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We provide a brief survey on the properties of submanifolds in metallic Riemannian manifolds. We focus on slant, semi-slant and hemi-slant submanifolds in metallic Riemannian manifolds and, in particular, on invariant, anti-invariant and semi-invariant submanifolds. We also describe the warped product bi-slant and, in particular, warped product semi-slant and warped product hemi-slant submanifolds in locally metallic Riemannian manifolds, obtaining some results regarding the existence and nonexistence of non-trivial semi-invariant, semi-slant and hemi-slant warped product submanifolds. We illustrate all these by suitable examples.

The projected image of a textured surface contains multiple texture cues to three-dimensional (3D) surface orientation. Previous studies have reported conflicting findings about the roles of various texture cues. We tested the influence of texture compression relative to other texture cues using a cue conflict paradigm. Observers viewed images of textured planar surfaces with varied slants (0°–70°) and estimated 3D slant by aligning their hand with the virtual surface. Conflicts between texture cues were created by stretching or compression the texture along the surface, which selectively changes the slant specified by texture compression. The texture distortions were relatively small (±10% or ±20%) to limit the size of the cue conflicts. Across three experiments, we varied the field of view (10° vs. 20°), texture regularity (circles vs. Voronoi), and availability of binocular cues. In monocular conditions, slant estimates were strongly affected by texture distortions. Analyses of cue weighting found that texture compression had more influence on slant settings than other texture cues and the relative influence of texture compression decreased with larger field of view and less regular textures. In binocular conditions, we also observed effects of texture distortion, and the influence of texture compression relative to information from stereo and other texture cues increased with slant. Our results provide evidence that texture compression contributes to perceived slant, in addition to other texture cues such as texture scaling. The observed effects of simulated slant, field of view, and texture regularity on cue weighting were all consistent with a model that integrates multiple sources of information according to their reliability.

Surface orientation is an important visual primitive that can be estimated from monocular or binocular (stereoscopic) signals. Changes in motor planning occur within about 200 ms after either type of signal is perturbed, but the time it takes for apparent (perceived) slant to develop from stereoscopic cues is not known. Apparent slant sometimes develops very slowly (Gillam, Chambers, & Russo, 1988; van Ee & Erkelens, 1996). However, these long durations could reflect the time it takes for the visual system to resolve conflicts between slant cues that inevitably specify different slants in laboratory displays (Allison & Howard, 2000). We used a speed-accuracy tradeoff analysis to measure the time it takes to discriminate slant, allowing us to report psychometric functions as a function of response time. Observers reported which side of a slanted surface was farther, with a temporal deadline for responding that varied block-to-block. Stereoscopic slant discrimination rose above chance starting at 200 ms after stimulus onset. Unexpectedly, observers discriminated slant from binocular disparity faster than texture, and for stereoscopic whole-field stimuli faster than stereoscopic slant contrast stimuli. However, performance after the initial deviation from chance increased more rapidly for slant-contrast stimuli than whole-field stimuli. Discrimination latencies were similar for slants about the horizontal and vertical axes, but performance increased faster for slants about the vertical axis. Finally, slant from vertical disparity was somewhat slower than slant from horizontal disparity, which may reflect cue conflict. These results demonstrate, in contradiction with the previous literature, that the perception of slant from disparity happens very quickly-in fact, more quickly than the perception of slant from texture-and in comparable time to the simple perception of brightness from luminance.

The main purpose of this article is to construct inequalities between a main intrinsic invariant (the normalized scalar curvature) and an extrinsic invariant (the Casorati curvature) for some submanifolds in a Sasakian manifold with a zero C-Bochner tensor.

Perceived three-dimensional shape can be systematically distorted by altering viewing conditions. Here we investigate whether such distortions are inconsistent or congruent among different surface properties (relative depth, orientation, and curvature). Participants matched two surfaces defined by different types of visual information (motion parallax, texture, or shading). Judgments of different surface properties elicited qualitatively different patterns of distortions. Adding multiple sources of information did not eliminate inconsistencies. This result indicates that 3D shape perception is not based on Euclidean representation. Moreover, models of cue integration that assume interpretation of information prior to fusion cannot account for such findings.

Estimating three-dimensional (3D) surface orientation (slant and tilt) is an important first step toward estimating 3D shape. Here, we examine how three local image cues from the same location (disparity gradient, luminance gradient, and dominant texture orientation) should be combined to estimate 3D tilt in natural scenes. We collected a database of natural stereoscopic images with precisely co-registered range images that provide the ground-truth distance at each pixel location. We then analyzed the relationship between ground-truth tilt and image cue values. Our analysis is free of assumptions about the joint probability distributions and yields the Bayes optimal estimates of tilt, given the cue values. Rich results emerge: (a) typical tilt estimates are only moderately accurate and strongly influenced by the cardinal bias in the prior probability distribution; (b) when cue values are similar, or when slant is greater than 40°, estimates are substantially more accurate; (c) when luminance and texture cues agree, they often veto the disparity cue, and when they disagree, they have little effect; and (d) simplifying assumptions common in the cue combination literature is often justified for estimating tilt in natural scenes. The fact that tilt estimates are typically not very accurate is consistent with subjective impressions from viewing small patches of natural scene. The fact that estimates are substantially more accurate for a subset of image locations is also consistent with subjective impressions and with the hypothesis that perceived surface orientation, at more global scales, is achieved by interpolation or extrapolation from estimates at key locations.

We report on the ability of human observers to discriminate local second-order shape of quadratic stereo-defined surfaces. Local second-order shape can be specified by two parameters: the curvedness (a scale-dependent quantity describing overall curvature of a shape) and the shape index (a scale-independent quantity describing the shape's appearance). We measured shape index discrimination thresholds of shapes subject to attitude and curvedness transformations. Results show that neither slant nor curvedness manipulations affect or bias observers' judgements of surface shape. Furthermore, ideal detector simulations show that observers do not perform optimally: they do not exploit all available information.

We measure the performance of five subjects in a two-alternative-forced-choice slant-discrimination task for differently textured planes. As textures we used uniform lattices, randomly displaced lattices, circles (polka dots), Voronoi tessellations, plaids, 1/f noise, "coherent" noise and a leopard skin-like texture. Our results show: (1) Improving performance with larger slants for all textures, (2) and some cases of "non-symmetrical" performance around a particular orientation. (3) For orientations sufficiently slanted, the different textures do not elicit major differences in performance, (4) while for orientations closer to the vertical plane there are marked differences among them. (5) These differences allow a rank-order of textures to be formed according to their "helpfulness"--that is, how easy the discrimination task is when a particular texture is mapped on the plane. Polka dots tend to allow the best slant discrimination performance, noise patterns the worst. Two additional experiments were conducted to test the generality of the obtained rank-order. First, the tilt of the planes was rotated by 90 degrees. Second, the task was changed to a slant report task via probe adjustment. The results of both control experiments confirmed the texture rank-order previously obtained. We then test a number of spatial-frequency-based slant-from-texture models and discuss their shortcomings in explaining our rank-order. Finally, we comment on the importance of these results for depth-perception research in general, and in particular the implications our results have for studies of cue combination (sensor fusion) using texture as one of the cues involved. Afdeling Experimentele en kwantitatieve psychologie. Laboratorium voor Experimentele psychologie. ispartof: Vision research vol:44 issue:13 pages:1511-1535 ispartof: location:SCOTLAND, GLASGOW status: published

Retinal images are perspective projections of the visual environment. Perspective projections do not explain why we perceive perspective in 3-D space. Analysis of underlying spatial transformations shows that visual space is a perspective transformation of physical space if parallel lines in physical space vanish at finite distance in visual space. Perspective angles, i.e., the angle perceived between parallel lines in physical space, were estimated for rails of a straight railway track. Perspective angles were also estimated from pictures taken from the same point of view. Perspective angles between rails ranged from 27% to 83% of their angular size in the retinal image. Perspective angles prescribe the distance of vanishing points of visual space. All computed distances were shorter than 6 m. The shallow depth of a hypothetical space inferred from perspective angles does not match the depth of visual space, as it is perceived. Incongruity between the perceived shape of a railway line on the one hand and the experienced ratio between width and length of the line on the other hand is huge, but apparently so unobtrusive that it has remained unnoticed. The incompatibility between perspective angles and perceived distances casts doubt on evidence for a curved visual space that has been presented in the literature and was obtained from combining judgments of distances and angles with physical positions.

By presenting oriented Gabor patches either monocularly or binocularly, we dissociated retinal orientation from perceived tilt and perceived slant. After adapting to binocular patches, with zero apparent tilt and non-zero slant, small tilt after-effects (TAEs) and large slant after-effects (SAE) were measured. Adapting to monocular patches with non-zero tilt and zero slant produced large TAEs and smaller SAEs. This pattern of results suggests that a common, low-level adaptation to monocular orientation is involved in slant and tilt after-effects. However, the incomplete transfer between slant and tilt makes it clear that higher-level adaptation is also involved, perhaps at the level of surface representation.

Texture information about surface shape can be decomposed into three constituents: compression, density and scaling. Blake, Biilthoff and Sheinberg's (1993,Vision Research, 33, 1723–1737) Ideal Observer theory of slant from texture predicts that the relative contribution of compression and density to the percept of planar surface slant should vary with field of view (FOV). The contribution of compression and density should both increase as FOV increases but statistical analysis shows that the reliability, and hence the expected contribution, of density increases at a greater rate than compression with increasing FOV. Specific predictions are that at FOV 20 deg compression should be more effective than density, at FOV ≈ 20 deg, compression and density should be equally effective, and at FOV > 20 deg, compression should be less effective than density. These predictions were tested by pitting these components of texture against one another. The method used was similar to that described in Frisby and Buckley [1992 In Orban, G. Nagel H.-H. (Eds)Artificial and biological visual systems. Berlin: Springer]: observers judged binocularly the slant of a large table on to which was projected a texture created using computer graphics to contain various texture component cues to slant. The size of the projected texture patch determined the FOV which was by this means set to either 10, 20 or 30 deg. It was found that compression was the dominant cue irrespective of FOV, even if it was perturbed by noise. Possible reasons for this divergence of human observers from predictions of the Ideal Observer theory of slant from texture are discussed.

The horizontal–vertical illusion, in which the vertical dimension is overestimated relative to the horizontal direction, has been explained in terms of the statistical relationship between the lengths of lines in the world, and the lengths of their projections onto the retina (Howe & Purves, 2002). The current study shows that this illusion affects the apparent aspect ratio of shapes, and investigates how it interacts with binocular cues to surface slant. One way in which statistical information could give rise to the horizontal–vertical illusion would be through prior assumptions about the distribution of slant. This prior would then be expected to interact with retinal cues to slant. We determined the aspect ratio of stereoscopically viewed ellipses that appeared circular. We show that observers’ judgements of aspect ratio were affected by surface slant, but that the largest image vertical:horizontal aspect ratio that was considered to be a surface with a circular profile was always found for surfaces close to fronto-parallel. This is not consistent with a Bayesian model in which the horizontal–vertical illusion arises from a non-uniform prior probability distribution for slant. Rather, we suggest that assumptions about the slant of surfaces affect apparent aspect ratio in a manner that is more heuristic, and partially dissociated from apparent slant.

Perspective views of textured, planar surfaces provide a number of cues about the orientations of the surfaces. These include the information created by perspective scaling of texture elements (scaling), the information created by perspective foreshortening of texels (foreshortening) and, for textures composed of discrete elements, the information created by the effects of both scaling and foreshortening on the relative positions of texels (position). We derive a general form for ideal observers for each of these cues as they appear in images of spatially extended textures, (e.g. those composed of solid 2-D figures). As an application of the formulation, we derive a set of `generic' observers which we show perform near optimally for images of a broad range of surface textures, without special prior knowledge about the statistics of the textures. Using simulations of ideal observers, we analyze the informational structure of texture cues, including a quantification of lower bounds on reliability for the three different cues, how cue reliability varies with slant angle and how it varies with field of view. We also quantify how strongly the reliability of the foreshortening cue depends on a prior assumption of isotropy. Finally, we extend the analysis to a naturalistic class of textures, showing that the information content of textures particularly suited to psychophysical investigation can be quantified, at least to a first-order approximation. The results provide an important computational foundation for psychophysical work on perceiving surface orientation from texture.