Review of Strip and Block Adjustment During the Period 1964-1967

Eleven th Congress of the International Society for Photogramrnetry Lausanne, Switzerland, July 8-20, 1968

Invited paper for Commission III Review of Strip and Block Adjustment During the Period 1964-1967

G. H. SCHUT Photogrammetric Research, Div. of Applied Physics, National Research Council, Ottawa, Canada

INTRODUCTION taneous adjustment of photographs in blocks HE PERIOD 1964-1967 is characterized by or strips. References [24] to [31] discuss the T the further development and the success­ solution of the large system of normal equa­ fuI completion of a number of com puter pro­ tions in this adj ustmen t. grams for the simultaneous adjustment of Already before the London Congress, H. H. aerial photographs in large blocks. Both the Schmid and D. C. Brown used the condition direct solution and the iterative solution of of collinearity of image point, perspective the resulting system of normal equations have centre, and object point for the simultaneous proved to be entirely practical. adjustment of a set of photographs. This Block adjustments in which models, sec­ condi tion leads to the linearized equation tions or strips are adjusted as units are still v + Bo = I: (1) far more common. They can be divided into In which two main groups: adjustment of models or sections by means of similarity transforma­ v is the vector of corrections to the photo­ tions, and adjustment of strips or parts of graph coordinates, strips by means of polynomial transforma­ o is the vector of corrections to the tions of higher than the first degree. parameters of camera orientation and The proponents of the adj ustmen t of photo­ to the coordinates of object points, graphs view this adj ustmen t as the most B is a matrix of coefficients, and rigorous solution and as the main trend in c is the vector of residuals of the photo­ computational photogrammetry. Others re­ graph coordinates in the non-linear con­ gard the similarity transformation of models dition equations. as the Irue or rigorous least sq uares adj ust­ ment and the ultimate solution, and they The normal equations become predict a decrease of interest in the poly­ -Pv + k = 0 nomial adj ustmen t. Nevertheless, the poly­ v + Bo = £ nomial adjustment has many adherents and B'k = 0 (2a) is much used because it is the easiest to pro­ gram and to use, and because it gives a very in which P is the weight matrix of the ob­ satisfactory accuracy for topographic map­ served photograph coordinates and is com­ ping. puted as the inverse of the covariance matrix. Any interest that may still exist in the Further, k is the vector of Lagrange multi­ analog adjustment of input data has not re­ pliers or correlates, and the superscript I indi­ sulted in more than one published paper. cates the transpose. Elimination of v gives

~lk+Bo SIMULTA~EOUS ADJUSTMENT = £ OF PHOTOGRAPHS Btk = O. (2b) 1. USE OF THS COLLINEARITY CONDITION Subsequently, elimination of k gives References [5] to [23] deal with the simul- BtpBo = Btpl:. (2c) 344 REVIEW OF STRIP AND BLOCK ADJUSTMENT DUR1NG 1964-1967 345

2. TREATMENT OF CONTROL POI TS and elimination of 02 gives

1 Both Brown [l1J and Schmid [5J to [7J have (N" - N,2'N22-IN,2)'6, = [I - N ,2'N22- [2. (2e) now introduced the use of a covariance matrix The components of 0, and of 02 can be par­ for the coordinates of the control points in a titioned into groups each of which contains block of aerial photographs. Unknown coordi­ only the corrections for one photograph or nates are given a large variance and for the for one terrain point, respectively. The cor­ sake of simplicity, although unrealistically, respondingly partitioned matt·ices Nil and one assumes that their estimated (observed) N contain non-zero submatrices along their values are uncorrelated. If a rectangular 22 main diagonal only, while the submatrix of coordinate system is used other than the one N'2 which corresponds to point i and photo­ formed from planimetric coordinates and ter­ graph j has non-zero elements only if point i rain heights, the proper covariance matrix in has been measured in photograph j. that system can be easily computed from the Accordingly, the inverse of N can be relation between the two coordinate systems. 22 computed by inverting its submatrices sep­ \Vith this procedure, the vector & contains arately and Equation 2e can be computed also the corrections to the coordinates, known directly from the Equation t by treating all as well as unknown, of all control points. To the condition equations for one terrain point retain the advantage that only one correction as a group and computing the contribution of to an observation occurs in each condition this group to the normal equations separately. equation, Equation 1 is supplemented with I n this way, no space need be reserved for the the equation matrices N 22 and N ,2. If two photographs (3) have no measured point in common, the cor­ responding off-diagonal submatrix in Equa­ in which vg is the vector of corrections to observed coordinates of the control points and tion 2e is equal to zero. 0" is the vector of corrections to their esti­ Each of the block adjustment programs mated or given coordinates. The equation ex­ that have been coded along these lines em­ presses that one identifies the observed values ploys a housekeeping routine, collapsing al­ with the estimated or given values. gorithm, or indexing technique to avoid com­ This procedure increases the size of the puting and storing zero-submatrices and to normal equations. However, it makes their keep track of the locations of the non-zero formation simpler because now each terrain ones. point receives three coordinate corrections 4. DIRECT SOLUTION OF THE NORMAL EQUA­ independent of whether it is completel; TIONS known, partially known, or unknown. Even for large blocks, the direct solution 3. PARTITIONING OF THE NORMAL EQUATIONS of the normal eq uations has proved to be The components of v, 0, and k may be par­ entirely practical provided that a block­ titioned into groups. If the condition equa­ elimination procedure is used. This means tions contain a set of constraint equations in that instead of the matrix elements the above­ ~vhich no observations occur, the correspond­ mentioned submatrices of Equations 2d or 2e Ing v-components will be equal to zero. In are used as units in the computations. addition, the sequence of the groups in the Both a Causs-Cholesky type of elimination normal equations may be rearranged. This ([91J and [21)) and a Causs-Jordan type [16J can lead to a variety of formulas. are used. If S is an on-diagonal submatrix It is customary to partition the vector 0 of which is to be used as pivot in the elimination, Equation 2c into a vector OL of corrections the Causs-Cholesky elimination involves the for improvement of the camera orientations simple computation of an upper triangular and a vector 02 of corrections to approximate matrix T such that Tt T = S and of its in­ coordinates of terrain points. The attitude verse. The Causs-Jordan eli mination requires corrections included in 01 can be ei ther the the computation of the itl\·erse of S. The two parameters of a correction matrix or correc­ elimination procedures perform equally well. tions to approximate values of attitude 5. ITERATIVE SOLUTION OF THE NORMAL parameters. EQUATIONS The corresponding partitioning of Equation 2c gives: The iterati ve sol ution of a system of equa­ lions has the basic disadvantage that it is N,,'6, + NI2'62 = [I difficult to know when a sufficiently good t N 12 '6 1 + N22'62 = [2 approxi mation of the exact sol ution has been 346 PHOTOGRA 1METRIC ENGINEERING reached. On the other hand, especially in the the exact solution. A number of ordinary case of a large block, it has the advantages Gauss-Seidel iterations is then required before that during the solution the required storage an acceleration procedure can again be used. space is not increased and that for a given A third acceleration method is called the absolute density of control the required com­ block-successive overrelaxation method [33]. putation time is only directly proportional to In theory, this method makes a sophisticated the number of photographs. Therefore, for computation of an optimum acceleration very large blocks the iterative solution will factor possible. In practice, such a computa­ be preferable. tion is too complicated. An educated guess as The principal iterative method of solution to the value of such a factor is therefore made is the Gauss-Seidel method. Although the and the sophistication consists mainly in the proof of (he convergence of this method can vocabulary that is used. This factor will be be found in mathematical textbooks, several larger than one, and it must be smaller than writers have deemed it necessary to include two. the proof in their paper. The asymmetry in­ A very different type of iterative solution herent in the method can be avoided by al­ is obtained if an orthogonalization method or ternately proceeding forth and back through a gradient iterative method is used. At the the vector of unknowns. lTC, Kubik [29. 301 has found the method of The rate of convergence is sufficiently fast conjugate gradients [34] to be the most useful only if a block-iterative procedure is used. one of these methods. I n theory, if no round­ The simplest submatrices for this purpose are off errors are introduced, this method con­ those obtained from the above partitioning verges to the exact solu tion after as many of the matrices in the Equations 2d and 2e. iterations as there are unknowns. Either the For large blocks with sparse control, one may matrix of coefficien ts and the constant terms then require from 50 to a few hundred itera­ of the correction equations or those of the tions. normal equations are stored and used in The unknowns in the normal equations can every iteration. Experience with a test pro­ be arranged in groups according to the natural gram showed that in general a satisfactory or to artificial strips such that the on-diagonal solution was obtained after some 60 itera­ submatrix of each group is connected via a tions. Accordingly, the method appears to non-zero off-diagonal submatrix to the on­ be com petitive wi th the Gauss-Seidel block­ diagonal submatrices of only the preceding iterative method. and the following group. If these much larger submatrices are used as units in the block­ 6. THE ESSA-COAST AND GEODETIC 5 RVEY iterative procedure, the required number of PROGRAM ([5]-[10]) itprations is of the order of ten [85]. However, From the beginning, the development of the com pu tations in each iteration step are computer programs for strip- and block­ then much more complicated. adjustment at the Coast and Geodetic Survey After a number of Gauss-Seidel iterations, has followed Schmid's approach. one will observe that successive corrections In the present system, input for the block to the obtained values of the unknowns tend adjustment is provided by a set of programs to form a geometric series. This fact may be or sub-programs for image coordinate refine­ used to accelerate the convergence at that ment, strip triangulation, polynomial strip stage. transformation, and resection of photographs. A simple and safe, but still rather slow, In the block adjustment, the Equations 2d accelerated method consists in computing the are formed. They are solved directly, by a ratio r of the corrections in two successive procedure of Gaussian elimination (forward iterations and multiplying the set of Gauss­ solution) and back substitution (back solu­ Seidel corrections in each iteration by 1 +r tion) which operates with the submatrices 2 or by 1 +r+r • for each point and those for each photograph A much faster acceleration method consists as units. With floating-point arithmetic and in computing the sum of all following terms l4-decimal digit word size, no round-off diffi­ of the geometric series and using this sum as culties are encountered even for the maxi­ the correction. The set of Gauss-Seidel cor­ mum size of block of about 180 photographs rections is then divided by 1-r. This is with six measured points across the centre of Luysternik's method [32]. If the Gauss­ each photograph. Seidel corrections do not form an exact geo­ The solution of the Equation 2d consists in metric series, the Luysternik acceleration corrections to the approximate values of the produces more or less random deviations from photograph parameters (that is, to three REVIEW OF STRIP AND BLOCK ADJUSTMENT DURING 1964-1967 347

coordinates and to three rotations) and to the words of core storage and four magnetic tape provisional terrain coordinates. Although the units and to adjust a block of 1,000 photo­ adjustment has been written as an iterative graphs (5 X200) on a large and fast computer. procedure, in practice one pass through the FRO~1 adjustment is found to be sufficient. 8. PROGRAMS EVOLVED 'THE HERGET Although computation costs will depend METHOD' ([14]-[19]) on the computer and on the time and care The Herget method was initiated in 1954 at spent on the preparation of the program, it is Ohio State University under contract with of interest to notice that a small block re­ the Aeronautical Chart and Information quired nine times more computer time when Centre. It has since gone through a sequence disk storage was used than when only core of modifications most of which were sponsored storage was used. The adjustment of a block by the U. S. Army Engineer research organi­ of 180 photographs took about eight times zation at Fort Belvoir (see [14] and [16]). longer than that of a block of 90 photographs. Herget used only one type of condition equation for all measurements: that of 7. THE PROGRAM OF D. BROWN ASSOCIATES, coplanarity of vectors. These vectors are the INC., ([11]-[13]) vectors from the projection centres to the Here, also, pre-edited input data for the image points and unit vectors through con­ block adj ustmen t can be provided by other trol points. One photograph at a time was programs, and the unknowns in the normal envisaged to be relaxed in an iterative pro­ equations are corrections to approximate cedure. coordinates and rotations. The block adjust­ In 1956, separate condi tion eq uations for ment program computes directly the normal partial control points and a rather unusual Equations 2e. An iterative solution is em­ scale constraint eqnation for three conj ugate ployed which, judging by the number of rays were added by McNair. Subsequently, iterations that is required, operates on small condition equations for two equal-height submatrices. poi nts in the same model and for known air­ On the basis of an investigation of iterative base were added and a simple simultaneous solutions, the method of successi\'e overrelax­ solution of the complete set of normal equa­ ation is considered to be the only one that tions was introduced. At this point, the converges sufficiently fast. The acceleration method became known as the U. S. Geologi­ factor 2/(1+...;{J-r)) is used, r being here cal Survey's Direct Geodetic Restraint j\lethod. the ratio between the largest corrections in A new program for the adj ustmen t of up to two successive iterations. Especially if r is 22 photographs and with undisclosed further close to unity, this leads to an even slower modifications was completed in 1965 [17J. convergence than is obtained with the factor A further series of modifications was ini­ 1+r. Luysternik's method is rejected because tiated in 1961. Weighting of the observations its use in every iteration causes divergence. was made possible and a search for an opti­ Equation 3 is used for control points and mum pivotal element was introduced in the for photographs with one or more known direct solution of the normal equations. parameters. In any second or following pass In the present version of the program [16]. through the adj ustment, the second part of for control points the collinearity equations this eq uation is replaced by the su m of the and Equation 3 are used. For pass points vectors of corrections obtained earlier. This (non-control points), the linearized coplanar­ makes the formation of the normal eq uations ity equation and a scale constraint equation more com plicated. I t is mean t to avoid the have been retained. The latter equation speci­ complications of the correlation that the pre­ fies that the distance from ground point to ceding adj ustmentintrod uces if corrected projection cen tre along the second of three approximations are used as new observations. rays must be the same for the two pairs of However, considering the facts that with a rays. No approximate coordinates of pass properly organized initial positioning one pass points are needed here. Becau e such coordi­ through the adjustment can be sufficient and nates can be easily computed from the coordi­ that the correlation of the initial estimates of nate of any two image points, this is only a unknown parameters of different points or small advantage. The need to select combina­ photographs is already neglected, this modifi­ tions of poi nts for the formation of the ob­ cation of Equation 3 can be dispensed wi tho servation equations and the resulting occur­ The most recent version of the program rence of a coefficien t matrix for the vector u [116] has been used to adjust a block of 162 in Equation 1 which differs from the unit photographs on a computer with only 8,000 matrix is a slight disadvantage. 348 PHOTOGRAMMETRIC ENGINEERING

I nstead of corrections to rotational param­ measurements of the fid ucial marks, sep­ eters of the photographs, here parameters of arately for each roll or even for each photo­ a correction matrix are computed by which graph. Further, there i room for disagree­ the matrix of the approxi mate orientation is ment as to whether the derived corrections premultiplied. This leads to a slightly simpler are the simplest and most suitable ones for formulation of the elements of the matrix B. the purpose. A direct solution of the normal Equations 2e with a block elimination technique is em­ LINEAR ADJUSTMENT OF MODELS ployed. Subsequent to the adjustment, AND OF SECTIONS ground coordinates are computed from the In the case of the triangulation of inde­ corrected image coordinates in two photo­ pendent models and of strips, a strip- or graphs. The program, developed by the block-adj ustmen t by similarity transforma­ Raytheon Co., is called the Simultaneous tion of individual models, or of two-model Multiple Station A nalytical Triangulation sections, can be performed. Such adjustments Program. are treated in ref. [20], [21], and [35] to [42] for three dimensions, in ref. [43] to [50] for 9. BLOCK ADJUSTMENT AT THE FRENCH INSTI­ planimetry only, and in ref. [51] to [54] fOI­ TUTE GEOGRAPHIQUE NATIONAL ([20]-[22]) hp.ights only. At the I.G.N., also, the simultaneous ad­ The Equations 1, 2, and 3 are used here too, justment of photographs (bundles) has long but with an appropriate redefinition of the been a subject of investigation. De Masson unknowns. H ere, v is the vector of corrections d'Autume [22] describes now a method in to the measured model coordinates, lh is the which, after an initial positioning of the vector of orien tation parameters of the models photographs, the observations are reduced to or of corrections to such parameters, and 02 quasi-observations valid for exactly vertical is again the vector of corrections to the ap­ photographs. The sum of squares of the cor­ proximate coordinates of terrain points. Con­ rections to these quasi-observations is mini­ sequently, the patterns that occur in the mized. This simplifies the computation of matrix of normal eq uations and the possible the condition equations and of the ground methods of solu tion of these eq uations are in coordi nates withou t, for approxi mately verti­ general the same as in the adj ustmen t of cal photography, perceptibly affecting the photographs. Howeyer, the size of the normal results. The collinearity Equations 1 and the equations can be much reduced by various normal Equations 2e are used. specifications as well as by the separate ad­ A direct solution of the normal equations justment of planimetry and of heights. with efficient use of fast-access storage is An alternati\-e to the solution of the set of ell\'isaged by arranging the bundles in groups, simultaneous equations consists in the trans­ each of which has points in common with formation of model after model in an iterative only the preceding and the following group. proced ureo Ki ng [35] and the presen t wri ter I n this way, the submatrices of no more than [39] have programmed this procedure but two groups need be in fast-access memory at compute transformation formulas for all the same time. models of one strip simultaneously. King In addition, a procedure is described to shows that one step in this procedure and the correct photograph coordinates for system­ corresponding step in the iterative solution of atic deformation before the block adj ustmen t the complete set of normal equations give the is performed. In this way, the complications same result. The transformation which fol­ which arise if deformation parameters are lows the computation of the transformation introduced as unknowns in the normal equa­ formulas corresponds to an updating of the tions are avoided. The procedure consists in coefficients of the normal equations. Although computing suitable polynomial corrections that updating (a Newton iteralicn) is some­ from the residuals of the adj ustment of a times advocated [27], it is of little or no im­ block with sufficient ground control. The cor­ portance if one starts from a reasonably good rections have been designed to eliminate the positioning. various systematic deformations which may The methods of adjustment can be divided occur in a triangulated strip. They are then into three groups: (i) adjustment of indepen­ applied to any other strip to block in which dent models or sections with seven parameters the same conditions apply. for each unit, (ii) adjustment with enforced Because different film rolls can have very coordinate connection in points at or near the different systematic distortions, it may be principal points, (iii) the same adjustment advisable to compute such corrections from with in addition correction for systematic REVIEW OF STRIP AND BLOCK ADJuSTMENT DURING 1964-1967 349 errors. A detailed descri ption of these formation and polynomial strip- and block­ methods can be found in ref. [38] and [39]. adj ustmenl. Ref. [55] to [64] treat the strip Most of the au thors use a method that be­ formation for that purpose_ longs to the first group. Especially simple is The strip formation consists in connecting the method of block adj ustmen t gi ven by each model to the preceding one by means of Roelofs [47J. Here, an internal block adjust­ a similarity transformation. In most cases, ment is performed in which scale, azimuth, an exact coordinate connection is made at the and shifts of the sections are adj usted sep­ common perspective centre. Very simple arately. Thompson [49] and Van der Weele formulas for this purpose are given by Thom­ [501 describe the use of base lines in this ad­ son [59], [60], and Schut [57]. j ustmen t. Thompson's paper is of addi tionaI Reference [62] gives the standard procedure interest because it exposes the often read fal­ for determining the model coordinates of the lacy that some errors in strip triangulation perspective cen tres from grid measurements are by their nature of the third degree in the made at two heights. Ref. [55] describes the x-coordinate. computation of these coordinates by resec­ References [38] and [39] describe a method tion, using measurements made at one height. of the second group in which the coordinate The latter computation requires pre-calibra­ connections between models are enforced by tion of the projection cameras. choosing the transformed coordinates of the Inghilleri and Galetto [55] perform only connecting points (or, rather, corrections to an approximate relative orientation in the their approximate values) as parameters. analog instru men t. The adj ustmen t of the This reduces the number of parameters from relative orientation, based upon recorded seven to just over four per model. parallaxes, is included in the strip formation. References [41], [42], [52], and [53], too, enforce the coordinate connection but they 2. TRIPLETS IN STRIP TRIANGULATION use the well-known double summation of the References [65] to [68] describe two effect of transfer errol-s. Although this reduces methods of analytical strip triangulation the number of parameters to those of one based upon the OI-ientation of triplets. Ander­ model of a strip, it produces condition equa­ son and McNair perform independent orien­ tions for each control point and for each tations of the triplets. The triplets are joined tie poi nt between stri ps in which corrections into strips by making the orientation of the to the transfers of scale, azimuth, and tilts centre photograph of a triplet and the bx of occur as corrections to quasi-observations. its first model eq ual to those obtained for Jerie, in the latter two references, has re­ this photograph and for this base component duced the complications which this causes in in the preceding triplet. Keller and Tewinkel the formation of the normal equations by the perform the triplet orientation while enforcing use of smoothing procedures and fictitious the orientation of its first photograph and points. the strip coordinates of the points whose Especially in the case of sparse ground con­ images lie across the cen tre of this photo­ trol, a provision for the elimination of system­ graph. atic errors in the strip tl-iangulation should Consequently, with both methods the man­ be included. Tn [38], [52], and [53] this is ner in which two successive triplets are con­ achieved by including second-degree terms in nected and as a result the strip deformation the transformation. If one wishes to avoid this caused by errors and by deformation of the contamination with the idea of polynomial photographs depends upon the direction of adjustment, the procedure in ref. [39] can be triangulation. This can be avoided by follow­ followed. Here, the conditions that the trans­ ing Mc Tair's recommendati::ll1 [66] to connect fer errors should be equal to zero are replaced successive independent triplets by similarity by the conditions that, at least in the case of transformations using all common points. equal model widths, the transfer errors at It has been claimed that triplet triangula­ each two successive connections should be tion results in a stronger or more rigid strip the same. than triangulation by independent relative orientation and scaling of successive models. STRIP TRIANGULATION Howe\'er, Moellman [69], using C&GS pro­ grams, reports that after second- or third­ 1. STRIP FORMATIO~ FROM Ii'lDEPENDENT degree polynomial strip adjustment there is MODELS no way to distinguish between the results of At several centres, the triangulation of the two triangulations. McNair [66] has ob­ independent models is followed by strip tained better results from his triplet triangu- 350 PHOrOGRAMMETRIC ENGINEERING lation than from a model-by-model triangu­ means of polynomials of higher than the first lation. However, for the latter he employs degree do not exist. Therefore, in practice the the modified Herget method in which the strip adj ustmen t of a strip is performed by means coordinates of points in the preceding model of various nearly conformal transformations. are enforced during the orientation of a photo­ It can be performed as a sequence of two­ graph. Besides, his conclusion is based upon dimensional transformations (see e.g. [85). the size of systematic errors and not upon the [86]) or of transformations of planimetry and residuals after polynomial strip adjustments. of heights (see e.g. [82). [88]), in each case I t is clai med as an ad"antage of tri plet with appropriate correction of the remain­ orientation that it is here possible to recog­ ing coordinate or coordinates. Alternatively, nize and eliminate errors in the x-coordinates after an initial positioning, it can be per­ of points in the triplet overlap. In the two­ formed as one three-dimensional transforma­ photo orientation recognition is possible by tion (see e.g. [36] and [84a]). A proper correc­ comparing the heights of these points in the tion for strip deformation cannot be guaran­ two models in which they occur. Errors are teed if independent polynomials are used for more evident than in the triplet orientation all three coordinates [99] or for planimetry because here the least squares adjustment and heights [69]. does not minimize them. Points with such Before a block adj ustmen t is performed, errors are here simply eliminated from use the strips must first be positioned approxi­ in the scale transfer and, if the x-error causes mately. In this positioning, a second-degree Y-parallax, also from the relative orienta­ correction for longitudinal curvature should tion. be included. This makes it possible to perform On balance, the triangulation by means of the block adjustment of planimetry and that independent relative orientation of each two of heights separately. successive photographs and scaling of the The transformation of planimetric ground resulting models has the advantage that it coordinates of control points to the axis-of­ uses simpler formulas and requires only half flight system of a strip is rather common the computation time [69J of the triplet tri­ (see e.g. [82]) but is rather awkward if a angulation. The latter, in the version used by block adjustment is performed. It can be Tewinkel and in the version recommended avoided either by using the known param­ by Mc Tair, has the advantage that a eters in the formulas for the initial position­ smoother fit between models is obtained ing directly in the correction equations and automatically. applying the polynomial transformations to axis-of-flight coordinates [36] or by applying 3. RADIAL TRIANGULATION a conformal transformation to the coordinates Numerical radial triangulation is discussed obtained after the initial positioning [85, 86]. in ref. [70] to [76]. Roelofs and Timmerman As in the cases of block adj ustmen t of describe results and the influence of errors if photographs and of models, normal equations the classical rhomboid triangulation is used. can be formed for the simultaneous adjust­ Van den Hout describes the use of triangles ment of all strips. The direct solution of the in a block adjustment by the A nblock method. equations is here rather simple [85). [87]. Al­ The lack of general availability of the exten­ ternatively, the iterative procedure can be sive literature on the rhomboid triangulation used in which strip after strip is transformed may explain recent interest in the numerical and tie points from overlapping strips are formulation of the old graphical radial tri­ used as additional control points [36). [78). angulation by means of alternate resection [85]. The iterative procedure is simpler to and intersection (ref. [74] to [76]). program than the simultaneous solution but it consumes more computer time. \Vith the POLYNOMIAL ADJUSTMENT OF STRIPS amount of control that is commonly available References [17). [36). [38J, [39). [69). and in topographic mapping, as a rule a sufficient [77] to [94] deal with the adjustment of strips, convergence of the iterative procedure is ob­ individually or in blocks, by means of poly­ tained after about ten iterations of the plani­ nomial transformations. The adj ustment metric adjustment and five iterations of the serves to correct the strips for deformation height adjustment [35). [85]. Tewinkel [9] and to obtain a reasonable fit of the trans­ and Jacobs [80J even state that after careful formed strips at the ground-control points positioning of the individual strips there is and at the tie points. often Ii ttle need for a block adj ustmen t. Reference [84 a to fJ have made it gradually Practice as well as theory have long since clear that conformal transformations by shown that it is advisable to keep the degrees REVIEW OF STRIP AND BLOCK ADJUST 1E T DURING 1964-1967 351

of the polynomials as low as possible. Restric­ means of similarity transformation of models tion to the second degree is possible by divid­ and, for the heigh t adj ustment, also the ITC­ ing long strips into sections. Such sections Jerie analog adju tment.. can be transformed by means of either inde­ I t is of particular interest that Ackermann pendent polynomials or composed poly­ [100) finds that in his investigations the sim­ nomials. ilari ty transformation of models and the An investigation of the present writer [39) second- or third-degree polynomial stri p has shown that with fairly sparse control the transformation gi ve abou t eq ui valen t results. block adjustment of models does not give a I n a practical test an adj ustmen t of a better absolute accuracy than the block ad­ block of 180 photographs wi th 60 percent justment of strips. Soehngen [87] has obtained longitudinal and lateral overlap by the U. S. better results with the height adjustment of Coast and Geodetic Survey [10) has produced models. However, he uses a larger number root-mean-square values of the residuals at and well-located control points. check points of only nine microns at photo­ graph scale. SUBBLOCKS, AND EXTERNAL The polynomial adjustment of strips with BLOCK: ADJUSTMENT normal lateral overlap gives values that Anderson [95, 96) describes the computa­ usually vary between 25 and 60 microns. [39). tion of subblocks of 3 X3 or m Xn photographs Jacobs (81), using analytical triangulation with 60 percent longitudinal and lateral over­ and the Coast and Geodetic Survey's correc­ lap and their assembly into a block by means tions for film deformation, obtains values of of similarity transformations. For the inter­ around 15 microns. nal adjustment of such an assembly, the method of Roelofs [47) would seem to be very CONCLUSION suitable. Prof. Schermerhorn's statement that the For the adjustment to ground control of methodical deveJopmen t of mathematics and an internally adjusted block, polynomial programming procedures in analytical tri­ transformations of the block coordinates angulation seems to be complete [2) can now could be used. However, with a low degree of be extended to stri p and block adj ustmen t. the transformations one can hardly expect to till, further refinements, modifications, and obtain a good fit at all ground-control points simplifications of present procedures will un­ and wi th high degrees one may obtain too undoubtedly continue to appear. large errors in uncontrolled areas. A second­ For instance, one can expect that more degree transformation may be suitable for an work will be done on the construction of initial positioning of the block and to enable economical direct and ite'"ative solutions of the final block adjustment to be performed the large systems of normal equations which separately for the three coordinates. occur in the adj ustmen t of photographs and Arthur [97) describes an interpolation of models. I n this field a more than four-year­ method for such an adjustment. Howe\'er, old claim by members of the ITC that an this method does not give a solution in the exceptionally economical direct solution is case of four ground-control points situated at possible by a sui table arrangemen t of the the corners of a square. Since this should be unknowns still awaits clarification. Brown a well-defined case, the suitability of the [llb] has recently made a rather similar method in other cases requires careful ex­ claim concerning the iterative solution. In amination. addition, the use of the method of conjugate Vlcek [98) and \Vainauskas [99) describe gradients and of related methods may war­ the use of orthogonal polynomials. The only rant further investigation. advantage of their use appears to be that it I n the field of analytical triangulation, the may be possible to identify and reject terms simultaneous triangulation of all photographs that do not contribute significantly to an of a strip in an arbitrary system should not improvement of the fit at the control points. be much more complicated than the triplet triangulation and could with advantage re­ ACCURACY OF STRIP AND place the latter. BLOCK ADJUSHIENT The adjustment of internally adjusted Ackermann and Jerie, [100) to [107), have strips and blocks to ground control should be investigated the theoretical accuracy of strip further investigated. This may provide a and block adj ustment, assu ming that sys­ very suitable procedure especially for small tematic errors are either absent or have been computers and where the utmost in accuracy eliminated. They deal with the adjustment by is not needed. 352 PHOTOGRAM 1ETRIC ENGI JEERI G

Of particular interest is the further de­ [12] R. G. Davis, Analytical adjustment of large velopment of methods which serve to elimi­ blocks. PHOTo ENG. 22: 1, Jan. 1966. [13] R. G. Davis, Advanced techniques for the nate the systematic errors which may occur rigorous adj ustment of large photogram­ in triangulated strips by means of corrections metric nets. Photogrammetria 22,5, July 1967. applied to the photograph coordinates. [14] R. A. Matos, Analytical simultaneous block A major problem that still remains to be triangulation and adjustment. PH0TO. ENG. solved is how to satisfy the need of especially 30, 5, Sept. 1964. [151 A. A. Elassal, Analytical aerial triangulation the many small photogrammetric organiza­ through simultaneous relative orientation of tions for suitable computer programs. Many multiple cameras. Civil Eng. Studies, of these organizations have neither the neces­ Photogr. Series No.2, Univ. of Illinois, Oct. sary experience nor the access to a sufficiently 1965. large computer to use large sophisticated [16] A. A. Elassal, Simultaneous multiple sta­ tion analytical triangulation program. Photo­ programs and they would be best served by grammetria 21,3, June 1966. small programs for specific purposes. As yet, [171 M. L. McKenzie and R. C. Eller, Computa­ only the U. S. Coast and Geodetic Survey tional methods in the USGS. PHOTO. ENG. and the National Research Council of Canada 31: 5, Sept. 1965. L. have made it their policy to publish their [181 M. McKenzie, Operational analytical aerotriangulation by the Direct Geodetic programs. Restraint Method. Paper presented at the I nternational Symposium on Spatial Aero­ BIBLIOGRAPHY triangulation, Urbana, March 1966. To be REVIEWS published in Photogra·mmetria. [191 A. A. Elassal, The use of modular computer [1] F. E. Ackermann, Development of strip and programs to edit data for very large simul­ block adjustment during 1960-1964. Int. taneous photogrammetric solutions. Paper Archives oj Photogrammetry, Vol. 15, part 5, presented at the semi-annual meeting of the 1965. Am. Soc. of Photogr., St. Louis, Oct. 1967. [2] W. Schermerhorn, Aerial triangulation at [20] Anonymous, Adjustment of large aerial the Lisbon Congress. Photogrammetria 20: triangulation blocks on an electronic com­ 5, Oct. 1965. puter of limited capacity. Fomth United Na­ [3] A. Verdin, Le Congres de Lisbonne. Compte tions Regional Cartographic ConJerence Jor rendu des activites de la Commission Ill. Asia and the Far East, Vol. 2, 1966. Ball de la Soc. Beige de Photo No. 80, June [21J G. de Masson d'Autume, Le traitement 1965. numerique des blocs d'aerotriangulation. [4] G. de Masson d'Autume, Le symposium in­ Esquisse d'une solution noniterative. Bul­ ternational sur I'aerotriangulation spatiaIe. letin de la Soc. Fran(aise de Photogr. No. 18, Bull. de lao Soc. Fran(aise de Photo. No 22, April 1965. April 1966. [22] G. de Masson d'Autume, The perspective bundle of rays as the basic element in aerial ADJUSTMENT OF PHOTOGRAPHS triangulation. Paper presented at the Sym­ [5] H. H. Schmid, Analytical aerotriangulation. po ium of Spatial Aerotriangulation, rbana, Fourth United Nations Regional Cartographic March 1966. To be published in Photogram­ ConJerenceJor Asia and the Far East, Vol. 2, metria. 1966. [23] J. Albertz, Blocktriangulation mit Einzel­ [6J H. H. Schmid and E. Schmid, A generalized bildern. Deutsche Geod. K0111.111. Reihe C, least squares solution for hybrid measuring Heft 92, 1966. systems. The Canadian Surveyor 19: 1, [24] Ch. Gruber, U ntersuchung der strengen March 1965. Verfahren zur AuRosung von Normalglei­ [7] H. H. Schmid, Ein allgemeiner Ausglei­ chungssystemen unter besonderer Beruck­ chungsalgorithmus zur Auswertung von hy­ sichtigung programmgesteuerter Rechenau­ briden Meszanordnungen. Bildmessunr, und Ragen. Deutsche Geod. KO"I"I11I1.Reihe C, LuJtbildwesen (B. und L.) 33: 3/4, Sept./ Heft 86, 1966. Dec. 1965. [25J S. W. Henriksen, Mathematical Photogram­ [8J M. Keller, Documented computer pro­ metry. PHOTo ENG. 31: 4, July 1965. grams. PHOTo ENG. 31: 5, Sept. 1965. [26J V. Knitky, On the solution of analytical [9J G. C. Tewinkel, Block analytic aerotriangula­ aerotriangulation by means of an iterative tion. PHOTo ENG. 32: 6, 1 OV. 1966. procedure. Photogram.111etria 22: 5, July 1967. ['10] M. Keller, Block adjustment operation at [27J V. KratkJl, Economical indirect solution of C&GS. PHOTo ENG. 33: II, Nov. 1967. the block adjustment of aerial triangulation. [lla]D. C. Brown, R. G. Davis, and F. C. John­ Geodeticky a Kartograficky Obzor 12: 54, son, Research in mathematical targeting: Aug. 1966. the practicaI and rigorous adj ustment of [28] V. Kratky, Contribution to the problem of large photogrammetric nets. U.S. A ir Force, convergence of indirect computing solution Griffith A ir Force Base report 133, Nov. 1964. in the analytical aerotriangulation. Presented [llb]D. C. Brown, The simultaneous adjustment paper, Lausanne Congress. of very large photogrammetric blocks. Paper [29] K. Kubik, Survey of methods in analytical presented at the Symposium on Computa­ block triangulation. fTC publication A39, tional Photogrammetry of the Am. Soc. of Spring 1967. Photogr., Potomac Region, Dec. 1967. [30] K. Kubik, A procedure for analytical block REVIEW OF STRIP AND BLOCK ADJUSTMENT DURING 1964-1967 353

triangulation. ITC publication A 40, Summer [49J E. H. Thompson, The computation of single 1967. strips in plan. The Photogranwletric Record [31 J iVI. Kusch, Beitrag zur analytischen Block­ 5: 29,ApriI1967. triangulation mit mittelsch nellen Reche­ [50J A. J. Van del' Weele, Relative accuracy and nautomaten. Presented paper, Lausanne independent geodetic control in strip triang­ Congress. ulation. Photogramme/ria 21,2, April 1966. [32J D. K. Faddeev and V. N. Faddeeva, Com­ [51J F. Ackermann, A method of analytical block putational methods of linear algebra. W. H. adjustment for heights. Photogranwletria 19: Freeman and Co., 1963. 8, 1962-64. [33] R. S. Varga, Matrix iterative analysis. Pren­ [52J H. G. Jerie, A simplified method for block tice-Hail, Inc., 1962. adjustment of heights. Ph%grarmnetria 19: [34] A. Ralston and H. . Wilf, Mathematical 8, 1962-64. Inethods for digital computers. Wiley, 1960. [53J H. G. Jerie, Eine Methode fiir die Hahen­ blockausgleichung von Aerotriangulationen LINEAR ADJUSTMENT OF MODELS AND OF SECTIONS unter Einbeziehung von Hilfsdaten. B. und [35J c. W. B. King, A method of block adjust­ L. 35: 1/2. March/June 1967. ment. The Photogra111111etric Record 5: 29, [54] A. D. N. Smith, j\/I, j. Miles, and P. Verrall, April 1967. Analytical aerial triangulation hlock adjust­ [36J C. \\T. B. King, Programming considerations ment; the direct height solution incorporating for adjustment of aerial triangulation. Paper tie-strips. The Photogrammetric Record 5: presented at the International Symposium on 29, April 1967. Spatial Aerotriangulation, Urbana, March 1966. To be published in Photogrammetria. STRIP TRIANGULATION [37] G. Kupfer, Eine umfassende Blockausglei­ [55J G. Inghilleri and R. Galetto, Further de­ chung unter Verwendung verkniipfter Hel­ velopment of the method of aerotriangula­ mert-Transformationen. B. lind L. 34: 4, tion by independent models. Photogram­ Dec. 1966. metria 22: I, jan. 1967. [38J G. H. Schut, Practical methods of analytical [56J H. Schkalziger, Programm Aerotriangula­ block adjustment for strips, sections, and tion. Die Auswertung von Triangulations­ models. The Can. Surveyor 18: 5, Dec. 1964. streifen und Blacken auf del' elektronischen [39J G. H. Schut, Polynomial transformation of Rechenanlage Zuse Z23. Deutsche Geod. strips versus linear transformation of models: K07/1.111. Reihe B, No. 121, 1965. a theory and experiments. Paper presented [57J G. H. Schut, Formation of strips from in­ at the rnternational Symposium on Spatial dependent models. Paper presented at the Aerotriangulation, Urbana, March 1966. To semi-annual meeting of the Am. Soc. of be published in Photogrammetr·ia. PhotogI'. St. Louis, Oct. 1967. Also publica­ [40J H. F. Soehngen, C. C. Tung, and L. W. tion NRC-9695 of the National Research Leonard, Investigation of block adjustments Council of Canada. of the ITC block using sections and the [58] K. Szangolies, Aerotriangulation mit unab­ iterative and direct solutions of the normal hangigen Bildpaarendas Verfahren del' Zu­ equation system. Civil Eng. Studies, Photogr. kunft? Ver1l1essungsteclmik 13: 3, March Series 8, Univ. of Illinois. 1965. [41J J. Somogyi, An interpolation method for [59J E. H. Thompson, Aerial triangulation by strip adjustment. The Can. Surveyor 20: I, independent models. Photogrammetria 19: March 1966. 7, 1962-64. [42] G. Alpar and J. Somogyi, Common adjust­ [601 E. H. Thompson, Review of methods of in­ ment of two parallel strips. Acta Geod., dependent model aerial triangulation. The Geophys. et Mont., Acad. Sci. Hung. I, 1966. Photogra1l1111etric Record 5: 26, Oct. 1965. [43J G. Galvenius, Principles of block adjustment [61J V. A. Williams and H. H. Brazier, Aerotri­ of aerial triangulation. Photogra1n111etria 19: angulation by the observation of independ­ 8, 1962-64. ent models. Photogra1n1'lletria 19: 7, 1962-64. [44J C. M. A. Van den Hout, The Anblock method [62J V. A. Williams and H. H. Brazier, The of planimetric block adjustment: mathemat­ method of the adjustment of independent ical foundation and organization of its models. H uddersfleld test strip. The Photo­ practical application. Photogrammetria 21: grammetric Record 5: 26, Oct. 1965. 5, Oct. 1966. [63J V. A. Williams and H. H. Brazier, Aerotri­ [45J K. Kraus, Untersuc~.ungen zur ebenen ver­ angulation by independent models: a com­ ketteten linearen Ahnlichkeitstransforma­ parison wi th other methods. P hotogra1l17/1.etria tion. Zeitschrift fur Vermessungswesen (Zf V) 21: 3, June 1966. 91: 4, April 1966. [64] H. S. Williams, Weight coefficient matrices [46] E. M. Mikhail, Horizontal aerotriangulation for unit model connections. Presented paper, by independent models using horizon camera Lausanne Congress. photography and B-8. Paper presented at [65J A. j. McNair, Triplets: a basic unit for the I nternational Symposium on Spatial analytical aerotriangulation. Photogram­ Aerotriangulation, Urbana, March 1966. To metria 19: 7,1962-64. be published in Photogramrnetria. [66J J. M. Anderson, R. L. Ealum, and A. J. [47] R. Roelofs, Une methode de compensation McNair, Analytic aerotriangulation using planimetrique de blocs par des equations de triplets in strips. Interim Technical Report, condition. Bull. de la Soc. Bdge de Photogram­ Surveying Dep., chool of Civ. Eng., Cor­ Inetrie No. 82, Dec. 1965. nell University, 1965. [48J W. Sander, Festpunkt\'erdichtung durch [67] M. Keller and G. C. Tewinkel, Three-photo photogrammetrische Blocktriangulation. ZfV aerotriangulation. ESSA-C&GS Technical 89: 8, Aug. 1964. Bulletin 29, Feb. 1966. 354 PHOTOGRAMMETRIC ENGINEERING

[68] M. Keller, A practical three-photo orienta­ [85] G. H. Schut, Development of programs for tion solution to the analytic aerotriangula­ strip and block adjustment at the National tion problem. Photogrammetria 22: 4, May Research Council of Canada. PHOTo ENG. 1967. 30: 2, March 1964. [69J D. E. Moellman, A comparative study of [86] G. H. Schut, Block adjustment by polyno­ two-photo versus three-photo relative orien­ mial transformations. PHOTo ENG. 33: 9, tation. Civil Eng. Studies, Photogram:metry Sept. 1967. Also publication 1 RC-9265 of Series No.7, niv. of lllinois. the National Research Council of Canada. [70] C. 1\1. A. Van den Hout, Analytical radial [87] H. F. Soehngen, Strip and block adjustments triangulation and 'Anblock'. Photogram­ of the lTC block of synthetic aerial triangu­ me/ria 19: 8, 1962-64. lation strips. Presented paper, Lausanne Con­ [71J R. Roelofs, Radial triangulation in moun­ gress. tainous country? Ph%gramrnetria 19: 8 [88] G. C. Tewinkel, Slope corrections in aero­ 1962-64. ' triangulation adjustments. PHOTo ENG. 31: [72] R. Roelofs, The influence of observational 1, Jan. 1965. errors and irregular film distortion on the [89] H. S. Williams and G. E. Belling, Hybrid and accuracy of numerical radial triangulation conformal polynomials. PROT. E G. 33: 6, Cartography 5: 3/4, 1964. June 1967. [73] J. Timmerman, The influence of instrument [90] M. E. H. Young, Block triangulation on the adjusting errors in nLllnerical radial triangula­ NRC-monocomparator. The Can. Surveyor tion. Photogranlrl1.e/ria 19: 8, 1962-64. 20: 4, Sept. 1966. [74] R. D. Turpin, Numerical radial triangula­ [91] L. P. Adams, Block adjustment by the di­ tion. PHOTo ENG. 32: 6, Nov. 1966. rection method. The Photograrnmetric Record [75] P. R. Wolf, Analytical radial triangulation. 5: 29, April 1967. PHOTo ENG. 33: 1, Jan. 1967. [92] R. E. Altenhofen, Analytical adjustment of [76] E. M. Mikhail, A study in numerical radial horizontal aerotriangulation. PHOTo ENG. triangulation. Paper presented at the semi­ 32: 6, Nov. 1966. annual meeting of the Am. Soc. of Photogr. [93] M. L. McKenzie, Adjustment of elevations St. Louis, Oct. 1967. ' derived from instrumentally bridged aerial photographs. PHOTo ENG. 30: 2, March 1964. POLYNOMIAL OF ADJUSTMENT OF STlllPS [94] G. H. Schut, A method of block adjustment [77] G. E. Bellings, On the application of poly­ for heights with results obtained in the in­ nomials in numerical block adjustment. ternational test. Photograllunetria 20: 1, The South African Journal of Photogram­ Feb, 1965. metry 2: 3, May 1965. [781 G. Gracie, Analytical block triangulation S BLOCKS, AND EXTERNAL BLOCK AD] USTMENT with sequential independent models. Photo­ [95] J. M. Anderson and A. J. iVI cNair, Analytic grammetria 22: 5, July 1967. aerotriangulation: triplets and subblocks. [79] C. T. Horsfall, Aerotriangulation strip ad­ Photogrammetria 21: 6, Dec. 1966. justment using FORTRAN and the IBM­ [961 J. iVI. Anderson, Coordinate transformations 1620 computer. C&GS publica/ion, March for basic subgroup assembly in dimplified 1965. systems of analytic aerotriangulation. Pre­ [80] 1. S. Jacobs, Practical analytical aerial tri­ sented paper, Lausanne Congress. angulation. The South African J oumal of [971 D. W. G. Arthur, Interpolation of a func­ Photogrammetry 2: 2, May 1964. tion of many variables. PHOTo ENG. 31: [81] 1. S. Jacobs, Some results of analytical aerial 2, iVI arch 1965. triangulation. The South African Journal of [98] J. Vlcek, Adj ustment of a strip using ortho­ Ph%grammetry 2: 3, May 1965. gonal polynomials. PHOTo ENG. 31: 2, [82] ]\1[. Keller and G. C. Tewinkel, Aerotriangula­ March 1965. tion strip adjustment. C&GS Technical Bul­ [99J \11./. Wainauskas, Uber die Ausgleichung der letin 23, Aug, 1964. raumlichen Bild triangulation und ihre [83] P. T. Koetsier and P. L. Meadows, An in­ Genauigkeit bei der Anwendung der Poly­ troduction to analytical aerial triangulation. nome von zwei Veranderlichen. Ve'rmes­ The South African Journal of Photogram­ sungstechnik, 13: 11/12, Nov/Dec. 1965. metry 2: 3, May 1965. [84a] E. M. Mikhail, Simultaneous 3-D transfor­ THEORETICAL ACCURACV OF STRI P AND BLOCK mation of higher degrees. PHOTo ENG. 30: ADJ STMENT 4, July 1964. [100] F. E. Ackermann, Fehlertheoretische nter­ [84b] D. W. G. Arthur, Three-dimensional trans­ suchungen ilber die Genauigkeit photo­ formations of higher degree. PHOTo ENG. 31: grammetrischer Striefen triangulationen. 1, Jan. 1965. Deutsche Geod. Komm. Reihe C, 87, 1965. [84c] H. H. Brazier, imultaneous three-dimen­ 1101] F. Ackermann, Some results of an investi­ sional transformation. (Discussion paper). gation into the theoretical precision of PHOTo ENG. 31: 5, Sept. 1965. planimetric block adjustment. Photogram­ [84d] J. Vlcek, Simultaneous three-dimensional m.etria 19: 8, 1962-64. transformation. (Discussion paper). PHOTo [102] F. Ackermann, On the theoretical accuracy ENG. 32: 2, :'I'larch 1966. of planimetric block triangulation. Photo­ [84e] P. L. Baetsle, Conformal transformations in metria 21: 5, Oct. 1966. three dimensions. PHOTo ENG. 32: 5, ept. [103] F. Ackermann, Photogrammetrische Lage­ 1966. genauigkeit streifenartiger iVI odellverbande. [84f] G. H. Schut, Conformal transformations and B. und L. 34: 3/4, Sept./Dec. 1966. polynomials. PI-IOT. ENG. 32: 5, Sept. 1966. [104] F. Ackermann, Theoretische Beispiele zur REVIEW OF STRIP AND BLOCK ADJUSTMENT DURING 1964-1967 355

Lagegenauigkeit ausgeglichener Blocke. B. [107J H. G. ] erie, Theoretical heigh t accuracy of und L. 35: 3, Sept. 1967. strip and block triangulation with and with­ [105] H. G. ]erie, Height precision after block out use of auxiliary data. Paper presented adjustment. fTC publication A24, Spring at the International Symposium on Spatial 1964. Aerotriangulation, Urbana, March 1966. [106] H. G. Jerie, Height precision after block To be published in Photogrammetria. adj ustment. Photograrnrnetria 19: 8, 1962-64.

THE PHOTOGRAMMETRIC SOCIETY, LONDON Membership of the Society entitles you to The Photogrammetric Record which is published twice yearly and is an internationally respected journal of great value to the practicing photo· grammetrist. The Photogrammetric Society now offers a simplified form of membership to those who are already members of the American Society.

APPLICATION FORM To. The Hon. Secretary, The Photogrammetric Society, PLEASE USE BLOCK LETTERS 47 Tothill Street, London, S.W.I I apply for membership of the Photogrammetric Society as, o Member-Annual Subscription-£2 2s. Od. (Due on application o Junior (under 25) Member-Annual Subscription-£! 5s. Od. and thereafter on o Corporate Member-Annual Subscription-£15 15s. Od. July I of each year.) (The first subscription of members elected after the 1st of January in any year is reduced by half.) I confirm my wish to further the objects and interests of the Society and to abide by the Consti· tution and By·Laws. I enclose my subscription. Surname, First Names Age next birthday (if under 25) Profession or Occupation Educational Status Present Employment Address

Signature of Date ...... •.•...... •..•. Applicant ..•...... Applications for Corporat" MelllOC!",!ujJ, which is open to Universities, lVlanufacturers and Operat· ing Companies, should be made by separate letter giving brief information of the Organisation's interest in photogrammetry.