Japanese Psychological Research 2008, Volume 50, No. 3, 117 127 doi: 10.1111./j.1468-5884.2008.00368.x Blackwell ORIGINAL Publishing ARTICLES rate to Asia fit to perception Magnification rate of objects in a perspective image to fit to our perception KAZUMI NAGATA Faculty of Engineering, Yamaguchi University, Japan ATSUSHI OSA Graduate School of Science and Engineering, Yamaguchi University, Japan MAKOTO ICHIKAWA Faculty of Letters, Chiba University, Japan TAKESHI KINOSHITA and HIDETOSHI MIIKE* Graduate School of Science and Engineering, Yamaguchi University, Japan Abstract: A landscape photograph may give a different impression from that formed at the real scene, with respect to the size and distance of objects. Researchers have reported that the perceived sizes and distances of objects in a photograph are not identical to those in a real space. In order to develop a method to create a graphic image that is close to our visual impression as seen in the real space, two experiments were conducted. In Experiment 1, we examined how the magnification rate of the perceived size to the object size on the retina varied with the viewing distance (range was from 1 m to 10 m). In Experiment 2, we examined whether transformation based on the magnification rate is effective for creating an image that matches the perceived size of the object at the scene. Our results indicate that the magnification rate is useful for transforming the perspective image to match our perception of the objects regardless of the viewing distance. Key words: perceived size, distance, location, perspective image, photograph. One might be disappointed by the landscape photographs taken by a camera because the objects in the photograph look very different from the ones seen at the real scene. For example, a mountain in a photograph is much lower and smaller than the real one seen in the countryside. Similarly, a straight road might appear much longer than a real one. This study attends to such discrepancies between the photograph and our perception. Previous studies have reported that the perceived size and distance of objects in a photograph are different from those perceived in a real space (Gibson, 1947; Smith & Gruber, 1958; Watanabe, 2004). These discrepancies tend to vary depending on photographic conditions, viewing distance between an observer and a photograph, size of a photograph, length of the camera lens used for taking a photograph, etc. For example, Gibson (1947) gave a remarkable account of how observers tend to overestimate the size of a stake by setting a distance of 421.6 m as compared to observers in a large open field. In contrast, the size of the stake in the photograph was underestimated, although its retinal size was identical to the one in real space. In addition, Smith & Gruber (1958) reported that the perceived * Correspondence concerning this article should be sent to: Hidetoshi Miike, Graduate School of Science and Engineering, Yamaguchi University, Tokiwadai Ube, 755-8611, Japan. (E-mail: miike@yamaguchi-u.ac.jp) 2008 Japanese Psychological Association. Published by Blackwell Publishers Ltd.
118 K. Nagata, A. Osa, M. Ichikawa, T. Kinoshita and H. Miike distance in the photograph was longer than the perceived distance in a real space. It was achieved when the visual angle of the photograph was smaller than that in the real space. Otherwise, the gained result was shorter under contrary circumstances. Such discrepancies in both the size and distance of objects are caused by differences in the rules that determine the relationship between the size and distance of the photographed objects and the ones viewed in the real space. On the one hand, in a photograph, the size of an object is determined in accordance with the perspective projection, thus proving that the image size of an object is inversely proportional to the distance between the camera and the object. On the other hand, the perceived size of the object in the real space does not vary in the same manner as does the perspective projection (Holway & Boring, 1941). Several studies concerning size constancy have shown that the perceived size remains at a constant level regardless of whether one considers distance changes (Boring, 1957; Ueno, 1968). These prior studies suggest that the specified object size with respect to the perspective projection always fails in representing our impression at the photographed scene. In order to obtain a photograph that matches our perception, the usual perspective projection rule should be converted to the rules that decide the perceived size of an object at a different viewing distance. The aim of the present study is to find a rule for developing a method that transforms the perspective image into a graphic image that matches our perceived size in the real space. To achieve this aim, a magnification rate needs to be determined to show how the size of objects in the photograph should be magnified in order to match the perceived size of the real space. The magnification rate is defined in terms of the ratio of the size of the object in the photograph (the visual angle of the object) to the perceived size in the real space. If we obtain the rule as a function based on the relationship between the magnification rate and viewing distance, we can create an image that represents the perceived size at different viewing distances. There have been several image-creating methods described that relate to the transformation of the perspective image into an image whose size and shape perception are favorable, that is, the perceptual perspective (Kuroda, 1992; Rauschenbach, 1985; Reggini, 1975). For instance, Reggini (1975) proposed an image-creating method by applying the magnification rate determined by Thouless (1931a, 1931b), which is derived from a study on size constancy. The Z-ratio was approximately 0.6 when each viewing distance was less than 20 m (Mori, 1961). This means that the perceived size in the real space is larger than the size determined by the perspective projection. In the case of two-stimuli comparison based on size constancy, the viewing distance to the comparison stimulus would influence the perceived size of the standard and comparison stimuli (Akishige, 1937; Kuroda, 1961; Mori, 1961). In previous studies, there have been several proposals for a method that transforms the perspective projection to our perception by using a function (Kuroda, 1992; Rauschenbach, 1985; Reggini, 1975). However, they did not investigate how the function is consistent with different viewing distances. In addition, they did not examine whether the images created by their methods would match the perceived size, distance, and location of the objects in real space. In order to realize the intended image-creating method, we need to find an adequate range of the magnification rate for different observers with different viewing distances. In this study, we conducted two experiments. In Experiment 1, we examined the perceived size of an object at different viewing distances in real space with 10 observers. In addition, we investigated the influence of the distance of the comparison stimuli on the perceived size of an object as the standard stimulus at different viewing distances (ranging from 1 m to 10 m). We derived a function based on the magnification rate and viewing distance from the results of Experiment 1. In Experiment 2, we examined if the function is effective and valid for creating images that fit to the perceived size, distance, and location of the objects. In addition, we
Magnification rate to fit to perception 119 investigated an adequate range for the parameters of the function. We conducted the experiment under the usual photographic viewing conditions, that is, the size of the photograph is palm-sized and the viewing distance is approximately 40 cm. We assumed that the usual size of a photograph is observed at arm s length. We applied the results of the experiment to creating an image that matches our perceived size in a real space. EXPERIMENT 1 It is known that the instructions given to the observers would strongly affect their judgment of the perceived size (Carlson, 1962, 1977; Epstein, 1963). An objective instruction requires observers to match the physical size of the comparison stimulus to the standard stimulus, whereas a projective instruction requires them to match the retinal size of the standard and comparison stimuli. The difference between the physical size and the estimated size resulting from objective instruction is smaller than that obtained from projective instruction. In Experiment 1, we used the instruction with respect to the projective size (retinal size, visual angle) because the relative size of the objects in a photograph is equal to the visual angle. Methods Apparatus and stimulus The standard and comparison stimuli were white circles on a black background. Each of the stimuli was presented at the center of a CRT display (17 in., Iiyama A702H). The standard stimulus was observed at 10 different viewing distances (Ds) of 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, and 10.0 m. The sizes of the standard stimuli were 15.3 cm or 8.3 cm in diameter. The visual angles of the standard stimulus ranged from 0.9 deg to 8.8 deg (15.3 cm condition), or from 0.5 to 4.8 arc deg (8.3 cm condition). The size of the comparison stimulus was increased or decreased by the observers manipulation of the computer mouse. The viewing distances to the comparison stimulus Figure 1. Location of the stimuli when Dc was 1.0 m. (Ds) were fixed at three different distances of 1.0, 2.0, and 4.0 m. The directional angle between the standard and comparison stimuli was 25 deg (see Figure 1). Both stimuli were approximately at the observer s eye level. The experiments were conducted in a well-lit room approximately 13.7 m long, 5.8 m wide and 2.7 m high. Procedure Observers viewed the stimuli binocularly. They were instructed to view both the standard and comparison stimuli alternately, and to adjust the size of the comparison stimulus until the apparent visual angles for the standard and comparison stimuli appeared to be equal to each other. Before the experimental sessions, the observers learned about the concept of the visual angle. They adjusted the size of the comparison stimulus to match the size of the standard stimulus. The matching procedure was repeated six times for each condition. The distances of the standard stimulus were chosen at random. Observers The observers were 10 university students, four males and six females, aged 20 24 years. All had either normal or corrected to normal visual acuity.
120 K. Nagata, A. Osa, M. Ichikawa, T. Kinoshita and H. Miike Figure 2. Dependence of the magnification rate on the viewing distance to the comparison stimulus. Figure 3. Dependence of the magnification rate on the normalized viewing distance to the comparison stimulus. The horizontal axis is Ds/Dc. Results and discussion We calculated the magnification rate f(ds), which was defined by the visual angle ratio of the comparison stimulus (the judged size) to the standard stimulus (the physical visual angle). A magnification rate that is greater than 1.0 indicates that the judged size was larger than the physical visual angle. Figure 2 shows the averages of the magnification rates for each Dc condition for all the observers with logarithmic abscissa and ordinate. It shows that the magnification rates rose systematically on increasing the distance from the standard stimulus. We obtained three different functions for three Dc, while the slopes of the functions were similar to each other. The purpose of this experiment was to obtain a function based on the relationship between the retina size of the stimulus and the perceived size in the real space for a given object, which is independent of the distance of the comparison stimuli. Figure 2 shows that the obtained magnification rates for the three Ds conditions are well fitted to the power-law functions, and that the slopes for the three Ds conditions were similar to each other. In addition, we noticed that the magnification rates for all conditions are approximately 1 when the distances of the comparison stimulus and standard stimulus are coincident, that is, Dc = Ds. Therefore, in order to unify the three functions into one, we adopted the following Equation 1. a Ds a f( d) = d, (1) Dc = where Ds is the distance of the standard stimulus and d (= Ds/Dc) is a rescaled distance. The exponent a indicates the slope of the line. Figure 3 shows the rescaled functions for each Dc condition. Table 1 tabulates the a and correlation coefficient for each Dc. The value of a in Equation 1 was approximately constant (a was 0.6) at different viewing distances to the comparison stimulus. These results are compatible Table 1. Values of a and correlation coefficients (r ) depending on the comparison stimuli Dc Experimental condition a r Dc = 1.0 0.59 0.98 Dc = 2.0 0.58 0.99 Dc = 4.0 0.62 1.00
Magnification rate to fit to perception 121 Table 2. Thouless ratio obtained from Experiment 1 and from Epstein (1963) Ds: Distance of the standard stimulus Experimental Condition 3 6 8 9 Dc = 1.0 0.64 0.63 0.6 Dc = 2.0 0.71 0.63 0.59 Dc = 4.0 0.74 0.69 0.63 Dc = 1.5 (Epstein, 1963) 0.49 0.63 0.61 with the study of Makino (1956), who also fitted Equation 1 to the ratio of the perceived size measured by method of transposition to size of the standard stimulus and showed that the value of a was 0.65 for the comparison stimulus at 4 m, although he didn t confirm that a is constant at different Dc conditions. We should note that this Equation 1 is mathematically equal to the equation which derives Thouless s Z-ratio (Thouless, 1931a, 1931b). Parameter a in Equation 1 is identical to Thouless s Z-ratio. Leibowitz (1956) reported that the apparent size measured by the use of the standard and comparison stimuli obeys Thouless s Z-ratio. Next, we compared our experimental results with those from a previous study on size constancy, in terms of Thouless s Z-ratio. Table 2 shows the Thouless s Z-ratios obtained in the present study and in one of the previous studies (Epstein, 1963), in which the measurement used the two-stimuli comparison method with projective instruction. In both studies, the ratios were close to approximately 0.5 0.7. We examined the individual difference between the value of a. Figure 4 shows a for each observer at each distance of the comparison stimulus. While a ranged from 0.40 to 0.87, the as for each observer were rather concentrated on similar values at all distances of the comparison stimulus. We confirmed that a was approximately constant for each observer despite different distances of the comparison stimulus Dc. Therefore, the relationship between the magnification rate and viewing distance could be described by Equation 1 using the standard stimulus s distance, which is normalized by the distance from the comparison stimulus. Figure 4. Individual values of a in Equation 1 for the 10 observers. EXPERIMENT 2 In Experiment 2, we investigated what values are appropriate for a in Equation 1 to create the images of objects that match the perceived size of the objects in a real space. In particular, we examined whether the average of a in Experiment 1 (approximately 0.6) can create images in which the apparent size and distance, as well as the location of the objects, match with those perceived in the real space. In addition, we examined how the Dc condition affects the perception of the objects in viewing the image. In the experiment, observers rated the fitness of the apparent size, distance, and location of the discs in the drawings of the discs with different as and Dcs
122 K. Nagata, A. Osa, M. Ichikawa, T. Kinoshita and H. Miike Figure 5. Location of each disc in Experiment 2. as compared to those of the discs presented in the real space. The rating with respect to the location of the discs was concerned with the impression of the spacing between the discs on the drawing. Methods Apparatus and stimulus Five white discs were placed at different viewing distances (ranging from 1 m to 9 m) in the same room as in Experiment 1. The discs were numbered from the left (see Figure 5). Each disc had a different diameter and was attached to a black iron pole (0.8 cm diameter). Table 3 shows the locations and the sizes of the discs. Because the wall of the room was white, the disc was rimmed with a 0.5-cm-long black line to distinguish between the disc and the wall. The width of the line appeared different, depending on the distance from the disc. The centers of all discs were adjusted to 96 cm above the floor. The eye level of the observer was fixed at a height of 96 cm from the floor by using a chin rest. Each drawing was presented at a leftward location of 55 deg and 41.5 cm from the center of the chin rest. The observers moved their neck while they compared the drawing with the discs in the space. This condition provided an easy situation for observers to carry out their comparisons. We printed 16 line drawings of the discs on a 12.9 cm 8.9 cm sheet of cardboard by combining eight conditions for a with two conditions for Dc in Equation 1 (when Dc was 1.0, a was 0.4, 0.2, 0.0, 0.2, 0.4, 0.6, 0.8, or 1.0; when Dc was 5.0, a was 0.2, 0.1, 0.0, 0.2, 0.4, 0.6, 0.8, or 1.0). The parameters a and Dc varied with the relative size of the discs in a drawing. When a was 0.0, the sizes of the discs were identical to those in a photograph taken using a 35-mm film camera fitted with a 35-mm focal lens (see Figure 6). Thus, in this case, the ratio in the disc sizes in a drawing was identical to that in the retinal size for the stimuli obtained in the real space observation. When a was 1.0, the relation between the disc sizes was identical Table 3. Location and size of the stimulus. Negative values of the angle indicate that the disc was located at the left side of the observer and positive values of the angle indicate that the disc was located at the right side Location Size Stimulus Angle (deg) Distance (m) Diameter (cm) Visual angle (deg) Disc 1 12.5 (left) 1 11 6.3 Disc 2 6.25 (left) 7 15 1.2 Disc 3 0 (center) 5 12 1.4 Disc 4 6.25 (right) 9 13 0.8 Disc 5 6.25 (right) 3 14 2.7
Magnification rate to fit to perception 123 Figure 6. The drawing (a = 0.0) in which the size and location of the discs was identical to the photograph taken by a single-lens reflex camera with a 35-mm focal lens. to the relation between their physical sizes. In eight drawings using different as, one disc located at distance Dc in the room did not vary its size in the drawing. In other words, when Dc was 1.0, the size of disc 1 (located at 1.0 m) in eight drawings was the same as in the photograph, regardless of varying a. We determined that the locations of the center points for the five discs in the drawing were the same as those in the photograph when Dc was 1.0 (see Figure 7). Procedure Observers viewed the discs binocularly in the real space and in the drawings, alternately. They rated the size, distance, and location of the discs in the visual field while viewing the drawings by using 7-point scales, ranging from the sizes (distances, or locations) are terribly different from those in the real space to they are just the identical to those in the real space. The observers were directed to rate how to match the apparent sizes, distance, and subsequently, the locations of the discs in the drawings to those in the real space. We presented the drawings to the observer in a random order. They were prevented from using any tools for measuring the image sizes during the experimental session. Observers The observers were 19 university students, 10 males and 9 females, aged 20 34 years. Figure 7. Examples of the drawings used in Experiment 2. The size of the disc located at Dc was identical to that in Figure 6. (a) Dc = 1.0, a = 0.4. (b) Dc = 5.0, a = 0.4. Ten observers participated in the condition in which Dc was 1.0, while the others participated in the condition in which Dc was 5.0. Four observers took part in Experiment 1. Results and discussion Figure 8 shows the averages and standard errors (SE) of the size rating for each a. The ratings were above the neutral line (3.0) for 0.2, 0.4, and 0.6 of a, while the ratings for 0.0 relating to the perspective projection were below the neutral line. We conducted a two-way ANOVA with repeated measures on one factor. In other words, the within-factor was the a condition ( 0.2, 0.0, 0.2, 0.4, 0.6, 0.8, or 1.0), and the between-factor was the Dc condition (1.0 or 5.0). The main effect of a condition was significant, F(6,102) = 12.40, p < 0.001, while the main effect of the Dc condition and the interaction of the two factors were not significant. Tukey s
124 K. Nagata, A. Osa, M. Ichikawa, T. Kinoshita and H. Miike Figure 8. Averages of the size ratings for each a. Figure 10. Averages of the location ratings for each a. Figure 9. Averages of the distance ratings for each a. post hoc HSD test showed that there were significant differences between a = 0.4 and the other four conditions (a = 0.2, a = 0.0, a = 0.8, and a = 1.0; p < 0.05). Figure 9 shows the averages and SE of the distance rating for each a. The ratings had their peaks at a = 0.2, both when Dc was 1.0 and 5.0, while the ratings for a = 0.0 were above the neutral line, when Dc was 1.0. We conducted the same two-way ANOVA that was conducted for the results of the size rating. The main effect of the a condition was significant, F(6,102) = 21.31, p < 0.001, while the main effect of the Dc condition was not significant. The interaction between the two factors was significant, F(6,102) = 2.39, p < 0.05. Tukey s HSD test for the interaction showed that when a = 0.4, there were significant differences between the two Dc conditions. In addition, when Dc was 1.0, there were significant differences between a = 0.2, and the conditions of a = 0.8 and a = 1.0 (p < 0.05). When Dc was 5.0, there were significant differences between a = 0.2 and the other four conditions (a = 0.2, a = 0.6, a = 0.8, and a = 1.0; p < 0.05). Figure 10 shows the averages and SE of the location rating for each a. The ratings for 1.0 of Dc had their peaks at the a = 0.4 condition while the ratings for 5.0 of Dc had their peaks at the a = 0.2 condition. The drawings in terms of the perspective projection (a = 0.0) would obtain a high rating for the location of the discs in the visual field when Dc was 5.0, while its rating would fall to the neutral level when Dc was 1.0. Again, we conducted the same two-way ANOVA. The main effect of the a condition was significant, F(6,102) = 25.04, p < 0.001, while the main effect of the Dc condition was not significant. The interaction between the two factors was significant, F(6,102) = 2.459, p < 0.05. Tukey s HSD test for the interaction showed that when a = 0.6, there were significant differences between the two Dc conditions. When Dc was 1.0, there were significant differences between a = 0.4 and the conditions of a = 0.8 and a = 1.0 (p < 0.05). When Dc was 5.0, there were significant differences between a = 0.2 and the other three conditions (a = 0.6, a = 0.8, and a = 1.0; p < 0.05).
Magnification rate to fit to perception 125 The conditions applied in Equation 1 with a ranging from 0.2 to 0.6 received a significantly higher size rating than those whose object size was determined in accordance with the perspective projection, as in the normal photograph (a = 0.0). These results are compatible with those in Experiment 1, thus revealing that the average of a from 10 observers was approximately 0.6 for different Dc. In addition, we found that there were conditions in which the distance and location ratings, as well as the size rating, were higher than the rating for the perspective projection (a = 0.0). For instance, for the distance and location, the ratings were the highest when a = 0.2 for both Dc conditions. These results suggest that the transformation of the perspective images in terms of Equation 1 would be effective and valid to create images that are more suitable than the perspective image to our perception concerning the size, distance, and location for the object viewed in a real space. Several researchers have described the relationship between perceived size and perceived distance as a positive correlation for the observation in a real space (Kilpatrick & Ittelson, 1953; Oyama, 1974). The results of the present experiment, showed that the size rating was positively correlated with the distance rating. The correlation coefficient was 0.66 (p < 0.001). Thus, we confirmed that the method for transforming object size in an image to perceived size in a real space would contribute toward representing perceived distance when viewing a photograph. In Experiment 1, we used the traditional method of size constancy. In this method, the observer changed the size of the comparison stimulus to match the size of the standard stimulus. However, Oyama (1959) argued that a method without a change in the stimulus size during the observation would be more preferable because the change in the stimulus during the observation might affect the perception. Consequently, he proposed the method of transposition (Oyama, 1959; Oyama & Sato, 1967). The method of Experiment 2 was similar to the method of transposition; there was no stimulus change during the observation. The small SE in Experiment 2 suggests that the method used in Experiment 2 had the effect of reducing intra- and interobserver variability (Makino, 1956). GENERAL DISCUSSION As described in the Introduction, several studies have shown the influence of the experiment stimulus distance on perceived size (Akishige, 1937; Kuroda, 1961; Mori, 1961). In Experiment 1, we confirmed that the relationship between the magnification rate and viewing distance could be described using Equation 1, which is independent of the viewing distance to both standard and comparison stimuli, as well as Thouless s Z-ratio. Although Makino (1956) fitted the apparent size obtained with the comparison stimulus at 4 m to the equation, which is mathematically equal to Equation 1, he did not examine whether the equation could be applied to the data from various distances of the comparison stimuli. This single equation, which is independent of the viewing distance, would be useful in transforming the perspective image into an image, which represents the perceived size in the real space by the use of a computer program. We need distance information to apply Equation 1 in transforming photographic images because the size of the object in the image should vary in accordance with the distance from the object. In three-dimensional computer graphics, the distance information is given in advance. Thus, we can apply Equation 1 to create an image that presents the perceived size using computer graphics. In a photograph, however, it is usually difficult to get the object distance information. It is expected that a stereo system for digital cameras will be developed to obtain this information. As mentioned in the Introduction, Reggini (1975) proposed a method to transform the perspective image into the perceptual perspective by using Thouless Z-ratio, which is independent of viewing distance. However, he did not conduct any experiments to prove the validity of his method. In our study, we found that Equation 1, which is also useful for obtaining the magnification
126 K. Nagata, A. Osa, M. Ichikawa, T. Kinoshita and H. Miike rate, is veridical for various viewing distances for both standard and comparison stimuli (Experiment 1). In addition, we proved that the drawing created by the parameters derived from the results of Experiment 1 was more suitable to our perception concerning size, distance, and location compared with the perspective image. There are still several challenging problems in our method, for future studies. First, the a obtained from the results of Experiment 1 did not seem to be the best one for the transformation of the perspective image into a suitable image for our perception of the size, distance, and location of the discs in Experiment 2. For example, the range of a caused a high rating for the disc size and distance that was slightly smaller than that obtained from 10 observers in Experiment 1, which ranged from a = 0.40 to a = 0.87. There were several differences in the stimulus and setting of the environment between Experiment 1 and 2, such as the number of discs, the luminance of the background, and the task of the experiments, among others. These differences might cause differences in the range of a between the two experiments. In future studies, we should find what factors determine the appropriate range of a for representing a suitable image for our perception. In addition, the significant interactions between a and Dc for the distance and location of the discs in the results of Experiment 2 suggest that a represents a suitable distance and location depending on the values of Dc. In order to obtain an appropriate a for all apparent sizes, distances, and locations of objects in a real space, future studies will have to find a method to take Dc into consideration when determining a. References Akishige, Y. (1937). Effekt der Entfernung des Darbietungsortes der Normalgrose auf den Grad der Grossenkonstanz. Mitteilung der Juristish- Literarischen Fakultät der Kaiserlichen Kyushu Universität, 4, 37 58. Boring, E. G. (1957). A history of experimental psychology, 2nd edn. Englewood Cliffs: N.J. Prentice Hall. Carlson, V. R. (1962). Size-constancy judgments and perceptual compromise. Journal of Experimental Psychology, 63, 68 73. Carlson, V. R. (1977). Instructions and perceptual constancy judgments. In W. Epstein (Ed.) Stability and constancy in visual perception: Mechanisms and processes. New York: John Wiley & Sons, pp. 217 254. Epstein, W. (1963). Attitude of judgments and the size-distance invariance hypothesis. Journal of Experimental Psychology, 66, 78 83. Gibson, J. J. (1947). Motion picture testing and research (Aviation Psychology Research Reports, No. 7). Washington, DC: U.S. Government Printing Office. Holway, A. H., & Boring, E. G. (1941). Determinants of apparent visual size. American Journal of Psychology, 54, 21 37. Kilpatrick, F. P., & Ittelson, W. H. (1953). The size-distance invariance hypothesis. Psychological Review, 60, 223 231. Kuroda, M. (1992). Kukan wo egaku enkinhou [Perspective drawing space]. Tokyo: Kaiseisya (In Japanese, translated by the author of this article). Kuroda, T. (1961). Experimental studies on size constancy. Bulletin of the Faculty of Literature of Kyushu University, 7, 59 102. Leibowitz, H. (1956). Relations between the Brunswick and Thouless ratios and functional relations in experimental investigations on shape, size and lightness. Perceptual and Motor-Skill, 6, 65 68. Makino, T. (1956). Mie no okisa to kyori tono kankei ni tsuite [The problem of perceived size and distance]. Studies in the Humanities [Jimbun Kenkyu]. Osaka: The Faculty of Literature, Osaka City University, 7, 235 250. (In Japanese, translated by the author of this article). Mori, T. (1961). Okisa no kojo to kansatsu kyori no kankei ni tsuite [Relationship between size constancy and viewing distance (1)]. Proceedings of the 25th Annual Meeting of the Japanese Psychological Association, 25, 36. (In Japanese, translated by the author of this article). Oyama, T. (1959). A new psychophysical method: Method of transposition or equal-appearing relations. Psychological Bulletin, 56, 74 79. Oyama, T. (1974). Perceived size and perceived distance in stereoscopic vision and an analysis of their causal relations. Perception and Psychophysics, 17, 175 181. Oyama, T., & Sato, F. (1967). Perceived size_ratio in stereoscopic vision as a function of convergence, binocular disparity and luminance. Japanese Psychological Research, 9, 1 13.
Magnification rate to fit to perception 127 Rauschenbach, B. V. (1985). Perspective pictures and visual perception. Leonardo, 18, 45 49. Reggini, H. C. (1975). Perspective using curved projection rays and its computer application. Leonardo, 8, 307 312. Smith, O. W., & Gruber, H. (1958). Perception of depth in photographs. Perceptual and Motor Skills, 8, 307 313. Thouless, R. H. (1931a). Phenomenal regression to the real object. I. British Journal of Psychology, 21, 339 359. Thouless, R. H. (1931b). Phenomenal regression to the real object. II. British Journal of Psychology, 22, 1 30. Ueno, T. (1968). Okisa no kojosei no kenkyu [A study of size constancy]. Studies in the Humanities [Jimbun Kenkyu]. Osaka: The Faculty of Literature, Osaka City University, 20, 65 150. Watanabe, T. (2004). Anisotropy in depth perception of photograph. Japanese Journal of Psychology, 75, 24 32 (In Japanese with English abstract). (Received May 15, 2006; accepted March 1, 2008)