A complete comparator calibration program

Photogrammetria, 29(1973):133-149 © Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

A COMPLETE COMPARATOR CALIBRATION PROGRAM L A W R E N C E W. FRITZ

Photogrammetric Research Branch, NOAA, National Ocean Survey, Rockville, Md. (U.S.A.) (Accepted for publication July 6, 1973) ABSTRACT Fritz, L. W., 1973. A complete comparator calibration program. Photogrammetria, 29:133-149. At the National Ocean Survey, routine comparator calibrations are performed to provide quality control for all comparator measurements. The mathematical model used for the isolation of linear systematic errors constitutes a new exact solution of the general affine linear-transformation problem, from which the calibration parameters are exactly derived functions. Polynomials are applied to define the systematic non-linear errors inherent in the comparator. The adjustment residuals are presented in the coordinate systems of both the measurements and the grid in order to isolate grid errors from random errors. Statistical analysis of all system components is performed to isolate the errors of the pointings, the grid, the comparator and the residual system noise (random errors).

INTRODUCTION

One of the primary components of the analytic photogrammetric system is the measuring comparator. At National Ocean Survey (NOS) (formerly known as the U.S. Coast and Geodetic Survey), we have several Mann monocomparators, Wild STK1 stereocomparators and a Zeiss PSK stereocomparator. Production from these instruments includes aerotriangulation bridging, satellite triangulation, camera calibration plus several other precision tasks. In order to maintain our high-quality production, we have established several qualitycontrol procedures. Comparator calibration is one of our primary quality controls. In the calibration all inherent systematic errors of the comparator are mathematically modeled to remove their detrimental influence from the measurements. Additionally, indications of the performances of the operator and of the comparator can be monitored. In many analytical photogrammetry problems the calibration of the comparator could be included as a portion of the coordinate refinement for other systematic errors such as film distortion. At NOS, we discourage this practice as our policy is to isolate each systematic error individually from the total analytical photogrammetric system. This policy provides for an insight into the magnitude of each contributing error source and helps prevent the modeling of quasi-systematic errors generated by false correlations of the error sources in a combined least-squares adjustment. The NOS computer program is written in Fortran IV and designed for the CDC-6600 computer. The routines are designed for calibration of any type of two-

134

I.. W. FRITZ

axis monocomparator. Calibration of stereocomparators is accomplished by linking the separate stage calibrations in an appropriate mathematical model. MEASUREMENT

PROCEDURES

Our experience has shown that the best standard for calibration is an engraved, 1/4 inch thick, optically clear and flat, glass grid plate. The grid intersections used should be evenly spaced, with the diagonal of the grid plate capable of extending across the maximum square-measuring area used on the comparator. A set of 25 grid intersections provides enough redundancy for the calibration parameters, while keeping measurement time to a minimum. Fig. 1 is the numbering scheme and coordinate system used for an engraved-side up grid plate. Each intersection should be clearly marked on the reverse side for indentification.

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Fig.1. The grid-plate system. Standard practice is to perform preventative maintenance, e.g., lubrication, lens cleaning, etc., before calibration. A complete calibration "set" consists of single operator measurements of the grid plate in four rotations ("cases") on the comparator. The designation of each case is defined by the relationship of the corner grid intersection nearest to the comparator x, y origin. Fig.2 is an example of the grid placements on the comparator which comprise a complete set. Under normal circumstances, one set of measurements is sufficient for a complete calibration. However, the computer program will accommodate up to five sets (twenty

A COMPLETECOMPARATORCALIBRATIONPROGRAM

135

cases) of measurements. For rectangular format comparators (e.g., 25 X 50 cm) the set (or sets) should include all of the measurement area, either with overlapping cases or, ideally, with overlapping sets. The grid should be rotated on the comparator to enhance the validity of the non-linear calibration parameters.

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Fig,2. Case numbersand grid placementsfor secondoperator. Multiple pointings are required for measurement of each intersection. An important criterion to be remembered in pointing is that, in both production and calibration measurements, the mean of the pointings of an object is used for data reduction. Therefore, the operator should strive to obtain measurements all of whose individual means are of equal quality. This procedure will validate the standard photogrammetric practice of assigning equal measurement weights to all (mean) measurements entered into least-squares adjustments. A calibration set requires approximately four hours of measurement.

POINTINGPRECISION Pointing precision is defined by a set of statistics which, when properly evaluated, can reveal several salient characteristics of the control state under which the calibration takes place. These statistics are a first approximation indicator of the quality of any measurements made on the comparator. Fig.3 is a sample of the statistics computed for a single case of measurements by the NOS program. Similiar statistics are computed and printed for all individual cases, for all

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A COMPI.ETE COMPARATOR CALIBRATION PROGRAM

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that all grid points provide the same appearance and may be pointed upon with equal precision. These statistics are also plotted as error ellipses for each case as shown in Fig.4. T h e p r o g r a m utilizes Cathode R a y T u b e ( C R T ) software routines incorporated in the C D C - 6 6 0 0 system to generate these plots on microfilm output. These plots are used to detect grid intersection irregularities, to evaluate com-

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parator zonal deficiencies, and to evaluate the pointing skills and techniques employed by different operators. LINEAR PARAMETERS

Comparator calibration is dependent on the measurement of a grid containing points whose calibrated positions are defined in an orthogonal coordinate system. The comparator-measurement coordinate system is non-orthogonal due to mechanical limitations created in the manufacturing process. The relationship between these two coordinate systems is defined by an affine transformation. Consider the simplest relationship between these two systems as shown in Fig.5. Y

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system; and ~l = lack-of-orthogonality angle of x, y system. By definition, the orthogonal grid system possesses uniform scale. However, each of the comparator axes possesses a unique scaler which is attributable to different measuring standards or temperature. These scale differences present them-

139

A COMPI.ETE COMPARATOR CALIBRATION PROGRAM

selves along the non-orthogonal axes of the measuring system. Therefore, they should be applied, before the lack-of-orthogonality angle as: Y

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Therefore, an exact solution for the general affine linear transformation exists from which the differential scalers (S,,, S.), non-orthogonality angle (.), and rotation angle (0) are explicitly derived functions. All previously published solutions to this problem have required iterative non-linear solutions. Assuming the given grid coordinates to be errorless and the comparator measurements to contain a variety of residual non-linear errors, the complete linear transformation between the systems is expressed as: Y

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140

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This set contains the same observations as eq.6 but different parameters. An interpretive comparison of eq.6 and 7 reveals that in 6 the comparator measurement system is transformed to the grid system, whereas in 7, the grid system is transformed to the measurement system. Obviously, the transformation of

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It can be shown that the exact least-squares solution for the a-f parameters of eq.7 produces a set o f inverse parameters for the non-linear least-squares solution of eq.6. Thus, an .explicit solution for the physically identifiable parameters of the calibration is obtained from:

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2n-u where n = number of grid points measured, u = six unknown parameters, and v<~.~,v,~ = residuals at each grid point (i) in the measurement system. Fig.7 is a computer printout of a single-case adjustment for the linear calibration parameters. In order to determine the total amount of the systematic errors in the measured data, a preliminary free-fit adjustment is performed. This adjustment allows only rotation and translations between the measurement and grid systems (no scale change allowed). The printout of this free fit, for the data of Fig.7, is shown in Fig.6. NON-I.INEAR "PARAMETERS

The non-linear errors introduced to measurements on comparators are present in a variety of forms, dependent on the brand or type of comparator used. Fig.8 is an exaggerated representation of the systematic error forms present in a typical two-axis comparator. Most treatments of non-linear errors involve individual component calibrations. Combining all these individual component parameters into a large math model limits the redundancy afforded by a 25-point grid. Therefore, rather than restrict the mathematical models to specific error patterns, which are traceable to specific error-causing components of the comparator, the NOS program uses polynomial models to describe the cumulative

143

A COMPLETE COMPARATOR CALIBRATION PROGRAM

effect of all of the non-linear sources. Most of the nonqinear error sources are attributable to manufacturing limitations in creating the built-in measuring standards of the comparator. The non-linear errors are usually quite small in comparison with the linear

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errors. Therefore, a two-stage approach is used to determine comparator parameters. With this approach, the user can choose to ignore the non-linear corrections, if their magnitude is small or insignificant, and only use the more significant linear correction parameters. The 4th-degree polynomials used, presented in a least-squares observation equation form, are: lx ~ v~ ~ A x 4 + B y 4 + Cx:; + D y "~ + E x 2 ~ F y 2 + G x + H y -~ J ly + v,j ~- A ' x 4 + B ' y a + C ' x 3 + D ' y :~ + E'x" ~ F'y'-' + G ' x + H ' y ~- J'

(9)

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144

L.W. FRITZ

transformation in the coefficients G, H, G', H', and a translation by terms J and J'. These coefficients are retained to absorb any additional affine transformation created by the removal of the higher-order systematic non-linear errors. The polynomials assume independence of comparator axes.

Fig.9. Evenly spaced samples of non-linear errors.

A rotated placement of the grid on the comparator is recommended to obtain a large representative sampling of the non-linear errors throughout the comparator range. Fig.9 demonstrates the even spacing obtainable by an l l.3°-rotation of a 25-point grid. This rotation provides a large enough sample to create a cumulative curve which contains the net effects of systematic errors--such as screw pitch, curve and weave of ways, etc. A rotation will also extend the range of the polynomial so that all subsequent comparator measurements will lie within its end points. This will prevent errors from extrapolation. The standard error of a single observation of unit weight after adjustment is computed by eq.8, with u ---- 18 unknown coefficients. An example of the adjustment printed output corresponding to the data of Fig.6 and 7 is given by Fig.10. A CRT-plot of the systematic corrections generated from the nonlinear model is shown in Fig.ll. MULTIPLE-CASE SOLUTIONS

As previously stated, a complete calibration set consists of single operator measurements of the grid plate in four rotated cases as shown in Fig.2. Up to twenty cases of measurements may be processed by this computer program. After all cases have undergone individual calibration solutions, they are all processed through a combined linear calibration solution. Rather than form a large simultaneous least-squares adjustment of all cases, the equivalent adjustment is per-

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1,2?S~W3WSE-OW 3,77576022E-05 1.27289512E-0; -E.~IBITBS6E-OS-1,gO331083S-O5 3.27882911E-05 -~.O0127593E-OS-6,7~BlOZ18E-OS 1.00881630E-G5 3.619kB~?9E°05 1.05577839E-OS 3,161~18~9E-O6-5,~8661~22E-O6-1.6755~690E-06 9,39159598E-07 3,5~11k516E-06 1,18356~5E-OS-l.7850kZ~BE-O6-6.W~63162~E°06 Z.965?W~39E-O? 1,17DT7181E-06 -6,22~33892E-OT-1,gz216120E-07 3.30636965E-07 1.02336257E-OT-S,8362~55E-O8-1.82375932E°08 3,79601388E-09 -2,35675~BE-OT-?.67~Z~SZg~-07 1.183k~989E-07 ~,ZS8Z~gSSE-OT°l,gsgoIBITE-OB-7o932~1089E-08 1,2098251~E-09 5,5705309~E-09 1,3805851~E-08 1,6915B~3/E-OB-7,ZB30~O33E-Og-9.523BO71~E-09 1,ZB81~639E°09 1.81678ZS~E-Og-8.66102352E-11-1.32680001E-IO

A

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VARIANCE-COVARIAN;E MATRIX OF ADJUSTED COEFFICIENTS ISAME FOR BOTH X AND Y)

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3.B3917~8~-O3-9.39037221E-OS-1.975116~E-03

C

NON-LINEAR 3DRR~CTION COEFFICIENTS FOR Y

2.12975385E-02

B

NON-LINEAR ADJUSTMENT NON-LINEAR 3ORR~CTION COEFFICIENTS FOR X

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PROGRAM

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m".~

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X~

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where X the meaned calibration parameters S,., S, and ~; j = the total number of cases combined in the adjustment; rn'-',,~ := the variance of unit weight of the ith case; X~ = the calibration parameters S,., S, and <~ of the ith case; and Zv~ -- the variance-covariance matrix of the ith case. Upon completion of the combined solution for the linear parameters, the program computes the residuals for each case. These residuals and the linearly adjusted observations are then combined in a simultaneous least-squares solution for a new set of polynomials (eq.9). The additional redundancy from multiple cases of measurements greatly enhances the validity of the final polynomials, as up to 100 points along each axis may be sampled from four cases of measurements. PARTIAL GRID CALIBRATION

The program computes and transforms all residuals to the given master grid coordinate system. The meaned residuals from any multiple of four 90°-cases exhibit biases which are attributable to errors in the given grid coordinates. Evaluation of these mean residuals from several calibrations on several brands of comparators enable the user to update (improve) the given grid coordinates. Through proper statistical analysis, an unbiased evaluation of the accuracy of the given grid coordinates is performed. Calibration of uncalibrated grids is possible except for limitations in absolute sealers. ANALYSIS OF VARIANCE

A summary page is printed as a final step in the comparator calibration program. Fig.12 is a typical summary page. In order to evaluate the calibration under a rigorous state of statistical control, a Model II Analysis of Variance is performed. This evaluation assigns statistics to the primary error source contributors. The evaluation is based on the following equations: (j21 =

(~21 Ar_ ~12nl _}_ (~2~ AU 02g _~ 02r

¢121 =

t121 -- CJ201

where

O2nl ~

Ci20I -- ¢~2011|

¢/'t = variance of all errors -- RMS from free-fit adjustment ~=0~ = variance after linear adjustment ,/-'.,,~-- variance after linear and non-linear adjustments

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MEAN SQUARE ERROR OF A SINGLE O~SERVAIIDN OF UNIT WEIGHT BEFORE CALIBRATION

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STANDARD ERROR OF SYSTEMATIC LINEAR ERRORS

COEFFICIENT~ FOR X

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VAKIANCE-COVARIANCE MATRIX SCALER X SCALER Y 3.15286707E-12 2.66515238E-20 3,15Z6932BE-12

j

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ALPHA W,I36I/B%BE-16 W,Z36tOO6BE-I6 b.3O568105E-IZ

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NON-LINEAR 3~RRECTION COEFFICIENTS FOR Y

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NON-LINEAR ;ORR~CIION

NON-LINEAR PARAMETERS

SIGMA L,//5631WoE-06 1,77558Z50E-Ob ; ~ .5183

~CALERS IN MICRONS/METER SCALER ~ = -~,7809 1,7756 SCALER ~ = -2Z,3~?8 I.T~S6 SX/Sf = 17.5674

PARAMETER VALUE = 9,9999522E-01 ; 9.9991765E-0% = 0 @ 10,210 : 1.0000176E+00

LINEAR PARAMETERS

AOJUSTEO COMPARATOR CALIBRATION PARAMETERS F~OM THE SIMULTANEOUS SOLUTION OF

,~W3 = SIANOARO ERROR OF RANDOM (IRREGULAR) COMPARATOR ERRURS

,d37 = STANOARO ERROR OF SYSTEMATIC NON-LINEAR ERRORS

2.151

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• 2W@ :

~ CASES

= ~TANBARO ERROR OF A SINGLE OBSERVATION OF UNIT WEIGHT AFTER LINEAR AND NON-LINEAR ADJUSTMENTS

= ~IANDARO ERROR OF A SINGLE OBSERVATION OF UNIT WEIGHT AFTER LINEAR ADJUSTMENT

• ~1] = ~IANDARD ERROR OF MEAN POINTING MEASUREMENTS

• ~5

I,~5)

2,WBL = ~o~r

ANALYSIS OF VARIANCE

SCALER X SDALER t N O N - O R T H O G . ANGLE A L P H A SCALER RATIO SX/SY

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• 93N

Y 2,620

°~39

X 2.355

COMPARATOR NUMBER B CALIBRATION OPERATOR NUMBER 1 DATE OF MEASUREMENT 7 1 1 7 / 7 ~ GRID AND CASE NUMBER %320 CALIBRATION SUMMARY

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An evaluation of these computed statistics from several calibrations provides the basis for monitoring the performance of all of the system components. SUMMARY

Complete comparator calibration for all 2-axis comparators (including those with rectangular formats) is available in a computer-coded program. The routines involved are exact and presented under a state of rigorous statistical control. The program has been thoroughly tested and proven satisfactory on over 100 comparator calibrations involving at least four different brands of comparators. The above discussion is a summary of an N O A A Technical Report No.57.* This report covers in complete detail all derivations and all computer program routines referred to in this text. The report is available upon request to the NOS Photogrammetric Research Branch in Rockville, Md. (U.S.A.).

* Fritz, L. W., 1973. Complete Comparator Calibration. NOAA Technical Report no.57, National Oceanic and Atmospheric Administration, National Ocean Survey, Rockville Md. (in press).