---
title: "Conjectures on the Perception of Elongation"
authors:
  - "Edward S. Deutsch"
doi: ""
url: "https://ijcai.org/proceedings/1971/22"
pdf: "https://ijcai.org/proceedings/1971/0022.pdf"
venue: "Proceedings of the Second International Joint Conference on Artificial Intelligence"
year: 1971
pages: "238-246"
track: ""
keywords:
---

# Conjectures on the Perception of Elongation

Edward S. Deutsch

*Proceedings of the Second International Joint Conference on Artificial Intelligence*, pages 238-246.

## Abstract

The papers start by discussino some of the background to the problem of the perception of length together with some known illusions in this field. The difference is then drawn between image contraction and image thinning, both processes of which serve to introduce the proposed conjecture. The suggested conjecture is that lenoth is perceived by a mechanism which first forms a minimal closed path (hull) around the object and then performs an isotropic image reduction operation on the hull of the imaoe. It is conjectured that the elongation of the image is exemplified in the resulting path so obtained; no path should result if the original image possesses no elongation. The conjecture is used to explain the known illusions discussed and a few new ones are suggested. An isotropic image reduction technique is described and experimental results testing the conjecture, obtained using a variety of images, are presented.

## Full text

## CONJECTURES ON THE PERCEPTION OF ELONGATinN by

E. S. Deutsch

Computer Science Center University of Maryland Colleoe Park, Maryland

U.S.A.

## ABSTRACT

The papers start by discussino some of the back­ ground to the problem of the perception of length together with some known illusions in this field. The difference is then drawn between image con­ traction and image thinning, both processes of which serve to introduce the proposed conjecture. The suggested conjecture is that lenoth is per­ ceived by a mechanism which first forms a minimal closed path (hull) around the object and then performs an isotropic image reduction operation on the hull of the imaoe. It is conjectured that the elongation of the image is exemplified in the resulting path so obtained; no path should result if the original image possesses no elongation. The conjecture is used to explain the known illusions discussed and a few new ones are suggested.

An isotropic image reduction technique is de­ scribed and experimental results testing the con­ jecture, obtained using a variety of images, are presented.

## INTRnDUCTION

In this paper we will address ourselves to the phenomenon of length perception in oeneral. In the first instance an attempt will be made at distinguishing between two types of image reduc­ tion operations, namely that of image contraction and that of image thinning or skeletonization. While both of these operations appear to be simi­ lar, the distinction we propose to make between their end results leads to the possibility of a conjecture upon how the human eye perceives whether a given image, whatever its shape, possesses the property of elongation or not. The author is mindful of the fact that his proposed conjecture falls short of explaining all the different types of experienced phenomena existino in the field of length perception. However, some interesting results have been obtained on apply­ ing the ideas to images in general and to some of

the optical illusions in particular.

It seems clear that for an image reduction pro­ cess to be meaninqful it must exhibit isotropic properties. In other words unless one is speci­ fically interested in the reduction operations exercising a certain bias in a particular direc­ tion (1) (see Fiqure 4, it will be explained in more detail later on), the reduction must take place uniformly throughout the entire imaoe. We shall indeed describe such an isotropic process in terms of a computer aloorithm; it is the very existence of such a process that gives substance to our conjecture.

Previous work of the author indicates that there are considerable advantages in workinq with images displayed on hexaoonal arrays (2). Accordinolv, all Processes to be described here ~ill uti~ize such arrays. This apparent restric­ tion .does nof limit the generality of the main discussion in any way; it may well emerge that what follows lends additional weioht to the arou­ ment that hexaoonal arrays are the most suitable for imaoe processino in general.

The layout of the remainder of this paper is as follows. ~e shall first aroue the distinction between the two types of image reduction opera­ tions mentioned above. A further related dis­ cussion then leads to the introduction of the conjecture. The imPlementation of an isotropic imaoe reduction techni~ue, the use of which has already been implied, is presented next. The compatibility of the conjecture with some of the observed optical illusions is then discussed. An attempt at introducino new optical illusions in light of the conjecture is also made.

## IMAGE REDUCT! nN: IMAGE CnNTRACTION AND IMAGE THINNING

We introduce the distinction between image con­ traction and imaoe thinning by means of an illustration. Assume that the imaoe to be con­ tracted or thinned is a solid disc of uniform gray level distribution. For the present arou­ ment we shall assume that the center of the disc coincides with a point on the display matrix. It will become apparent at once that if reduc­ tion takes place uniformZ~ throuohout, and, that provided continuity is maintained, the result of this imaoe contraction ooeration should reduce the disc to a sinqle dot. An imaoe thinning operation should oive the same result under simi­ lar conditions. However, it is quite unnatural to sugoest that a thinned disc is a dot . A simi­ lar aroument applies to the case when we consider the imaoe to be a uniform square. If the point of intersection of the diaqonals coincides with a point on the matrix then . the result of both a contraction and a thinnino or skeletonizing operation will be that single dot on the matrix. Aoain, it is quite in order to call that point, in the limit, the contracted form of the original s~uare, but that Point is not the skeleton of the square. It is in fact ~uestion­ able whether the term thinnino or skeletonizing operation is aoolied to two dimensional imaoes similar to those just described is at all applic­ able.

Let us now aoolv a deformation to the disc so that for a given set of perpendicular diameters, one diameter is very s 1 i ohtl v shorter than the other, such that the end points of this diameter are equi-distant from the center of the disc. The disc deforms into an ellipse. Consider the result of contraction and thinnino ooerations on the deformed disc, as the ratio of the lenoths of the lonoer to the shorter diameter increases. When the ratio of these Principle axes is near one or when ~ (9 ~ eccentric angle) 1s near n/2 both these operations will yield an image in the form of a single point, However, as ~ decreases a continuity of points emerges i.e . , a skeleton. T~e or~entation of this skeleton will be in the d1r~ct1on of the major axis of the particular ~ll1pse. The nu~er of points emerging will be 1nverse~y proport1onal to the value of~. In these c1r7u~stances it becomes possible to speak more s~ec1f1cally of a thinning or skeletonizino operat1on rather than a mere contractinq operation f?r the contraction operation may or may not ' y1eld a skeleton dependino upon its design.

A similar argument holds with reference to the square be~ng defo:med, as one of its perpendicu­ lar axes 1s curta1led. A skeleton wi 11 emerge as the ratio of the longer to the smaller axis in­ creases, the number of points in the skeleton being proportional to this ratio. Note that in both of the examples just given, the skeleton em~rqed along the longer center line of the obJects, that is, it coincided with a geometric axis of the object.

In both the cases just discussed, the deformation ~rod~c~d a degree of elongation - if only in the 1ntu1t1ve sense - such elongation being register­ ed with· the emergence of a skeleton. Our inten­ tion, in part, is to generalize this notion to­ gether with its converse and suggest that the ex­ lstence of elongation within an image implies ttle emergence of a skeleton and vice versa. In other words, given t~at an image reduction operation is appl1ed to an 1mage; should a skeleton emerqe then it can be said that the original image'ex­ hlblted elongation (i.e., image thinnino has occured). Should, however, only a single point or a clusterinq collection thereof be obtained then no such property is attributable to the original imaqe (i .e., image contraction has occurred). This would then lead to a qualitative rather than an intuitive definition of elongation.

The distinction we have Just proposed is not merely one of semantics. The distinction is between kinds of end results obtained from an image reduction operation; a mere contracted image for some types of images and a skeleton for others.

An additional factor involved in the above specu­ latlon would be the information borne by the actual skeleton itself. Intuitively speaking, the degree of elongation exhibited by an ellipse of a given eccentric anqle 0 2 would exceed that of another ellipse of eccentric angle 0 1 if 0 1 > 0 2 . (Assume that their major axes are of equal length . ) Application of a reduction operation to two such ellipses should, provided our previous discussion is valid, yield two skeletons (or a point and a skeleton) one of which, that derived from the ellipse ~f 0 ~ 0 2

being longer in dot count than the other. In F1gure 1 are shown two such ellipses their skeletons, superimposed, having been obtained usinq the image reduction mechanism which is de­ scribed later .

We are thus proposino both a qualitative and a quantitative measure of elongation within an image. In surrmary, the production of a skeleton by a uniform reduction operation, signifies the existence of the elonqation property within the imaoe, the actual number of points forminq the said skeleton beino an estimate of its degree of elonqation.

The discussion so far has been restricted to images of reoular shape . '~hat about the elici­ tation of the elonqation Property of irreqular shapes? Consider the images shown in Fioure 2. An isotropic imaoe reduction mechanism applied to the rioht-hand image would yield two almost per­ pendicular skeletons. Yet this imaoe clearly ex­ hibits elongation in the vertical sense. Further remarks concerning this difficulty are deliber­ ately postponed until after the ensuing dis­ cussion, whence its solution will become apparent. We now return to our proposition re­ garding the qualitative measurement of lenqth.

It would be interestino to see how compatible our proposition regarding the attainment of a quali­ tative estimate of length, is with some of the observed visual phenomena. '~e have seen: its application in Fioure 1 in determinino the ~longat~on of two ellipses, and the same argument 1~ appl1cable to rectanoles of differing dimen­ Slons. Yet would the proposition apply to line drawinos? Consider the Mueller Lyer illusion, Figure 3(a), in which the top vertical line looks longer, althouqh the vertical members of each of the drawinos are of the same length. Further­ more, consider the two rectanoles and the two sections in Figure 3(b) and 3(c) respectively . They are of the same 1 enoth in each case, yet the upper member apoears slightly longer. It was considerations such as these which led one to speculate on how elonoation, in oeneral , i s perceived. To assume that there are a number of length perceivino facilities within the human mind each of which aopertains to a different · kind of image, say, regular shaped ones, stroke com­ prising images, etc., is not impossible; it can be neither substantiated nor denied in view of our scanty knowledqe about human oerceotion. It would be exceedingly useful to arrive at a oeneralized mechanism by means of which the elonoation of any image is perceived.

Accordingly, we propose the following conjecture on the overall oerceotion of lenoth. That the elonoation orooerty of an image is perceived by the eye (?) first formino the (convex) hull around the imaoe and then, treating the hull as an imaoe with uniform qray-level distribution, applying isotrooic imaqe reduction to this image. By the arguments above the resultino skeleton exemplifies the elonoation both in size and direction.

The conjecture states that the residual skeleton obtained from an isotropic imaoe reduct i on operation yields a meausre of elongation. The skeleton itself is derived by drawino the envelope around the object whose length is to be measured. It is the length of the resulting skeleton rather than the extent of the envelope which indicates object elongation. (One can introduce many small undulations in the contour of the envelope thus increasing its extent, without altering the final result.) Once the envelope is formed and the en­ closure is 'filled-in', the skeleton should lie along, or very close to the longest center line of the 'filled-in' object. This, provided that the reduction takes place isotropically.

The solution to the problem relating to the images in Figure 2 should now be evident. For the formation of the envelope around each of these images yields objects whose longest axis is near vertical. The resulting skeleton accordino to the conjecture closely approximate this axis.

For the conjecture to be meaningful it must, not only comply with ordinarily observed phenomena but also, cope with, and possibly give rise to the creation of examples of such phenomena. We will devote our attention to both these aspects later. For the moment we digress in order to discuss the concepts of an isotropic image reduction mechan­ ism. We will return to the main problem later on. In the next few paragraphs we review the re­ levant literature.

It is pointed out that in no way does our previous discussion imply that because all regular objects are contractable to a dot, that they will be re­ cognized as being identical. The individual shapes of such objects could be identified as being different by means of an edge-detection mechanism, a visual facility for whose existence there is considerable evidence. It would indeed be interesting to evaluate experimentally the scope of an image reduction mechanism together with an edge-detection process.

A very interesting explanation has been proposed by Blum (4) who has used his Medial Axis Trans­ formation to explain some of the optical illusions. The skeletons, termed Medial Axis Functions, pro­ duced by this transformation differ from those obtained using the algorithm presented in the next section. His argument, however, is also based upon the determination of the properties of the skeleton, i.e., the Medial Axis Functions. He suggests that the visual process tries to line up these Medial Axis Functions rather than the outline of the objects themselves; it is the speed of formation of these Functions which gives vise to the observed illusions. Figure 3(d) de­ picts the formation of the Functions for Figure 3(a), the velocity of their generation is repre­ sented by, and is proportional to the thickness of the dots. The Functions associated with the converging arrows figure coalesce faster thus producing the illusion that this vertical line is shorter.

## TOWARDS AN ISOTROPIC IMAGE REDUCTION PROCESS

We have already mentioned that we will concern ourselves with hexagonal display structures only.

Below we oresent an isotrooic imaoe reduction al­ gorithm appertainino to such arrays. For a fuller treatment the reader is referred to (1).

Consider the six perioheral neiohbors of any element x on the hexaoonal arrav, labelled x(l) throuoh x(6) as shown. Since we are dealing with imaoes of uniform gray-level distribution, the

![Image](https://ijcai.org/proceedings/1971/fulltext/22/image_000000_2bb982ad17d356da934c7dee6214000797e93e9140fd6ffa23009eac4a6a13ef.png)

value of any element in the array can be either 1 (gray image) or zero (background). We shall also refer to the value of an element by its name and say that x(l), i.e., ·element x(l) has the value of either x(l) = 1 or 0, x(l) = 0. This will not lead to a confusion between the name and the value of an element since the meanino will be clear from the context.

Given below are the conditions, all of which must be fulfilled, under which the element x is delet­ able from the original image during the image re­ duction process.

![Image](https://ijcai.org/proceedings/1971/fulltext/22/image_000001_7611d728c93be4576bd149c9699cba2215ce3ece532e0d9f178775a38329285b.png)

2. x(p)·x(p+l)·x(p-1) 0
3. x(p)·x(p+l)·x(p+2) 0

<!-- formula-not-decoded -->

<!-- formula-not-decoded -->

where pis fixed at 1, or 2 or .... 6, and n has a period of 6.

The subscript to the right vertical bar in con­ ditions 4 and 5 sionifies the name of the x ele­ ment the peripherai elements of which are in­ volved in the summation.

Each element is scrutinized independently of all the others; the orocess takes place in oarallel. This process, which maintains connectivity, is carried out until no further erasure occurs. Condition 1 preserves end points of the skeleton, conditions 2 and 3 cause the deletion of certain points whose neighbors observe certain connecti­ vity situations, condition 4 preserves the points in on an already derived skeleton and condition 5

## preserves intersection points.

The six-way symmetry of the hexagonal array lends generality to the above conditions and renders the latter independent of position within the array of peripheral elements, in the sense that the conditions "look the same" whatever fixed value is assigned top.

Consider the skeleton of the letter A shown in Figure 4(a). It was derived using one value of p, p = 1, throughout each cycle. The bias of the skeleton towards the lower left portions of the original image is evident. The skeleton will occupy different positions with respect to the original image for different values of p chosen, and it is this facet which facilitates isotropic reduction, For if the value of p is continually changed during the reduction process so as to compensate for a bias in a particular direction, then hopefully, a uniform reduction will result.

Whereas the skeleton in Figure 4(a), obtained with p = 1, is confined to the lower left por­ tions of the image, the skeleton obtained using the value p = 4 was found to occupy the upper right portion of the image (approximately). In­ deed, experimental evidence seems to indicate that the bias produced as a result of using the value p = pi during one pass over the ima9e would be neutralized, during a followinq pass, usinq the value p = pi+3. Thus, in order to

attain isotropic reduction, one would simply per­ form each pass over the image with a different value of p thus making sure that any inbalance caused during one pass is taken care of in the next. More, or less, than six passes may be necessary but the sequences of the values of p would consist of a repetitive group of six differing p values : - P 1 P 2 P 3 P4P5P6·

A factor influencing the final position and shape of the skeleton within the image is the initial value of p and the remaining values of p in the sequence . We have· found that smoother and more centrally disposed skeletons were obtained if the contraction operations were actually per­ formed six times with p. as the initial value 1

of p at the beginning of the ith pass followed by the remaining p values in order. Thus using the basic sequence p 1 p 2 p 3 p 4 p 5 p 6 , p 1 would be the

first value of p followed by p 1 .·· p 6 in the first pass; p 2 would be the first value of pat the beginning of the second round followed by p 3 ... p 6 p 1 in that round . The third time round

the sequence would be p 3 ... p 6 p 1 p 2 and so on. The resulting skeleton is obtained by forming the logical OR of the previous six skeletons and re­ peating, etc. In practice, the following sequence yielded best results: - P 1 = 1, Pz =

p 1 +3, p 3 = 2, p 4 = p 3 +3, etc. Figure 4(b) shows the result obtained usinq this sequence; the skeleton is centrally disposed alona the limbs of the letter. · Note that where there is no true center line, that is, where the local oart of the limb is an even-number of elements thick, two contiauous skeleton ooints are aenerated. Here, the true aeometric center line is situated between these two points. l·le return now to the main discussion .

## THE CnNJECTURE - DISCUSSinN

In our attemot at establishino credence in our conjecture we sha 11 try to interpret some of the ootical illusions usina the conjecture, and at the same time, suagest how some new ones might be constructed. We shall be investiaatinq some of the better known illusions, and in doing so, pre­ sent some of the theories suqgested by others.

The most popular illusion in the field of lenath perceotion is the ~ueller-Lyer lines, or the "arrow" illusion, shown in Fiaure 3(a). Why in­ deed should the vertical line flanked by the out­ aoina arrow heads aopear lonaer than an equally lona vertical line with inaoina arrow heads? The most recent attempt at exolainina this phenomenon has been made by Greqory (3). It is suqqested that the illusion is produced by our general per­ ception of depth in three dimensions, and by virtue of the fact that there exists an interest­ ing inverse relation - known as Emmert 's Law - between the size of an object and its distance from the viewer . According to this theory, the line with the outaoina arrows is beina associated with the three dimensional representation of a far corner, whereas the line with the ingoinq ar~ows is reminiscent of the representation of a nearer corner. Neqlectina the lenqthy discus­ sion, it is finally araued that the inverse re­ lation causes the imaae which "looks" as thouqh it were furthest away, (the line reoresenting the far corner) to aooear laroer.

The same argument is used to explain why, in the "railway-line" illusion, the upper horizontal rectanale, Fiaure 3(b), looks longer than the lower one. The two converainq lines introduce perspective effects and hence the top rectanale, which appears now further away, is magnified by\_ virtue of the inverse relation . Support for th1s approach is provided by some experimental evidence. The illusion shown in Figure 3(c), the Jastrow illusion, is not discussed.

We now discuss the arrow illusion in the light of our conjecture. The two lines are displayed as thouah they were reproduced on an hexagonal array in Figure S(a). By our conjecture, the lenath of an object is perceived by the eye, in its overall view, formina an imaginary closed hull around the object and then "filling in" the entire enclosure, thus treatina it as a uniform image, Figure S(b). For the proposed conjecture to be valid, the application of isotropic image reduction to these images should yield two skeletons, the larger one of which should be derived from the imaae repre­ senting the line with the outqoina arrows. The result of applyina the alaorithm discussed in the previous section is shown in Fi9ure S(c). (The arms of the ingoina arrows are sli9htly longer, this is because the difference in shape between the two uniform images in Figure S(b) would hardly be noticable otherwise.) It is seen that this is in fact the case.

Two arguments in favor of the way our conjecture treats the above illusion are now presented. The first deals with the possibility of accomplishing this very illusion in a different way. Consider the two vertical lines shown in Fiqure 6. Which one appears longer? If these two lines are just another way of representing the arrow i 11 us ion then it must also be explicable in terms of our conjecture. Now, it is easy to see that the hulls of both the images of Figure 6 are rect­ angles in which the two shorter sides have been made circular. Isotropic image reduction will yield two skeletons of different lengths and hence the illusion. It would be interesting to see how the above notions of three dimensional depth perception and the inverse size relation would cope with this representation.

The second argument runs as follows. Assuming our conjecture to be valid, is it possible to find an "arrow" whose hull would yield an image which, in turn, would be reducible to a skeleton whose size ranked between those of the other two? Should such an "arrow" be found, then it too must form part of the illusion in that it must appear to be shorter than the line with the outqoing arrows and longer than the line with the ingoing arrows. We suggest that the third "arrow" shown in Figure 7(a) is such a possibility. The author has not as yet performed extensive tests with these drawings, but tentatively, of the numerous persons questioned almost all without exception ranked the arrows in the predicted order. The result of applying the reduction operation to these arrows is shown in Figure 7(b).

We now draw our attention to the two vertical lines shown in Figure 8(a). The upper line, with the smaller horizontal line through it, appears slightly longer. This too can be explained by means of our conjecture, as Figure 8(b) indicates. Figure 8(b) also explains why generally, a narrow rectangle of a given length gives the impression of looking slightly longer than a rectangle of the same length but of increased width. (c. f., our early discussion regarding the skeletons obtained from rectangles of varying side ratios.)

Our conjecture does not explain the "railway line" illusion of Figure 3(b) and the Jastrow illusion of Figure 3(c) fully. We feel however, that it has a bearing on the phenomenon. Treatinq the "railway" illusion first; the reason the upper rectangle appears longer is because there is a confusion between the actual boundary of this rectangle and the exceedingly close local boundary dictated by the converging lines. The eye in forming the enclosure, includes the outer lines too, which by our conjecture, results in the upper rectangle looking longer. No such confusion arises in the case of the lower rectangle since the distance between the railway lines and the actual rectangle is fairly large. It is also noteworthy that the illusion only takes place provided one of the obiects is very near the converging lines; the illusion will not occur if both objects are drawn far away from them.

A convenient way of testino this idea would be to ascertain whether an increase in the width of the upper rectanole (or a decrease in the width of the lower rectanqle) causes the illusion either to be less pronounced or to disappear entirely. By a orevious arqument, the aoparent decrease in length caused by increasino the width of the upper rectangle (or the aoparent increase in lenoth caused by decreasinq the width of the lower rectanole) should comoensate for the elonoa­ tion influence produced by the two bounding lines. Preli~inary tests have certainly shown that altering the width of one of the rectanoles reduces the degree of length discrepancy observed between the rectanoles.

We now discuss the bearin9 our conjecture has uoon the Jastrow illusion. This illusion is re­ produced in Figure 9 with the two sections "filled in". The two vertical lines complete the hull of the entire image, the remainder of the hull beino formed by the edoes of the sections themselves. The bottom section apoears as if it were diveroing from the boundary and at the same time converging into the soace between the two lines. Note that this effect does not arise in the case of the too section because the corres­ ponding oortion which gives rise to this impres­ sion actually forms part of the enclosure. It is this diverging effect, interactino between section and boundary which gives rise to the section looking smaller, locally.

This aroument does not refute our comments con­ cerning the "railway" illusion, since in the case of the Jastrow illusion, the divergent effect is overwhelmino. He have tried the rai 1way illusion using inverted triangles instead of the rectangles. The trianoles were drawn so that their apex pointed away from the point of inter­ section of the converoino lines. This was done in order to obtain the diverging effect of the Jastrow illusion. The illusion seems to dis­ appear under these circumstances (see Figure 10). Comoare the imoression obtained from Fioure 3(b) with that obtained from Fioure 10. ·

Finally we present an instance which seems to contradict the conjecture. Consider the two images with their unequal vertical lines shown in Fioure 11. The extremities of the squares flanking both of these lines are the same. The envelope of each of the imaqes is therefore the same; a rectangle whose shorter sides corresponds to the outer side of the small squares and whose larger sides are equal to the distance between these squares' outer side. The coniecture implies that since the resultino skeletons for each of the hull is of the same lenqth, the elongation, or rather the extent of the vertical lines should aopear similar. Yet this is not what is observed.

## CONCLUSION

As usual, the kind of material described in this paper is particularly vulnerable to criticism and counter arguments. The author is fully aware of this and particularly so on the following two counts. First, because most of the material discussed is, to the author's knowledQe, new, and second, because the author is knowingly encroach­ ing upon a domain which is a little alien to his own.

Chronologically, an isotropic image reduction technique was all that we were after originally. This then gave rise to the speculations we de­ scribed above. The fact that we could actually simulate some of the ideas on a computer seemed particularly attractive. It is of course realiz­ ed that the investigations have been limited in their scope - for example, we have not mentioned the Herring illusion at all - but they seem in­ teresting enough to warrant further attention.