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National Institute of Standards and Technology

NIST Technical Note 1370

Software for Performing Gray-Scale

Measurements of Optical Fiber End Faces

CM. Wang

Dominic F. Vecchia

Matt Young

Nathan A. Brilliant

QC

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NO. 1370

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NIST Technical Note 1370

Software for Performing Gray-Scale

Measurements of Optical Fiber End Faces

CM. Wang

Dominic F. Vecchia

Matt Young

Nathan A. Brilliant

Statistical Engineering Division

Computing and Applied Mathematics Laboratory

National Institute of Standards and Technology

325 Broadway

Boulder, Colorado 80303-3328

November 1994

U.S. DEPARTMENT OF COMMERCE, Ronald H. Brown, Secretary

TECHNOLOGY ADMINISTRATION, Mary L. Good, Under Secretary for Technology

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, Arati Prabhakar, Director

Software for Performing Gray-Scale

Measurements of Optical Fiber End Faces

C. M. Wang and Dominic F. Vecchia

Statistical Engineering Division

Matt Young and Nathan A. Brilliant

Optoelectronics Division

National Institute of Standards and Technology

Boulder, Colorado 80303

The video microscope, or gray-scale, method is the most frequently used technique on

the m.anufacturing floor for measuring critical dimensions of an optical fiber end face. Grayscale images of optical fibers and hence their corresponding edge tables, or locus of points

that represent the edge of the fiber in digitized form, can easily be contaminated by dirt or distorted by faulty cleaves. Analysis of such edge tables can be difficult. We present a method for performing optical fiber dimensional quality control which allows for end face

damage and accounts for the special structure of measurement errors in fiber edge tables.

The new approach adheres to the industrial standard test procedure by fitting an ellipse

to the edge table to obtain geometric measurements. But, to create high breakdown resistance to outliers, a data filter based on the least-median-of-squares criterion is used. Some

computational issues and a brief description of a computer program that takes the digitized

image, locates and filters the edge points, and estimates the geometric parameters of interest are given. Its operation is also described.

Key words: edge points; gray-scale analysis; least-median-of-squares regression; outliers;

orthogonal-distance regression

1. Introduction

The optical fiber industry, with a U.S. market that amounted to $2.1 billion in 1992

(Department of Commerce statistics), is very measurement intensive. About 20 percent of

the cost of manufacturing fibers is in process and quality measurements. As more buildings, homes, and offices are wired with optical fibers, the need to accurately measure fiber

geometry becomes even greater. This is because fiber networks require many connections

between fibers; poor connections could result in degraded light signals if the geometry of

the two connecting fibers differs. For example, a typical single-mode fiber is nearly circular

with a glass core of about 10 /zm in diameter surrounded by a glass cladding whose outer

diameter is about 125 fim. If the cladding diameters differ, the inner cores probably would
fail to align precisely when joined. For an offset of 1 jim of the cores, the loss is about 0.2

dB, or about 5 percent in transmitted power.

To minimize these coupling losses, the industry needs to measure the cladding diameter

with an accuracy of 0.1 /xm. To support this effort, the National Institute of Standards

and Technology (NIST) has developed instruments that can make diameter measurements

accurate to 0.04 /xm [1]. These instruments are expensive and delicate, and are used primarily

to generate Standard Reference Material (SRM) fibers by which the industry can calibrate

their own measuring systems and hence are not suitable for routine industrial quality control.

The gray-scale method, which uses a video m.icroscope in conjunction with a frame

digitizer, is the most frequently used technique on the manufacturing floor since geometric

parameters of interest, such as the cladding diameter and circularity, are easily and quickly

measured by this method. Very briefly, the fiber is illuminated with light, and the microscope

is focused onto a cleaved end face; that is, the axis of the microscope is parallel to that of the

fiber and perpendicular to the end face. In our instrument, the image, an array of 512 x 440

pixels, is acquired and digitized. Each pixel receives a value between and 255, representing

the image of the fiber end in 256 gray levels. The left panel of Figure 1 shows the grayscale image of a fiber end. Also, calibration is used to determine the correspondence between

positions of pixels and distance, that is, the relationship between pixel units and micrometers.

For a more detailed discussion of video microscopes dedicated to fiber geometry, see Refs.

[2] and [3].

100 200 300

Pixel Number

400 500 50 100

Distance, Micrometers

150

Figure 1. The left panel is the 512 x 440 pixel gray-scale image of a pristine optical fiber end in reverse

contrast; that is, black represents the highest intensity. The right panel shows edge points (•) computed

from a 64 X 55 image by averaging 8x8 blocks of pixels.
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100 200 300

Pixel Number

400 500 50 100

Distance, Micrometers

150

Figure 2. The left panel is the 512 x 440 pixel gray-scale image of a damaged optical fiber end. The

right panel plots edge points (•) computed from lower resolution 64 X 55 image.

An edge table, or locus of points that represent the edge of the fiber in digitized form,

is then obtained. The right panel of Figure 1 shows the cladding edge points for the fiber

image in the left panel. The edge points were computed by estimating the inflection points

of the intensity distribution across respective horizontal or vertical cross sections of the

image. (This method gives a small systematic error in the calculated value of some fiber

dimensions, but that is not our concern here, see Ref. [2].) Finally, an elHpse is fitted to the

edge table to obtain dimensional measurements useful in quality assurance, such as cladding

diameter and noncircularity, and cladding-core decentering (or concentricity error). The

main reason for modeling the edge points by an ellipse, rather than a circle, is that fiber

cladding noncircularity is one of the primary geometric measurements of interest, and most

fibers are approximately elliptical in cross section.

Gray-scale images of optical fibers can easily be contaminated by dirt or distorted by

faulty cleaves. The left panel of Figure 2 shows the gray-scale image of a fiber end that was prepared by deliberately striking it against a microscope cover slip to simulate damage

routinely encountered in an industrial setting. The damage appears in the lower right corner; the upper left corner may show damage due to cleaving the fiber. The computed edge points

for the damaged fiber are shown on a micrometer scale in the right panel.

Analysis of images like that in Figure 2 can be difficult unless the influence of outliers

in the edge table can be eliminated from the computations. In this context, the term "outliers" refers to points that do not truly lie on the circumference of the undamaged fiber, but

are artifacts of cleaving or imperfection in the end of the fiber and can be safely ignored.
This outlier problem prompted the Telecommunications Industry Association (TIA) to propose Fiber Optics Test Procedure-176 [4]. This FOTP is part of the series of recommended

standard test procedures proposed by TIA. FOTP-176 documents guidelines for measuring

optical fiber cross-sectional geometry by gray-scale analysis. The basic algorithm proposed

in FOTP-176 consists of three steps: edge table filtering, edge table fitting, and parameter estimation. The algorithm calls for removing outliers before calculating the best-fit

ellipse. There are, accordingly, many possible fits, depending on the criterion for omitting

outliers from calculation. Furthermore, the protocols recommended in FOTP-176 depend

on iterative outlier rejection rules which are difficult to automate reliably.

In this Technical Note, we present a method for filtering edge tables and allowing for end

face damage. This method accounts for the special structure of "measurement" errors in fiber

edge tables. The approach adheres to the industrial test procedure by fitting an ellipse to the

edge table to obtain dimensional measurements. But, to create high breakdown resistance

to outliers, we have implemented an edge table filter based on the least-median-of-squares

(LMS) criterion [5, 6]. We also discuss certain aspects of the computation and describe an

integrated program that takes the digitized image, locates and filters the edge points, and

estimates the geometric parameters of interest.

2. Equation and Fitting of the Ellipse

An elliptical model for fiber edge points must be parameterized to allow for an arbitrary

angular orientation. The general equation of an ellipse (assumed not to pass through the

origin) is given by

f{x,y) = Ax^ + Bxy + Cy' + Dx + Ey + 1 = 0, (1)

where B^ — AAC < 0. Figure 3 is a graphical illustration of the elliptical fitting problem

for the edge table shown in Figure 1. The ellipse in Figure 3 is more conveniently described

by its center (q:,/3), semimajor axis M, semiminor axis m, and the angle 9 formed by the

major axis with the x-axis. The five parameters in the geometric representation are simple

functions of the parameters in eq (1):

2CD - BE

" =^ B^ - AAC ' ^^^ 2AE-BD ,^.
M

^] A-]-C + B/sm2e

m =

2{a^A + a^B + P^C - 1)

\ A + C-B/ 26

(5)

(6)

where the definition of M and m above correspond to the geometry in Figure 3. In general,

the semimajor axis is defined to be the larger of the quantities in eq (5) and eq (6) and

the semiminor axis the smaller value, depending on the sign of sin 2^. We define the naean diameter of an ellipse as the sum M^m^ while noncircularity is defined as [M — m)/{M + m)

and is expressed as a percent. Given an edge table like that in Figure 1, the mean diameter

and noncircularity are estimated by substituting appropriate parameter estimates A, B, C,

D, E in eq (2) to eq (6).

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Distance, Micrometers

Figure 3. Optical fiber edge table with embedded ellipse to be fitted.
2.1 A "Convenient" Least-Sum-of—Squares Solution

One procedure commonly used in industrial systems (e.g., see Ref. [4]) is to fit eq (1)

to a set of edge data {{Xi, Yi),i = 1,2, ... ,n} by computing estimates of the five elliptical

parameters as the solution of the following minimization problem:

min f^ (ax; + BX,Y, + CY^ + DX, + EY, + l)'. (7)

The solution of eq (7) may be easily obtained in a one-step, closed-form equation using

linear least-sum-of-squares (LSS) calculations. We will denote the computed estimates by

A, B, C, D, E and call this result the convenient LSS solution for the following reason.

While the solution of eq (7) is computationally convenient, the pseudo-residuals associated with eq (7) have no useful interpretation as deviations of the computed edge points

from a fitted ellipse. In other words, if the terms being squared in eq (7) are interpreted, in

the usual sense of regression analysis, as deviations of a dependent variable from an assumed

model, then the ith residual from the convenient LSS fit would be expressed as

r, = -l- {Ax; + BX,Y, + CY^ + DX, + EY^ . (8)

In the terminology of regression analysis, by choosing the convenient LSS solution, we are, in

effect, computing the regression equation of a dependent variable w = —1 on the independent

variables Xf, X^Y^, Y^ , Xj, and Yi. Forbes [7] has shown that rj is proportional to the

difference of two areas. One is the area of the ellipse that has center (a, 13), orientation 9,

and eccentricity m/M, and passes through the point [Xi,Y)', the other is the area of the

ellipse with the same center and orientation, and semimajor and semiminor axes M and m.

These "residuals" are purely a mathematical convenience that have no physical significance

in the usual sense of Euclidean distance from measured data to the fitted curve. Accordingly,

we cannot expect the convenient LSS solution to have any of the "good" statistical properties

of least-squares estimates in general. In fact, a simulation study, given by Vecchia, Wang,

and Young [8], shows that the convenient LSS fit is not accurate for noisy data. In our

application, however, it is expected to be "accurate enough" for fitting edge tables of the

quality shown in Figure 1.

2.2 Orthogonal-Distance Regression

A LSS solution that properly incorporates the natural residuals in the gray-scale image application is given below. The method presented, called either errors-in-variables or orthogonal-distance regression, applies to many other model-fitting problems as well.
The orthogonal-distance LSS solution to our problem can be formulated as follows ([9], p. 238). We let {zj = (si, iji), i = 1,2, ... ,n} denote the true, or noise-free, edge points of

the fiber. The true values are defined implicitly by eq (1); they are assumed to fall exactly on an ellipse for some values of the unknown parameters A, B, C, D, E. The observed data,

however, are perturbed by measurement errors; that is, they are noisy measurements of the

true edge points. If Zj = (Xj, Yi) denotes the measurement of Zj, the observed edge points

may then be described by

Zj = Zj + e^, t - 1, 2, . . . , n, (9)

where the errors {ei,i = 1,2, ... ,n} will be assumed to be independent and come from a two-dimensional distribution with mean and covariance matrix Sg = cr^V, where V is known and nonsingular.

The measurements {Z^, i = 1,2, ... ,n} and the equations satisfied by the true values

{/(z^; A, .6, C, Z), £^) = 0, i = 1, 2, . .

. , n\ (from eq (1)) jointly specify the model to be fitted

to fiber images. A LSS fit of this model is obtained by minimizing the Lagrangean

f](Z,: - z,)S;^(Z, - zO' + J2 Az/(z^; i, B, C, b, E) (10)

1=1 1=1

with respect to A, B, C, D, E and {z^, 2 = 1,2,..., n}, where the Aj are Lagrange multipliers.

The first sum in eq (10) is the sum of squared "statistical" distances between the actual

measurements and estimated true values on the fitted ellipse, while the second sum in eq

(10) constrains the estimated true values to fall on an ellipse. If errors in the measured data

are small enough, the first part of eq (10) (the sum of squared residuals) becomes negligible,

and the problem reduces to the convenient LSS solution in eq (7). In other words, the simple

linear fit of eq (7) is strictly correct only when the data are exactly on an ellipse.

Owing to the procedure for digitizing gray-scale images, there is a special error structure

that we can take advantage of when solving the optimization problem in eq (10). Specifically,

the edge-finding routine obtains the edge tables by splitting the image into two sections.

The first section includes the pixels from approximately —45° to 45° and 135° to 225°. In this

section, the pixels are scanned horizontally line by line. The second section includes the pixels

from 45° to 135° and 225° to 315°, and the pixels are scanned vertically column by column

(Figure 4). Thus an edge table Hke that in Figure 1 can be partitioned into two groups of

edge points: a collection of points {Z[^' = (x^, Yi), i = 1,2,. . . ,ny} estimated by processing

columns of pixels (fixed x-coordinate, noisy y-coordinate) and the complementary set of

points {Zj"'^ = {Xj, yj), j — 1,2, .. . ,n^} obtained from rows of pixels (fixed y-coordinate,

noisy ^-coordinate). This special structure can be used to simpHfy eq (10). For example, if
Figure 4. Directions of scanning and processing pixels.

we assume that Sg = cr^l2, and without loss of generality, we also assume that sequence of

n = Tix + Uy points in eq (9) have been arranged so that the first n^ measurements correspond

to edge points with y fixed, then eq (10) reduces to

rix

t=l 1 =712+1 1=1

1=712+1 (11)

The minimization is with respect to A, 5, C, -D, -B, {xj, i — l,2,...,nx} and {^i, i —

n^- + 1, . . . , n}. This reduces the number of unknowns in the optimization from 5 4- 2n in eq

(10) to 5 + n in eq (11). We call this result the gray-scale LSS solution. Iterative methods

are used to obtain the solution. Further remarks on computational issues are presented

in Section 4. Moreover, in contrast to the convenient LSS approach, the gray-scale LSS

solution provides approximate standard errors for the parameter estimates A, 5, (7, -D, and

E. Using the relationships in eq (2) to eq (6), we can also derive the approximate standard

errors for the geometric parameter estimates a, /3, ^, M, and rh by the propagation of errors technique. Specifically, if we denote the approximate variances of A, 5, C, Z), and £", which

are part of the gray-scale LSS solution, by a^, o"^, o"^, a|), and o"|;, respectively, then the
approximate variance of d is given by

where the partial derivatives are evaluated SitA = AjB = B,C = C,D = D, and E = E.

Thus

2 16C^(2CD-BEy . UBCD-AACE-B^EY , [B^-AACy {B^-4ACy

4:B^{2AE - BPy ^2 4(7^ ^^ __?l___-2

(52 - 41(7)4 "^^ (52 - 4i(7)2

""^ (^2 - 41(7)2

"^^^

The approximate variances of /3, ^, M, and m are similarly obtained.

Other authors have also considered the problem of fitting circles and ellipses to data,

from both numerical and statistical points of view; several references, for example, are given

by Mamileti, Wang, Young, and Vecchia [10].

3. Data Filter for Fiber Edge Tables

Because they are least-squares procedures, both the convenient and the gray-scale LSS

will perform poorly when outliers are present. Thus the outliers must be removed before

calculating the best-fit ellipse. We use least-median-of-squares (LMS) regression developed

by Rousseeuw [5] to identify and eliminate outliers.

LMS regression is a robust method obtained by replacing the sum in LSS by the median.

That is, while LSS produces parameter estimates which minimize the sum of the squares of

the residuals, LMS chooses parameter estimates which minimize the median of the squares of

the residuals. It can tolerate a high proportion of contaminated data without degrading the

accuracy of the fitted equation. In the terminology of robust regression, the LMS solution

has a breakdown point of 50 percent (the highest possible value), whereas the LSS method

has a breakdown point of percent. A single aberrant value can cause LSS to give an

arbitrarily bad answer, whereas nearly half of the data could be corrupted without affecting

the validity of the LMS solution. In the optical fiber application, there is no obvious way

to implement the LMS criterion in the (impHcit model) form of the objective function in

eq (11). However, robust residuals obtained from an approximate LMS solution, based on

pseudo-residuals associated with equation eq (1), are rehable indicators of bad edge points.
Specifically, the robust solution based on pseudo-residuals is

min median{(AX,' + BX,Y, + CY^ + DX, + EY, + 1)', i = l,...,n}. (13)

A,B,C,D,E

The solution of eq (13) may be easily obtained using existing software for linear LMS calculations [6, 11, 12]. We call this result the convenient LMS fit for the reason given earlier in

connection with the convenient LSS fit. An edge table filter based on the convenient LMS regression consists of following steps:

first obtain estimates of the regression parameters A, B, C, D, E, using the criterion in eq

(13), together with a corresponding error scale estimate a ([6], p. 44). Then for each edge

point (Xj, Yi), compute the associated absolute standardized pseudo-residual

R, = \AX; + BXiY, + CYl + DX, + EY, + l|/a, (14)

and remove that point from the edge table if its value of Ri is larger than a threshold, such as

2.5 or 3. Figure 5 shows the edge table of the damaged fiber as well as the ellipses fitted by

convenient LMS (solid curve), the analogous one-step convenient LSS (dotted curve), and

the gray-scale LSS (dashed curve). The difficulty of designing an automatic outlier-rejection

scheme based on standardized residuals from iterated LSS fits, as suggested by FOTP-176, is evident: some good edge points miay be identified as outliers and some outliers may appear

to be valid edge points. The LMS fit, by contrast, produces small (robust) residuals for

virtually all good edge points and a clear indication (by large residuals) of the bad edge

points. The "o" points, which have standardized residuals larger than 2.5, are outliers and

are removed from the edge table.

4. Computational—Related Issues

We have developed an integrated software package for fiber geometry measurement. A

brief description of the program and its operation is given in the next section. The programi

reads the digitized image and locates the cladding edge points for rows and columns. The

edge-finding routine is flexible and can find the edge at a specified fraction of the maximum

intensity or at the inflection point. We use an efficient algorithm developed by Savitzky

and Golay [13] to calculate the second derivative of the intensity distribution. A detailed

discussion of edge-finding and related problems can be found in the paper by Mechels and

Young [2]. Having found the cladding edge table, the program locates a tentative center of the image.

An area around this center is then used to scan for the core intensity contour. This step is required because the image is not necessarily located at the center of the frame.

10
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50 100 150

Distance, Micrometers

¥iguTe 5. Ellipses fitted to the damaged-fiber edge table. The convenient LMS, the convenient LSS, and

the grayscale LSS ellipses are denoted by the solid, the dotted, and the dashed curves, respectively. The

"o" points are outliers according to the LMS outlier-rejection scheme.

The program then uses the LMS fit to filter the edge tables. To fit an ellipse by LMS, we

use the same procedure as the program PROGRESS described by Rousseeuw and Leroy [6]. That is, the program randomly chooses five points, fits an ellipse to these five points, and

calculates the median of the squared residuals of all edge points. It performs this operation

a number of times and estimates the five parameters as those corresponding to the ellipse

that displays the lowest median. Since the purpose of this step is to remove the potential

outliers, the number of resampling runs we used is much smaller than the 2,500 recommended

by Rousseeuw and Leroy [6] for a five-parameter model. We use 250 in our program.

Next, we fit an elHpse to the remaining "good" points using the gray-scale LSS solution.

The general orthogonal-distance LSS solution of eq (10) can be obtained by a Gauss-Newton

11
type of iterative algorithm described by Fuller ([9], p. 239). A brief description of the

algorithm follows. Let w = [A, B, C, D, E)' and z^, z = 1, 2, • • • ,n be the initial estimates.

Calculate

w = w

Zi Z.

where

(Z, -z,)' + /(z,;w),

dz) '

and use the resulting (w, Zj) as initial estimates (w, z^) for the next iteration. The expression

9fi/dw is a row vector of size 5 containing the partial derivatives of /(z^; A, B, C, D, E) with

respect to A, B, C, D, and E evaluated at (Zj-,w), and dfi/dz is an 1 x 2 vector containing

the partial derivatives of f{Zi\ A, B,C, D, E) with respect to x and y evaluated at (zj,w).

The iterations continue until the relative change in the sum of squared residuals and /(z^; w^)

are within a pre- determined tolerance.

The gray-scale LSS solution is a special case of the orthogonal-distance LSS regression.

Specifically, dfi/dz is reduced to

dz \dx' dy dx ,0 2-1,

= 0,

dy

i = rio; + 1, • • ,n,

and only Xi, i = 1, • • • ,na;, and yi, i = Uj: -\- 1,- • ,n need to be updated in each iteration.

By using initial values obtained from the convenient LSS fit, the program usually takes only

a few iterations (four or five) to converge.

5. Program Description and Operation

The program is written in ANSI FORTRAN and has been tested on a 386 personal

computer. (Appendix contains a list of the files necessary to perform the analyses described

in this document.) The program prompts the user for the following information that is

12
needed to carry out the analyses (see Figure 6).

• A brief description of the job.

• The data type: either the digitized image or edge points. In our installation, the

image data consist of 512 X 440 2-byte hexadecimal values, corresponding to setting the

parameters IPXLS and JPXLS to 512 and 440, respectively, in the subroutine OFIBER.

To read image data other than hexadecimal format, just replace the OPEN and READ

statements in the subroutine HEXGET with the appropriate input routines. For the

edge table, the data consist of X and Y coordinates of edge points in micrometers.

The maximum allowable number of cladding edge points is 1216, and the maximum

allowable number of core edge points is 88, corresponding to setting the parameters

NMOBSl and NM0BS2 to 1216 and 88, respectively, in the subroutine OFIBER. If the

input contains both the cladding and core edge tables, the first 1216 records contain

the cladding edge points and the remaining 88 records contain the core edge points.

• The name of the file containing either the digitized image or the edge points.

• If a digitized image is specified as the data input, a file that contains the calibration

coefficients must also be given. The calibration is used to determine the correspondence

between pixel units and micrometers. The file consists of four records. The first record contains the degree of the calibration polynomial equation for the horizontal

pixel. The second record contains the coefficients of the polynomial. The third record

contains the degree of the calibration polynomial equation for the vertical pixel. The

fourth record contains the coefficients. For example, suppose it is determined from the

calibration that the relationship between the horizontal pixel number and distance (in

micrometers) is X = 0.273054 P + 9.332 x 10"* P^

where P is the pixel number, and relationship between the vertical pixel number and

distance is Y = 0.320220 P + 3.9825 x 10"^ P\

Then the file would contain the following four records

2

0.273054 9.332E-8

2

0.320220 3.9825E-6

13
•

If the data input is a digitized image, an adjustment factor that is used in the edge

detection must also be given. If is entered, the edge points are located by estimating

the inflection points of the intensity distribution. If a number between and 1 is entered, the program finds the edge points at that specific fraction of the maximum

intensity.

If the data input is a digitized image, it is possible to write the edge points (in micrometers) to a file.

The selection of the method(s) of geometric parameter estimation. The input can be

1, 2, 3, or any combinations of these three digits. For example, "123" requests that all three methods be used to estimate the geometric parameters. For the gray-scale LSS

method, the program assumes that the first half of the edge points have the structure

of fixed y-coordinate and noisy cc-coordinate.

• The name of the file to capture the output.

In this example, the digitized image stored in the file "yaw5.dat" was used, and the

calibration coefficients were contained in the file "cal.dat". The edge points were found as the inflection points of the intensity distribution. It also requested that the edge tables be

written to the file "yaw5edge.out" . The geometric parameters were obtained using both the

convenient and the gray-scale LSS methods. The output of the example, saved in the file "yaw5.out", is given in Figure 7. The program first prints the description of the run and the method employed in edgefinding. It then prints the parameter estimates for each method requested. In this example,

since the input was the digitized image, the results from both the cladding and the core data

are given. The gray-scale LSS solution also provides the approximate standard errors of the

estimates for the cladding data (values inside the parentheses). The estimated cladding diameter is 125.0209 /v.m and the noncircularity is 0.0776 percent, based on the convenient LSS

solution; and the cladding diameter is 125.0217 /zm and the noncircularity is 0.0772 percent,

if the gray-scale LSS method is used. Thus, these two methods are in good agreement for

the fiber YAW5. The standard error of the estimated cladding diameter is calculated to be

0.0247 lira. We can use the distance between the cladding and the core centers as a possible

measure for the concentricity error, which is 0.243 fim based on the gray-scale LSS solution.

14
Enter fiber ID

This is an example with fiber YAW5

Enter type of data

12

3

Raw image data

Previously saved edge table (both cladding and core)

Previously saved edge table (cladding only)

Previously saved edge table (core only)

Enter file neotie containing data

yaw5 . dat

Enter file name containing calibration coeff

cal . dat

Enter adjustment factor for edge detection

Amount of data to save

12

3

1

No save

Both cladding & core edge tables

Cladding edge table only

Core edge table only

Enter file name to save edge table(s)

yaw5edge.out

Enter method(s) of parameter estimation

12

3

13

Convenient LSS

Orthogonal-distance LSS

Gray-scale LSS

Enter file name to save results

yaw5 . out

Figure 6. The input for the example run.

15
This is an example with fiber YAW5 , Use inflection point s for edge detection.

Units are: micrometers for lengthIS and radicals for angles.

Method: Convenient LSS

Data: Cladding

Major Axis = 62.55896

Minor Axis = 62.46192 , Angle = -1.47398

Center X-Coord = 70.64283

Y-Coord = 73.75232

Method: Gray-scale LSS

Data: Cladding

Major Axis = 62.55914 ( 0.01880)

Minor Axis = 62.46261 ( 0.01605)

Angle = -1.46443 ( 0.01638)

Center X-Coord = 70.64265 ( 0.00951)

Y-Coord = 73.74964 ( 0.01040)

Method: Convenient LSS

Data: Core

Major Axis = 3.68977

Minor Axis = 3.58388

Angle = -1.13511

Center X-Coord = 70.74800

Y-Coord = 73.95351

Method: Gray-scale LSS

Data: Core

Major Axis = 3.75637

Minor Axis = 3.60715

Angle = -1.45926

Center X-Coord = 70.76023

Y-Coord ~ 73.96218

Figure 7. The output for the example run.

16
6. References

[1] Young, M.; Hale, P.D.; Mechels, S.E. Optical Fiber Geometry: Accurate Measurement

of Cladding Diameter. Journal of Research of the National Institute of Standards and

Technology 98: 203-216; 1993.

[2] Mechels, S.E.; Young, M. Video Microscope with Submicrometer Resolution. Applied

Optics 30: 2202-2211; 1991.

[3] Brilliant, N.A.; Young, M. Video Microscopy Applied to Optical Fiber Geometry Measuremients. Natl. Inst. Stand. Technol. Technical Note 1369; 1994.

[4] Fiber Optics Test Procedure FOTP-176: Measurement Method for Optical Fiber Geometry by Automated Gray-Scale Analysis. Telecommunications Industry AssociationElectronic Industries Association, 2001 Pennsylvania Avenue, N.W., Washington, DC

[5] Rousseeuw, P.J. Least Median of Squares Regression. Journal of the American Statistical Association 79: 871-880; 1984.

[6] Rousseeuw, P.J.; Leroy, A.M. Robust Regression and Outlier Detection. New York:

Wiley; 1987.

[7] Forbes, A.B. Fitting an Ellipse to Data. National Physical Laboratory (U. K.) Report

DITC 95; 1987.

[8] Vecchia, D.F.; Wang, CM.; Young, M. Outlier-Resistant Fitting of Gray-Scale Images Illustrated by Optical Fiber Geometry. Proceedings of the Measurement Science

Conference, Los Angeles, CA; 1993.

[9] Fuller, W.A. Measurement Error Models. New York: Wiley; 1987.

[10] Mamileti, L.; Wang, CM.; Young, M.; Vecchia, D.F. Optical Fiber Geometry by GrayScale Analysis with Robust Regression. Applied Optics 31: 4182-4185; 1992.

[11] S-PLUS User's Manual Volume 2, Version 3.0. Statistical Sciences, Inc., Seattle, WA;

1991.

[12] Hawkins, D.M.; Simonoff, J.S.; Stromberg, A.J. Distributing a Computationally Intensive Estimator: The Case of Exact LMS Regression. University of Kentucky Technical

Report No. 339; 1992.

[13] Savitzky A.; Golay, M.J.E. Smoothing and Differentiation of Data by Simplified Least

Squares Procedures. Analytical Chemistry 36: 1627-1639; 1964.

17
Appendix

The accompanying disk contains the following files:

• FIBER.FOR: the Fortran source file,

• FIBER.EXE: the program that can be executed on a 386 (or better) personal computer,

• YAW5.DAT: the raw image data used in the example,

• CAL.DAT: the file containing the calibration coefficients used in the example,

• YAW5EDGE.0UT: the file containing the edge tables (output from the example),

• YAW5.0UT: the file containing the estimation results (output from the example).

Trade names are used to allow the reader to use the program effectively; no endorsement by

NIST or the authors is implied.

18


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