PHILIPPINE-NUMERICAL PHOTOGRAMMETRIC CADASTRE 283

more accurate assessment of the sources ofrevenues for the government; this when chan­neled to the different development programsof the country leads to more improved facil­ities and conditions of living for all citizens;(2) the utilization of every available piece ofland to production which together with theintense application of scientific methods inagriculture will lead to a greater agriculturalproduction and thus, to national self-suffi­cIency.

PROPOSED LOI\G RANGE PROGRA:\{ OLePHOTOGRAMMETRIC SURVEY

As a result of our experience in the BulacanPilot Photogrammetric CadastraL Project, ourDirector of Lands has drawn up a long rangeplan for a nationwide survey of the Philip­pines. He proposes a IS-year period within

which to finish the remaining two-thirds ofthe country's area. Under this plan we hopeto achieve a unit cost of less than $6.00 perhectare. To implement this program, we arerecommending to our Congress the passageof a bill appropriating $25 million for thispurpose. \Ye are also looking forward to pos­sible aid from the UN Special Fund which weare negotiating through the local UNTABrepresen tati ve.

Thus we have shown how we are utilizingscience and technology in our cadastral sur­veys. And in proposing to do it in a nation­wide scale we are striking at the heart of ourcoun try's socio-economic program of alleviat­ing the living conditions of the masses of ourpeople and seCLu-ing continued prosperity forthem, their children and their children'schildren.

Development of Programsfor Strip and Block Adjustmentat the National Research Council of Canada*

G. H. SCHUT, PhotogrammetricResearch,

Div. of Applied Physics,

National Research Council, Ottawa, Can.

ABSTRACT: A description is given of recently developed programs for strip andblock adjustment. The strip adjustment consists in three-dimensional trans­formation of strip coordinates by means of various polynomials. Conformaltransformations are used systematically. Block adjustment consists in trans­formation of the coordinates of the individual strips by means of polynomials.It is performed separately for horizontal coordinates and for heights. Use ismade of a direct solution of the complete system of normal equations for o.llstrips. Results obtained with a block of RC9 super-wide angle photography areshown.

AT THE National Research Council, the use1"1.. of analytical methods in photogram­metry has been a subject of research since19S3.

First, a method of analytical aerial tri­angulation was developed and programmedfor electronic compu tation. In addition, athree-dimensional strip transformation wasprogrammed for the conversion of strip co­ordinates to a geodetic coordinate system.

This transformation did not incl ude correc­tions for strip deformation. Subsequently,methods were developed for the adjustmentof the horizontal coordinates of single stripsand of blocks of overlapping strips. Here,stri p deformation was corrected by means ofconformal transformations. Reports aboutthese methods were pu blished in PHOTO­GRAMMETRIC ENGINEERING and in The Cana­dian Surveyor [1, 2, 3] in 1960 and 1961.

* Presented at 1953 Semi-Annual Meeting of the Society.

284 PT-TOTOGRAMMETRIC ENGINEERING

The present paper is a report on the workthat has been done since then, and on plansfor the near future. This work concerns amethod for three-dimensional adjustment ofsingle strips with corrections for various typesof strip deformation, and furthcr de\"e!op­ments in block adjustment.

CONFORMAL TRANSFORMATIONS

1. For the adjustment of strip coordinateswith an electronic computcr, it is rather con­venient to use transformation formulas thatgive the transformed coordinates as poly­nomials with respect to the stt-ip coordinates.

Restricting ourselves for the moment tothe adjustment of horizontal coordinates, itis found that various types of polynomialtransformations are used. Of these, the for­mulas for conformal transformation are themost sui table since these transform anysmall area in the strip coordinate systemwithout deformation. From this property, thetransformation derives its name.

Because of the importance of the conformaltransformation, a somewhat detailed de­scription of it is given in this papeL In termsof complex numbers, this transformation canbe written:

(X" + iY,,) = (al + ia2) + (a3 + ia4) (X + iy)

+ (a5 + ia6) (X + iY)2

+ (a7 + ias) (X + i Y)3 + . . . (1)

Separating real and imaginary terms oneobtains:

X" = a, + a3X - a4 Y + a5()(2 _ Y2)

- a62X Y + a7(X3 - 3X Y2)

- as(3X2Y _ Y3) + ...Y" = a2 + a4X + a3 Y + a6(X2 - Y2)

+ a52XY + a,(X3 - 3Xy2)

+ a7(3X2Y - ya) + . . . (2)

The effect of small changes in the stripcoordinates X and Y upon the transformedcoordinates X" and Y" is found by differen­tiation:

ax ax,dX,,=--"dX+--' dY

ax aYaY" aY"

dY =-dX+-dY (3)"ax (lY

For the conformal transformation, onefinds that for any value of X and Yand forany degree of the equations

and

Therefore, Formulas (3) are of the same typeas the linear parts of Formulas (2), which arethe \\"ell-knoll"n formulas for transformationwithout deformation" This proves that For­mulas (2) transform any small area lI"ithoutdeformation" Consequcn tly, any two Ii nesthat intersect cach other at right angles be­fore transformation will do the same aftertransformation. Also, the change in scale thata small area incurs through transformationis the same in any direction.

No other polynomial transformation ispossible that has this property. If, for in­stance, the terms with Y' are omitted in thesecond-degree transformation, the trans­formation is not any more conformal. As aresult, when the conformal transformation isnot used, care must be taken that the stripcoordinates are in a region of the X, Y coor­dinate system where the transformationchanges the angle between any two lines bya negligible amount" For instance, if in Equa­tions (2) the terms with Y' are omitted, thestrip axis should either coincide with or beclose to the X-axis.

THREE-DIME:\SlO:-IAL STlHP ADJusTME:\T

2. For three-dimensional strip adjustmentby means of polynomials, one can attempt todevelop formulas for conformal transforma­tion as follows:

x" = a,o + >.(a"X + a 12 Y + a13Z) + .Y" = a20 + >.(a2tX + a"Y + a23Z) + .Z" = a30 + >.(a31X + a32Y + a33Z) + . .. (4)

The linear terms in Formulas (4) producea linear conformal transformation (change ofscale, rotation, and shift) if and only if, dis­regarding the factor A, the matrix of coef­ficients of X, Y, and Z has the propertiesthat the sum of squares of the elements ineach column is equal to 1 and that the sum ofproducts of corresponding elements in eachtwo columns is equal to zero. A matrix thathas these properties is called orthogonal.

If one includes in (4) terms of higher thanthe first degree \\"ith respect to X, Y, and Z,and differentiates X", Y,,, and Z", with re­spect to X, Y, and Z, the coefficients ofdX, dY, and dZ are the nine partial deriva­tives. The differentiated formulas give the ef­fect of small changes in X, Y, and Z upon thetransformed coordinates, as do Equations (3)for the two-dimensional transformation.Therefore, they define the character of thetransformation of a small area.

PROGR.\MS FOR STRIP AND BLOCK ADJuST 'lENT 285

[n order thal the transformation be con­formal it is necessary that any small area betransformed withou t deformation. Conse­quently, disregarding a scale factor, the ma­trix of the nine differential quotients must beorthogonal for any values of X, Y, and Z.This leads to a number of conditions whichthe coefficients in Formulas (4) must satisfy.

Unfortunately, it turns out that the condi­tions can be satisfied only if the coefficien tsof all terms that are of higher than the firstdegree with respect to X, Y, and Z are equalto zero. This proves that a conformal trans­formation in three dimensions which includesstrip deformation is not possible by means ofpolynomials.

3. The above discussion can be summarizedin two statements:

a) For three-dimensional strip adjust­ment, it is rather convenient to useformulas that give the transformed co­ordinates as polynomials with respectto the strip coordinates.

b) Of the different types of polynomialtransfonllations, conformal transfurma­tions are the most sui table. However,these are possible in two dimensions butnot in three dimensions.

These statemen ts lead directly to the fol­lowing specifications for transformation for­mulas:

I. The adjustment must be done in steps,in each of which a suitable sectionthrough the strip is subjected to a two­dimensional conformal transformation.

11. In each step where the conformal trans­formation includes an appreciablechange of scale in the area of the strip,the third dimension must receive thecorresponding scale correction.

In this way, each step can correct for onetype of strip deformation, such as verticalcurvature in the strip direction, torsion alongthe strip axis, vertical curvature across thestrip, and gradual changes in scale andazimuth. The resulting three-dimensionaltransformation cannot be exactly conformalbu tit will be very close to it.

4. An alternative solution can he found in ageometric approach.

One of the requiremen ts of a three-di men­sional strip adjustment is that the strip mustbe corrected for tilt and for vertical curva-

ture. DUl'ing this correction, differences interrain elevation must be taken into account.For this purpose, it is possible to com pu te areference surface in the strip coordinatesystem which after adjustment shall be ahorizontal plane. Differences in terrain ele­vation can be taken in to accoun t by specify­ing that any perpendicular on the referencesurface must after transformation be per­pendicular to the horizon tal plane.

Recently, papers have been published byAckermann and Perks which describe meth­ods based upon this principle [7, 5). In bothmethods vertical sections through the stri pare made, one along the strip axis and oneacross the strip. In these sections, the line ofin tersection wi th the reference surface andthe perpendicular from a point on the line ofintersection are drawn. The adjusted coor­dinates are computed as distances measuredalong the line of intersection and along theperpendicular.

Ackermann, in a method employed at theInternational Training Centre, uses a torusas reference surface: the strip is assumed tohave constant CUl'vature in the strip dil'ectionas well as across the strip. The method in­cludes leveling of the strip and is intended asthe first step in a more comprehensive stl'ipor block adjustment.

Perks, in a method used at the Surveys andMapping Branch of Bdtish Columbia, de­fines the reference surface by means of apolynomial with respect to the X- and Y­coordinates. Therefore, the lines of intersec­tion with the reference surface are poly­nomials of a specified degree.

In Ackermann's procedure, the computa­tion of lengths in the strip coordinate systemrequires the use of trigonometric functions,In Perks' procedure, the length along theline of in tersection in the longi tudinal sectionwas computed first as an integral, later as asu m of chords.

\\'hen these methods are programmed forelectronic computation, one will replace theabo\'e functions by their series developmentwith respect to the strip coordinates. Sinceat any point in a strip the radius of curvatureis very large compared wi th the length of thestrip, this series developmen t will requireonly a few terms. Thus both methods end upwith a fOl'mulation in which the adjustedcoordinates are expressed in polynomials withrespect to the stl'i p coordinates.

This raises the question "'hether it IS not

286 PHOTOGRA1\IMETRIC ENGI EERING

simpler to define the adjusted coordinates di­rectly as such.

5. The approach of successive two-dimen­sional conformal transformations has beenfollowed in a three-dimensional strip adjust­ment developed at the National ResearchCouncil and programmed for the IBM 1620.

The computation, which is performed inone pass of the computer, contains the follow­ing steps:

The strip coordinates are first trans­formed to an axis-of-f1ight coordinate sys­tem (X, Y, Z) with origin in the centre ofthe strip.

The strip is then brought to terrain scaleand leveled. At the present time, the level­ing does not include strip deformation buta correction for vertical curvature in thestrip direction by means of a second-degreeconformal X, Z transformation can easilybe included.

Vertical curvature in the strip directionand residual tip are corrected by means ofa conformal X, Z transformation of speci­fied degree.

Torsion and residual cross-tilt are cor­rected by means of a linear Y, Z transfor­mation which is a rotation abou t theorigin:

Y,,=qY-pZ

Z" = pY + qZ

The coefficient p is defined as a polynomialof specified degree with respect to X. Thecoefficient q is found as v'1-p2.

Vertical curvature across the strip iscorrected by means of a second-degree con­formal Y, Z transformation. The coefficientof the second-degree terms may be com­puted either from a specified radius ofcurvature or from the height control.

Finally, the horizontal coordinates areadjusted by means of a conformal X, Ytransformation of specified degree.

The formulas are given in more detail inreference [5].

6. In the three steps where a correction forheight deformation is applied, vertical crosssections are subjected to conformal trans­formation. Therefore, a small area in a crosssection is rotated but not deformed. In thisway, the effects of tilt and height differencesupon the horizontal coordinates are auto­matically taken into account.

Further, in the case of height corrections,those coefficien ts which do not con tribu tea height correction if z=o are assumed to beequal to zero. Thus, for instance, the correc­tion formulas for vertical curvature in thestrip direction are:

X .... = X - 2b 6 XZ - bs(3X 2Z - Z3) - ...

Z" = Z + b6(X2 - Z2) + bs(X3 - 3XZ2) + . .. (5)

Because, in addition, the remaining coef­ficients are computed from the Z equationsonly, while the origin has been shifted to thecentre of the strip and the strip has first beenleveled, the height corrections leave the scalepractically unchanged. Therefore, no scalecorrection of the third coordinate is neces­sary.

However, in the the case of the correctionfor vertical curvature in the strip direction,any variation in scale has a cumulative effectupon the X coordinate. \\"hen the strip is 100km long, has horizontal-control at both ends,and a curvature equal in size to the earthcurvature, this curvature correction causeslargest X residuals of 0.2 meter. For longerstrips, the errors increase proportionally tothe third power of the strip length.

If one wishes to a void these errors, it isnecessary to re-introduce the terms that wereomitted in Equations (5) and to computetheir coefficients. One can now specify thatthe curve which in the X, Z coordinate systemhas the equation Ztr = 0 must be transformedat true length. For this curve, the second ofthe Equations (5) gives Z as an implicit func­tion of X. If Z is solved from this equationand the obtained polynomial in X is sub­stituted into the first of the Equations (5),X tr is obtained as a polynomial in X. If thepolynomial for Z is differentiated with re­spect to X, distances S along the curve canbe obtained from dS= v'dX2+dZ2 and canbe written as a polynomial in X. Since thecurve Ztr = 0 must be transformed at truelength, the polynomial for X tr and the poly­nomial for S must have identical coefficients.In this way, the coefficients with odd-num­bered indices are found as functions of thecoefficients with even-numbered indices. Inthe case of a second-degree correction,b1 = -tb 6

2 ; in the case of a third-degree cor­rection, in addition bg = -ib6bs.

7. The scale cOlTection that is included in theadjustment of horizontal coordinates shouldbe applied also to the Z-coordinate. For thispurpose, the scale factor can be computed foreach point with the help of Equations (3).

PROGHAMS FOR STRIP AND BLOCK ADJUSTMENT 287

The scaled Z-coordinates should be usedin the computation of the formulas for correc­tion of heigh t deformation. On the otherhand, the strip should be corrected forheigh t deformation before the horizon tal ad­justment is performed.

These two contradictory requirementsseem to make necessary performing the ad­justment for height deformation and thehorizontal adjustment in turn, in an iterativeprocedure. However, if this scaling of the Z­coordinate is omitted, one height adjustmentfollowed by one horizontal adjustment hasproved to be sufficient. Appreciable heighterrors will then occur only in the case oflarge variations in terrain height combinedwith a poor scale propagation through thestrip. Because in this case the accuracy re­quirements can be less strict, it has not beenfound worthwhile to make the adjustmentan iterative procedureo

BLOCK ADJUSTME:\,T, 1962

8. In a block adjustment of strips, each in­dividual strip is adjusted so as to agree asfully as possible with the existing ground con­trol and with the other strips.

This computation can be performed insteps. First, each strip can be approximatelyleveled and positioned with respect to thegeodetic coordinate system. As a rule, su ffi­cient height-control will be available to in­clude here a second-degree correction forvertical curvature in the strip direction. Forpractical purposes, this makes the effects ofresidual tilt and of height differences uponthe horizontal coordinates negligible. Afterthis, therefore, the adjustment can be sub­divided in an adjustment of horizontal co­ordinates and an adjustment of heights.

9. The program for three-dimensional striptransformation is used for the approximateleveling and positioning. It provides thepossibili ty of storing the transformed coor­dinates of tie-points of a strip in memoryand of using these coordinates as addi tionalcontrol for a following strip. This featuremakes possible transforming all stri ps of ablock in one pass on the compu tel', eyen ifonly one strip has sufficient ground-controlfor independent positioning. Also, if one striphas sufficient control for second-degree trans­formation of horizontal coordinates orheights, all strips can be transformed in thisway.

For horizontal block adjustment, an IBM650 program in which second-degree con­formal transformations of individual stripswere used in an itera ti \'e proced ure gave veryfavorable resul ts. This was a reason to re-codethis program wi thou t modification for theIBM 1620.

A program for block adjustment of heightswas coded also, using the same iterative pro­cedure. Here, the heights are adjusted bymeans of polynomials with respect to thestrip coordinates.

These two programs were coded by Topo­graphical Survey of the Department of Minesand Technical Surveys and by the ArmySurvey Establishment in close cooperationwith the National Research Council.

REMARKS ON PROGRAMMING

10. \Vhen the program for analytical aerialtriangulation was first coded for the Feru telectronic computer, everything that wasreasonably possible \\'as done in order to re­duce the computation time to a minimumwithout relinquishing accuracy, A simple setof formulas was derived, fixed-point arith­metic \\'as used throughou t the compu ta­tions rather than floating-point subroutines,and the shortest possible formulation in termsof computer instructions was sought. \\'henthe program had to be re-coded for the IBM650, the same policy \\'as adopted in the ex­pectation that this computer would be avail­able for a long time to come. That made theexpenditure of the necessary time and etTortworthwhile.

The results of this policy became visible inan investigation by GIMRADA, reported onby Matos [6], who stated that the NRCmethod of analytical triangulation provedto be about 5 to 10 times faster than the othermethods tested.

However, two years after completion ofthe program, an IBM 1620 was installed atthe N.R.C. laboratories. A year later, the lastof the IBM 650 to which N.R,C. had accesswas withdrawn from the Ottawa area. Atabout the same time, N,R.C.'s IBl\1 1620 wasprovided \\'ith the optional floating-pointhardware. Also, the ever increasing demandfor computer time has raised the questionwhether this com pu ter will not sooner orlater be replaced by a faster one. These cir­cumstances and the continuing actiyity in thefield of computer development give rise to

288 PHOTOGRAMMETRIC ENGl EERING

the speculation that it may become necessaryto re-code our programs periodically.

In view of this situation, it does not seemworthwhile anymore to spend an appreciableamount of time on the writing of programsin fixed-point arithmetic and on looking forother economies in terms of computer time.At the present time, the best policy appearsto be to use floating-point arithmetic, tospend the necessary time on writing an eco­nomical set of general purpose computationalsubroutines for the current computer, andto spend a minimum of time on the frame­work of instructions that surrounds the actualcomputations. In this way, a better balancebetween programming time and computationtime will be reached. As a result, the pro­grams may become considerably slower thanthey could be, but they can become opera­tional before the compu ter for which theywere written is modified or even withdrawn.

BLOCK ADJusTMEKT, 1963

11. Having accepted the regular use offloating-point arithmetic, a new approach toblock adjustment becomes possible.

An adjustment requires the solution of asystem of normal equations with as many un­knowns as there are coefficients in the trans­formation formulas that must be solvedfor simultaneously. The iterative adjustmentwas adopted partly because a fixed-point pro­gram for single strips had been written al­ready and could be used as a subroutine inan iterative block adjustment; partly be­cause it was felt that with fixed-point arith­metic it might not be possible to solve agreat number of normal equations with suf­ficient accuracy. The use of floating-pointarithmetic makes it feasible to solve large sys­tems of linear equations and therefore to at­tempt a direct solution of the complete sys­tem of normal equations for all strips of ablock.

In the case of block adjustment by strips,the solution is facilitated by the fact that thematrix of coefficients in the normal equationscontains a relatively large number of zeros.

If in this matrix the unknowns, which arethe coefficients in the transformation for­mulas, are grouped according to the strips,each ground-control poin t provides contribu­tions only to one sub-matrix on the maindiagonal. To each strip corresponds one suchsub-matrix. If one type of transformationformula is used for all strips, these sub-

matrices are all of the same order. They canbe used as a starting point for subdividingthe matrix into square sub-matrices of thesame order.

Each tie point between two strips providescontributions to the two corresponding sub­matrices on the main diagonal and to the twooff-diagonal sub-matrices that lie with thediagonal ones on the corners of a square.

Therefore, in the case of a block of paralleland uninterrupted strips contributions aremade only to the sub-matrices on the maindiagonal and to those immediately adjoiningit. All others are equal to zero.

If, instead of with sub-matrices, one had todo with real numbers, the solution of thesenormal equations through successive elimina­tion and back substitution would be verysimple. In the case of matrices, exactly thesame procedure can be used. It is only neces­sary to replace each operation with realnumbers by the analogous operation inmatrix algebra.

12. As a check on the feasibility of the directsolution, a program for horizontal block ad­justment of parallel and uninterrupted stripshas been coded following this procedure. Theformulas for second-degree conformal trans­formation have been used.

The computations are performed in float­ing-point arithmetic with 10-digit mantissas.I n order to reduce the requiremen ts on thenumber of significant digits in the computa­tions, the origin of the coordinate system istemporarily shifted to a point inside theblock, and the transformation formulas areset up to give, primarily, corrections to ap­proximate coordinates.

Since the transformations are conformal,all computations can be performed with com­plex numbers. In this case, the sub-matricesare of order three and have complex elements.In the case of real numbers, the matrix ofcoefficients in the normal equations is sym­metric. Here, this matrix and the sub­matrices on the main diagonal are hermitian.These properties have been used to economizeon storage space for the normal equationsand on computation time.

Rather than storing the ground-con trolpoints and tie-points in memory, the com­pu tation has been made a two-pass process.

PROGRAMS FOR STRIP AND BLOCK ADJUSTMENT 289

During the first pass of the dala, the normalequations are computed and solved, and thetransformation equations are stored. Duringthe second pass, all points are transformed,and transformed coordinates, residuals onground-control points, and half-differencesin tie-points are punched. In this way, suf­ficien t storage space for the normal equationsis available to adjust 60 strips simultaneously,and an unlimited number of ground-controlpoints and tie-points can be used.

The data deck con tains one card for eachpoint with the input coordinates. The cardsare sorted in groups according to strip and totype of point, and each group is preceded bya card that gives strip number, type of points,and their weight in the adjustment. No fur­ther preparations are required.

A test of the program with a theoretical ex­ample of 20 strips with the minimum of threeground-control points and three tie-points ineach overlap, and covering an area of 200 by200 km, gave an exact fit on ground-controlpoints and tie-points. Two computations withopposite sequence of the strips in the forma­tion of the normal equations gave results thatdiffered by at the most 1 cm. The computa­tion time for this adjustment, including pro­gram read-in, was 8 minutes.

13. Because of the excellent internal accuracyand speed of this program and its simplicityof operation, its usefulness has been increasedby introducing a modification which allowsthe adjustment of blocks of strips with breaksin the triangulation. J ow, each strip mayhave tie-points in common with the stripswith the three next lower strip numbers andthe strips with the three next higher stripnumbers.

This affects the storage space lhat is neededfor the normal equations. Space has been pro­vided for the off-diagonal sub-matrices thatare one, two, and three places removed fromthe main diagonal. As a resul t, not more than28 strips can be adjusted simultaneously.

\'Vhen an off-diagonal matrix has non-zeroelemen ts as a resul t of a tie-poi n l condi tion,the elimination procedure that is used duringthe solution of the normal equations may pro­duce non-zero elements in other off-diagonalsub-matrices down the same column, untilthe main diagonal. The resulting increase inthe amount of computation, compared withthe case of uninterrupted strips, makes theaccuracy req uiremen ts more critical. For

instance, in an extreme case of three parallelstrips with lWo cross strips on one side of theblock and a distribution of tie-points andground-control points that was designed tolax the capabilities of the program, slightlyinaccurale results were obtained if during thecomputation the origin of the coordinatesystem was shifted to the centre of the block.For centimeter accuracy, the origin had to benearer the centroid of the control points andtie-points.

14. A similar program for vertical block as­justment is planned also. Here, a complica­tion is caused by the fact that one type ofpolynomial cannot be used for the adjust­ment of all blocks, and possibly not alwaysfor the adjustment of all strips in one block.

The program for analytical aerial triangu­lation has not been re-coded for the IBM1620. Instead, the IBM 650 program is usedwith a simulator on the IBM 1620. Thismakes the program 4 or 5 times slower thanit was on the 650. Although this is rather un­satisfactory, lhe limited use that is as yetmade of the program does not warran t theeffort of re-coding.

AN ApPLICATION

The programs for strip and block adjust­ment were used in the adjustment of a blockof aerial photography taken with an RC9camera over an area in Southern Rhodesia.The block contains 5 strips with 22 photo­graphs each and measures 140 X 65 km. Thefocal-length of the camera is 88.48 m m, thephotograph-scale is abou t 1: 80,000, and theaverage forward-overlap is 63%. Ilford HRAfil m was used.

The distribution of ground control pointsand of tie points between strips is shown inFigure 1. The ground-con trol points were nottargetted.

The pholographs were measured by theRhodesian Departmen t of Trigonometricaland Aerial Surveys in a Nistri TA3 stereo­comparator and the obtained coordinateswere correcled by that Departmen t for fil mdistortion, asymmetric and symmetric lensdistortion, refraction, and earth curvature.The correction for fil m distortion was basedupon measurements of the four cornerfiducials, and was performed with a second­degree non-conformal polynomial transfor­mation. For relative orientation and scaletransfer, eight points were measured per

290 PHOTOGRAMMETRIC ENGINEERrNG

Rt 9 BLOCK

......... Phofo Cenlers

........ Tie Poinls

........ Ground Con'rol

STRIP I +-+-+-+-+-+-+_+_+_+_+_+_+_+_+_+_+_+_+_+_+_+

STRIP 2+-+-+-+-+"_+_+_+_+_+_+_+_+_+_+_+_+_+_+_+_+_+.. . . · .STRIP

STRIP

3 +-+-+-+-+_+_ ... _+_+ _+_ ... _+_+ _+_+ _+_ +_+_+ _+._._..·.

4 +-+_+_+_+_+_+_.+_+_+_+_+_+_+-+~+_+_+_+_+_+_+.. .' , . ·.STRIP 5

+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-r+

, ,

.L9 ~20 4,LO 6.L,O ~8,O ~19_O '2~O ~I~O km

FIG. 1

model: six points in the usual POSl tlOns andtwo near the centres of the two squaresformed by the first six points.

The analytical triangulations gave satis­factory results except for one model in strip 1which showed exceptionally large residualparallaxes and height deformation. The meansquare value of the residual parallaxes inthe points used for relative orientation is 4.8micron at photograph scale.

The strips were first transformed inde­pendently, using second-degree transforma­tions in order to correct vertical curvature inthe strip direction, scale, and azimuth.Figure 2 shows the discrepancies in tie pointsafter these transformations. They give a clearindication of third-degree deformation in Xand Z. The strips had little or no torsion.

Subsequent third-degree transformationsshowed systematic height errors of a fewmeters in ground-control points and sys­tematic discrepancies on tie-points of thesame size which could be reduced by a fourth­degree correction of vertical curvature.Finally, therefore, strips 4 with the most con­trol, and strips S, 3, and 2 were transformedin this sequence by means of third-degreeX, Y transformations and fourth-degree Ztransformations. For the latter three strips,a few tie-points with an already transformedstrip were used as additional control. Strip 1was adjusted in two parts by means of second-

degree transformations. This procedure madea subsequent block adjustment unnecessary.

Because in this block the third-degree X, Y

RC 9 BLOCK Coordlnale differences oller second degree Irons forma !Ion

0- X

FIG. 2

PROGRAMS FOR STRIP AND BLOCK ADJUSTMENT 291

transformations give significantly better re­sults than the second-degree transformations,and the horizontal block adjustment appliessecond-degree transformations only, it is ofinterest to see what can be achieved withthis block adjustment. For that purpose, eachstrip has been divided into two parts of aboutequal length and a block adjustment hasbeen performed using the same ground-con­trol points and tie-points as in the final striptransformations. In order to obtai n a strongconnection between strip halves, two "wingpoints" at the break were used as tie-pointsand given a ten times greater weight than thetie-points between strips and the ground­control points.

The results of both adjustments are listedin Table 1. The largest value in the table, 2.1m, equals 26J.L at photograph scale. Evidently,the result of the block adjustments is some­what better than that of the strip adjust­ments. Both are quite satisfactory for the re­quired mapping at scale 1 :50,000.

TABLE 1

MEAN SQUAllE VALUES OF RESIDUALS IN GROUNDCONTROL POINTS, AND HALF-DIFFER~;NCES

IN TIE POINTS, IN METERS

Strip Blocktransformations adjustment

117 X my 111Z mx 111]'

All ground control 1.9 2.1 1.9 1.6 2. t..-\11 tie points 1.4 1.2 1.7 1.0 1.2

The ti me required for analytical triangula­tion, using the program to simulate the IBM650 on the IBM 1620, was between 55' and 60'per strip. The final strip adjustments tookaltogether about 30' with the latest programwhich employs the floating-point hardware.The block adjustment required 20'. Thetimes for these two adjustments include thetransformation not only of the used controlpoi n ts bu t also of abou t a thousand addi tionalpoints. These points are the eight points usedfor relative orientation in each model, theprojection centres, one of each two groundcontrol points that occurred in pairs, andunused tie points.

REFEREKCES

[I] G. H. Schut, "Experiences with AnalyticalMethods in Photogrammetry." PHOTOG RAM­METRIC ENGINEERING, September 1960.

[2] ---, "A Method of Block Adjustment forHorizontal Coordinates." The Canadian Sttr­veyor, March 1961.

[3] ---, "Transformation et Adjustement desCoordollnces d'une Bande par Calcul Elec­tronique." The Canadian Surveyor, ="!O\'"ember1961.

[4] M. Perks, "A ='IumeriCc"ll Adjustment Procedurefor Aerotriangulation Programmed for IBM 650Computer." The Canadian Surveyor, May 1962.

[5] G. H. Schut, "The Use of Polynomials in theThree-Dimensional Adjustment of Triangu­lated Strips." The Canadian Surveyor, May1962.

[6] R. A. Matos, "Analytic Triangulation withSmall or Large Computers." PHOTOGRAM­METRIC ENGII\EERING, March 1963.

[7] F. Ackermann, "Zur 'Entkrlimmung' vonStereomodellen und Triangulationsstreifen."Bild1llessung und Luftbildwesen, June 1963.

ANNOUNCEMENTTENTH CONGRESS OF THE

INTERNATIONAL SOCIETY FOR PHOTOGRAMMETRY

LISBON, PORTUGAL

Headquarters at:

SEPTEMBER 7-19, 1964

Feis das Industriak de Portugal

For information write to

M r. Rupert B. Southard, Jr., 229 Burke Road, Fairfax, Va.