Vertical Exaggeration andPerceptual Models

ROBERT SINGLETON, Consultant, Pennington, N. J.

ABSTRACT: By computing the vertical exaggeration in the geometric stereo­scopic model, it is shown that the geometric model and the perceptual modelexperienced by the observer are radically different. It is concluded that anexperimental study to determine the metric of the perceptual model isrequired before accurate knowledge can be had as to subjective estimates ofthe model's characteristics.

T HE properties of the stereoscopicmodel have almost always been ex­

amined in photogrammetry by a geometrictreatment. This has been satisfactory solong as we dealt with properties subject tophysical measurement in the model, aswith floating mark and scales. When sub­jective properties are to be treated, re­quiring measurement or characterizationof the model by estimation, such as occurin interpretive applications, the geometricmodel is quite the wrong thing to use.Instead, a model termed here the perceptualmodel must be used. This is the model asperceived by the operator, and is quitedifferent from the geometric model.

This has been illustrated by the volumeof papers in the last few years dealing withvertical exaggeration. It so happens thatthe concept of vertical exaggeration is anideal tool for demonstrating that there is adifference. Hence the balance of this paperis concerned with vertical exaggeration.The point, that there is a difference be­tween the geometric and perceptualmodels, having been made, however, it isapparent that the difference will extendto many properties other than verticalexaggeration. The perceptual model is notcharacterized in this paper; unfortunately,not enough is known about it yet. Anexperimental investigation of the per­ceptual model is called for, and this shouldprove to be an interesting and fruitfulfield.

In photogrammetric practice there areseveral useful concepts of vertical exaggera­tion. Three which come to mind im­mediately are:

1. Gross vertical exaggeration. Ratio of

vertical to horizontal scale for largeobjects (e.g., mountains) or for largedistances in the space as a whole.

2. Local vertical exaggeration. Ratio ofvertical to horizontal scale for smallobjects (e.g., buildings), or differentialscales.

3. Slope exaggeration. Ratio of verticalangles, or of their tangents, or otherfunctions.

Undoubtedly there are more concepts.Each of these concepts is useful and prac­tical in some phase of photogrammetricpractice. Under some conditions two ormore concepts may be the same. For ex­ample, if the Jacobean (or the indicatrix)of the transformation from space to modelis a constant, then gross and local verticalexaggeration are the same by any reason­able measure. But slope exaggeration maybe different, and it may not be possible tofind any suitable measure which wouldmake it the same as the other two.

Because photogrammetry includes alarge number of different applications anda variety of taking and viewing conditionswhich seldom result in a simple space-to­model transformation, we must expect nosingle formula for the general concept ofvertical exaggeration, but instead a differ­ent formula for each specific concept. This,of course, should cause no difficulty, and isjust what one might expect.

The trouble comes when one starts to bespecific and to define measures for thedistances, angles, etc., that enter into thegeneral concept. There is just one case thatpresents no difficulty. This occurs whenphysical measurements in both the space(survey methods) and the model (floating

175

176 PHOTOGRAMMETRIC ENGINEERING

mark with scales) are valid for the applica­tion. Most ordinary mapping applications,such as flight .planning to get the desiredscale and contouring interval, lie in thiscategory. Here a straight-forward geomet­ric computation may be made to comparethe actual motion of the floating mark inthe model with its corresponding fictionalmotion in real space, and the result will bevalid for the purpose intended regardlessof the appearance of the model to theoperator.

Trouble arises when one or both of thespaces is not subject to physical measure­ment-that is, when one of the entitiesbeing considered is neither real space nor ageometric model of real space, but insteadis a subjective space formed by the percep­tion of an operator. Two examples of sucha pplications are:

a) vertical exaggeration of culture orterrain in a stereoscopic model as comparedwith the actual view of an unaided observ­er at the same altitude as the pictures,such as may occur in certain navigation orintelligence problems,

b) estimation of slopes in a stereoscopicmodel, to compare with true ground slopes,such as may occur in geological studies.

In each case at least one of the spaces isperceptual, and the measurement made issubjective.

As of now the metric for perceptualspaces is unknown. We do know the fol­lowing things about them:

1. The distance at which an objectseems to lie in stereoscopic viewing isnot given by the intersection of theprincipal rays from the eyes. This issimply common experience-witnessthe ability to fuse a model even underdivergence.

More elegantly, it is known that conver­gence is only a rough clue to distance, iseasily over-ridden by other clues, and isaccepted to the exclusion of stronger forcesonly if extreme enough to be difficult. Itdoes contribute to and sharpen other non­conflicting clues.

2. vVhen a strange scene is presentedvisually, other sensory clues beingexcluded and closer examination (asby motion of the observer through theobject space) prevented-all of whichapply to photogrammetric viewing­the operator's perception is deter-

mined by his experience, and he tendsto give the scene the most familiarinterpretation, even though this maysometimes result in some intellectual(but not visual) absurdities.

Since no one has ever seen the earth from20,000 feet with eyes 12,000 feet apart, it isquestionable just what the perception maybe. Probably it is different for differentobservers, but there may be enough com­mon characteristics to derive, eventually,a "standard observer's" perceptual space.

3. Whatever the metric of a perceptualspace may be-and there have beenattempts to metrize such spaces-itis almost surely different from theEuclidean metric of the geometricmodel, especially if head motion isprevented, as in photogrammetry.

Witness the frontal plane horopters andsimilar phenomena. Items 2 and 3 abovemake it extremely unlikely that anytheoretical studies will be able to charac­terize the subjective appearance of themodel to the observer at present, for ver­tical exaggeration or f~r many other usefulproperties of models. What we need rightnow is carefully collected, controlled ex­perimental data. We can then begin tolearn the metric of perceptual space. Thisseems to be a proper and necessary activ­ity for a field so depenaent on the proper··ties of vision. .. h ..

4. Although we do not know the actualsubjective metric, it appears fromexperiment that the perception isapproximately invariant underchanges of scene which cause allparallax angles to change by anadditive constant, and leave polarangles unchanged.

The normal variables in photogrammetricviewing practice are viewing distance,photo separation and lateral position.Changes in the last two change parallaxangles by an additive constant, approxi­mately. Viewing distance does not.Changes in lateral position of photos whichdo not change the centering of the pairleave polar angles unchanged. Hence weshould expect the perceptual model to besensitive only to viewing distance andcentering-but not, perhaps, completelyindifferent to large changes of the othervariables.

VERTICAL EXAGGERATION AND PERCEPTUAL MODELS 177

in a space coordinate system. The imagepoints, in the usual coordinate systems ofphotographs, of a space point (x, y, z) are:

Left: _f_ (x + ~, Y)Il-z 2

Right: _f_ (x _~, Y)Ii - z 2

Let these photographs be arranged inthe u-v plane of a coordinate system forthe viewing space, with their centers at

In order to show the marked differencebetween geometric and perceptual modelsthe transformation between real spaceand the geometric model is needed. Ob­viously, according to what has been said,this model will not provide any informationas to the subjective model exaggeration.The transformation is given for verticalphotos only. It may be made general byinserting the tilt transformation but thisis not necessary now. Details of derivationare omitted since this is not new, and thework can easily be checked by the reader.

Let two vertical photos with principaldistances f be taken from stations

( ~ , 0, h) and (- +, 0, h)

and viewed with the eyes at

(+, 0, k) and (- ~ , ~, k) .J'he point of intersection of rays from theeyes through the photo images is

f:~+(~)(It-z)11 = e

fli - (c - d - e) (It - z)

fyl'=e (1)

fli - (c - d - e)(h - z)

fb - (c - d)(h - z)1(1 = k .

fli - (c - d - e) (It - ~)

Of course, if the photos are presented tothe eyes with a change of scale, the quantityf used in (1) must be the principal distancefor the projection.

Equations (1) represent the geometricmodel. If a, physical model were to beconstructed using the transformation (1)and viewed from the specified eye-points,the visual coordinates of all points wouldbe the same as in the binocularly viewedphotographic images. This does not mean

Left: (d, 0, 0) Right: (c, 0, 0)

that (1) is a valid representation of theperceptual model. But we proceed to de­rive the scale properties of this model asif it were.

The horizontal scale for two points at thesame elevation is

lbl - II, 112 - v, eH=--=--=-

~:2-.r, y2-y, bD

where

(c - d - e) (11 - z)D=I- .

bf

The vertical scale for two poi nts atelevations z" Z2 is

W2 - WI ek]1=---=---,

Z2 - z, bfDID2

D, and D2 being the expression D at Zl andZ2 respectively.

The gross vertical exaggeration is anexpression of the general form V/H. SinceH is a function of z unless c-d-e=O(which means the principal points of thephotos are separated by the i. d. of theoperator-a condition seldom used) thequestion arises as to what H to use. Sup­pose we use some mean scale

e D, + A(D2 - D,)H* = All, + (1 - A)H. = - ----:::-::::---

b D,D.

Then the gross vertical exaggeration is

V k 1E=-=- .

H* f D, + A(D2 - D1)

Evaluating the denominator we find that if

bfIi = (AZ2 + (I - A)Z,) = d (2)

c- -e

then the denominator is zero and E is in­finite.

But c-d-e is the photo-base corres­ponding to stereoscopic viewing with zeroconvergence. In much usual photogrammet­ric practice (except multiplex and Kelch,for example) equation (2) is nearly satis­fied, and it requires only a very slight ad­justment of the print separation c-d tocause it to be satisfied. Thus, (2) says

whatever mean horizontal scale is used toto define vertical exaggeration, adjustmentof the print separation for zero conver­gence, at a corresponding mean altitude,causes vertical exaggeration, as defined inthe geometric model, to become infinite.

Since it is a m'ati:er J~f experience that

178 PHOTOGRAMMETRIC ENGINEERING

small adjustments of the photo separationdo not cause large changes in apparentvertical exaggeration, and infinite verticalexaggeration is never observed, this is aproof that

the geometric model does not represent theperceptual model.

If for local vertical exaggeration E isdefined as

oW oWoz oZ

---= ---ott ovax oy

the proof goes through exactly the same.A similar proof for slope exaggeration re­quires more manipulation but comes tothe same result.

To show the point made in item 4 above,

for models which are viewed with nearzero convergence, a very good approxima­tion to the parallax angle is

_e_= ~(~-c+d+e).k-w k h-z

Thus changes in c, d and e change all paral­lax angles by the same amount. Changes ink cause proportional changes in parallaxangles, under which the perceived model isnot invariant.

The tangents of the polar angles are

k~W= T(h~ z + d~ C)

_y_ = ~ -.l!_ .k-w k h-z

Hence the perceived model should be rela­tively insensitive to changes in e, and tochanges in c and d which keep d+c fixed.

NEWS NOTES

RELIEF MAP OF SEATTLE AREA

A big, 6X7! ft. relief map of the Seattlearea will help officials of that city withtheir planning problems. The new map is alightweight plastic model produced re­cently by Aero Service Corporation. Mapscale is 1 inch = tmile. The map covers1,656 square miles. Below 1,000 feet,elevations have a vertical exaggeration of3: 1, and above that height a vertical exag­geration of 2: 1. Highest point is five inchesabove the lowest point on the relief map

The corporate limits of Seattle includesome 88 square miles of land and 3 squaremiles of water surface. Its water supplyand waste channels reach into the moun­tains and tides beyond the city limits.Therefore, when a new water supply line orpower line is brought into the city, theproblem is complex and difficult. Successfulplanning calls for a good visual presenta­tion of the geographic areas and physicalproblems involved. The new relief mapwill help greatly to solve such problems.With it an engineer can demonstrate waterflow problems by pouring water on amountain divide and watching it flowdown creek and river into Puget Sound.

The map was produced in 75 workingdays and is made of a durable non-inflam­mable plastic. Map information is shownin 4 colors. The surface of the map has beenplastic coated to protect these inks. Weightof the map is only 65 pounds, including arigid plywood mounting.

For more information write to RobertSohngen, Aero Service Corp., Philadelphia20.

GEOPHOTO ANNOUNCES SOILS AND

ENGINEERING GEOLOGY SECTION

Geophoto Services, Denver, Colorado,has recently expanded its services to in­clude a soil~ and engineering section.

GeophotQ was organized in 1946, andduring the past decade has specialized inphotogeolo~icevaluation and detailed sur­face mappipg in the field of petroleumexploration. The Geophoto Group includesGeophoto Services for projects in theUnited States; Geophoto Services, Ltd.,Calgary, Canada, for Canadian work; andGeophoto Explorations, Ltd. for foreignwork. Combined staff is over 100 andincludes geologists, professional engineers,draftsmen, and other technical personnel.Geophoto has operated in 15 countriesbesides the United States.

The new soils and engineering geologysection is headed by Mr. James G. John­stone, formerly Assistant Professor ofEngineering Geology and Highway Engi­neering, Purdue University.

The new section will be concerned withnumerous applications of air photo inter­pretation in soils and engineering studiessuch as highway, pipe-line and transmis­sion line route locations, dam sites, indus­trial site selection, water supply problems.and general geological engineering.