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A Mathematical Expression for Stereoscopic

Humberto Rosas, Watson Vargas, Alexander Cerón, Darío Domínguez and Adriana Cárdenas.

Abstract Prade, 1972). On the one hand, the projectionists believed The metric nature of stereoscopic depth perception has that the perceived model was formed at the intersection of remained an enigma. Several mathematical formulations visual rays, although they faced the difficulty of explaining proposed for measuring the stereoscopic effect have not the fact that -vision can also be performed under shown to be reliable. This may be due to the lack of a conditions of parallelism and even divergence of eye axes. conceptual distinction between the 3D model geometrically The second group, called fixacionists, assumed that even in obtained by intersection of visual rays (geometric model), conditions of parallelism or divergence of the eye axes, the and the 3D model perceived in the observer’s mind (percep- perceptual model is placed at a “virtual fixation point” by tual model). Based on the assumption that retinal virtue of some telemetric cues such as convergence and is the only source of information on depth available to the (Raasveldt, 1956). In any case, the idea of brain, we developed an equation that shows real and visual intersection of rays on a real or virtual point pre- perceptual space to be connected by a logarithmic function. vailed in the investigations. At the end, the lack of a This relationship has allowed us to formulate the vertical solution for measuring vertical exaggeration caused this exaggeration for all sorts of stereoscopic conditions, includ- problem to be considered a “mystery” (Collins, 1981), and ing natural stereo-vision. The obtained formulations might the search for a mathematical expression, such as a involve possibilities of technological applications, such as “Quixotic effort” (Yacoumelos, 1973). Since then, investiga- the artificial recreation of a natural stereo-vision effect, and tions on this subject fell in an evident stagnation and the design of stereoscopic instruments with a desired degree created an atmosphere of skepticism (Rosas, 1986). This may of vertical exaggeration. be the reason why no significant literature on the metric aspects of depth perception has been published lately. More recently, it has been suggested that some Introduction monocular and binocular cues to depth perception are combined with parallax in depth perception. Those cues The theme of depth perception, which causes vertical are ordinal configural cues (Burge et al., 2005), induced exaggeration in stereo-vision, focused the attention of pho- vertical-shear and vertical-size effects, motion parallax, togrammetrists by the middle of the Twentieth century. At , occlusion, blur, , and accommoda- that time, several equations for determining vertical exaggera- tion(Landy et al., 1995; Mon-Williams et al; 2000; Allison tion were proposed (Stone, 1951; Aschenbrenner, 1952; et al., 2003). The idea of considering vergence and Goodale, 1953; Raasveldt, 1956; Singleton, 1956; Miller, 1958; accommodation as binocular cues for distance and depth Miller, 1960; Yacoumelos, 1972; La Prade, 1972; Yacoumelos, may seem reasonable in natural stereo-vision. However, in 1973; La Prade, 1973; Collins, 1981). Invariably, the investiga- artificial stereo-vision of photographic images under tions were centered on observations of aerial , vergence has no importance since stereo- with the aid of a stereoscope. scopic perception can be obtained under conditions of Within the considerable number of proposed solutions, parallelism and even divergence of visual axes. Bearing two main groups prevailed in irreconcilable conflict (La this in mind, the field of our investigation is based on the hypothesis that stereoscopic depth perception is solely determined by retinal parallax. On the other hand, considerable research on depth Humberto Rosas is with the School of Engineering, perception is recently being focused on physiological Universidad Militar Nueva Granada, Calle 141 7B-37 Bogotá, studies at a neural level. However, at present, neural Colombia ([email protected]). encoding of depth is seen as a complex question that Watson Vargas is with the School of Engineering, continues being a mysterious subject (Backus, 2000; Universidad Militar Nueva Granada, and formerly with the Backus, et al., 2001; Chandrasekaran, et al., 2007). Accord- School of Engineering, Department of Chemical Engineering, ing to Quian (1967), the neural approach does not exclude Universidad de los Andes the possibility that depth perception can also be treated as a mathematical problem. And, in the opinion of Marr Alexander Cerón is with the School of Engineering, Department of Multimedia Engineering, Universidad Militar Nueva Granada. Darío Domínguez is with the Department of Mathematics, Photogrammetric Engineering & Universidad Militar Nueva Granada. Vol. 76, No. 3, March 2010, pp. 301–306. Adriana Cárdenas is with the School of Engineering, 0099-1112/10/7603–0301/$3.00/0 Department of Civil Engineering, Universidad Militar Nueva © 2010 American Society for Granada. and Remote Sensing

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(1982), physiological details do not become absolutely b¢D ‹ ¢P ϭ . (4) necessary for understanding the process. D In our investigation, we believe that a mathematical connection between object (input) and its corresponding From the geometric point of view, there is no question perceptual image (output) can be established regardless of about the validity of the above relationships, and on this the intricate mechanisms of the brain. basis, several authors developed formulations concerning stereoscopic depth perception. However, experience has shown that when aerial photographs are viewed stereo- Geometry of Stereoscopic Vision scopically, the stereo-model perceived by the observer Stereoscopic vision becomes a case of stereoscopic photog- (perceptual model) appears exaggerated in depth relative raphy where eyes behave as two photographic the one yielded by intersection of visual rays. This feature separated by the eye base b. Therefore, the geometry has led us to establish a substantial difference between involved in the obtainment of aerial photographs is perceptual and real magnitudes. We will use an apostro- applicable to the stereoscopic vision. This analogy is phe to designate perceptual magnitudes, and symbols illustrated in Figure 1 where a vertical object, represented without apostrophe for real magnitudes. by a tree, is photographed stereoscopically by two cameras Geometrically, parallax yields depth information enough (eyes) separated by the eye base b, that are located at a for building a 3D model out of two plain retinal images. We viewing distance D from the reference plane. can say that parallax gives the measure of depth perception, The parallax difference between base and top of the which makes it reasonable to think of a direct proportionality tree is ⌬P. If the object is located on the reference plane, existing between parallax ⌬P and its corresponding perceptual as it is shown in Figure 1, the following well known depth ⌬D'. That is, expression is used: ⌬D' ϭ K ⌬P. (5) D¢P ¢D ϭ . (1) ⌬ b ϩ ¢P Substituting P for its value obtained in Equation 4: Kb¢D If the object is located below the reference plane, the ¢D' ϭ . (6) expression is: D ¢ Differentiating: ¢ ϭ D P D Ϫ ¢ . (2) b P KbdD dD' ϭ . (7) When the object is located at an average viewing D distance, that is, the reference plane lies at the middle of the object, we have: Integrating:

D¢P ϭ dD ¢D ϭ (3) dD' Kb (8) b L L D

‹ D' ϭ Kblog D ϩ C. (9) For values of D' starting from 0 we have: D' ϭ Kb log D (10) where b is eye base, and K is a constant characteristic for the stereoscopic vision. At this point, the key step is to find K. In Equation 3, by substituting the real values ⌬D and D for their corresponding perceptual values ⌬D' and D', we have:

D'¢P ¢D' ϭ . (11) b Substituting D' for its value given in Equation 10: Kb log D¢P ¢D' ϭ (12) b

¢D' ‹ K ϭ (13) logD¢P Theoretically, constant K could be calculated form Equation 13 by making experimental measurements of ⌬D', D, and ⌬P. However, perceptual depth ⌬D' is not susceptible to be measured directly with a desired level of precision. Figure 1. Geometry of stereoscopic In order to overcome this limitation, we developed an vision. indirect method for obtaining reasonable estimations of perceptual depth.

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Method for Measuring Perceptual Depth Because no mathematical connection between real and perceptual depth is known, no scale for measuring percep- tual depth is available. However, some geometric figures allow us to make metric estimations of perceptual depth. For example, Figure 2 shows an inclined equilateral triangle whose base b (equal to s) rests on a horizontal reference plane. The horizontal projection of its height h on the reference plane equals half its base s/2. Because of being an equilateral triangle, the elevation e of the top vertex above the horizontal plane is: s e ϭ . (14) 12 This equation is valid for any equilateral triangle, real or perceptual. Therefore, if an inclined triangle is stereoscop- ically perceived to be equilateral, and its base s is known, e becomes a perceptual depth that can be calculated by applying Equation 14. The stereoscopic perception of triangles can be performed by means of stereograms where different levels of depth perception are produced by parallax variations. Figure 3 shows a stereogram to be seen with the aid of a pocket stereoscope, where two sets of upward and downward triangles are viewed stereoscopi- cally. They will be used for determining the numerical value of K (Equation 13).

Experimental Determination of K Figure 3. Stereogram of triangles to be viewed with a In the stereogram of Figure 3a, the observer perceives, pocket stereoscope: (a) Triangles dipping upward, and under a 12 cm stereoscope, six upward (b) Triangles dipping downward triangles of 0.5 cm base, each one higher and steeper than the preceding one, as the result of varying their top vertex that are shown on the right of the stereogram The projection of each triangle height on the horizontal we chose the one perceived as equilateral. Must of the plane of the paper is equal to half their base, that is, 0.25 observers who participated in the experiment (16 of 20) cm. Among the six upward triangles viewed stereoscopically, viewed Triangle Number 4 as an equilateral one. Conse- quently, the values of Triangle 4 are determined by Equation 14. In this equation, being triangle side s ϭ 0.5 cm, it follows that e ϭ 0.35 cm. In fact, e becomes the perceptual elevation ⌬D' above the reference plane, or depth perception between base and top of Triangle 4, so that ⌬D' ϭ 0.35cm. The corresponding parallax ⌬P is 1 mm as indicated on the right of the stereogram. In Equation 13, substituting the values obtained for ⌬D'ϭ 0.35 cm, ⌬P ϭ 0.1 cm, and D ϭ 12 cm (stereoscope focal length), the constant Ka for upward Triangle 4 is obtained:

0.35 K ϭ (15) a log 12 * 0.1

ϭ Ka 3.24. (16) Figure 2. Trigonometric relation- ships used for estimation of depth It is convenient to note that each of the triangles of perception. In a dipping equilateral Figure 3a undergo a reduction in its horizontal scale progres- triangle (real or perceptual) whose sively upwards when it is viewed stereoscopically. In the base b (equal to s) rests on a calculation of Ka, such scale reduction was not considered. horizontal plane, the perceptual To compensate this error, we made similar calculations by elevation e of its upper vertex using triangles dipping downward (Figure 3b), whose hori- above the horizontal plane can be zontal scale increases progressively downwards, in order to expressed in terms of its side average upward and downward data. In Figure 3b, downward length s. This means that an Triangle 3 is perceived to be equilateral, with ⌬P ϭ 0.6 mm. interval of depth perception e can A similar procedure followed for determining Ka was used in be calculated when a triangle is the calculation of constant Kb for downward Triangle 3. For stereoscopically perceived as this triangle, we have that ⌬D' ϭ 0.35 cm, ⌬P ϭ 0.06 cm and equilateral. D ϭ 12 cm. Then, substituting these values in Equation 13, the value of Kb for downward Triangle 3 is obtained:

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TABLE 1. EXPERIMENTAL CALCULATION OF K BY STEREOSCOPIC OBSERVATION OF TRIANGLES

ϭ ⌬ ⌬ ⌬ D fS D' Pa Pb Ka Kb Km K (cm) (cm) (cm) (cm) (cm)

Pocket stereoscope 12 0.5 0.35 0.1 0.625 3.24 5.19 4.21 12 1.0 0.70 0.24 0.12 2.70 5.4 4.09 Mirror stereoscope 25 1.0 0.70 0.175 0.87 2.86 5.75 4.30 4.2 25 2.0 1.41 0.375 0.175 2.89 5.76 4.23 25 4.0 2.82 0.7 0.45 2.88 5.04 3.96

D ϭ f Viewing distance to the stereogram, or focal length of the stereoscope. S Base length of the triangle, measured on the stereogram ⌬ Pa Parallax between base and vertex of the upward equilateral triangle. ⌬ Pb Parallax between base and vertex of the downward equilateral triangle. ϭ⌬ ⌬ Ka Constant calculated for the upward triangle. Ka D'/log D Pa ϭ⌬ ⌬ Kb Constant calculated for the downward triangle. Kb D'/log D Pb ϭ ϩ Km Average K of upward and downward equilateral triangles. Km (Ka Kb)/2 K Average constant for the five experimental triangles and for the stereoscopic vision in general.

0.35 B D K ϭ (17) E ϭ * . (21) b log 12 * 0.06 H b

The perceptual exaggeration E', instead, has remained ϭ Kb 5.41 (18) an enigmatic subject, and it is the central theme of our ϭ ϭ investigation. In the following sections, we will explain The mean Km of Ka 3.24 and Kb 5.41 is 4.32. We how to determine the vertical exaggeration E' for three made similar experiments with a mirror stereoscope of focal cases of stereoscopic vision: stereoscopic viewing of length f ϭ 25 cm, by using triangles of 1 cm, 2 cm, and 4 cm photographs, stereoscopic viewing of 3D objects, and sides. For each of them (upward and downward), we calcu- stereoscopic natural vision. lated its average Km and then the final average K. Table 1 shows the data obtained in the experiments. With this procedure, we find that K approaches the Stereoscopic Viewing of Photographs value of 4.2. Additional estimations by comparing aerial Equation 21 shows geometric exaggeration E to be a product photographs with maps corroborated 4.2 to be a quite of two factors: Factor B/H which refers to the geometry accurate value for K. Then, under which photographs are taken, and D/b which deals K ϭ 4.2. (19) with the geometry under which photographs are viewed through the stereoscope. The perceptual exaggeration E', is Substituting this value of K in Equation 10: obtained by substituting, in Equation 21, the actual viewing distance D for the perceptual viewing distance D' found in D' ϭ 4.2blogD. (20) Equation 20. Then, Equation 20 shows that the perceptual space, repre- B E' ϭ * 4.2 log D. (22) sented by D', is logarithmic relative to the real space, H represented by D. This equation gives the value of vertical exaggeration (E') Calculation of Vertical Exaggeration when a pair of photographs is viewed stereoscopically. Vertical exaggeration is defined as the ratio of vertical scale to When vision is performed with the aid of lenses, D corre- horizontal scale. In stereoscopic vision, the vertical scale sponds to the focal distance f of the ocular lenses. In lens corresponds to that measured in the depth direction. It is systems, objective lenses have no effect upon E'. It is worth while to add that the term “exaggeration” connotes the convenient to remark also that, according to Equation 22, idea of increment, like it normally occurs when a relief is and in contrast to other proposed formulations (Stone, 1951; viewed vertically exaggerated in aerial photographs. However, Aschenbrenner, 1952; Raasveldt, 1956; Miller, 1958; in other instances, the perceived stereo-model may appear Yacoumelos, 1972; La Prade, 1972), vertical exaggeration E' flattened, rather than exaggerated, in which case we might is not influenced by eye base. Our approach also establishes, talk of a vertical exaggeration less than 1. It may also happen in disagreement with other hypotheses (La Prade, 1978), that that the stereo-model is perceived with no deformation in E' is not affected by the photographs’ enlargement. depth, which means that its vertical exaggeration is 1. In fact, Besides, in some proposed formulations (Stone, 1951; two types of vertical exaggeration take place in stereoscopic Goodale, 1953), prints’ separation is considered as a determi- vision: (a) the vertical exaggeration E of the external 3D model nant variable of E'. From our point of view, prints’ separation produced geometrically by intersection of visual rays, or modifies slightly the incidence angle u of visual rays on the geometric exaggeration, and (b) the vertical exaggeration E'of plane of photographs. As a result, the actual photographic the perceptual 3D model created in the observer’s mind, or parallax ⌬P is projected onto the as an apparent ⌬ ⌬ ϭ⌬ perceptual exaggeration. The geometric exaggeration E can be parallax Pa given by the expression Pa P cos u. In a ⌬ ⌬ geometrically calculated and is normally explained in books strict sense, P should be substituted for Pa; however, in ⌬ ⌬ (Gupta, 2003; Paine and Kiser, 2003). It is defined as the most practical cases, the difference between Pa and P base to camera distance ratio (B/H), times the viewing becomes negligible. In extreme instances, for example, when distance to eye base ratio (D/b). That is: the eye axes do not remain parallel but converge on the

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photographs like in anaglyphic vision, a slight difference in vertical exaggeration is perceived. However, if the eye axes remain parallel, perpendicular to the photographs, and at a constant viewing distance, real parallax ⌬P and apparent ⌬ parallax Pa become the same one, and vertical exaggeration remains unchanged independently of the prints’ separation.

Illustrative Example A pair of aerial photographs was taken with B/H ϭ 0.5. Find the vertical exaggeration E' when the photographs are viewed under a 12 cm focal length stereoscope. Answer: Assigning data in Equation 22: E' ϭ 0.5 ϫ 4.2 log12, (23) Figure 4. Variation of vertical exaggeration E' with E' ϭ 2.27. (24) viewing distance D, for an average eye base b equal to 6.5 cm.

Stereoscopic Viewing of 3D Objects Usually, in stereoscopic instruments for viewing 3D objects, such as microscopes and telescopes, the B/H ratio is given in terms of convergence angle a, being B/H ϭ 2tan a/2. E' ϭ 6.5/500 ϫ 4.2 log 500, (29) Viewing distance D corresponds to oculars’ focal length f. Hence, substituting in Equation 22: E' ϭ 0.15. 30) E' ϭ 2tan a/2 ϫ 4.2 log f. (25) In stereo-vision, we use the expression realistic viewing distance Dr to designate the distance at which a person This equation gives the value of the vertical exaggeration perceives a 3D object in its real shape, that is, with a vertical (E') perceived when a 3D object is observed through a exaggeration E' ϭ 1. In natural vision, this condition takes stereoscopic viewing instrument. The equation shows that place at a viewing distance Dr which we can calculate from ϭ ϭ ϭ the focal length f of the ocular lenses affects the vertical Equation 28, by making E' 1, D Dr and b 6.5 cm exaggeration E'. As to the objective lenses, they influence (average).Then, image enlargement but have no effect on E'. ϭ ϫ 1 6.5/Dr 4.2 log Dr (31) Illustrative Example ‹ ϭ In a stereoscopic microscope, the convergence angle ␣ ϭ 12°, Dr 45 cm. (32) and the oculars’ focal length f ϭ 2.5 cm; Find E'. Answer: ϭ Distance Dr 45 cm is the distance at which a 3D object is Assigning data in Equation 25: viewed, under naked eye, in its right shape. For greater E' ϭ 2tan 6 ϫ4.2 log 2.5, (26) distances, vertical exaggeration E' decreases progressively. Curiously, 45 cm is the average distance at which many ϭ manual tools are handled by humans. E' 0.35. (27) The curve of Figure 4, obtained from Equation 28, shows the variation of vertical exaggeration E' with viewing distance, for an eye base of 6.5 cm. It can be observed, that Stereoscopic Natural Vision for a viewing distance of 45 cm, vertical exaggeration is equal to 1. A topic that has not been studied in detail is the behavior of vertical exaggeration E' when 3D objects are perceived in natural . Norman et al. (1996) observed, from Conclusions multiple ingenious experiments that perceived intervals in Equation 20 shows that the perceptual space is logarithmic depth become systematically compressed with increasing relative to the real space. In addition, a mental image viewing distance. This is equivalent to say that vertical (perceptual space) can be considered as a visual sensation exaggeration decreases with viewing distance. In general terms, caused by the stimulus of a 3D object (real space) when this this idea agrees with our observations as will be shown below. one is viewed stereoscopically. Therefore, it is not surpris- From our point, eyes act as cameras, so that the camera ing that the logarithmic relationship found to exist between B/H ratio corresponds to the eye b/D ratio. Making this perceptual space (sensation) and real space (stimulus) substitution in Equation 22, the following expression is agrees with the psychophysical law proposed by Fechner, obtained: 1889, according to which the magnitude of a sensation E' ϭ b/D ϫ 4.2 log D. (28) increases proportional to the logarithm of the stimulus intensity. The role of the Fechners’s Law in stereoscopic This is the equation that gives the value of the vertical vision had been suspected to occur (Rosas et al., 2007). In exaggeration (E') perceived when a 3D object is observed in this context, the value we have obtained for K ϭ 4.2 would stereoscopic natural vision. correspond to the psychophysical constant that rules the stereo-vision sensorial mode. Illustrative Example On the other hand, the possibility of calculating vertical Find the vertical exaggeration obtained by an observer exaggeration may involve some technological applications. For having b ϭ 6.5 cm, when a 3D object is viewed at a distance example, a 3D effect obtained in natural vision can be recre- of 5 m. Answer: Assigning data in Equation 28: ated by means of a photographic stereo-pair, without changing

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the three-dimensional shape of the object. This simulation is Gupta, R.P., 2003. Remote Sensing Geology, Springer, 680 p. possible by correlating camera base/distance ratio with focal La Prade, G.L., 1973. Stereoscopy - Will data or dogma prevail?, length of the stereo viewing instrument. Among other applica- Photogrammetric Engineering, 39(12):1271–1275. tions, stereoscopic viewing instruments could also be designed La Prade, G.L., 1972. Stereoscopy - A more general theory, with a desired degree of vertical exaggeration. Photogrammetric Engineering, 38(10):1177–1187. La Prade, G.L., 1978. Stereoscopy, Manual of Photogrammetry, American Society of Photogrammetry, Fourth edition, Chapter Acknowledgments X, pp. 519–534. We thank G. Konecny, Emeritus Professor, University of Landy, M.S., L.T. Maloney, E.B. Johnston, and M. Young, 1995. Hannover, Germany, for his valuable comments. The paper Measurement and modeling of depth cue combination: In was significantly improved thanks to the clever suggestions defense of weak fusion, Vision Research, 35(3):389–412. of anonymous reviewers. This research was supported by Marr, D., 1982. Vision: A Computational Investigation into the the Project ING-2004-017 of Universidad Militar Nueva Human Representation and Processing of Visual Information, Granada, Colombia and by The Colombian Fund for the W. H. Freeman and Company, New York, 397 p. Development of Science and Technology (COLCIENCIAS). Miller, C.L., 1960. Vertical exaggeration in the stereo space-image and its use, Photogrammetric Engineering 26(5):815–818. Miller, C.L., 1958. The stereoscopic space- image, Photogrammetric References Engineering, 26(54):810–815. Allison, R.S., B.J. Rogers, and M.F. Bradshaw, 2003. Geometric and Mon-Williams, M., J. R. Tresilian, and A. Roberts, 2000. Vergence induced effects in binocular and motion parallax, provides veridical depth perception from horizontal retinal Vision Research, 43:1879–1893. image disparities, Experimental Brain Research, 133:407–413. Aschenbrenner, C.M., 1952. A review of facts and terms concerning Norman, J.F., J. Farley, J.T. Todd, V.J. Perotti, and J.S. Tittle, 1996. the stereoscopic effect, Photogrammetric Engineering, The visual perception of three- dimensional length, Journal of 18(5):818–825. Experimental Psychology, 22(l):173–186. Backus, B.T., 2000. Stereoscopic vision: What’s the first step?, Paine, D.P., and J.D. Kiser, 2003. Aerial and Image Current Biology, 10:R701–R703. Interpretation, Second edition, John Wiley and Sons, 648 p. Backus, B.T., D.J. Fleet, A.J. Parker, and D.J. Heeger, 2001. Human Qian, N., 1997. Binocular disparity and the perception of depth, cortical activity correlates with stereoscopic depth perception, Review, Neuron 18:359–368. Journal of Neurophysiology, 86:2054–2068. Raasveldt, H.C., 1956. The stereomodel, how it is formed and Burge, J., M.A. Peterson, and S.E. Palmer, 2005. Ordinal configural deformed, Photogrammetric Engineering, 22(9):708–726. cues combine with metric disparity in depth perception, Journal Rosas, H., 1986. Vertical exaggeration in stereo-vision: Theories and of Vision, 5:534–542. facts, Photogrammetric Engineering & Remote Sensing, Chandrasekaran, C., V. Canon, J.C. Dahmen, Z. Kourtzi, and A.E. 52(11):1747–1751. Welchman, 2007. Neural correlates of disparity-defined shape Singleton, R.,1956. Vertical exaggeration and perceptual models, discrimination in the , Journal of Neurophysiology, Photogrammetric Engineering, 22(9):175–178. 97:1553–1565. Stone, K.H., 1951. Geographical air-photo interpretation, Collins, S.H., 1981. Stereoscopic depth perception, Photogrammetric Photogrammetric Engineering, 17(5):754–759. Engineering, 47(1):45–52. Yacoumelos, N.G.,1972. The geometry of the stereomodel, Fechner, G.T., 1889. Elemente der psychophysik, Vol. 1, Breitkopf and Photogrammetric Engineering, 38(8):791–798. Härte,Leipzig, Germany, Translated into English by H E Adler, Yacoumelos, N.G., 1973. Comments on Stereoscopy, Photogrammetric 1966. Elements of Psychophysics Vol. 1, New York: Holt, Engineering, 39(3):274–283. Rinehart, and Winston. Goodale, E.R., 1953. An equation for approximating the vertical exaggeration of a stereoscopic view, Photogrammetric Engineering, (Received 05 December 2008; accepted 03 March 2009; final version 19(4):607–616. 17 June 2009)

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