Application of Photogrammetric Techniques to the Measurementof Historic Photographs

David A. Strausz19 March 2001

Oregon State UniversityGEO 522 - Reconstructing Historic Landscapes

Prof. Ronald Doel

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

REPHOTOGRAPHY AS A RECORD OF LANDSCAPE CHANGE . . . . . . . . . . . . . . . . . . . . . 2

TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

ANALYTICAL ORIENTATION SOLUTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

PROPOSED DEMONSTRATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

APPENDIX A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

APPENDIX B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

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INTRODUCTION

The science and art of photography has provided a record of past landscapes throughoutthe 150 years since its development. This record documents change due to the natural successionof vegetation, cultural development, environmental evolution, and other inexorable processes,both uniformitarian and catastrophic. The legacy of these images of past landscapes can be a richresource for the study of the processes and effects of change in the environment, at least forprocesses that operate in the time scale of hundreds of years. An effective tool to utilize thisresource is the technique of rephotography or repeat photography, in which historic photographsare recreated so that changes in the landscape that have occurred in the intervening time can bereadily detected and studied (Figs. 1, 2).

This technique involves taking a new photograph as closely as possible from the samepoint in space, with the same camera axis orientation, as the original photograph. This can bechallenging when, as is often the case, the original camera position, and such parameters ascamera focal length, or even image format size, if the original negative is not available, are allunknown. Fortunately, much of this information can be derived directly from the geometricproperties of the image itself. In addition, it is not necessary for comparative purposes toreproduce the original focal length and image format size, since the same geometric propertieswill be inherent in any photograph taken from the same point in space and with the sameorientation.

It is valuable also in the repeated photograph to recreate other environmental parametersof the original photograph, such as season of the year (important for vegetation comparisons)andtime of day (affecting shadow length and orientation) in order that the new image duplicates asmany original parameters as possible so that true change in the environment is more evident. This is not difficult to do once the spatial orientation of the photograph is established, althoughrecreating the precise time of day might involve considerable patience. Fortunately, theprovenance of much archival photography does record the date of exposure; further clues can bederived from the geometry of shadows in the image.

There are many advantages in using historic photographs as evidence. The popularity ofphotography as a recording medium almost since its inception means there is an abundance ofimages taken for personal, commercial, and scientific purposes that have survived and areavailable for study. It is a remarkable landscape indeed that has not been recorded. A valuablecharacteristic of a photographic image is the immense amount of data that is recorded in theemulsion. At the moment of exposure a permanent record is made of the reflectance values ofevery object in the field of view. This record exists in great detail as microscopic chemicalchanges in the emulsion that can be resolved to at least 10 microns in the original image scale. The emulsions of even the earliest photographs have this level of resolution, which combinedwith the large formats common in earlier times, recorded incredibly fine detail. This detail canallow researchers to glean information from these images that was unrecognized by the originalphotographer.

REPHOTOGRAPHY AS A RECORD OF LANDSCAPE CHANGE

Repeat photography over time has been recognized for over 100 years as a technique torecord and study landscape change. Sebastian Finsterwalder pioneered the application ofphotogrammetric techniques to the measurement of Alpine glaciers as early as 1889 (Hattersley-Smith). The sometimes rapidly changing morphology of glaciers has inspired many researchers

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to advocate the use of repeated photographs since then. G.K. Gilbert, in the 1899 Harrisonexpedition to Alaska, made photographs of glaciers with the specific purpose of recording scenesfor future comparison. He also repeated in 1903 a glacier photograph in the Sierras taken twentyyears earlier (Rogers, Malde, and Turner 37).

There have since been many studies of landscape change utilizing repeat photography. Rogers, Malde, and Turner provide a bibliography of over a hundred pages of referencescollected during their own repeat photography research. Hart and Laycock have published anupdated (1996) annotated list of references of studies that have utilized repeat photography forthe study of vegetation change, cross referenced by area of study and type of vegetation.

Several examples of these studies are Hastings and Turner, Rogers, and Sallach,documenting landscape change in different habitats of south east Arizona, the Central GreatBasin of Utah, and in New Mexico, respectively. These studies all utilized historic photographsgleaned from many sources, such as museums and archives, recreated the photos, and used themto deduce environmental changes occurring in the intervening years. In these studies, the onlyuse of quantitative measurement of change was by Sallach, who covered the photos with a grid tomeasure relative coverage of different vegetation types.

Several studies have been made to assist in the management of landscapes byinvestigating the effects of potentially manageable influences, such as fire, disease, grazing, andother human interactions. Two examples are Skovlin and Thomas, and United StatesDepartment of Agriculture, which use repeat photography to document changes in the BlueMountains of Oregon and the Boise National Forest, respectively. The Blue Mountain studyrecognized the value of recording the positions of the repeat photographs with the idea ofcontinuing them in the future. Many of these repeat photographs included calibrated stakes inthe foreground, which could be used in scale measurements, although most of the comparisonswere made through subjective description.

Some studies have used the rich legacy of early photographic coverage of manyscenically spectacular areas of the western United States. Happily for posterity, much of the westhas inspired considerable photographic coverage since it was first explored, during a time whenphotographic field techniques, while cumbersome, were well developed. Stevens and Shoemakerand Webb are two examples that cover the spectacular landscape of the Colorado River. Stevensand Shoemaker recreated, as a centennial project, many of the photographs taken on the secondPowell expedition. Webb utilized an expedition retracing the steps and recreating photographsfrom the Stanton expedition of 1889-90 as a basis for a study of existing and past ecosystemsalong the Colorado River. The photographs by Meagher and Houston of Yellowstone NationalPark are also in this category. The interest in the park at the turn of the twentieth century led tovery extensive photographic coverage which provides many opportunities for repeatphotography. The authors have gone one step further in this study by continuing their ownearlier repeat photography made in the 1970's, into the 1990's. This allowed comparison ofchange rates across two different time intervals and recorded the change wrought by the largeintervening episodic event of the Yellowstone fires of 1988.

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Fig. 1. Little Holecampground, fromStephens andShoemaker.

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Fig. 2. Virginia City, Nevada, 1868 and 1979, from Klett, Manchester, and Verburg

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TECHNIQUES

The primary challenge in recreating a photograph is to determine the exterior orientationof the original image, that is, the point in space occupied by the camera (strictly speaking thefocal node of the lens) and the orientation in space of the image (emulsion) plane. Note that thefocal length is not strictly necessary to this orientation since any image taken from the sameperspective point in space, at the same orientation, will be identical to any other image with thesame exterior orientation, as long as they are scaled identically and cropped to the same field ofview. The usefulness of the repeat photography directly depends upon the accuracy of therecreation of the original, particularly if the images are to be compared in any quantitativemanner. Especially if an area of interest exists in the near foreground of the photos, a slightchange in orientation can significantly change the proportions in the respective fields of view(Harrison 471). Some studies have only been concerned with approximating the cameraposition, which limits their comparisons to subjective analysis (Goin).

Several methods have been used to reproduce the original image orientation. By far thesimplest, and most often used, judging from the studies reviewed, is to merely move the cameraposition (or initially, the eyeball of the photographer) around until the image in the viewfinder oron the ground glass at the image plane matches that of the original image. A further step insophistication is to compare actual measurements of parallax between identifiable objects in bothimages. This method has been applied as an iterative process, where after a photograph is taken,measurements between objects in the image are compared to the original, any differencesindicate needed changes in the orientation. A third method, common to terrestrialphotogrammetric applications, is to plot lines from objects that lie on lines of intersection withthe camera station on a plan view, to locate the camera station by the intersection of such lines, amethod that is only available when the image contains a very rich field of identifiable objects(Borchers 16). Another method, which is proposed in this paper, is to subject measurements ofthe original image to photogrammetric analysis, in order to directly solve for the exteriororientation.

The trial and error, “eyeball,” method has certainly been the most popular, probablybecause of its simplicity, but can still involve hours of moving about in search of the illusivepoint of perspective (U.S. Dept. of Agr. 48). This method can yield very accurate results,particularly for images that contain a good mix of identifiable objects throughout the depth offield of the view, and particularly, when objects are lined up in one or two ranges that defineunique lines terminating at the point of perspective, a condition which is not often encountered. In the above conditions, once near the original perspective point, deviations of only a fewcentimeters may be readily perceived. This process can be systematized by first lining up withobjects along the original image center line, then moving in and out along the center line until theparallax of objects to either side match (Malde 198).

The method proposed by Harrison and clearly explained and applied by Klett,Manchester, and Verburg is a considerable improvement in terms of accuracy and repeatability. Harrison points out that there are six unknown elements of the orientation of the original image: three coordinates in three dimensional space to define the camera position, and three angulardisplacements about three orthogonal axes of the image plane. These unknown elements can bereconstructed theoretically by comparing six or more independent measurements. Harrison usesthe measurements of image distances between identifiable points. To satisfy the requirement forat least six independent measurements, at least five points are needed (since independent lengthsbetween n points equals 2n-3). To remove the element of scale between photos, the distances

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can be normalized by dividing all measurements in each photo by a common measurement in therespective photo. If both photos have the same exterior orientation, all normalized measurementswill be the same.

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Fig. 3. Pyramid Lake, Nevada, from Klett, Manchester, and Verburg.

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Harrison further recommends constructing axes in the photos perpendicular to several ofthe photo distances (Fig. 3). In this way the ratios between the line segments created by theperpendicular axes provide a quantifiable measurement of the relative parallax of differentobjects, which can be readily compared between photos. If the axes are constructedapproximately perpendicular and parallel to the image horizon, the differences in the ratiosbetween photos can clearly indicate what relative movement is necessary.

ANALYTICAL ORIENTATION SOLUTION

Analytical photogrammetry is the application of mathematical principals tomeasurements in photographic image planes in order to model and reconstruct both the originalspacial orientation of the image and the relative and absolute locations of imaged object points. While the principals of spacial geometry basic to analytical photogrammetry were seriouslyinvestigated as early as the 15th century by such polymaths as Leonardo da Vinci and AlbrechtDurer, their application to the measurement of photographs largely began with the work ofSebastian Finsterwalder at the beginning of the twentieth century. The development of analogmeasuring devices that recreated the geometry of aerial photographs for the purposes oftopographic mapping dominated the practical application of photogrammetry for the first half ofthe twentieth century. The solution of the same problems analytically requires an amount ofcomputation that precluded practical application before the advent of computer processing. Ottovon Gruber, who derived the projective equations fundamental to modern analyticalphotogrammetry, is famously quoted as saying that their application was “a waste of time and isof minor importance.” (Thompson 461-64).

In photogrammetry the orientation of an image is defined by several parameters. Theinterior orientation elements are the focal length of the camera, which is the perpendiculardistance from the film plane to the focal node of the lens, and the location on the image wherethat perpendicular line intersects the image plane, called the Principal Point (PP in Fig. 4). Knowledge of these parameters is important if the image is to be used for photogrammetricmeasurements. In a non-metric camera, defined as any camera not specifically designed forphotogrammetry (Atkinson 64), the interior parameters are not necessarily known with anyprecision (Faig, “Photogrammetric;” Karara, ”Simple;”) . This is certainly the case for nearly allhistoric photographs. The exterior orientation is defined by six parameters, the three spatialcoordinates of the camera station and three angular measurements that define the spatialorientation of the image plane. In the United States it is conventional to use the terms w, f, andk to indicate the rotation, around respectively and in order, the x, y, and z axes of an orthogonalcoordinate system parallel to the space coordinate system in use in order to transform a planeparallel to the x, y axes of the space coordinate system into a plane parallel to the image plane. These rotations are considered to be positive in a right handed sense relative to their respectiveaxes (Fig A1). These six parameters of exterior orientation need to be known to recreate anhistoric photograph.

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Fig. 4. The Collinearity Condition

The basic problem in analytical photogrammetry is to mathematically relate the positionsin space of imaged objects to the positions of their image points in the plane of the image, andvice versa. The most basic geometric concept utilized in developing solutions to this problem isthe collinearity condition which is that the camera station, the image point, and the imaged objectall lie on a straight line (Fig. 4). This concept is used to develop the collinearity equations,usually in the form where the image coordinates of an object are expressed as functions of theinterior orientation parameters, the exterior orientation parameters, and the space coordinates ofthe object. These equations are derived in Appendix A and shown below:

(1)

(2)

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In these equations xa, ya are an object A’s image coordinates; XA, YA, ZA are the object’scoordinates in object space; XC, YC, ZC are the object space coordinates of the camera position;m11-m33 are the coefficients of the orthogonal transformation between the image plane orientationand object space orientation, and are functions of the rotation angles w, f, and k; xo, yo, and f arethe interior orientation parameters of the image, the image coordinates of the photo principalpoint and the camera focal length, respectively.

In the conventional situation where the elements of interior orientation are known thecollinearity equations can be used to solve for the six unknown elements of exterior orientationby developing two equations for each of at least three control points of known location. Sincethese equations are non linear they are first linearized by the use of a Taylor’s series expansion ofthe function with respect to each of the unknown elements. This results in a linear function ofthe general form, for equation (1):

(3)

where FO is the evaluation of the original function F with estimated values of the unknowns, the

coefficients of the form are partial derivatives of F relative to the unknown elements

“a” and evaluated using original estimates of the unknowns, and the Da’s are corrections to theoriginal estimated values for unknowns “a”and which are solved for as unknowns for thecombination of linear equations. The solution moves through several iterations, applying theresulting corrections from each iteration to the previous estimates and resolving, running untilthe corrections, hopefully, approach zero (Burnside 331-34; Gullberg 767). This approach alsousually incorporates an unknown variance factor in the linear functions, so that in the case wheremore equations can be developed than there are unknowns, a least squares solution can beperformed, by first normalizing the system of equations, to solve for a best fit with the givenmeasured control point coordinates.

Several reduction methods have been developed to deal specifically with the non-metriccamera situation, where the interior orientation is unknown (Karara, Handbook 35-38; Atkinson64-72; Faig, “Calibration;” Bopp and Krauss). The Direct Linear Transformation of Abdel-Azizand Karara (Photogrammetric; “Direct”) provides a direct transformation between measuredimage coordinates and object space coordinates, bypassing entirely the interior and exteriororientation elements of the image. While these methods have advantages of computationalefficiency when used for photogrammetric measurements using non-metric cameras in general,they do not directly address recovering the exterior orientation elements that are needed forrephotography.

In adapting the collinearity solution to the situation of recovering the original orientationof historical photographs, it is necessary to add to the linearized equations terms for theadditional unknowns of the original interior orientation, xo, yo, and f. This results in thefollowing linear versions of equations (1) and (2) (derived in Appendix B):

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(4)

(5)

These equations use the shorthand notation of dab to indicate the partial derivatives of the originalequations for the respective unknowns. In addition, a variance term v has been added to allowthe least squares solution when there are more equations than unknowns. (In the situation of aunique solution, the v terms become zero.)

A system of linear equations based on the development of (4) and (5) for our 9 unknownsbased on n control points can be described in matrix notation as follows:

2nL1 + 2nV

1 = 2nD9

9X1 (6)

where L is the matrix of the values of differences between calculated and measured imagecoordinates (xa - FO and ya - GO), V is the matrix of variances, D is the matrix of the partialderivative coefficients, and X is the matrix of corrections to the estimated unknown values. Itcan be demonstrated that the 9 normal equations necessary for a least squares solution can beexpressed as (Wolf 510-12; Thompson 57-59):

(DTD)X = DTL (7)

where DT is the transpose of matrix D. Solving for X yields:

X = (DTD)-1(DTL) (8)

with (DTD)-1 indicating the inverse of matrix (DTD).Initial estimates are made of all of the 9 unknown elements, which are used to calculate

the elements of matrices D and L in equation (8), which is then solved for the X matrix, whichgives the corrections to apply to the initial estimated values. This process is iterated, using thecorrected estimate values from each successive iteration, continuing until the correction valuesapproach zero. The process of estimating the initial values is important, since values differingtoo greatly from actual could lead to a divergent solution. Dewitt has proposed a method ofcomputing initial values addressing the often difficult visualization of orientation angles interrestrial photography. It is anticipated that estimates of sufficient accuracy may be made fromempirical inspection of the image and initial location of the camera station.

The described solution assumes that measurements on the image accurately represent truepositions on the focal plane of the image at the time of exposure. This is almost surely notstrictly the case. Factors that could affect the accuracy of these points include departure fromtrue flatness of the emulsion at exposure, subsequent distortion of the emulsion, and lensdistortions. Further disruption to the collinearity situation results from atmospheric distortionand the departure caused by the curvature of the earth in true space versus that modeled by thespace coordinate system in use. While not addressed in this project, it should be understood thatall of these factors could be modeled and added as corrections to the xa, ya image coordinates in

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the collinearity equations. These can be treated as additional unknown elements which can besolved for, given a sufficient number of control points.

Karara and Abdel-Aziz have demonstrated that for a wide range of non-metric cameras(including a Kodak Instamatic!) unless very high precision is needed, most of the corrections toimage coordinate measurements may be ignored. Therefore, for this demonstration, themeasurements from the photographs will be used without correction.

PROPOSED DEMONSTRATION

Several photographs of the Columbia River Gorge area have been located to serve indemonstrating the feasibility of recreating their original orientation parameters throughphotogrammetric measurements. The photographs chosen are from the Benjamin Giffordcollection of the Oregon State University Archives. The photographs, in the form of diapositivecontact glass prints of the original 6.5 X 8.5 inch negatives, have been scanned using a flat bedscanner at a resolution of 300 pixels per inch. This will allow measurements to be made usingimage processing software, to within the accuracy of one pixel. The accuracy of thesemeasurements of course ultimately depend upon the accuracy of the flat bed scanner, which isunknown, and needs to be investigated

Photographs need to be chosen of views that can be identified in today’s landscape so thatthe approximate original camera location can be found. When this is done objects imaged in thephotograph that can be identified in the current landscape will be noted that have a highlikelihood of not having moved in the intervening years (Fig. 5). The space coordinates of thesepoints will then be surveyed, relative to an appropriate space coordinate system. For theaccuracy anticipated in this demonstration an appropriate system would be Universal TransverseMercator coordinates for the X, Y space coordinates, and a corresponding vertical datum for the Zdimension. The surveying will be accomplished using a frequency mode GPS receiver, with postprocessing, for control points that can be directly occupied. Control points difficult to accesswill be tied to GPS derived control points by triangulation using a theodolite. Distant controlpoints, such as mountain peaks, may be located in the space coordinate system with sufficientaccuracy by plotting from map coordinates, since overall accuracy of the solution depends not onabsolute accuracy of the control point coordinates, but on their angular displacement relative tothe perspective point of the camera station. Such distant points may need to have theircoordinates corrected to account for discrepancies between the horizontal projection used and theactual surface of the earth.

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Fig. 5. Shepherd’s Dell Bridge, Columbia River Highway, showing proposed controlpoints, from Benjamin Gifford Collection, Oregon State University.

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Once sufficient control point coordinates are collected (a minimum of 5 for ourdemonstration situation), they will be used to solve for the unknown interior and exteriororientation parameters using equation (8). Calculating the variance values for the measuredimage coordinates, by applying the resulting values for the unknown parameters to equation (6),will indicate how well the solution fits the measured coordinates. A large remaining variance forany of the image coordinates would indicate a possible problem with either the image or spacecoordinates for that respective control point, which could be measured again or eliminated toimprove the overall solution.

The resulting orientation parameters will then be used to reoccupy the original cameraposition with a camera in the original angular orientation. A photograph taken under theseconditions should have identical geometric perspective properties to the original image. This canbe analyzed from measurements taken on the repeat photograph, which can be compared tonormalized comparable measurements on the original image, as in the Harrison method.

APPLICATIONS

The most direct application of a photogrammetric solution of historic photograph locationand orientation would be in repeating the original photograph. A modern camera set up withidentical orientation parameters will duplicate the original photographic perspective and field ofview on the first attempt, at least theoretically, without having to take several photographs,progressively homing in on the orientation at the original point in space. In exchange, theresearcher would have to spend time in the field identifying and surveying control points. Whether this would practically take more or less time than traditional methods would depend onthe individual situation, but certainly, the progress being made in the precision, accuracy, andspeed of GPS surveying instruments will contribute to the practicality of applying this directmethod. In addition, the calculations involved, which are dauntingly tedious, can easily beprogrammed in a personal computer; the matrix operations involved can even be handled bymany handheld calculators, a consideration for practical field application.

Once the interior orientation of the original image is known with some precision thehistoric image can be used to take direct measurements of objects in the field of view. This isaccomplished through the obverse solution to the collinearity situation, using the now knownelements of exterior and interior orientation to solve for the unknown space coordinates of pointswhose image coordinates can be measured. The linear form of the two collinearity equations forthis situation are:

(9)

(10)

in which the all of the terms involving the known parameters which are held fixed drop out sincetheir respective D terms become zero. The new unknown terms are the X, Y, and Z spacecoordinates of the object in question. In these equations, b has been used as shorthand for thepartial derivatives of the original equations with respect to the new unknowns in order to avoidconfusion with the d coefficients of equations (4) and (5). Since the two image coordinates of apoint contribute two equations, a solution for all three unknown space coordinates is not possiblefrom one photo image. The two equations solve for the collinearity line in space that satisfies theequations; the object position of the point could lie anywhere on the line. Without some other

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information, such as some scale dimensions in different points of the depth of field of the image,where the object lies on the collinearity line cannot be resolved.

The situation is completely soluble, however, if another overlapping image of the sameobject is available. In this case two more equations can be developed from the collinearitysituation of the second image. In this way any point in the field of view of both photographs canbe resolved in all three dimensions of object space. There are many situations where more thanone contemporary image exists with the required overlapping fields of view. In such cases, athree dimensional model of the overlapping area can be constructed and measurements takenanywhere in that space.

Images that would provide such coverage could be very valuable to researchers in manyfields that are concerned with changes in the landscape over time. Several studies surveyeddemonstrated possible applications. McDonald’s survey of mass changes to Collier Glacier,Oregon, over nearly one hundred years utilized both aerial and terrestrial historic photographs. This study might have benefitted from accurate measurements of the terrestrial images used. Inthese images there may have been sufficient fixed control points to provide fairly accurate spacecoordinates for changing glacial dimensions even though the coverage was limited to a singlephotograph. Graf’s study of the changing morphology and dynamics of a stream channel in amining environment also used historic photographs, but mainly as a subjective guide to mappingvegetation patterns. It is possible that the photographs could also have been applied to accuratestream channel shape measurements had the orientation geometry of the photos been calculated.

The utility of applying non metric camera images to accurate stream channelmeasurement was demonstrated by Welch and Jordan, using a 35mm camera and enlarged anddigitized images, which yielded results comparable in accuracy to measurements developed onimages from metric cameras measured on a conventional photogrammetric comparator. Theimprovements in digital technology since this study (1983) have made these photogrammetrictechniques even more available to researchers without access to technical photogrammetricequipment and software.

Many research applications can benefit from extracting precise landscape measurementsfrom the wealth of detail preserved in historic photographs. Inventories of vegetation, withmeasurements of spatial extent, tree bole sizes, and species distribution, can increase confidencelevels of statistical analyses. Geomorphologists can quantify mass movements, creep,landslides, erosion, and other landscape changes in the temporal dimension, at least as far back intime as photography records. Archeologists can measure for reconstruction structures andartifacts that have either been displaced or no longer exist. All these applications can benefitfrom very precise positioning in the cases where two or more overlapping photographs areavailable. Even with one photograph, however, precise orientation in space of the collinearityline upon which an object lies can still allow fairly precise positioning of that object if otherexisting geometric elements, such as the ground surface, which may be modeled from theexisting landscape, can be used as another constraint upon the object’s position.

CONCLUSION

The principles of photogrammetry, applied to the study of historic photographs, can yieldmany benefits to researchers interested in studying past landscapes. These techniques are alsobecoming more practical and accessible to non specialists with the continuing development ofprecision digital scanners, which can render historic photographs into a medium in which theycan easily be measured with high precision. The voluminous calculations necessary in the

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application of analytic photogrammetry, so long an impediment even in the profession ofphotogrammetry, can now practically be applied by anyone with a personal computer. Thepractice of repeat photography can be facilitated by directly calculating the original image’sspatial orientation. Precise knowledge of this original orientation can enhance the value ofhistorical images much beyond their recognized importance as subjective snap shots of long pastlandscapes.

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APPENDIX ADerivation of the Collinearity Equations

The ultimate goal is to express the coordinates of the image point, xa and ya in Fig. 4, inan arbitrary coordinate measurement system in the image plane, in terms of the space systemcoordinates, XA, YA, and ZA, of the object. In this derivation the measurements are related to thepositive image plane, analogous to the original negative plane on the other side of the focal node,but geometrically more straightforward. Note that it is important to relate all measurements inthe image plane to the Principal Point, PP, which is where a line perpendicular to the image planeand passing through the focal node of the lens, C, intersects the image plane. The arbitrarycoordinate system used is related to the coordinate system parallel to the arbitrary system butcentered on the Principal Point by the translation offsets xo, yo, which are the arbitrary systemcoordinates of the Principal Point.

The first step is to transform the image point a coordinates from the coordinate systembased on the image plane to an intermediate system with the same origin, the camera position inspace, but with axes parallel to the object space system (Fig. A1). This is accomplished byrotating the camera system axes around the intermediate axes, x’,y’,z’, in order, by the angles w,f, k, until the original axes are parallel to the spatial system coordinate axes. This isaccomplished mathematically by the transformation equations below, the derivation of which isdetailed in Wolf (535-39), Thompson (46-48), or Soffit (89-92):

(A-1)

(A-2)

(A-3)

Note that z is equal to -f, the focal length of the original image. The m coefficients are shorthandfor the transformation matrix coefficients, listed below:

m11 = cosfcoskm12 = sinwsinfcosk + coswsink m13 = -coswsinfcosk + sinwsinkm21 = -cosfsinkm22 = -sinwsinfsink + coswcoskm23 = coswsinfsink + sinwcoskm31 = sinfm32 = -sinwcosfm33 = coswcosf

From Fig. A1 we see that the line connecting the camera station C, the image point a, andthe object point A forms the common side of similar triangles so that:

(A-4)

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Expressing the coordinates in terms of z’ results in:

(A-5)

(A-6)

(A-7)

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Fig. A1. Orthogonal image coordinate system

Substituting into the rotational transformation equations, (A-1)-(A-3):

(A-8)

(A-9)

(A-10)

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Dividing (A-8) and (A-9) by (A-10), and cancelling the common term z’a/(ZA - ZC) on the rightsides:

(A-11)

(A-12)

more commonly expressed as below, recognizing that z = -f :

(A-13)

(A-14)

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APPENDIX BLinearizing the Collinearity Equations

Since the collinearity equations are non-linear functions with respect to most of thevariables, it is necessary to linearize them so that they can be simultaneously solved for theseveral unknowns. In addition, in their linear form they can be developed into a set of normalequations enabling a least squares solution.

The Tayler power series expansion can be used to expand a continuous monotonicfunction of the general form, for a function of a variable x, f(x):

(B-1)

where c is the point around which the series is centered and f(k) denotes the kth derivative of f(x),evaluated at c. It would be rather tedious to evaluate this series out to infinity. Fortunately, ascan be seen, for values of x very close to c, all but the first couple of terms become insignificant. Therefore a very good approximation can be made using only the first two terms:

(B-2)

For a function of more than one variable, the same approximation applies, with the firstderivative term being replaced by terms representing the partial derivative of f(x) with respect toeach variable:

(B-3)

where ao, etc. represent the convergent, or initial, estimated values, the partial derivatives beingevaluated at those values (Burnside 331; Gullberg 767).

The two collinearity equations, (A-13) and (A-14), can be similarly expanded, using F todenote the expression for xa and G for ya, with terms for the 9 unknowns of the interior andexterior orientation:

(B-4)

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(B-5)

Here each of the partial derivatives is evaluated using the convergent or initial values of thevariables and each of the terms of the form (w - wo) can be considered the corrections to beapplied to the initial values, and are the unknowns that are solved for. FO and GO represent thefull respective functions evaluated using all of the initial values, both fixed and estimated. Notethat only unknown variables of the functions appear in the partial differential terms, since anyknown values are held fixed, and so their correction factors become zero. These linear equationscan more simply be expressed, including a variance factor to allow for a least squares solution,as:

(B-6)

(B-7)

where we use the d’s as shorthand for the derivative expressions in (B-4) and (B-5) and the D’s torepresent the differences to be solved for.

It will be convenient in evaluating the partial derivatives to use the following expressionsin representing the collinearity equations:

Let

So that the collinearity equations become:

(B-7)

(B-8)

Strausz 25

In order to solve the equations we need to evaluate the d terms, which are the partialderivatives of the collinearity equations with respect to the unknown variables in question. As anexample, we will evaluate the d11 term, which relates to the unknown w.

by the derivative of quotient functions, r and q being the only functions that contain w, but

similarly we can find that

substituting and simplifying

The remaining partial derivatives are similarly developed. The following is a list of all thenecessary coefficients for the equations solving for the nine unknowns, using the shorthand melements where possible:

Strausz 26

Strausz 27

Strausz 28

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Strausz 29

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