Stereopsis is the capacity of the visual system to obtain three-dimensional information from the exterior world by the processing of disparity. ^{1–3} This is a complex task for our visual system and, for accurate execution, requires the precise functioning of the motor and sensory pathways. Stereopsis provides a vivid sensation of depth and object disposition that is superior, in most cases, to that possible with monocular cues.

Given the paramount importance of spatial information derived from stereopsis, numerous studies have been conducted on this phenomenon, using different experimental parameters and conditions. Within the practice of optometry, many factors influence binocular vision and stereopsis, one of which is the decentration of ophthalmic lenses. ^{4, 5} Such decentration changes fusional convergence and can thereby alter certain aspects of binocular vision, such as fusion, and can cause such dysfunctions as eye fatigue or headaches. ^{4–5} It is known that the prismatic effects of refractive lenses can distort information concerning three-dimensional perception, such as the position of objects. Fogt and Jones ^{6} published a study on the perceived distortion created by the prismatic effects of refractive lenses, demonstrating that the prismatic effect of the spectacle lenses results in inappropriate extra-retinal eye-position information in a dark environment. Some effects that changes in convergence exert on stereopsis have been analyzed by different authors (see Howard and Rogers ^{3}). In terms of the influence of vergence variations on disparity limits, the work of Fry and Kent ^{7} is notable; in this work with vertical rods, losses in stereoacuity are experimentally demonstrated for some observers when the stimulus to accommodation is kept constant and changes in convergence are induced by means of base-in and base-out prisms.

The aim of our study is to analyze whether variations in fusional convergence induced by the differential prismatic effects of decentered ophthalmic lenses influence maximum disparity. For this, we decentered (both horizontally and vertically) the spectacle lenses of the subjects and calculated the differential prismatic effect. We evaluated the effect of decentration determining the parameter known as the disparity range, using random-dot stereograms (RDS). We used the RDS instead of other tests with recognizable monocular information ^{1–3} because, in this type of test, ^{1–3, 8} disparity is the only information available for the observer to obtain three-dimensional spatial information. Thus, when the two images forming the stereogram are presented, the observer receives no monocular information that would indicate the shape of the object, which could thereby mask the processing of the disparity from which stereoscopic information is derived.

As indicated above, we used the disparity range to examine the influence of convergence variations in stereopsis. Essentially, with this parameter, it is possible to measure the maximum disparity possible within the central square that generates stereopsis in the RDS; as the displacement increases, so does the sensation of depth. With this parameter, different issues and experimental parameters related to stereopsis have been studied. For example, by varying the size of the RDS dot and analyzing the variations in the disparity range, Marr ^{9} carried out theoretic studies to gather information on stereoscopic correspondence at different spatial frequencies. Jiménez et al., ^{10} studying the effect that associated phoria exerts on stereopsis, found a decrease in the disparity range as the associated phoria increased in value. In addition, this parameter has been studied experimentally to clarify further the relationship between color and stereopsis. ^{11–13} With Gabor stimuli, maximum disparity has also been used to explore the influence of size and spatial frequency content. ^{14} From a practical standpoint, the importance of disparity range (or maximum disparity) resides in the fact that a greater disparity range indicates more efficient stereoscopic perception, because the stereoscopic correspondence involves a larger area of the retina and thus a greater region of the physical space perceived stereoscopically.

#### METHODS

##### Decentration of Lenses

Mounted ophthalmic lenses were horizontally decentered, both nasally and temporally, with respect to the center of the subject’s pupils. In addition, the lenses were decentered in both vertical directions. The horizontal decentrations were made in the same frame; in the vertical case, the lenses were given the desired decentration in mm. In the horizontal case, we made four decentrations in 2-mm increments, each from 1 mm-decentration, and the vertical decentrations were 1 and 2 mm. More horizontal displacements were tested because the binocular visual system has more tolerance for horizontal prismatic effects, as we will see when we compare the results with the ANSI Z80 Standards and the British Standards. ^{5} In any event, our principal objective was to identify the minimum value of the prismatic effect that significantly decreases the disparity range. However, had the maximum value of 2 mm in the vertical case proved insufficient, we would have increased the decentration. Thus, each observer was tested for 24 horizontal decentrations (8 for the right eye and 8 for the left and 8 for both eyes in opposite directions) and 12 vertical decentrations (4 for each eye and 4 for both in opposite directions). In the case of observer AP, the frame as well as the observer’s interpupillary distance restricted the number of horizontal decentrations to 7 (a total number of 21), rather than 8 as in the other observers. The values of the disparity range were compared with those obtained for lenses with a centration that was correct for the distance used.

In all cases, the decentration of the lens in relation to the position of the subject’s pupils caused a prismatic effect with a module given by Prentice’s rule ^{4} :MATHMATHwhere P_{V} = vertical prismatic effect at the point where the prismatic effect is calculated in prism diopters (Δ; P_{V} is base “up” when positive); P_{H} = Horizontal prismatic effect at the point where the prismatic effect is calculated in prism diopters (P_{H} is base “out” when positive); F_{S} = power of the spherical component of the lens; F_{C} = power of the cylindrical component of the lens; θ = axis direction of cylinder reckoned from the nasal side of the lens for each eye; x = horizontal displacement of point where the prismatic effect is calculated from the optical center (m) (x is positive if the optical center lies on the temporal side of the point where the prismatic effect is calculated for each eye); and y = vertical displacement of point where the prismatic effect is calculated from optical center (m) (y is positive if the optical center lies above the point where the prismatic effect is calculated for each eye).

Equation U1 Image Tools |
Equation U2 Image Tools |

The horizontal and vertical differential prismatic effects, δP_{H} and δP_{V}, respectively, are determined through the expressions:MATHMATHwhere P^{R}_{H} and P^{R}_{V} are the horizontal and vertical prismatic effects induced in the right eye and P^{L}_{H} and P^{L}_{V} are the ones induced in the left eye.

Equation U3 Image Tools |
Equation U4 Image Tools |

Finally, the total (module) differential prismatic effect, δP, is given by:MATH

##### RDS and Disparity Range

The stereo images of the RDS were generated with a 50% density and presented on a NEC Multisync monitor connected to a PC. A Wheatstone stereoscope device with front surface mirrors was attached to the front of the display monitor. This device, fully described in the literature, ^{1–3} enabled the left- and right-hand images of the stereo pair to be projected to the left and the right eye, respectively, resulting in a central stereo image. Head position was stabilized with a chin and forehead rest. The block or dot size used in the RDS patterns subtended 2′ from the observation point. Disparity increments were generated by displacement in the small central square of both images and had a minimum value of 4′. The size of the central square stereoscopically perceived was 80′, the separation between squares that generated the RDS was 1°, and the size of each pattern was 4°.

As indicated above, the parameter used to evaluate stereoscopic depth perception is the horizontal disparity range. The term “disparity range” is used to signify the maximum horizontal displacement (disparity) possible in the small central square in the RDS that still allows depth perception. Any influence of lens decentration on stereoscopic perception can be detected by determining this parameter. As we indicated above, larger disparity ranges under certain experimental conditions seem to indicate more efficiency in the processing of disparity information by carrying out the stereoscopic correspondence over a greater disparity range. ^{9–13}

The disparity range for each decentration was determined using the constant-stimulus method. ^{1, 10, 12, 13} There are two possible spatial configurations for the stereogram: in front (or crossed disparity), where the central square appears closer to the observer and behind (or uncrossed disparity) where the central square appears farther from the observer. When a spatial configuration is set, stimuli with random disparity ranging in 4′ steps from 0′ to the next disparity value up to the maximum disparity range are presented to the observer.

The maximum disparity range, which is a preliminary determination carried out before the constant stimulus method is applied, was determined as follows: the RDS was presented with a value of 0′ and the disparity was successively increased in 4′ steps until the observer no longer perceived the square stereoscopically. This operation was repeated three times, and we chose the most repeated value as the final value of the maximum disparity range. The maximum disparity range was determined for each decentration tested and for the crossed and uncrossed situation. Each disparity was presented a total of 30 times and was characterized by a weight factor equal to the number of times that stereoscopic depth perception was recorded. ^{10, 12, 13, 15}

The stimulus presentation time was 0.5 s, ^{10, 12, 15} and the time between presentations was 1.25 s. The duration of the RDS presentation was sufficient to initiate the vergence movements to compensate for the prismatic effect.

The observer’s task was to press a button when the square was not stereoscopically perceived, in front or behind, depending on the case, and this information was stored in a program. Disparity value zero was also presented to confirm the validity of the method with regard to false-alarm-rate analysis, ^{16} although the observer’s response was right in practically 100% of the cases.

Sessions included 3 min of prior adaptation to the dark screen background. In darkness adaptation, the computer monitor frame was not visible. Afterward, the stimuli generated with different disparities were presented randomly, discarding the first stimuli (randomly from 3 to 7) of the session, these were considered preparatory trials intended to maintain the observer’s concentration. Between presentations, one of the stimuli generating the stereogram was used to fill in the RDS area completely to aid fusion ^{15}; at this moment, the observer perceived only a 4° square. Therefore, after the time between presentations, the observer directly perceives the RDS without needing to start fixation and fusion, which could induce undesired prismatic effects. ^{2} The observers were instructed to fixate on the complete RDS.

For a given decentration, we estimated the (arithmetic) disparity mean from the distribution of weights associated with the disparities corresponding to each spatial configuration (crossed and uncrossed). Because each experimental condition had two configurations (in front and behind), we considered the disparity range as the average of the two disparity means. Two disparity ranges (or a set) were not considered significantly different when their corresponding disparity means for the two spatial configurations did not significantly differ. Significant difference analysis was performed by analysis of variance with a level confidence of 95% and posthoc comparisons. ^{17} For each observer, once the type of decentration (horizontal or vertical) and the decentered eye were selected, an analysis of variance was conducted for the sets of disparity means calculated for all the decentrations assayed as well as for nondecentration. We tested whether the set of disparity ranges differed significantly and, most importantly for the aim of our study, whether the disparity range without decentration was greater than the disparity means calculated with decentration. Figures with results also include the average of the two mean estimation intervals with a 95% confidence level.

##### Observers

Six trained ametropic observers (4 myopes and 2 hypermetropes) with normal stereoscopic acuity (stereo-fly tests) took part in the experiments (Table 1). This number of observers is higher than usual in this type of experiment; Jiménez et al. ^{12–13} used 2 and 4 observers in different experimental phases, and Isono and Yasuda ^{11} tested 3 observers. To avoid any possible conditioning of observer response, no observer, except for JR, who manipulated the experimental device, knew the amount of decentration at which the test was conducted or received any information concerning the partial results of the experiment.

#### RESULTS

For all the observers in all the decentrations tested, and for low disparity values, fusion and stereopsis persisted. Despite the high degree of decentrations tested, the two halves of the stereogram were fused with clear stability in most cases. The observers AA and JL, with strong ametropia, initially showed a certain difficulty perceiving the test in the extreme cases of decentration, 2 mm in the vertical and 5 and 7 mm in the horizontal cases, a situation caused by the strong prismatic effect generated in these cases (for example: right eye, 7 mm horizontal decentration: 3.91Δ for AA and 4.34Δ for JL). With respect to fusion, these results are to be expected because of the tolerance that fusion shows under prismatic effects as Ogle ^{18} showed. In addition, we should take into account that this particular type of test, the RDS, provides a broad tolerance to experimental conditions that could induce fusion difficulties. Thus, for example, the RDS tolerates a high aniseikonia (in some cases as high as 15%) above the level tolerated in the fusion of real images, severe defocusing of one of the images, as well as a contrast reduction of some of the images. ^{9}

Table 2 presents the weights corresponding to one of the conditions tested (in front, no decentration, observer JL) in the determination of the disparity range. The observer detected the small disparity values, and, as the disparity was increased, the weight progressively decreased, thereby justifying the choice of the constant-stimulus method. The value of the maximum disparity range (36 min arc for the case shown in Table 2) would not be a numerically relevant parameter, because random presentation of the stimuli reduces memory effects ^{9, 10, 12} and the high disparity values are less likely to be detected, as their lower numerical weight values indicated. Therefore, the disparity mean constitutes an appropriate numerical parameter to quantify the stereopsis in each spatial configuration (in front or behind), because the weight distribution is taken into account. Later, as indicated in Methods, we take the disparity range (the average of the two disparity means) as the representative parameter for evaluating the stereopsis in each decentration case.

Table 3 presents for all the observers the values of the minimum differential prismatic effects caused by decentration, both in vertical and horizontal, for which significant statistical differences were found in the disparity ranges (p < 0.05, case in front or behind) with respect to the disparity range found without decentration. For decentrations higher than the prismatic effects shown in Table 3, the stereoscopic perception diminished as shown by the reduced disparity range. The region where stereoscopic correspondence was possible was therefore reduced.

#### DISCUSSION

These results lead, as Fry and Kent pointed out, ^{7} to the conclusion that the convergence, in some way, influences the stereo-correspondence problem. Concerning minimum disparity, these results could be expected. Howard and Rogers’ book ^{3} describes experiments ^{19–21} that show some deterioration in stereoacuity and depth-discrimination thresholds caused by changes or noise in the vergence angle. As indicated above, Fry and Kent ^{7} found a deterioration of stereoacuity for some observers because of the prismatic effects induced by prisms, although we should indicate that these experiments were not performed with RDS but with different types of tests including rods. With reference to maximum disparity, we have found no results in the literature from researchers working with RDS that are similar to ours. Using Gabor stimuli, Wilcox and Hess ^{14} determined maximum and minimum disparity and found important differences between the processing of fine (minimum) and coarse (maximum) disparity, proposing that stereopsis is achieved at large disparities by way of nonlinear processing. In addition, it is accepted that separate disparity estimates are generated on each spatial scale, ^{22} in which convergence information is taken into account in determining the overall perceived depth. Thus, our results could be compatible with the possibility that convergence influences the nonlinear processing that determines the maximum disparity.

Here, we should emphasize the sensitivity of the disparity range (maximum disparity) to certain experimental conditions. For example, small variations—even 1Δ—in associated phoria can diminish the average stereoscopic capacity by reducing the disparity range. ^{10} Low values of associated phoria that do not affect binocularity aspects, such as fusion or the fact of having or not having stereopsis, do influence the quality of stereopsis by decreasing the expected maximum disparity.

Figs. 1 to 6 show some results for the six observers under certain decentration conditions (we did not show the average results for all the observers, because each observer has a different prescription and therefore the prismatic effects corresponding to each decentration differ in each case). Fig. 1 (vertical decentration, right eye, JA) shows the decline in the disparity range with respect to null decentration. The decentrations corresponding to ± 1 and ± 2 mm caused prismatic effects exceeding 0.23Δ (Table 3), and their corresponding disparity ranges differed significantly (p < 0.05, case in front or behind) with respect to null decentration. The same situation applies to Figs. 2 to 5, which show that for decentration causing prismatic effects greater than those shown in Table 3 (see each decentration for each observer) the disparity ranges are significantly smaller than for the nondecentration situation. For observer AP (Fig. 3), the prismatic effect corresponding to 1 mm vertical decentration of the right eye is 0.54Δ, which significantly decreases the disparity range. Table 3 indicates that for this observer, the minimum vertical decentration that causes a significant decline is 0.39Δ, not 0.54Δ; this is explained by the fact that 0.39Δ results for 1 mm vertical decentration, but for the left eye. For observer JR (Fig. 6, right-horizontal decentration), the horizontal power corresponding to the lens of the right eye was very low and therefore caused a slight prismatic effect for the decentration tested (a maximum prismatic effect of 0.18Δ). Therefore, the differences found for the disparity ranges are not significant (p > 0.05, in front and behind). One salient aspect in the figures is the great variability between the results for the observers; the values of the disparity ranges differed notably from one observer to the other, as is normal in any type of stereoscopic depth-perception experiment and particularly in relation to disparity ranges. ^{11–13} It is worth noting that stereopsis deteriorates with respect to null decentration for decentrations with prismatic effects greater than those given by Table 3. But if we continue to increase the decentrations, the disparity ranges do not generally continue to decline (Fig. 1 and 5), although under some experimental conditions for some observers, such a decline does continue (Fig. 2 - 4), although slightly. This fact could not be confirmed by the study of Fry and Kent, because no data were given in this regard. ^{7}

Figure 1 Image Tools |
Figure 2 Image Tools |
Figure 3 Image Tools |

Figure 4 Image Tools |
Figure 5 Image Tools |
Figure 6 Image Tools |

To provide practical applications for our results, we compared the values of the prismatic effects for which the disparity range significantly decreased (Table 3) with the tolerances ^{5} to prismatic effects given by the rules ANSI Z80 Standards (1/3Δ for the vertical decentration and 2/3Δ for the horizontal) and in the British Standards (0.25Δ vertical and 1.0Δ horizontal). For horizontal decentration, we detected a decline in stereoscopic vision corresponding to values of prismatic effects lower than the tolerance level in both rules. Thus, for the six observers, the horizontal prismatic effect was clearly less than that given by the rule of the British Standards and also for five observers using the ANSI rule. JR appeared at the limit of the ANSI rule.

For vertical decentration, we detected a worsening of the stereoscopic vision for prismatic effects lower than the ANSI rule for three observers (including observer JR, who appeared at the limit for the horizontal case for this rule) and AP appeared at the limit. In the case of the British Standard rule, only one observer would be below the limit and another at the limit. With respect to the results, because of the individualized correction for each observer and the discrete decentrations tested, an exact continual variation in the prism effect was not possible. Therefore, detecting the exact value at which statistically significant differences appeared was not possible, either. It would be expected that the value was even lower than the data given in Table 3. However, in any event, the results are sufficient to demonstrate that, in general, prismatic effects even lower than in the ANSI and British Standards may diminish stereoscopic vision, at least in the horizontal case. These results indicate the marked sensitivity of stereopsis to decentration; therefore, it is fundamental from a practical standpoint to ensure, as far as possible, the precision of lens centering in the frame, even though the observer, with the sensation of clear fusion, may not be conscious of the decentration.

##### Experiments with Figural-Stimuli Stereograms and Prisms

At this point, our interest was to generalize our results to a wider set of experimental conditions. On the one hand, as prismatic effects caused by decentration of ophthalmic lenses generated the convergence variations, we also proceeded to evaluate the disparity range by generating the prismatic effects with prisms. On the other hand, our experiments were made with random-dot stereograms, because, as indicated above, the observer receives no monocular information that could thereby mask the processing of the disparity. In any case, under normal conditions for an observer **,** the stereoscopic information is acquired through situations in which information on disparity is presented simultaneously with recognizable monocular information. To ascertain whether our results with recognizable monocular information is valid, we performed our experiments using a test with recognizable monocular information (figural-stimuli stereograms). For this, we used wallpaper stereograms, this being a widely used figural-stimuli test. ^{13, 23, 24}

In the first experiment, we used prisms 0.5Δ in the vertical case and 1Δ in the horizontal. Four observers (AP, AA, JM, JL) participated in these experiments **,** and for each observer tested, the prisms were located in the frame with the lenses correctly centered. We tested the prisms using the same conditions as with the lens decentrations, situating the prisms in each eye and in both with opposite deviations. We did not use the prisms of 0.5Δ in the horizontal case because, according to previous results, this would be at the limit of significant decrease of the disparity range. In any event, it was not our aim to calculate the exact value of the significant decrease, and this is not possible with discrete variations of 0.5Δ. Rather, the intent was to test our results by means of other optical elements (prisms) widely used in studies of optometry and visual experiments. The results confirm the previous ones on detecting statistically significant decreases (p < 0.05, in front or behind) in the disparity range; the four observers detected a significant decrease in the disparity range for all the cases tested. It is noteworthy that 0.5Δ in the vertical case significantly reduced the disparity range for observer AA, whereas decentration of ophthalmic lenses gave a value of 0.59Δ in the vertical case (Table 3) for a significant reduction in the disparity range. As indicated above, this fact is explained by the discrete values tested in the case of decentration with ophthalmic lenses. As for decentration of ophthalmic lenses, our results indicate the sensitivity of stereopsis to prismatic effects.

With respect to figural-stimuli stereograms, as indicated above, the test used was the wallpaper stereograms. Basically, this test is composed of various groups of regularly spaced vertical bars in each of the stereo pair. A group of bars in the center of each image of the monitor (fixation bars) ensure fusion, whereas the upper and lower bars in each image are shifted in opposite directions with respect to the fixation bars to generate disparity when the bars are fused. When the disparity is generated the observer perceives a group of bars in front of and another behind the fixation bars (for a more detailed explanation of this test see Jiménez et al., ^{10} Jordan et al., ^{23} and Woodworth ^{24}). The experimental procedure for determining the disparity range was the same that for random-dot stereograms. Three observers (JM, JL, AA) took part in this experiment.

Disparity ranges were determined with no decentration of ophthalmic lenses and for the minimum horizontal and vertical decentrations (Table 3) for which significant differences in the disparity ranges were detected using the random-dot stereograms. The results were the same for the three observers; a significant decrease in the disparity range appeared for the decentrations tested with respect to the null decentration. Thus, the dependence of maximum disparity with convergence variations is maintained with stereograms containing recognizable monocular information, confirming the sensitivity of stereopsis to a wide variety of experimental conditions and type of tests.

#### CONCLUSIONS

In the present study, we have experimentally tested the effect on stereopsis exerted by convergence variations generated by decentration of spectacle lenses. Our study indicates that low prismatic values cause a decline in stereopsis reducing the disparity range. This result, in accord with those reported by other researchers, shows that the phenomenon of convergence can influence the process of stereo-correspondence, reducing under certain experimental conditions the maximum disparity processed. These results are confirmed with figural-stimuli stereograms and when prisms are used to generate the prismatic effects. Our results suggest that this reduction in disparity range with respect to correct centration could have practical implications, because for distances of less than 1 m, where disparity is critical, the manipulation of objects could be less efficient because the disparity range is lower; thus, the physical region with stereoscopic vision is diminished.

#### ACKNOWLEDGMENT

We wish to thank the anonymous reviewers who have helped to improve the manuscript.

This research was supported by the DGICYT, Ministerio de Educación y Ciencia (Spain), grant no. PB96–1454.