Alleys in visual space

An approach to geometry of visual space with no a priori mapping functions: Multidimensional mapping according to Riemannian metrics

Results of experiments made under the scheme of the Luneburg's model in the author's laboratory were reviewed and were also discussed two fundamental postulates: Frameless binocular perceptual space as a Riemannian space of constant curvature K and the simple and rigid mapping functions between the physical space and the Euc1idean map of the perceptual space.Then, experimental results to embed perceptual configurations into a Euclidean space with the aid of the metric multidimensional scaling method were reviewed.In the last part, possibilities of testing the two fundamental postulates independently with each other and of incorporating into the model perturbation as to K and/or to the mapping functions were discussed.

The metrics of visual and haptic space based on parallelity judgements

Visual space perception has been a topic of sustained research since the nineteenth century. Much of this research into the geometry of visual space, however, required observers to make judgments about spatial relationships between isolated points in total darkness. While a sizeable number of previous investigations have now explored visual space perception in outdoor natural environments, nearly all of the previous investigations evaluating the curvature of visual space have utilized only small numbers of observers. In the current experiment, a large number (30) of observers adjusted triangular configurations of markers in an outdoor field until they appeared either as equilateral or right triangles in depth. There was a wide range of outcomes, such that the observers’ judgments were consistent with elliptic, Euclidean, and hyperbolic geometry. There is thus no single consistent relationship between physical space and perceived space. Furthermore, the geometry of visual space frequently changes as the size of spatial configurations is varied—for many observers, judgments for small configurations are consistent with elliptic or Euclidean geometry while judgments for large configurations are frequently consistent with hyperbolic geometry.

A convergent series of studies in monkeys and man suggests that the computation of visual space is performed in several brain regions for different behavioural purposes. Among these multiple spatial areas, the ventral intraparietal cortex, the putamen and the ventral aspect of the premotor cortex (area 6) contain a system for representing visual space near the face (peripersonal space). In these cerebral areas some neurons are bimodal: they have tactile receptive fields on the face, and they can also be driven by visual stimuli located near the tactile field. The spatial correspondence between the visual and tactile receptive fields provides a map of near visual space coded in body-part-centred co-ordinates. In the present study we demonstrate for the first time the existence of a visual peripersonal space centred on the face in humans. In patients with right hemispheric lesions, visual stimuli delivered in the space near the ipsilesional side of the face extinguished tactile stimuli on the contralesional side (cross-modal visuotactile extinction) to the same extent as did an ipsilesional tactile stimulation (unimodal tactile extinction). Furthermore, a visual stimulus presented in the proximity of the contralesional side of the face improved the detection of a left tactile stimulus: i.e. under bilateral tactile presentation patients were more accurate to report the presence of a left tactile stimulus when a simultaneous visual stimulus was presented near the left side of the face. However, when visual stimuli were delivered far from the face, visuotactile extinction and visuotactile facilitation effects were dramatically reduced. These findings are consistent with the hypothesis of a representation of visual peripersonal space coded in bodypart-centred co-ordinates, and they provide a striking demonstration of the modularity of human visual space.

Integrable systems associated with separation of the variables in real Riemannian spaces of constant curvature are considered. An isomorphism between all such systems and the hyperbolic Gaudin magnet is established. This isomorphism is used in a classification of all coordinate systems that admit separation of the variables, the basis of which is the classification of the correspondingL operators of the Gaudin magnet.

Hypersurfaces in spaces of constant curvature satisfying a particular Roter type equation

The morphology of city visual space is examined using Komsomolsk-on-Amur as an example. This study is currently relevant considering the need to understand the visual perception of space and the uniqueness of the urban environment. The visual morphology of the city is defined as a part of visual urban studies. The morphology of the visual space of Komsomolsk-on-Amur within the framework of a visual model of the urban structure is revealed through the fundamental category of time, which determines the nature of the historical, cultural and visual space. The history of the city is considered focusing on the main stages in the formation of the socio-cultural space that influenced the image of the city. The current urban space of Komsomolsk-on-Amur is analyzed. Cultural historical analysis allowed us to identify the positive and negative aspects of its visual space. Recommendations are offered for the formation of a visually informative urban environment of Komsomolsk-on-Amur using student works as examples.

The mammalian visual cortex harbors a number of functional maps that represent distinct attributes of stimuli in the visual environment. How different functional maps are accommodated within the same cortical space, especially in species that show marked irregularities in one or more functional maps, remains poorly understood. We used optical imaging of intrinsic signals and electrophysiological techniques to investigate the organization of the maps of orientation preference, ocular dominance, and visual space in ferret. This species shows striking nonuniformity in the arrangement of ocular dominance domains and disruption of the mapping of visual space along the V1/V2 border. We asked whether these irregularities would be reflected in the organization of the map of orientation preference. The results show that orientation preference is mapped consistently within both V1 and V2, and across the interareal boundary, with no reflection of the irregularities in the other maps. These observations demonstrate the accommodation of multiple functional maps within the same cortical space without systematic geometrical relationships that necessarily constrain the organization of each representation. Furthermore, they imply that the structure of the map of orientation preference reflects the architecture and activity patterns of cortical circuits that are independent of other columnar systems established in layer 4.

On each of three frontoparallel planes at distances of about 98, 186, and 276 cm from the subject, two series of light points running horizontally in the dark, one above the other, were constructed by six subjects: a P-alley in which the two series appear as two straight lines that are parallel, and a D-alley in which each pair of points appear equally separated in the vertical direction. Luneburg's mapping functions between visual space and physical space have been used to obtain estimates of K (curvature) and sigma (a constant related to depth perception). In addition, ratios of perceptual distances were assessed by four out of the six subjects with eleven triplets of points on the alleys and values of K were estimated (and the fits of theoretical equations were tested) without the use of the mapping functions. In contrast to experiments with P-alleys and D-alleys on the horizontal plane, K was not unequivocally negative in either of the two experiments.

Luneburg's model for computation of the curvature K of visual two-dimensional space (horizontal visual surface) was tested with equidistant and parallel alleys in large open spaces. Forty-six subjects used stakes to produce 406 experimental alleys of variable sizes (from 5 x 1 to 240 m x 48m). The results show that, contrary to results obtained under laboratory conditions with small alleys and light spots, the individual curvature of visual space does not have negative constant value. K varies in the interval - 1 to + 1 in ninety computed settings: K greater than 0 (N = 38); K less than 0 (N = 52). Therefore the Lobachevskian gemetry currently attributed to visual space ought to be replaced by a Riemannian geometry of variable curvature. Moreover K is an individual function dependent on the size of the alley (distance from the subject), and visual perception would be better understood as scale-dependent. Independently of Luneburg's model we have tested the constancy of the curvature hypothesis in experiments with horopters and visual triangles. The results obtained invalidate Luneburg's hypothesis also.

We compare temporal growth of the mean magnetic energy ℰ driven by a small-scale dynamo in Euclidean and Lobachevsky spaces. The governing parameters of the dynamo, such as the rms turbulent velocity and the correlation scale of turbulence, are presumed to vary randomly in space so the dynamo growth rate is a Gaussian random field. Since such a field is unbounded in unbounded space and can achieve, with a low probability, very large values of ℰ, it can grow super-exponentially in both cases. The super-exponential growth of ℰ in Euclidian space, known since the 1980s, can be considered as a statement that the mean energy growth rate is determined up to a weakly growing factor proportional to . We demonstrate that the super-exponential growth of ℰ in Lobachevsky space is a much more radical phenomenon, where ℰ grows as exp (const × t 5/3). We stress that extrapolating the properties of small-scale dynamos in Euclidean space to curved geometries such Lobachevsky space is not straightforward and requires some care. The effects under discussion becomes however important only if the spatial scale of domain in which the small-scale magnetic field is excited exceeds the radius of curvature.

A new formulation concerning the Riemannian space of constant curvature is preposed. Our formula, which is based on the fundamental quantity γ·lλ in place of the fundamental Riemannian metric tensor, is applied to discuss the wave equations for particles with low spin values. It is shown that the theory concerning the Riemannian space of constant curvature can be constructed in a unified manner in terms of the quantity γ·lλ.

We consider the shape of “visual space”. One immediate complication is that there are numerous “visual spaces”, often confused in the literature. Here we focus on visual awareness, rather than optically guided behavior, and “contemplative”, rather than “enactive” vision. Even so, the number of conceptually different “visual spaces” is appreciable. Important distinctions are between visual space as a rigid image of physical space, and visual space as an entity that depends critically upon an observer’s present state. For instance, in judging the relation between two locations, an observer may fixate one or the other. Conceivably, “visual space” will change the moment the observer swaps fixations. If that is indeed the case, then many of the conventional ways to address visual space, or to analyze raw psychophysical data, become effectively invalid. Here we re-analyze a number of available data-sets from such a perspective. We suggest that our notions of “the shape of visual space” need revision and re-investigation.KeywordsVisual SpaceVisual AwarenessSpace PictureFixed View PointObservational PictureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Visual space is, according to Luneburg, a non-Euclidean hyperbolic space of constant negative curvature. Two personal constants enter into the geometrical description of binocular subjective space. One is σ indicating the degree of depth perception characteristic to the observer. The second is K, the curvature; a value which denotes by how much the visual space of an individual departs from Euclidean space. The constants σ and K were determined experimentally for 30 untrained observers by Luneburg's 3- and 4-point tests. All values found for K were negative and support the hypothesis that the geometry of binocular space is hyperbolic. Some discrepancies between the theory and findings were observed, while the sensitivity of the experiments to the lack of training and even a slight measure of aniseikonia is evident.