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Map Projection Transformation with Digitally Controlled Differential

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Map Projection Transformation with Digitally Controlled Differential G. E. BORMANN Wild Heerl~ruggLtd. CH-943.5 Heerbrugg, S~~;itzerlatid E. VOZIKIS Techt~iculUtli~ersit!/ Vienna A-1040 Viet~nu,Austria Map Projection Transformation with Digitally -Controlled Differential Rectifiers An unconventional application of modern orthophoto equipment. INT-RODUCTION The navigation charts of central Europe ancl HE CONTENT of existing maps is very often the elsewhere are, however, on the Lambert confor- T basis for new cartographic products. In mal (angle-true)conical projections of the Interna- many cases the extracted information will be used tional (Hayford) or Bessel Ellipsoids with either for various different purposes. It happens quite one or two length-true parallels at scales of 1 : 1 frequently, however, that a geometric projection million or 1 : 500 000. The meridians are straight other than the original one is desired or needed. lines converging at the (North) Pole and parallels Cartographers are then hced with a problem are imaged as concentric circles about the (North) which could in the past only be solved in either a Pole. In this particular case the 1 : 500 000 scale crude and not very accurate procedure or in a very Aeronautical Chart ICAO 2253-B of Switzerland costly way. Meshwise re-projection with a rectifier (one length-true parallel, Bessel Ellipsoid) had andlor re-drawing was necessary. consequently to be transformed into a (special) Today digitally controlled instruments and sys- rectangular Cassini projection of 1 : 1 million tems, which were basically developed for the pro- scale. Assr~~c'r:The geometrically correct tr(lr~sforr?lcltiot~of the entire mu))content from one curtogruphic projection to another is tlr~oltl problon. It curl be soloetl economicully with digitally-controllet1 dfirerlticll rectifiers st~cll(1s the Wild AVIOPLAN OR1, using udditioilul softr~tlrernotlttles. Esmnz))les illt~strtltethe potential of this method and sltorc; ulso the gretrt udoc~tltclgesof the color cupu- bility of the OR1 system. duction of orthophotographs, offer a new eco- The work was successfully carried out on the nomic solution of the problem. The Wild AVIO- AVIOPLAN OR1 in Heerbrugg using additional PLAN OR1, a differential rectifier, belongs to this and modified software originally developed at the category of equipment. Institute of Photogrammetry of the Technical Uni- versity in Vienna, Austria. The converging merid- ians became parallel and the latitude circles were rectified to straight lines with high accuracy. The A practical case in the field of aviation naviga- entire map content was transfor~nedsimultaneous- tion, approximately two years ago, led to a closer ly into the new projection in its original colors. investigation of the new possibilities. In connec- Although the differences of the two cartographic tion with some of the data supplied by an inertial projections are obvious and considerable at the navigation system, a special navigation chart was map scale, even over such a small area as Switzer- needed very quickly. The specifications asked for land, another more spectacular example was se- a rectangular grid formed by the geographic me- lected for presentation in a publication where the ridians and parallels (with additional constraints). scale of reproduction is limited by the page size. PHOTOGRAMMETRICENGINEERING AND REMOTESENSING, 0099-1 112/83/4!309-1317$02.25/0 Vol. 49, No. 9, September 1983, pp. 1317-1323. O 1983 American Society of Photogrammetry The choice fell on the physical map of Europe CENTER OF PROJECTION taken from the Swiss Atlas for High Schools, a creation of the well known Swiss cartographer Prof. Ed. Imhof. The principle of orthophoto production is well known and the operation of the AVIOPLAN OR1 has been described in detail in several publica- tions (Kraus, 1976; Stewardson; 1976). We will therefore restrict the discussion of these items to some background remarks which seem to be nec- essary for the appreciation of the map transforma- tion method. In general, orthophotos cannot be obtained di- rectly with a camera. A subsequent process is nec- essary to remove the perspective distortions from, for instance, an aerial photograph. The perspec- tive effects are caused by central projection when the photographed terrain surface is not flat and/or the camera axis was not perpendicular to the datum plane. //// For the elimination of these effects it is neces- ORTHOPHOTO sary to resolve the content of the original photo- t graph into small area or line elements. They have s-l to be changed in scale and rotated as well as shift- FIG.1. Basic relations between central and or- ed into the correct geometric position before re- thogonal projections. composition to a new photograph, the orthophoto, can take place. Figure 1 illustrates the principle of taking an The OR1 applies the principle of differential (aerial) photograph. A regular, rectangular grid in rectification of an infinite number of line elements the (horizontal) datum plane (x,y) is assumed to of finite (selectable) length which are arranged in undergo vertical parallel projection onto an undu- adjacent profiles. The orthophotograph or rec- lating terrain surface (or a model of that surface). tified image is composed on a rotating film drum There it forms a spatial network of irregularly which is shifted in the direction of its axis at the shaped quadrangles. By central projection this end of each profile scan. The amount of shift corre- grid net is consequently deformed by the shape of sponds to the length of the line element, i.e., the the terrain and by the spatial orientation of the length of the selected slit mask through which the camera axis. photographic image has to pass. The production of an orthophoto corresponds to The input data to the OR1 process-computer are the inverse process. The irregular grid (u,a) has to image coordinates (u,~)supplied from magnetic be transformed into the regular rectangular grid tape. With reference to Figure 1, this means that (x,y) or, multiplied with a scale factor, into (X,Y) as the projection scan width S can be chosen be- is shown in Figure 2. This transformation is, how- tween 3 and 15 mm. The Y-interval (AY) in the ever, only possible when the shape of the terrain, rectified image is always 1 mm; however, it is lin- the orientation of the camera axis, the spatial posi- tion of the projection center, and the principal dis- tance of the camera are known in some digital fox-in. Normally, this is not the case u priori. The necessary information has to be obtained either from measurements in stereoplotters, or by com- putation, for instance, from an existing digital ter- rain model. The Wild AVIOPLAN OR1 performs the re- quired transformation. It is an instrument with op- tical projection. The optical devices such as a double zoom system for continuous change of I- U in; scale, a dove-prism for image rotation, and the ORIGINAL IMAGE RECTIFIED IMAGE (u,v) cross-slide of the photo-carrier for position- FIG.2. Grid transfer from aerial photograph to ortho- ing are servo-driven by a process computer. photo. MAP PROJECTION TRANSFORMATION The principle is very simple. The "new" (trans- formed) projection shall be given in cartesian co- ordinates (x,y) as functions of longitude h and latitude cp: i.e., x = fl (hcp), Y = f, (hcp). (1) The same shall be the case for the "old" (to be transformed) projection with the caltesian coordi- nates (u,v): i.e., u = h, (h,cp), o = h, (h,cp). (2) Next a cartesian grid (x,y) is defined in the "new" projection (Figure 4). With the inverse for- mulas of Equation 1, corresponding geographic coordinates (h,cp) are obtained: i.e., A = g, (x,Y), cp = g, (x,Y). (3) FIG.3. Wild AVIOPLAN OR1 with projection unit, process computer, magnetic tape station, and printer. By substitution of Equation 3 into Equation 2, it follows that early interpolated to 0.01 mm by the process com- puter. In mathematical terms the OR1 performs a The result of this numerical transformation bilinear transformation for each grid mesh. (Equation 4) is illustrated in Figure 5. As was mentioned above these image coordi- The regular rectangular grid of the "new" (trans- nates have to be established first either by mea- formed) projection appears deformed and is super- surement or by computation in an external imposed on the "old" (to be transformed) map. computer. For the latter case a FORTRAN computer Consequently, a "rectification" of the grid of program SORA was developed at the Technical Figure 5 according to the orthophoto method will University at ViennaIAustria some years ago. lead to Figure 4 and will at the same time trans- (SORA stands for Software for Offline Rectification, form the entire map content into the new projec- Avioplan). (Otepka and Loitsch, 1976). tion. In analogy to the orthophotographic process, the GENERALMETHOD OF MAPTRANSFORMATION WITH "old" projection corresponds to the aerial photo- THE OR1 graph and the "new" one to the orthophoto. From the explanation above, the following pre- There are some additional points to be ob- sentation should very easily be understood. What served, however. A reproduction of the "old" map can be rectified by means of differential transfor- must be made on film (either negative or positive) mation can obviously also be distorted. The not larger than 9 by 9 inches. Even if this is done "distortion" may be a desired change of an original in a good reproduction camera, small projective if the appropriate mathematical nlodel can be for- displacements cannot be excluded. It is therefore mulated and is then imposed. The transformation advisable to measure the rectangular coordinates of a map with its whole content from one projec- (ur,u') of 10 to 15 well distributed points in the tion to another type of map projection is such a case.
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