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A Mathematical Expression for Stereoscopic Depth Perception

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A Mathematical Expression for Stereoscopic Depth Perception 301-306_08-090.qxd 2/16/10 3:12 PM Page 301 A Mathematical Expression for Stereoscopic Depth Perception 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 stereo-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 parallax virtue of some telemetric cues such as convergence and is the only source of information on depth available to the accommodation (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 perspective, occlusion, blur, vergence, 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 photographs stereoscope, 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 & Remote Sensing 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 Photogrammetry Granada. and Remote Sensing PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING March 2010 301 301-306_08-090.qxd 2/16/10 3:12 PM Page 302 (1982), physiological details do not become absolutely b¢D ‹ ¢P ϭ . (4) necessary for understanding the visual perception 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 cameras 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. 302 March 2010 PHOTOGRAMMETRIC ENGINEERING & REMOTE SENSING 301-306_08-090.qxd 2/16/10 3:12 PM Page 303 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).
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