731

FIG. 1. Planimeter, coupled to the stereo instrument.

and then measuring with the planimeter,are the foIIowing.

1. By a suitable adjustment of thelength of the planimeter arm, the totalvolume is made directly readable on theplanimeter. This is the difference betweenreadings before and after the contouring.Drawing paper and numerical calculationsare unnecessary; time ·is thereby gained.

2. By reducing the contour intervalbelow the practical limit for contouring onpaper, the approximation error is lessened,

and a smoothing effect on unsystematiccontour errors is at the same time obtained.

3. The error in working the planimeterby hand is eliminated, that is deviationsfrom the contour do not occur.

4. Time is also gained by making thecontouring and planimeter operationssimultaneous.

REPORTS OF EXPERIMENTS ON VOLUME DETERMI-NATION BY THE AID OF STEREO INSTRUMENT

AND A PLANIMETER*

P. Hjelmstrom, Civil Engineer, First Assistant, Division of Photogrammetry,Royal Institute of Technology, Stockholm

I N THIS paper a description is given ofa method for measuring volumes auto-maticaIIy in a stereo instrument with aconnected planimeter. The experimentsconcerned sweIIings on the face of dentalpatients and were conducted at the Divi-sion of Photogrammetry at the RoyalInstitute of Technology, Stockholm.

The experiments were based on the fol-lowing considerations. _

If a stereo model is defined by a closedsurface, it is possible to determine the vol-

ume within this surface, provided the con-tours are traced with the moving point ofthe planimeter connected to the drawingpencil of the stereo instrument as illus-trated in Figure 1. The advantages of di-rectly connecting the planimeter to thestereoscopic instrument, instead of theusual practice of first drawing the contours

* This paper is a part of the Symposium bn on-Topographic Pho'togrammetry, See the SePt.19.'>3 issue. It was decided to postpone printing of this paper until illustrations could be obtained.

732 PHOTOGRAMMETRIC ENGINEERING

The planimeter computes the volume bymeans of numerical integration. Differentformulas give different measuring pro-grams, but the general (orm of the ap-proximate volume is some weightedl meanof the contour areas, and therefore, it ispossible 'to perform integration accordingto any suitable formula. The ordinaryaverage multiplied by the contour intervalis the most simple formula, and since theinterval can be chosen arbitrarily small(no plotting), this formula was used in thepresent application. The fact that only dif-ferences in volume were required; made a

o

FIG. 2. Illustrating how a closed sur/ace isdefined and~measured.0= Perspective center (corresponds to camera

lens). '-!B = Plate holder. IoS = Surface of the stereo model.7·10·8·9·13· 7=S'=Part of S. The deforma-

tion lies within S'.1· 2·3·1 Contour drawn on plate holder.

This contour and the point 0 define the refer-ence cone in the following manner: 0 is thetop of the cone and the generatrix describesthe contour.

4 ' 5 . 6 . 4 Bottom surface of the reference cone.The part of the reference cone between S'and the bottom surface (4·5' 6· 4) constitutethe closed surface ,the volume of which is tobe measured.

10 ·11· 12·13 ·10 levelling contour, followed by"the floating mark." The planimeter meas-ures the area of this closed contour and addsthe value obtained to the sum of all previouscontour areas, beginning with "the bottomsurface" (or the top surface).

refinement of the approximation to truevolume ,unnecessary. The choice of thissimple formula was intended to lessen therisk of gross errors.

The difficulties which arise when usingthis method are the following.

1. A stereoscopic view of the closed sur-faces must be possible. To produce astereoscopic model of an arbitrary form ofsurface, the space within the surface mustbe transparent, and the front surface mustalso' be partly transparent so that theback surface will not be entirely hidden.

This requirement was not met. Howevereven though the conditions of transparencydo not exist, the direct method for volumedetermination is still applicable providedcertain limitations wi'th reference to thegenerality of the form of the surfa'ce areallowed. -

The most simple case is that of the backsurface being plane and limited by a closedcurve, which at the same time forms thelimitation of the front surface. When con-touring on planes parallel to this "bottomsurface," the "lower curve" will thus con-sist of the limitation curve. As an example,this would be- the case if a heap of non-transparent material, placed on a flatplate, were stereoscopically photographed,so that, on the one hand, the whole heapis stereoscopically covered and, on tht:other hand, the surface of the heap is nothidden on any of the photographs. Byabsolute orientation on points on the plate,the contouring would be more easily exe-cuted in planes parallel to this plate.

An estimate of the volume variations ina locally deformable closed surface issomewhat more complicated when not allof the front surface is stereoscopicallycovered. This case can be brought back tothe most simple one, described above, ifthe orientation is left untouched and thewhole of the deformed part is stereo-scopically covered. This procedure, whichis diagrammed in Figure 2, is based on acontour, 1.2.3.1., drawn in one of the plateholders so as to include the deformed sur-face,S'. \iVhen drawing contour lines thiscontour is followed wherever it lies beneath(10.8.9.13) the deformed surface, S'; else-where, the surface contour, 13.10, is tracedto form a closed plane curve whose areais measured by means of the planimeter.It is evident from Figure 2 that the solidwH.ose volume is to be measured is thetruncated portion of a cone whose vertex

EXPERIMENTS ON VOLUME DETERMINATION 733

is the center of projection, 0, and whosegeneratrix is the contour, 1.2.3.1, drawnon one of the plate holders. The top ofthis solid is the deformed surface, 5',and the bottom is the plane surface. 4.5.6.4.placed at least as low as the lowest curveof the top surface. Thus, if it is possible todraw the contour on the picture holder asabove described, the total volume vari-ation, due to surface deformation of thespace within the surface originally closed,but not stereoscopically covered, manifestsitself as a variation of the volume meas-ured.

2. The planimeter mu~t be moved be-tween the different contour lines along thesame line. If the floating mark is shiftedfrom one contour to the next, the two sep-arate marks move on straight lines, whichin the normal case lead through the princi-p"!-I point. When one of the picture holdersis furnished with a contour to limit the por-tion of the stereoscopic model that is tobe contoured, it is advisable to introducethis straight line on the picture holder-ifpossible as a part of the contour.

'Beside these two chief difficulties, cer-tain practical details claimed special atten-tion in making the experiment. The photo-graphs were taken with Wild's stereocamera with a focal length of about 9 cmand a base of 40 cm. Ancillary lenses were

. used to shorten the working distance. TheWild Autograph allows a minimum focallength of 10 em, thus making it necessaryto modify the inner orientation throughaffine transformation unless the plates areenlarged. The distortion introduced bythe ancil.lary lenses, on the one hand,causes disturbing parallaxes and, on theother hand, causes deformations of themodel. The model deformation and affinetransformation were of no importance,because the outer orientation was unvariedand the photographs were taken as in thenormal case: The disturbing verticalparallaxes were most easily eliminatedwi th one of the camera constan ts. I t is truethat this gives rise to both an incompleteparallax elimination and extra model de-formation, but neither of these was a verygreat disadvantage in making the experi-ment. The alternative of eliminating thevertical para,llaxes with cI>, thus divergingfrom the normal case, was tried at an earlystage of the experiment but was' aban-doned.

As the picture holder of the Autograph

has no marks for adjusting negatives fromthe stereoscopic camera, these had to beintroduced. Through use of a coordinato-graph, this was done by scratchingcrosses on a layer of ink on the glass of thepicture holder.

Certain additional experiments weremade to measure the accuracy of themethod. The most important of these aredescribed in the following. The behavior ofthe planimeter was investigated by a seriesof measurements of known surfaces. Theplanimeter showed a good consistency foraverage size of surface, when the anglebetween the moving arm and the rollingdirection was not less than a certainminimum. The standard deviation, com-puted from the divergences from the aver-age of 10 measurements with differentinitial angles all surpassing the minimumangle, amounted to 1 pro mille (0.25 percent) of the surface (25 cm2). The system-atic error, due to sliding, was investi cgated by changing the direction of rota-tion, but this error was without impor-tance in the case in question.

The affinitely transformed elevationscale of the stereoscopic model required aspecial adj ustment of the length of theplanimeter arm., This was performed aftercomputation and was controlled throughseparate plane and height measurings. Acrude control was effected by cQntouringand determining the volume of a clod ofputty, which had b~en photographedunder the same conditions as the objectsof investigation. Variations of volume,caused by weaknesses in the 'instrumentsand in the contouring procedure, werestudied through repeated plotting of thesame pair of stereoscopic pictures. Volumevariations caused by variations on the pa-tients and, in general, variations of theouter orientation were investigated byrepeated photographing and plotting ofthe same patient. The average errors de-rived from these calculations was of theorder of magnitude of 1 and 2 cm3 respec-tively, the model of reference chosen being60 to 130 cm3• To decide the reliability ofthe approximation, the same model wasmeasured by a series of successively re-duced equidistances. In our case 1 mm inthe Autograph, that is about 2 mm in.reality, appeared to be a division which wassmall enough.

Concerning the instrumental work thefollowing must b~ noted: The reciprocal

734 PHOTOGRAMMETRIC ENGINEERING

orientation was left untouched, and theabsolute orientation was executed oncontrol points attached to the object, bymeans of a matrix of the jaw of suitableimpression compound, which sets to re-quired hardness. Each model was measuredtWIce, since one omitted contour linewould introduce an unacceptable error.This procedure naturally admits of argu-ment. The contouring work is increasedand of course it would be possible t usethis additional work for lessenirlg theapproximation error by using half thecontour interval.

The importance of each volume determi-

nation separately decided us in favor ofthe first of the above mentioned proce-d ures. When deali ng wi th a large series ofexperiments with a slow variation, wherelarge errors give rise to inexplicable jumpsin the results, the latter method is prefer-able. Our case was different, and so themeasuring was doubled. This lengthenedthe total time of work by about 20 percent. The average time used for contour-ing one model was about 30 minutes. Thechief part of the work was executed in theAutograph A6, belonging to the Photo-grammetric Division at the Royal Insti-tute of Technology in Stockholm.

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