ANIMATED TOOLS FOR ILLUSTRATION, DEMONSTRATION AND STUDY OF GEOMETRIC RELATIONS IN PHOTOGRAMMETRY AND REMOTE SENSING USING MS-EXCEL

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1 ANIATED TOOLS FOR ILLSTRATION, DEONSTRATION AND STDY OF GEOETRIC RELATIONS IN PHOTOGRAETRY AND REOTE SENSING SING S-EXCEL K.A. Grabmaer, Internatonal Insttute for Geo-Informaton Scence and Earth Observaton (ITC), P.O.Box 6, 7500AA The Netherlands, KEY WORDS: Vsualzaton, Anmaton, Geometry, Orentaton, Teachng, Photogrammetry, Remote Sensng. ABSTRACT: For teachng about the geometry of Remote Sensng mages, Orentaton and Georeferencng problems spreadsheet graphcs were used for vsualzaton of stuatons n three-dmensonal space. They vsualze the stuaton durng acquston of Remote Sensng mages and demonstrate the effect of changes n the orentaton parameters onto the mage geometry, and show the nfluence of errors n the orentaton parameters on the geometry of the reconstructed object by stereo-resttuton. The relevant parameters (vewng, orentaton and errors n the orentaton) can be changed by macros n a contnuous fashon and the graphc s contnuously updated. In ths way a move-lke effect s obtaned. Also other spreadsheets were made, e.g. one showng the rotatng earth and an orbtng satellte as n a move (wth sphercal globe and crcular orbt, but all other parameters can be changed by macros). RESE : Pour ensegner la géométre des mages télédétecton ans que des problèmes d orentaton, on utlse les graphques des feulles des calculs pour vsualser la stuaton en 3 dmensons. On peut observer la stuaton des prses de vues des mages télédétecton, et démontrer les effets des changements des paramètres d orentaton sur la géométre des mages. Les graphques montrent l nfluence des erreurs des paramètres d orentaton sur la géométre des objets reconstruts par méthode de resttuton stéréoscopque. Les paramètres les plus mportants (présentaton, orentaton des mages et leurs erreurs) peuvent être changés de manère contnue par des «acros». En même temps le graphque est modfé. De cette façon, on obtent un effet «cnéma». On a auss développé d autres feulles des calculs, par exemple un satellte qu orbte un globe tournant. (Le globe est sphérque et l orbte est crculare, mas des «acro» permettent de changer tous les autres paramètres.) KRZFASSNG: Für den nterrcht über de Geometre von Fernerkundungsbldern und Orenterungsprobleme wurden Tabellenkalkulatons- Graphken engesetzt zur Darstellung von 3-dmensonalen Stuatonen. Se zegen de Aufnahmestuaton und den Enfluss der Parameter auf de Bldgeometre, sowe de Auswrkung von Orenterungsfehlern auf de Geometre des wederhergestellten Objektes. De relevanten Parameter (der Darstellung, der Orenterung der Blder und deren Fehler) können mttels akros kontnuerlch verändert werden, wobe de Darstellung folgt und de Veränderung we n enem Flm abläuft. Es wurden auch andere derartge Graphken gemacht, z.b. en Satellt, der de sch drehend Erde umkrest (mt kugelförmger Erde und kresförmger Satelltenbahn, aber alle anderen Parameter mt akros veränderbar). INTRODCTION For most of my teachng I normally prepare the teachng ads by myself. Wth the change from overhead projector to dgtal sldes I ddn t scan my old graphs, but prepared new ones (n S-PowerPont). Frst I supported my drawngs wth smple calculatons, whch I dd n S Excel. (I often use Excel, when others would use a calculator.) From ths pont t was a small step to vsualze the calculated lnes as a graph n Excel. When I also succeeded to mport the graphcs nto S PowerPont, soon the functonalty grew, as wth every realzed mprovement the desre arouse to add yet another functonalty. Orgnally I used a smple parallel projecton, but for a teacher of mage geometry the formulas for a perspectve vew should not gve any problem, and a perspectve vew gves a much more natural appearance. In addton to a smple terran object, whch s shown as wre frame from a lst of coordnates, the spreadsheet shows two cameras above the object wth the mage of the object n the mage plan. Especally for teachng about geometry and orentaton of mages good vsualzatons are very essental, but t s often dffcult to choose the rght composton and vewng angles for a scene one wants to show. The spreadsheet was thus organzed to allow easy changes of the vewng parameters, but also the orentaton parameters of the two cameras were changeable. Soon ths was consdered not convenent enough and the parameters were made changeable by macros. Also the mages n the mage planes of the cameras were shown n separate graphcs sheets. Next changes to the orentaton parameters were ntroduces and the mages projected back to the object space and the reconstructed object shown. Typcal objects shown n these graphcs are:

2 A box connectng ponts n a rectangular pattern of 3 x 3 x ; A group of 9 hgh rse, flat roof buldngs wth a road along them; A landscape wth a road, a house and a lake; A regular grd DT pattern. A verson wth the same functonalty for push broom scanner mages s not yet fnalzed but a varety of spreadsheets wth smlar prncples was produced to show: A satellte orbtng a rotatng globe, The prncple of a laser scanner, Geometrc transformatons of square grd patterns, A geocentrc and a local coordnate system. The spreadsheets can be used drectly to show the changng graphcs n Excel, but the graphcs can also be mported nto other software, to prepare dgtal sldes or llustratons n a text fle. S Excel, S Vsual Basc and S PowerPont are products of crosoft Corporaton. Perspectve Vew REALIZATION Wth the usual (photogrammetrc) camera parameters the functonalty was dsappontng, as rotatons of the camera made the object move out of the feld of vew. I therefore changed the calculatons such, that the rotaton axes dd not pass through the projecton center (O), but through the vewed pont (), whch one can best choose n the mddle of the object area. The projecton center (O) s calculated such, that pont s n the camera axs, at a dstance (D) from O, whch s varable by the user. In addton to the three angles of rotaton, the coordnates of and the dstance D, the user can specfy: The mage coordnates of the mage of pont (xm, ym), beng the coordnates of the prncpal pont, The mage scale (s), beng the rato between the prncpal dstance and the object dstance (D). Ths gave the functonalty, that the object remaned n vew wth angular changes, and the object dstance controlled the amount of perspectvty n the mage, wthout changng ts sze. r ( ) + r (V V ) + r3 (W W ) x = D s + xm () r ( ) + r (V V ) + r (W W ) + D r3 ( ) + r3 (V V ) + r33 (W W ) y = D s + ym r ( ) + r (V V ) + r (W W ) + D x, y = mage coordnates xm, ym = coordnates of prncpal pont D = object dstance s = mage scale r j = elements of rotaton matrx, V, W = coordnates of vew pont (), V, W = object coordnates of any pont In the case of zero rotatons (as the vew from O n fgure.) the vewng s n the V-drecton, whle x s parallel to and y s parallel to W. Dfferent from the usual stuaton n terrestral photogrammetry, Kappa rotates the vew around an axs parallel to the W-axs, thus allows to dance around the object, whle Omega nclnes the drecton of vewng (up or down). ove lke moton To fnd good vewng parameters by typng new values was far from optmal, so I developed a few macros to allow easy change of the parameters. Not famlar wth Vsual Basc, I smply recorded a macro to add (and another to subtract) the value of a partcular cell from the selected one by Paste Specal. The letter I assgned to the macro was smply the one n the left upper corner of the keyboard, the letter Q. Later I changed the macros for more robustness, but the prncple remaned the same: the value of a partcular cell, the ncrement was added to selected one by pressng Ctrl-Q and t was subtracted by pressng Ctrl-Shft-Q. When the keycombnaton s held, then the ncrement s added (or subtracted) repeatedly. The graphcs, whch are vsble n other wndows are contnuously updated, whch creates an almost contnuous moton. The speed of the moton depends on the ncrement. Also for changng ths ncrement a macro was made, usng the next letter n the row, the W. Ctrl-W wll multply t by 0 and Ctrl-Shft-W wll dvde t by 0, but the macros wll not set values outsde the range from 0.00 to 0. In ths stuaton the sheet contanng the parameters must be open n wndow number of the spreadsheet and the parameter to change must be selected there. Another approach usng much more macros, n order to allow selecton of parameters, ncrementng them and changng the speed wthout havng the parameters vsble on the screen s presently under constructon. 3 3 O O W V P Camera(s) above the object In order to show the relef- and tlt-dsplacement, whch has to be corrected n orthophoto producton I wanted to show a DTgrd together wth ts mage n a camera above t, so I had to calculate the magng of the DT grd nto the mage plane of the camera. The orentaton parameters of the camera were added to the parameters sheet, thus they were varable usng the same macros. The typcal photogrammetrc parameters of nteror and exteror orentaton were used, but object coordnates of the mage ponts had to be calculated rather than mage coordnates. Fgure. Two vews to the same vew pont () The formulas used are:

3 the object pont and the projecton center wth the mage plane. There s a crtcal plane, whch s parallel to the mage plane and passes through the projecton center. Object ponts n ths plane are theoretcally maged nfntely far away, practcally a Dvson by zero error happens and they wll be maged to poston (0, 0). Ponts just a lttle n front of ths plane wll be maged very far away to one sde, whle ponts just behnd t wll be far away to the other sde. A perhaps short jonng lne of two such ponts (on ether sde of the crtcal plane) can thus result n a very long mage lne! Fgure. DT-grd wth two cameras above. To vsualze the camera a square mage frame was shown, and the corners of the square were connected wth the projecton center of the camera. Addng a second camera was smple, so ths was done as well. Also the vertcal projecton of the DTgrd onto a horzontal plane s shown. A typcal graph obtaned wth ths sheet s shown n fgure. The resoluton s better than shown here, but the shape of the deformed grd n the mages dd not show up well enough. To show ths n a better way, separate graphcs were created for each mage to show the mage tself wth good resoluton. Now however, the coordnates of the mage ponts had also to be calculated n the mage coordnate system. Fgure 4. landscape wth mage partly outsde. Ths only happens, f the crtcal plane passes through the object area. The effect was consdered rather curous than nasty, as such extreme stuatons were not meant to be used. For more normal stuatons the fact that part of the drawng s outsde the mage frame can sometmes be consdered llustratve for the danger of errors n the coverage by errors n the camera nclnaton. Resttuton To demonstrate the effect of orentaton errors on the geometry of the resttuted object another set of cameras (resttuton cameras) was ntroduced. The dfferences between the parameters of the resttuton cameras and those of the orgnal cameras are kept as parameters n the parameter sheet, beng the changes to the orgnal parameters. Fgure 3. mage graphc of a DT-grd. No attempt was made to lmt the drawng of the mage to the nsde of the mage frame or n any other way. In a new calculaton sheet the changed parameters were calculated, then for each mage pont ts locaton n the object coordnate system was calculated usng those changed parameters of the camera. Ths gves rather strange stuatons n extreme cases, as mathematcally the mage plane s unlmted and mage ponts are calculated as the ntersecton of a straght lne defned by

4 (,W)-plane. For horzontal camera axes or for a base n V- drecton t s not sutable. Strct formulas were tred, but consdered too cumbersome. Two dfferent V-coordnates were calculated, one for each one of the straght lnes. The mean of them s used - together wth the - and W-coordnate from the ntersecton calculaton - to show the pont n the graph. The dfference of the two V-values s used as Y-parallax. The sheet calculatng the actual graph uses the calculated coordnates to show the resttuted locatons and allows also to show the y-parallax wth a user selectable exaggeraton factor. Ths can also be used to show the prncple of analogue orentatons. One can exaggerate the y-parallaxes so much, that even small ones are vsble, and then one can vary the changes to the parameters to elmnate or reduce those parallaxes. One can also add a sheet showng only the values of the y-parallaxes of selected ponts (e.g. the 6 standard ponts) and show ths n an addtonal wndow. Fgure 5. Orgnal and resttuted object and the cameras. u v = R w x y c xh yh () u, v, w = vector from projecton center to mage pont n the object coordnate system R = rotaton matrx from the changed angles x, y = mage coordnates of mage ponts xh, yh = mage coordnates of the prncpal pont = ndex of the mage ( or ) For the calculaton of the pont of ntersecton the smplfed approach, whch s mostly used for sngle models n photogrammetry was used: ntersecton of the two straght lnes only n D-space (,W). = t u + ( t = o, ( t = o, V = (V + V o, W = t w + W o, o, V = t v + V o, o, ) / = t u +, y ) w (W o, ) w (W o, o, = t w + W u w u w u w u w o, W W o, o,, V = t v + V p = V V ) u o, ) u (3), V, W = object coordnates of the resttuted ponts u, v, w = vector from projecton center to mage pont n the object coordnate system o,, V o,, W o, = coordnates of projecton center () t = scale factor V = V calculated from mage = ndex of the mage ( or ) Push broom scanner(s) Fgure 6. Analogue Common Ph. The functonalty of the graphcs for push broom scanner mages s lmted. One verson shows cameras wth parallel projecton n one drecton and central projecton n the other. Ths lmts the sensor to a straght flght path and does not allow varaton of the angles wthn an mage. The overvew graph as well as vew(s) of the mage(s) are avalable, but resttuton wth changed parameters s not yet possble. Ths works fne as long as the jonng lnes of the two projecton centers and the two camera axes do not devate much from the

5 to show t at the congress. The transformaton from object to mage s not straght forward n that case, because the orentaton parameters to be used depend on the mage lne, n whch the pont s maged. Ths requres teratve calculaton or hgher order ratonal polynoms. Iteratve calculaton wth three teratons wll be used. Ths should gve satsfactory results. To allow all possble constellatons, the resttuton wll make use of concse formulas for the ntersecton of the rays such, that the jonng lne of the resultng ponts on each lght ray s perpendcular to both lght rays. Fgure 7. Two push broom vews (forward and backward). Another verson shows a seres of lne magng events. Each event s shown by four lnes only: A straght and horzontal lne n the object space, Two magng rays for the extreme ponts The sensor lne. In ths case the flght path and the rotaton angles are modelled by a second order polynomal. There s no object shown and no mage. o V = Vo W Wo t u = t v t 3 w,,, u + t v w u v w ( v w w v) o ( w u u w ) Vo ( u v v u) Wo,,, V o, W o, o, (4), V, W = object coordnates of the resttuted pont, calculated from mage u, v, w = vector n mage from projecton center to mage pont n the object coordnate system o,, V o,, W o, = coordnates of projecton center () t = scale factors = ndex of the mage ( or ) OTHER GRAPHICS Orbtng Satellte Fgure 8. Push broom magng events and parameters There were no macros made to handle the parameters n ths verson, as the speed of changes n zero order, frst order and second order terms s ncompatble. A better choce of the unts could probably overcome ths problem, but there are much more changes foreseen: It should become an magng verson and also resttuton wth changed parameters should become possble. The concepts and layouts are ready snce some tme, but the tme to mplement t was not avalable so far. I hope to be able Fgure 9. orbtng satellte and globe (showng The Netherlands) Ths graphc shows a sphercal globe and a crcular satellte orbt. The followng parameters can be changed contnuously by macros: all vewng parameters, the radus of the globe and ts speed of rotaton, the heght and the nclnaton of the orbt, the poston of the satellte at tme zero, the (angular) speed of the satellte, the swath (angle), the forward and the sdeward look angle, the tme (n mnutes). When the tme changes, then the globe rotates and the satellte moves, both accordng to the speeds specfed.

6 Laser Scannng Ths graphc shows the deflecton of a laser beam by a rotatng prsm. For output drven resamplng the grd pattern shows the pxels, whch should be created and the crcles show the postons of the pxel centers of the exstng mage transformed to the new geometry. When changng the parameters usng the macros one can ncely demonstrate the effect of each one, lke shft, change of scale, shear etcetera. Local and global coordnates For the llustraton of the relatons between a local and a global Cartesan coordnate system another spreadsheet was made. It shows a (sphercal) globe, the axes of the geocentrc coordnate system, the local merdan and the axes of the local coordnate system together wth a horzontal square around the local orgn. Fgure 0. laser scannng prncple The user can rotate the prsm, change the number of facets (between 3 and 0), and the speed of moton (30 to 3 steps per facet). After a change of the number of facets the reflecton may be wrong such, that a reflecton s shown outsde the lmts of a facet. After the rotaton starts, ths s corrected. Besdes the vewng parameters one can change the sze of the globe, the longtude and the lattude of the local orgn and the sze of the horzontal square by macros. The sze of the coordnate axes shown depend on those settngs. The global axes are 0% longer than the radus of the globe, whle the local axes are equal to the sdes of the square. Transformatons of square grd patterns Ths graphc was made to llustrate the prncples of resamplng. Blnear transformaton s used here: x = a0 + a c + a r + a3 c r (5) y = a4 + a5 c + a6 r + a7 c r c, r = column and row coordnates x, y = output coordnates a0 to a7 = transformaton parameters Fgure. local and global coodnates Conclusons and Outlook Spreadsheet graphcs offer an easy tool to vsualze objects and relatons n 3D-space to anyone famlar wth the spreadsheet software. Ths can be used to make nce teachng ads, but they can also be used as a teachng ad by themselves. The use of macros to easly ncrement or decrement some parameters can make t easy to select sutable parameters to use the graphc else, but they are even more valuable when usng the spreadsheet tself as teachng ad, as t allows to show the effect of changes n a contnuous mode. Fgure. mage transformaton The parameters (a0 to a7) can be vared usng macros. For affne transformaton one can smply set the parameters a3 and a7 to zero, but for conformal transformaton a modfcaton of the sheet would be useful to couple a5 to a and a6 to a. PowerPont sldes I have made wth these tools are apprecated by students as well as colleagues, I have found these graphcs n numerous presentatons of colleagues. Besdes the completon of the push broom functonalty a verson s envsaged to gve two vews from dfferent postons, one n red and one n cyan, for stereoscopc vewng usng the anaglyph prncple.