Physical Oracles: The Turing Machine and the Wheatstone Bridge

Earlier, we have studied computations possible by physical systems and by algorithms combined with physical systems. In particular, we have analysed the idea of using an experiment as an oracle to an abstract computational device, such as the Turing machine. The theory of composite machines of this kind can be used to understand (a) a Turing machine receiving extra computational power from a physical process, or (b) an experimenter modelled as a Turing machine performing a test of a known physical theory T. Our earlier work was based upon experiments in Newtonian mechanics. Here we extend the scope of the theory of experimental oracles beyond Newtonian mechanics to electrical theory. First, we specify an experiment that measures resistance using a Wheatstone bridge and start to classify the computational power of this experimental oracle using non-uniform complexity classes. Secondly, we show that modelling an experimenter and experimental procedure algorithmically imposes a limit on our ability to measure resistance by the Wheatstone bridge. The connection between the algorithm and physical test is mediated by a protocol controlling each query, especially the physical time taken by the experimenter. In our studies we find that physical experiments have an exponential time protocol; this we formulate as a general conjecture. Our theory proposes that measurability in Physics is subject to laws which are co-lateral effects of the limits of computability and computational complexity.     

The stereoscopic sliver: a comparison of duration thresholds for fully stereoscopic and unmatched versions

Just as positional disparities of image features seen with both eyes provide depth information, the presence of an area visible to one eye but not the other within a binocularly viewed scene can indicate an occlusion at a depth discontinuity. The close geometrical association between these two kinds of cues suggests they may both be exploited by stereopsis. To investigate this, we developed a novel binocular stimulus entirely lacking in classical disparity that contains an unmatched vertical sliver which elicits a warping of the surrounding surface to accommodate a depth discontinuity. We measured depth-discrimination performance at a range of stimulus durations, correcting for variations in stimulus visibility, to characterise the decline of the efficacy of the depth signal with limited integration time. Results show a close correspondence of performance for similar stimuli with unmatched features and classical binocular disparity across a sixtyfold range of viewing durations, supporting the notion of a close association between the two types of cues in human stereopsis. Control experiments excluded simple eye-of-origin cues and long-range false matches as explanatory factors.     

The effects of stereoscopic depth on completion

Stereoscopic depth has a critical effect on completion of partially occluded figures. However, it has not strictly been distinguished whether the effect is direct or indirect through alteration of contour segmentation or parsing. Here, I report that stereoscopic depth does not influence completion of partially occluded figures when parsing is unambiguous from motion cues. This is consistent with the present proposal that stereoscopic depth does not have a unique role in completion and that it is one of the cues to contour segmentation or parsing, which in turn influences completion and surface representation, like motion, shape, or transparency.     

The discriminability of smooth stereoscopic surfaces.

In this study the sensitivity of human vision to the smoothness of stereoscopic surface structure was investigated. In experiments 1 and 2 random-dot stereograms were used to evaluate the discrimination of smooth versus 'noisy' sinusoidal surfaces differing in the percentages of points on a single smooth surface. Fully coherent smooth surfaces were found to be much more discriminable than other less smooth randomly perturbed surfaces. In experiment 3 the discrimination between discontinuous triangle-wave surfaces and similarly shaped smoothly curved surfaces obtained from the addition of the fundamental and the third harmonic of the corresponding triangle-wave surface was evaluated. The triangle-wave surfaces were found to be more accurately discriminated from the smoothly curved surfaces than would be predicted from the detectability of the difference in their Fourier power spectra. This superior discriminability was attributed to differences between the curvature and/or discontinuity of the two surfaces. In experiment 3 the effects of incoherent 'noise' points on the discrimination between the two surface types were also evaluated. These randomly positioned noise points had a relatively small effect on the discrimination between the two surfaces. In general, the results of these experiments indicate that smooth surfaces are salient for stereopsis and that isolated local violations of smoothness are highly discriminable.     

The stereoscopic anisotropy: individual differences and underlying mechanisms

Observers are more sensitive to variations in the depth of stereoscopic surfaces in a vertical than in a horizontal direction; however, there are large individual differences in this anisotropy. The authors measured discrimination thresholds for surfaces slanted about a vertical axis or inclined about a horizontal axis for 50 observers. Orientation and spatial frequency discrimination thresholds were also measured. For most observers, thresholds were lower for inclination than for slant and lower for orientation than for spatial frequency. There was a positive correlation between the 2 anisotropies, resulting from positive correlations between (a) orientation and inclination thresholds and (b) spatial frequency and slant thresholds. These results support the notion that surface inclination and slant perception is in part limited by the sensitivity of orientation and spatial frequency mechanisms.     

Resolving discrepant results of the Wheatstone Experiment

Summary Soon after Wheatstone published his results obtained with the twelve earliest stereograms. Brewster disputed the result obtained with one of them. This stereogram consisted of a thick vertical line presented to one eye and a thin vertical and thick inclined line to the other. Wheatstone reported seeing depth with this figure but Brewster did not. We conducted three experiments to explore the possible reason for the disagreement. Experiment 1 indicated that the stereogram produces a variety of perceptions including the depth as reported by Wheatstone and the rivalry as reported by Brewster. Experiment 2 showed that Brewster's procedure of keeping the eyes fixed while viewing the stereogram reduces the proportion of subjects reporting depth. In Experiment 3, the top half and the bottom half of the stereogram were presented in different trials. The top of the stereogram that contains no conflicting cues produced reliable depth, but the bottom that does, did not. It is concluded that the difference in the procedures they used and the nature of the stimulus were responsible for the conflicting results.     

The Poggendorf illusion in stereoscopic space.

The stereoscopic depth separation between the bisecting rectangle and the oblique line of a Poggendorf configuration was manipulated by varying the direction and magnitude of disparity carried by the rectangle. Based upon data of 6 subjects, the magnitude of the illusion decreased with increasing depth separation regardless of the direction of disparity. Depth separation varied directly with disparity. These findings make plain that depth adjacency can operate symmetrically in stereoscopic space.     

The role of monocular regions in stereoscopic displays

Random-dot stereograms of an object standing out from a background always contain a monocular region at the side of the foreground object. This is equivalent to the monocularly occluded part of the background in the real-life viewing of one object in front of another. The role of these monocular regions in the stereoscopic process has not been investigated previously, although it is generally assumed that they are a source of difficulty in stereoscopic resolution because of the unmatchable texture within them. The basis of the present study was a prediction that the presence of texture within these regions would facilitate rather than retard stereoscopic processing. This prediction follows from a hypothesis that stereoscopic processing is initially located at disparity discontinuities. Unmatched regions are only found at such discontinuities, and could serve to locate them.     

The induced effect,vertical disparity,and stereoscopic theory

The induced effect is an apparent slant of a frontal plane surface around a vertical axis, resulting from vertical magnification of the image in one eye. It is potentially important in suggesting a role for vertical disparity in stereoscopic vision, as proposed by Helmholtz. The paper first discusses previous theories of the induced effect and their implications. A theory is then developed attributing the effect to the process by which the stereoscopic response to horizontal disparity is scaled for viewing distance and eccentricity. The theory is based on a mathematical analysis of vertical disparity gradients produced by surfaces at various distances and eccentricities relative to the observer. Vertical disparity is shown to be an approximately linear function of eccentricity, with a slope or gradient which decreases with observation distance. The effect of vertical magnification on such gradients is analyzed and shown to be consistent with a change in the eccentricity factor, but not the distance factor, required to scale horizontal disparity. The induced effect is shown to be an appropriate stereoscopic response to a zero horizontal disparity surface at the eccentricity indicated. However, since extraretinal convergence signals provide conflicting evidence about eccentricity, they may attenuate the induced effect from its mathematically predicted value. The discomfort associated with the induced effect is attributed to this conflict.     

The relation of lightness and stereoscopic depth in a simple viewing situation

The effect of stereoscopic depth on perceived lightness was studied using a simple, achromatic stimulus arrangement. In Experiment 1, depth/lightness interactions were sought between a single test field and a single induction field. In Experiment 2, depth/lightness interactions were looked for between a single test field and two induction fields. Stimuli were presented on a computer screen and viewed with a stereoscope. The subjects reported perceived lightness of the achromatic test field by rating its apparent blackness along a dimension of 0%–100%. In Experiment 1, they reported lightness judgments of the test field across 13 perceived depth levels and 8 contrast levels. In Experiment 2, they gave lightness judgments of the test field across 7 perceived depth levels and 16 contrast levels. We were particularly interested in observing the generality of Gilchrist’s coplanar ratio hypothesis. The results showed that when stereopsis and contrast levels are the available cues, depth and lightness percepts are independent, and it is retinal ratios, not coplanar ratios, that dictate lightness perception. We conclude that before the relative depth location of an object is determined, its lightness value is known through sensory-level processes.     

Planarity and segmentation in stereoscopic matching

The matching of stereograms which contain periodic patterns suggests ways in which the stereo correspondence problem may be solved in human vision. The stereograms seem to be segmented by coarse-scale features. Within each segment a set of matches approximating a plane is chosen. In regions with periodic patterns there may be many such planar sets, and the disparity of coarse-scale features seems to guide the choice of a particular set. This emphasis on planarity may reflect the occurrence of correlation-like operations in cortical neurons. An attractive possibility is that segmentation effectively delimits areas of the visual field within which disparities are likely to be slow changing (eg local tangent planes to surfaces) so that the correlation sums evaluated in a segment can give the best estimate of depth. A mechanism of this kind cannot account for all of stereo matching, since not all visual objects are well described by ensembles of planes. But it is likely to be a component of the matching system which is particularly important where images are 'noisy' and averaging is needed to extract reliable disparities.     

Binocular rivalry and stereoscopic depth perception

An investigation was made of stimulus factors causing retinal rivalry or allowing stereoscopic depth perception, given a requisite positional disparity. It is shown that similar colour information can be “filtered” out from both eyes; that stereopsis is not incompatible with rivalry and suppression of one aspect of the stimulus, and that the strongest cue for perception of stereoscopic depth is intensity difference at the boundaries of the figures in the same direction at each eye. Identity of colour can also act as a cue for stereopsis. The brightness of different monocular figures seen in the stereoscope in different combinations was estimated by a matching technique, and it is suggested that the perceived brightness is a compromise between the monocular brightness difference between figure and ground seen in relation to the binocular fused background, and the mean brightness of the figures. The results are discussed in terms of neurophysiological “on,” “off” and continuous response fibres.     

Geometrical structures of photographic and stereoscopic spaces

Two experiments were conducted to investigate the geometrical structures of photographic and stereoscopic spaces. In Experiment 1, it was investigated how accurately photographic space reproduces real physical space, and the geometrical structure of photographic space was compared with that of visual space. As a result, the mapping function of distance between photographic and physical spaces (delta = ad(b)) shows that a and b range from 0.96-1.1 and 0.69-0.78. The mapping function of angle between photographic and physical spaces (phi = g phi(h)) shows that g and h range from 2.37-5.29 and 0.74-0.97. Further, photographic space has larger anisotropic property than visual space and photographic space may be hyperbolic. In Experiment 2, the geometrical structure of stereoscopic space was compared with that of visual space. It was found that stereoscopic space was almost the same as visual space.     

Aftereffects and the representation of stereoscopic surfaces

The structure of human disparity representation is examined through (i) adaptation experiments and (ii) model simulations of the data. Section 3 presents results of adaptation experiments designed to illuminate the structure of human disparity representation. Section 4 presents model simulations of three different disparity representation schemes. In the experiments, participants adapted to a 0.133 cycle deg-1 sinusoidally corrugated surface with 10 min of arc peak-to-trough disparity. A flat test surface was briefly presented, in which the aftereffect surface was perceived. Adapt and test surfaces were placed on disparity pedestals and thus presented in front of or behind the plane of fixation. The adapt surface could be offset from the fixation plane by +/- 8 to 24 min of arc. The test surface could be offset from the fixation plane by +/- 8 to 48 min of arc. The depth aftereffect was measured in different disparity planes by a nulling method and 'topping-up' procedure. Aftereffect tuning functions were obtained whose bandwidths, magnitudes, and tuning depended on the disparity planes of both the adapt and test surfaces. These parameters were used to constrain the models tested in section 4. On the basis of the two studies, it is argued that the human stereoscopic system encodes spatial changes of disparity using channels localised within disparity planes. A localised disparity-gradient model of the human representation of disparity is proposed.