The Shape of Space

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.

Alleys in visual space

Human vision is extremely sensitive to equidistance of spatial intervals in the frontal plane. Thresholds for spatial equidistance have been extensively measured in bisecting tasks. Despite the vast number of studies, the informational basis for equidistance perception is unknown. There are three possible sources of information for spatial equidistance in pictures, namely, distances in the picture plane, in physical space, and visual space. For each source, equidistant intervals were computed for perspective photographs of walls and canals. Intervals appear equidistant if equidistance is defined in visual space. Equidistance was further investigated in paintings of perspective scenes. In appraisals of the perspective skill of painters, emphasis has been on accurate use of vanishing points. The current study investigated the skill of painters to depict equidistant intervals. Depicted rows of equidistant columns, tiles, tapestries, or trees were analyzed in 30 paintings and engravings. Computational analysis shows that from the middle ages until now, artists either represented equidistance in physical space or in a visual space of very limited depth. Among the painters and engravers who depict equidistance in a highly nonveridical visual space are renowned experts of linear perspective.

In this paper I shall investigate the geometrical structure of phenomenal visual space. I shall assume at the outset that visual perception is indirect, and thus that we are never immediately aware of even the surfaces of physical objects. Instead I will hold that our immediate visual awareness consists of a phenomenal field of colors, visual space, whose geometrical and qualitative features are determined by causal connections with the physical objects being seen; that is by light rays reflected from these objects being focused on the retina, and the various subsequent neural events which take place in the brain. I have defended the causal theory of perception in detail elsewhere (French, 1987), but in order to keep this paper within manageable length, while still doing justice to the geometrical arguments involved, I shall not summarize those arguments here. While there may be various structural resemblances between portions of visual space and the corresponding physical objects being seen, many of which being given by the laws of projective geometry, it remains possible that both the topological and metric structure of visual space is quite different from that of physical space. Thus, an attitudinal shift is required here away from direct realism, which numerically equates the two, to a more neutral attitude where a geometrical analysis can be made of our visual experience in and of itself. I shall attempt to undertake such a geometrical analysis in this paper. Thus, I shall now turn to an investigation of the topology of visual space, to be followed by an investigation of its metric structure.

Our ability to explore our surroundings requires a combination of high-resolution vision and frequent rotations of the visual axis toward objects of interest. Such gaze shifts are themselves a source of powerful retinal stimulation, and so the visual system appears to have evolved mechanisms to maintain perceptual stability during movements of the eyes in space. The mechanisms underlying this perceptual stability can be probed in the laboratory by briefly presenting a stimulus around the time of a saccadic eye movement and asking subjects to report its position. Under such conditions, there is a systematic misperception of the probes toward the saccade end point. This perisaccadic compression of visual space has been the subject of much research, but few studies have attempted to relate it to specific brain mechanisms. Here, we show that the magnitude of perceptual compression for a wide variety of probe stimuli and saccade amplitudes is quantitatively predicted by a simple heuristic model based on the geometry of retinotopic representations in the primate brain. Specifically, we propose that perisaccadic compression is determined by the distance between the probe and saccade end point on a map that has a logarithmic representation of visual space, similar to those found in numerous cortical and subcortical visual structures. Under this assumption, the psychophysical data on perisaccadic compression can be appreciated intuitively by imagining that, around the time of a saccade, the brain confounds nearby oculomotor and sensory signals while attempting to localize the position of objects in visual space.

The relationship between visual space and physical space was investigated by means of the apparent fronto-parallel plane percept. A set of curves in physical space, corresponding to fronto-parallel planes perceived in front or behind a given fixation point, was determined. The curvature at the apex of these curves follows a hyperbolic law that depends on the distance between the perceived plane and the fixation point. Our results are interpreted by distinguishing absolute disparity, which sets the position of the perceived plane, from relative disparity measured by adjusting in this plane eccentric vertical lines. We show that Foley's model, which interprets the shape of the apparent fronto-parallel planes on the mis-evaluation of egocentric distance, works for absolute disparity judgements, but seems to be inadequate for relative disparity judgements in our experimental conditions.

Different geometries are used to model perceived psychological space. Affine geometry is one such model which can be used to understand how we map physical space to perceived space. In such affine transformations, properties like coplanarity of points, parallelism of a pair of lines and ratio of lengths of line segments in the same direction, remain invariant. One approach to study such a visual space is to consider its intrinsic structure, that is, how the spatial judgments that people make, relate to each other. According to action-specific approach to perception, perceived distance to targets is scaled by effort required to walk that extent. In this experiment, we tested whether effort required to walk affected the intrinsic structure of perceived space, by using Varignon’s theorem. According to Varignon’s theorem, if we join the bisection points of the four edges of an arbitrary quadrilateral, it forms a parallelogram in affine space. Also, bisection points of the line joining bisections of opposite edges will coincide with each other in the case of an intrinsically consistent affine structure. Here, for half of the participants, we manipulated their effort required to walk, by using a treadmill. The participants were shown two adjacent vertices of a quadrilateral in each trial and were asked to blind-walk to the midpoint. They also performed bisection judgments of the bisections of opposite edges in the last two trials and we checked the distance between these two final points for intrinsic consistency. Although we did not find a significant effect of required effort on the intrinsic consistency of perceived affine space, we observed a trend that there was more distortion when effort required to walk was manipulated using a treadmill. The results suggest that perceived space might be affected by effort required to act on physical space.

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.

Pictures, perception, and mental models

In each of five experiments Ss made direct assessments of interpoint distances within a stimulus configuration. A multidimensional scaling method was applied to make explicit the form of the subjective configuration. The stimulus configurations consisted of small light points arranged in a two- or three-dimensional array in a dark room, and of real stars in the sky. All the data were adequately accounted for by a configuration constructed in Euclidean space of the appropriate dimensionality. That was true even in the situation where alley experiments with the same Ss gave the result that is usually regarded as evidence for a hyperbolic binocular space. The Euclidean interpretation entails a more complicated form of correspondence between physical and visual spaces than the hyperbolic interpretation.

Curvature of binocular visual space a modified method of right triangle

Spatiotemporal flows of neural activity, such as traveling waves, have been observed throughout the brain since the earliest recordings; yet there is still little consensus on their functional role. Recent experiments and models have linked traveling waves to visual and physical motion, but these observations have been difficult to reconcile with standard accounts of topographically organized selectivity and feedforward receptive fields. Here, we introduce a theoretical framework that formalizes and generalizes the connection between 'motion' and flowing neural dynamics in the language of equivariant neural network theory. We consider 'motion' not only in physical or visual spaces, but also in more abstract representational spaces, and we argue that recurrent traveling-wave-like dynamics are not just useful but necessary for accurate and stable processing of any signal undergoing such motion. Formally, we show that for any non-trivial recurrent neural network to process a sequence undergoing a flow transformation (such as visual motion) in a structured equivariant manner, its hidden state dynamics must actively realize a homomorphic representation of the same flow through recurrent connectivity. In this "spatiotemporal perspective on dynamical computation", traveling waves and related flows are best understood as faithful dynamic representations of stimulus flows; and consequently the natural inclination of biological systems towards such dynamics may be viewed as an innate inductive bias towards efficiency and generalization in the spatiotemporally-structured dynamical world they inhabit.

Conventional approaches to relevance feedback in content-based image retrieval are based on the assumption that relevant images are physically close to the query image, or the query regions can be identified by a set of clustering centers. However, semantically related images are often scattered across the visual space. It is not always reliable that the refined query point or the clustering centers are capable of representing a complex query region.In this work, we propose a novel relevance feedback approach which directly aims at extracting a set of samples to represent the query region, regardless of its underlying shape. The sample set extracted by our method is competent as well as compact for subsequent retrieval. Moreover, we integrate feature re-weighting in the process to estimate the importance of each image descriptor. Unlike most existing relevance feedback approaches in which all query points share a same feature weight distribution, our method re-weights the feature importance for each relevant image respectively, so that the representative and discriminative ability for all the images can be maximized. Experimental results on two databases show the effectiveness of our approach.KeywordsLocal Binary PatternRelevance FeedbackQuery PointQuery ExpansionRelevance ScoreThese 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.

The place in the creative space of poetry has always been a focus of sight, and a subject that occupies the pioneers and scholars of this art, who have counted the place as an integral part of the identity of both the poetic text and the poet alike. The concept of 'place' is not exclusive to the physical space or geographical space in which events take place but an important element that demonstrates the relationship between man and place, identity, belonging, and the origins that bind man to the past and present, and the characters with their concerns, feelings, pains and hopes. In my talk about the place in the poem, I do not deal with these concepts only; I actually Igo beyond it to talk about the space that has geometric dimensions over which words are arranged, how these words are designed, the relationship of blackness to whiteness above this visual space, or what we might call the visual formation of the poem. This study seeks to discuss the features of the thematic and visual location in two selected poems of the poet Anwar Saba, to monitor the relationship between the spatial expressions mentioned in each poem and their visual spatial formation, and how they serve or complement each other, and their visual spatial formation, and how they serve or complement each other.

Image reconstruction plays a critical role in the clinical use of magnetic resonance imaging (MRI). The MRI raw data is not acquired in image space and the role of the image reconstruction process is to transform the acquired raw data into images that can be interpreted clinically. This process involves multiple signal processing steps that each have an impact on the image quality. This review explains the basic terminology used for describing and quantifying image quality in terms of signal-to-noise ratio and point spread function. In this context, several commonly used image reconstruction components are discussed. The image reconstruction components covered include noise prewhitening for phased array data acquisition, interpolation needed to reconstruct square pixels, raw data filtering for reducing Gibbs ringing artifacts, Fourier transforms connecting the raw data with image space, and phased array coil combination. The treatment of phased array coils includes a general explanation of parallel imaging as a coil combination technique. The review is aimed at readers with no signal processing experience and should enable them to understand what role basic image reconstruction steps play in the formation of clinical images and how the resulting image quality is described.