixel areas and all intersections between these areas and the pixel grid. The areas are related by intensities by Eq.(3) and intersection points with the measurand edges, as previously seen. Accordingly, many lines are traced from one side to another side of 3x3 pixels area to obtain the desired values, Fig. 4 centre. More than a million lines were used, so about nine million of pixel intensities were saved. At first sight these data can hardly be used. By this reason some kind of filter is necessary to extract useful information from data: Bayesian probability is liable to do this. But before, areas have to be proper defined i.e. the areas must be properly located into the pixel. To do it, eight distances are defined as can see at right of the Fig. 4 . These distances, lijk , always have their origin in one vertex ..

Fig.2. Pixel classification based in how area may cover it.

Fig.3. (a) Nine pixel model to study central and neighbours relations; (b) lijk definitions, from vertex.

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The reverse model performs the following operations too: Determine the centre of mass of the initial pixel area, using the “polygeom.m”algorithm [18]. Establish the closest vertex to the centre of mass which agrees with the minimum mean in intensities of the three neighbours closest to any vertex. The opposite vertex with the minimum mean in intensities mín is the maximum mean in intensities máx . Finally, the lengths lijk are obtained, whereby the position of the edge at this pixel with subpixel approximation is completely determined from one vertex position. From definition of lengths and from the observation of the possible types of pixels, a table is made which can be used as a new criterion to pixel selection just based in lijk . Once all area and intensity relationship have been determined in the initial 3x3 model, these data are filtered using Bayesian probability. The output of this process is a PDF of lijk that depend on the pixel intensities, I and the pixel types, t pix (T290, T3 and T4): p(lijk |t pix , I) 

p(I |t pix , lijk ).p(lijk |t pix ) p(I |t pix )

 

(4)

Some suppositions are needed for determination of probabilities values in Eq.(4): the type of pixel is related with lengths, lijk  l121 , l122 , l232 , l233 , l343 , l344 , l414 , l411 ; p  t pix  is almost the same in any case; and p  I |t pix  , p  I |t pix , lijk  and p  t pix | lijk  can be calculated using the results of the 3x3 pixels model. Should take into account, that variables in probability function, Eq.(4), have different ranges and belongs to distinct number sets: I  0,255 , y lijk  0,1 . The Bayesian filter is applied to obtain the lijk PDF for one intensity and for one type of pixel. These PDFs will be fundamental to estimation of uncertainty using the Monte Carlo method that in general needs long computing times. To speed up these algorithms a simplified database of PDF is created in this way: the PDFs from T290 and T4 are fitted as double Gaussian, Eq.(5), and in T3 case, as a rectangular PDF.   x  b 2    x  b 2  1 2 f (x )  a1 exp       a2 exp         c1     c2      



Fig.4. (a) PDFs for different pixel type and intensities: p lijk |T 290, I

(5)

 163  ; (b) p  lijk |T 3,I  144  ; (c) p  lijk |T 4, I  72  .

5. Validation The previous algorithms are validated using synthetics images created by the direct method, Eq. (3). After a synthetic triangle image is generated, the edge coordinates are obtained by Canny and Trujillo Pinoalgorithms . However Canny's algorithm yields the edge values in pixel range so is not presented in this study. A fitting procedure has to be used [14, 15] to evaluate measurement result. On the basis of the equation of the straight line, the minimum objective function  given by Eq.(7) in term of line equation parameters A, B, and C 1 subject to the constraint A2  B2  Ax  By  C  0 

(6)

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M. Berzal et al. /Manufacturing Procedia Manufacturing 13 (2017) 442–449 M. Berzal / Procedia 00 (2017) 000–000 n

  Ax  By  C 

  A, B, C

i

2

i

447

(7)

i 1

Minimizing Eq. (7), parameter C is calculated, holding A and B (8)

C  Ax  By   n

  A  x  x   B  y   y 

 A, B 

2

i

i

qxx A2  2qxy AB  qyy B2

(9)

i 1

where�̅ and �� are sample means. In matrix form:

  A 

q  A  q B   qxx qxy    A T QA   yy   B   xy

A

(10)

Minimizing Eq.(10) subject to A  1 : min A T QA  min (Q )   A 1

(11)

Angle of line is obtained by: B   tan    A

1



 2

(12)

 

Uncertainty is obtained using squares sums of residues Qe and mean squared error:  Qe

m

e  i

2

eT e  

(13)

i 1

se2 

Qe mn

(14)

where m an n rows and column of data. Covariance matrix is determinates as follow:  x1 D 0     xm

y1

 ym



1     1

(15)



(16)

Cov  A   se2 DoT Do

1

In this case, the triangular measurand is reconstructed by the equations of three edge lines. The angles between them and the measurement uncertainty due to fittings are collected in Table I. The result of comparison is positive owing to the angles calculated from Trujillo Pino et al. algorithm and the proposed reverse method, both in subpixel range, can reproduce nominal angles with an uncertainty close to pixel size. 6. Conclusions This paper has set out a set of procedures that simulate the process of light transformation in the CCD sensor when being transformed into digital data. The light conversion process replaces the pinhole camera model, thus

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changing one black box model for another in order to apply a new testing procedure for accuracy. These code lines developed the direct and reverse method to transform real dimensions into [mm], then to grey levels and vice versa, as explained in Eq.(2) and (3) . The direct method was applied to obtain synthetic images that were used as a code evaluation test, therefore, the relationship between coordinate systems were the starting point for the reverse method. The reverse method transformed intensities, using natural numbers, from images into intersections over a pixel grid, thus obtaining a real number. This set of algorithms was based on a few basic suppositions about edge pixels: T290, T3 and T4,Fig. 3. Additionally, a more general definition was provided using another model, the 3x3 pixels,Fig. 4, which explored millions of linear relationships between pixels and their first neighbours. With the former model, these links between pixels, areas, intensities and pixel grid intersections were characterized and the measurand edge could be located. The mathematical description of these pixel relations were formulated using Bayesian probability developed in Eq.(4) , herewith applied as a tool to calculate probability density functions that obtain the most probable intersections over the pixel grid that was defined in parameter lijk previously. To simplify and speed up the code, PDFs were saved in a rectangular PDF format for T3 pixel, and double Gaussian for T290 and T4 pixel, as shown in Eq.(5). At this point in the procedure, the edge positions were known, thus enabling the calculation of the measurand’s dimensions using an least squares fitting. By using equations from 6 to 16 principal lengths and angles, the uncertainty estimation of fitting process was determined. The estimation of uncertainty could be evaluated using the Monte Carlo method included in Supplement 1 of GUM using the PDFs related with parameter lijk . Table I. Data and fitting uncertainty obtained from measurement of synthetic triangle as zoom function, using Trujillo Pino (TP) algorithm and proposed algorithm (PROP). Fitting and nominal values (Z=i)

1

U 1 

2

U 2 

3

U  3 

o   

o   

o   

o   

o   

o   

Z=1

15.17

-

82.42

-

82.42

-

PROP

15.14

0.06

82.45

0.020

82.41

0.019

TP

15.13

0.06

82.40

0.022

82.47

0.021

Z=1.5

15.17

-

82.42

-

82.42

-

PROP

15.14

0.05

82.42

0.017

82.44

0.016

TP

15.15

0.05

82.41

0.018

82.44

0.017

Z=2

15.17

-

82.42

-

82.42

-

PROP

15.17

0.04

82.42

0.015

82.41

0.014

TP

15.16

0.04

82.42

0.016

82.43

0.015

Z=2.5

15.17

-

82.42

-

82.42

-

PROP

15.17

0.04

82.42

0.012

82.41

0.013

TP

15.17

0.04

82.42

0.013

82.41

0.014

Z=3

15.17

-

82.42

-

82.42

-

PROP

15.17

0.03

82.42

0.012

82.41

0.012

TP

15.17

0.04

82.42

0.012

82.42

0.013

Z=3.5

15.17

-

82.42

-

82.42

-

PROP

15.17

0.03

82.42

0.011

82.42

0.011

TP

15.17

0.03

82.42

0.011

82.42

0.012

Z=4

15.17

-

82.42

-

82.42

-

PROP

15.17

0.03

82.41

0.010

82.42

0.010

TP

15.17

0.03

82.41

0.011

82.41

0.011

Z=4.5

15.17

-

82.42

-

82.42

-

PROP

15.17

0.03

82.42

0.009

82.42

0.009

TP

15.17

0.03

82.41

0.010

82.41

0.010

8

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In summary, the above procedures demonstrate the application of a general framework that characterizes measurand edges and evaluates their uncertainty using intensity as the sole input variable being generated from an image captured by an optical measuring machine. Through these means, the usefulness of a new tool, the parameter lijk , has been demonstrated to be as useful as the pinhole cameramodel for estimating uncertainty of measurements of pixel images.

Acknowledgements The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Supercomputing and Visualization Centre of Madrid (CeSViMa). Thiswork has beencarriedoutpartlywithintheframework of Project DPI2016-78476-P "Desarrollo colaborativo de patrones de software y estudios de trazabilidad e intercomparación en la caracterización metrológica de superficies", belongingtothe 2016 Proyectos de I+D correspondientes al Programa Estatal de Fomento de la investigación Científica y Técnica de Excelencia, Subprograma Estatal de Generación del Conocimiento. References [1] F. Franceschini, M. Galetto, G. Genta, Precis. Eng. 42 (2015) 133–142. [2] M. Idir, K. Kaznatcheev, S. Qian, R. Conley, Current, Nucl. Instruments Methods Phys. Res. Sect. A Accel. Spectrometers, Detect. Assoc. Equip. 710 (2013) 17–23. [3] F. Siewert, in: F. Canova, L. Poletto (eds.), Optical metrology - on the inspection of ultra-precise fel - optics, Optical Technologies for Extreme-Ultraviolet and Soft X-ray Coherent Sources, Heidelberg, 2015, pp. 137-149. [4] W. Sudatham, H. Matsumoto, S. Takahashi, K. Takamasu, Measurement. 78 (2016) 381–387. [5] J. Caja, E. Gómez, P. Maresca, Precis. Eng. 40 (2015) 298–304. [6] J. Caja, E. Gómez, P. Maresca, Precis. Eng. 40 (2015) 305–308. [7] A. Lazzari, G. Luculano, Measurement. 36 (2004) 215–231. [8] R. Hartley, A. Zisserman, Multiple view geometry in computer vision, Cambridge University Press, 2004. [9] K. Kanatani, Statistical optimization for geometric computation: theory and practice, Dover Publications, 2005. [10] C. Harris, M. Stephens, Alvey Vision Conference, 15, 50, 1988. [11] J. Canny, IEEE Trans. 6 (1986) 679–698. [12] A. Trujillo, K. Krissian, M. Alemán-Flores, D. Santana-Cedrés, Image Vis. Comput. 31 (2013) (1) (2013) 72–90. [13] JCGM 101:2008, Evaluation of measurement data - Supplement 1 to the “Guide to the expression of uncertainty in measurement” Propagation of distributions using a Monte Carlo method, 2008. [14] N. Chernov, Circular and Linear Regression: Fitting Circles and Lines by Least Squares, CRC Press, 2011. [15] J. de Vicente, A. M. Sanchez-Perez, M. Berzal, P. Maresca, E. Gomez, Meas. Sci. Technol. 25 (2014). [16] J. Janesick, M. Blouke, Sky Telescope. 74 (1987) 238. [17] D.G. Bailey, Proceedings of the New Zealand Image and Vision Computing Workshop, 37–42, 1995. [18] H. Sommer, “polygeom.m,” MatWorks. File exchange. Available at: http://es.mathworks.com/matlabcentral/ /fileexchange/319‐polygeom‐m?requestedDomain=www.mathworks.com (2017, April).