Multidimensional scaling in riemannian space

A metric which is a function of position is proposed for the analysis of the intrinsic geometry involved in preference or similarity judgments. Variation in the distance function or metric is characteristic of the Riemannian spaces and may be interpreted as curvature, stress or distortion in distance estimates and thus in the subjective perceptual space. It is possible to find the coefficients in the distance function at selected points by fitting a least-squares Riemannian surface to the Euclidean plane. The functional form of the distance can then be obtained by an application of the Laplace equation. Several examples are worked out for the two-dimensional solution but extension to higher spaces appears to be quite feasible.     

A Riemannian geometry theory of human movement: The geodesic synergy hypothesis

Mass-inertia loads on muscles change with posture and with changing mechanical interactions between the body and the environment. The nervous system must anticipate changing mass-inertia loads, especially during fast multi-joint coordinated movements. Riemannian geometry provides a mathematical framework for movement planning that takes these inertial interactions into account. To demonstrate this we introduce the controlled (vs. biomechanical) degrees of freedom of the body as the coordinate system for a configuration space with movements represented as trajectories. This space is not Euclidean. It is endowed at each point with a metric equal to the mass-inertia matrix of the body in that configuration. This warps the space to become Riemannian with curvature at each point determined by the differentials of the mass-inertia at that point. This curvature takes nonlinear mass-inertia interactions into account with lengths, velocities, accelerations and directions of movement trajectories all differing from those in Euclidean space. For newcomers to Riemannian geometry we develop the intuitive groundwork for a Riemannian field theory of human movement encompassing the entire body moving in gravity and in mechanical interaction with the environment. In particular we present a geodesic synergy hypothesis concerning planning of multi-joint coordinated movements to achieve goals with minimal muscular effort.     

Alleys in visual space

Geometry of frameless visual space is dealt with. First, parallel and equidistant alleys, horopters in the horizontal plane of eyes' level are discussed within the framework of the Luneburg's model that the frameless visual space is a Riemannian space of constant curvature. That basic postulate and the specific mapping functions assumed by Luneburg between the euclidean map of visual space and the physical space are kept separate, and efforts are directed to make the model applicable to more natural conditions of our visual space. A possibility is pointed out to remove the constraint “frameless” in the sense that perceptual geometrical properties are primarily determined by the convergence of optic axes. So far, only alleys in the horizontal plane extending from us toward infinity have been studied, but more often we perceive parallel lines, horizontal or vertical, in front of us like shelves of a bookcase. Hence, equations are derived for horopter plane appearing fronto-parallel in the three-dimensional visual space and alleys running horizontally or vertically on the horopter plane. It is shown that parallel and equidistant alleys are not the same in the horopter plane as in the horizontal plane, if the visual space is not euclidean. A method to evaluate the discrepancy between the two alleys without using any mapping functions is stated with some numerical examples.     

A critical review of Luneburg's model with regard to global structure of visual space

Visual space (VS) is a coherent self-organized dynamic complex that is structured into objects, backgrounds, and the self. As a concrete example of geometrical properties in VS, experimental results on parallel and (equi) distance alleys in a frameless VS were reviewed, and Luneburg's interpretation on the discrepancy between these 2 alleys was sketched with emphasis on the 2 hypotheses involved: VS is a Riemannian space of constant curvature (RCC) and the a priori assumed correspondence between VS and the physical space in which stimulus points are presented. Dissociating these 2 assumptions, the author tried to see to what extent the global structure of VS under natural conditions is in accordance with the hypothesis of RCC and to make explicit the logic underlying RCC. Several open questions about the geometry of VS per se have been enumerated.     

Alleys on an extensive apparent frontoparallel plane: a second experiment

Small light points were presented, in the dark, around a point in the center which was fixed at a distance of about 3 m from the subject. In experiment 1, the subject adjusted the positions of points so that all were frontoparallel and in three horizontal series, each consisting of five points, with the middle series level with the eyes, to satisfy the following conditions: (i) the three series must appear straight and horizontally parallel; (ii) the points of each of the five triplets must appear equally separated vertically; (iii) the three points of each triplet must appear to move horizontally along straight and parallel paths; (iv) the three points of each triplet must appear to move horizontally with a constant vertical separation. The most distant points were about 0.51 rad to the left and right of center, and about 0.22 rad above and below. In experiment 2, with the configuration of points obtained in experiment 1, the subject assessed ratios of all perceptual distances between points and also from the subject to all points. From experiment 1 (three subjects used), Gaussian curvature K and a constant related to depth perception (sigma) were estimated under the assumption that the frontoparallel plane is a Riemannian plane of constant curvature K and that Luneburg's mapping functions between visual space and physical space hold. The analysis was made according to equations different from those used previously. The results of experiment 2 (two subjects used) were analyzed by a new computer program in which no preassumed mapping functions are necessary for the estimation of K. From both analyses it is clear that there is no need to assume any other value of K than 0 (Euclidean) to describe the geometry of the frontoparallel plane. This presents a striking contrast to the results from experiments on parallel and equidistance alleys running toward the subject on the horizontal plane.     

Short-term visual memory for contour curvature in a delayed discrimination task

Abstract: A two-interval forced-choice of constant stimuli was used to measure the point of subjective equality (PSE) and discrimination threshold for standard contour curvature (1.91, 3.24 deg⁻¹) held in short-term visual memory (STVM). At both standard curvatures, the PSE for remembered curvature was nearly constant for standard curvature from 2 s to 16 s retention intervals, while the discrimination threshold increased as a linear function of retention interval. These results show that the decay in STVM for contour curvature is due to the noisy representation of curvature, neither to fading of the represented curvature nor to converging to the constant curvature. Furthermore, the Weber fraction was nearly constant for both standard curvatures at any delay from 2 to 16 s.     

Prospective strategies underlie the control of interceptive actions

The purpose of this study was to test whether a constant bearing angle strategy could account for the displacement regulations produced by a moving observer when attempting to intercept a ball following a curvilinear path. The participants were asked to walk through a virtual environment and to change, if (deemed) necessary, their walking speed so as to intercept a moving ball that followed either a rectilinear or a curvilinear path. The results showed that ball path curvature did indeed influence the participants' displacement kinematics in a way that was predicted by adherence to a constant bearing angle strategy mode of control. Velocity modifications were found to be proportional to the magnitude of target curvature with opposing curvatures giving rise to mirror displacement velocity changes. The role of prospective strategies in the control of interceptive action is discussed.     

Riemannian Newton and trust-region algorithms for analytic rotation in exploratory factor analysis

In exploratory factor analysis, latent factors and factor loadings are seldom interpretable until analytic rotation is performed. Typically, the rotation problem is solved by numerically searching for an element in the manifold of orthogonal or oblique rotation matrices such that the rotated factor loadings minimize a pre-specified complexity function. The widely used gradient projection (GP) algorithm, although simple to program and able to deal with both orthogonal and oblique rotation, is found to suffer from slow convergence when the number of manifest variables and/or the number of latent factors is large. The present work examines the effectiveness of two Riemannian second-order algorithms, which respectively generalize the well-established truncated Newton and trust-region strategies for unconstrained optimization in Euclidean spaces, in solving the rotation problem. When approaching a local minimum, the second-order algorithms usually converge superlinearly or even quadratically, better than first-order algorithms that only converge linearly. It is further observed in Monte Carlo studies that, compared to the GP algorithm, the Riemannian truncated Newton and trust-region algorithms require not only much fewer iterations but also much less processing time to meet the same convergence criterion, especially in the case of oblique rotation.     

Parallel—and distance—alleys with moving points in the horizontal plane

Stimulus points were presented on the horizontal plane of eye level under both dark and illuminated homogeneous spaces. When two apparent movements towards the subject were generated and positions of points were so adjusted that the two movements appeared straight and parallel (P alley) or with a constant lateral distance (D alley), the D alley lay outside the P alley, as traditionally shown with stationary sets of points. The two alleys were constructed with various patterns and velocities of movement, and the Lüneburg formulas were used as experimental equations to describe the results. The equations have two parameters: K (curvature) and o (sensitivity in depth perception). Values of K and o obtained with stationary points in previous experiments are shown too. Predominantly, K < 0 (hyperbolic), and the same is true in the present study. No first-order effect of patterns and velocities of the movement upon K and o was found.     

Are curves detected by 'curvature detectors'?

Five experiments which attempted to evaluate the relationship between orientation and curvature selectivity in human vision are described. In the first two experiments, threshold elevation for curved gratings was measured after exposure to similar gratings, with the use of either an adaptation (experiment 1) or a masking (experiment 2) paradigm. In both experiments threshold elevation occurred which was selective for both the degree and the direction of curvature of the adapting pattern. Experiment 3 compared the effects of adapting to tilted rectilinear or vertical curved gratings upon threshold for a vertical rectilinear grating. Threshold elevation declined systematically as the adapting gratings were either tilted or made more curved. Experiment 4 measured curvature selectivity as a function of the orientation of a curved adapting grating. Threshold elevation declined as the adapting grating was tilted more, but curvature selectivity remained. Experiment 5 measured the orientation tuning for curved gratings directly. Threshold elevation declined to 50% of its maximum value at an adpating orientation of about 28 degrees. This was constant for all values of curvature used. The resulsts are discussed with reference to the question of whether the human visual system contains 'curvature detectors' or linear-contour detectors which respond to the tangents of curves.     

Detection of collision events on curved trajectories: optical information from invariant rate-of-bearing change

Previous research (Andersen & Kim, 2001) has shown that a linear trajectory collision event (i.e., a collision between a moving object and an observer) is specified by objects that expand and maintain a constant bearing (the object location remains constant in the visual field). In the present study, we examined the optical information for detecting a collision event when the trajectory was of constant curvature. Under these conditions, a collision event is specified by expansion of an object and a constant rate-of-bearing change. Three experiments were conducted in which trajectory curvature and display duration were varied while time to contact, speed, and initial image position of the collision objects were maintained. The results indicated that collision detection performance decreased with an increase in trajectory curvature and decreased with a decrease in display duration, especially for highly curved trajectories. In Experiment 3, we found that the presentation of a constant rate-of-bearing change in noncollision stimuli resulted in an increase in the false alarm rate. These results demonstrate that observers can detect collision events on curved trajectories and that observers utilize bearing change information.     

The haptic perception of curvature

Ninety-two subjects, schoolchildren and undergraduate and postgraduate students, took part in a series of experiments on the haptic perception of curvature. A graded series of surfaces was produced using piano-convex lenses masked off to produce curved strips that could be explored without using arm movements. Thresholds were measured using the constant method and a staircase procedure. Experiments 1 and 2 yielded data on the absolute and difference thresholds for curvature. Experiment 3 demonstrated that the effective stimulus for curvature is represented by the overall gradient of a curved surface. Using this measure, it was shown that the present absolute thresholds for curvature are lower than those previously reported. In Experiment 4, absolute thresholds were compared using spherical and cylindrical curves: the results showed that, at least with the narrow strips used, the type of curvature does not exert a significant influence on performance. In Experiment 5, the subjective response to curvature was assessed using a rating procedure. Power functions are reported, although the relationship between stimuli and responses had a strong linear component. This suggests that haptically perceived curvature may be a metathetic rather than a prothetic continuum.     

Identity imposition and its role in a stereokinetic effect

Lines of constant curvature, a circle or a straight line, have no distinguishable parts. Yet they are perceived as if they did. When they move and intersect, they are perceived to slide across each other as if one of them had parts that can be seen to move in relation to the other line. With no such parts present in stimulation, they are products of perception. It was found that a third line of constant curvature, the helix, is also seen to slide when two helixes intersect and are in motion. Another manifestation of perceived identical parts is that rotating circles and similar shapes are perceived not to rotate even when cues for rotation are present. Furthermore, changes between merely perceived identical parts can result in apparent depth. Evidence is presented that such depth, known as the stereokinetic effect, results from kinetic depth-effects that are based on perceived identical parts instead of on actually identical parts, and that depth is seen when the intervals between such perceived parts change length and orientation simultaneously.     

Exocentric pointing to opposite targets

We use an exocentric pointing task to study exocentric visual directions to targets that are opposite to a pointer relative to the observer. (The apparent distance between the target and the pointer always exceeded 90 degrees of visual angle.) All pointing takes place in the horizontal plane at eye height. Observers could not see both target and pointer at a single glance. They had to look back and forth between them, using combinations of eye movements, head turns, twists at the waist and turning on the feet. In the limit of diametrically opposite targets we find that the observers pick either one of two distinct orientations of the pointer as equally "visually correct". Which one results depends on the stance assumed by the observer. The difference between the two equally acceptable pointings is between 5 degrees and 10 degrees. Such a result is predicted from earlier measurements in the context of a model that describes the geometry of the horizon as a Riemannian space with varying intrinsic curvature. The present results thus fit--perhaps surprisingly--very well in such a picture.     

The role of constant curvature in 2-D contour shape representations

In early cortex, visual information is encoded by retinotopic orientation-selective units. Higher-level representations of abstract properties, such as shape, require encodings that are invariant to changes in size, position, and orientation. Within the domain of open, 2-D contours, we consider how an economical representation that supports viewpoint-invariant shape comparisons can be derived from early encodings. We explore the idea that 2-D contour shapes are encoded as joined segments of constant curvature. We report three experiments in which participants compared sequentially presented 2-D contour shapes comprised of constant curvature (CC) or non-constant curvature (NCC) segments. We show that, when shapes are compared across viewpoint or for a retention interval of 1000 ms, performance is better for CC shapes. Similar recognition performance is observed for both shape types, however, if they are compared at the same viewpoint and the retention interval is reduced to 500 ms. These findings are consistent with a symbolic encoding of 2-D contour shapes into CC parts when the retention intervals over which shapes must be stored exceed the duration of initial, transient, visual representations.     

Cellular spaces

This paper is an introduction into the theory of cellular spaces. From the more general model of nets of abstract cells which are interpreted by finite automata, it is shown how the model of cellular spaces is achieved by specialization. Cellular spaces are extremely homogeneous in function and in geometry. The relation between local and global behavior is regarded as the main topic of the theory. After a formal definition of cellular spaces, it is shown that not all functions of the configuration space are induced by cellular spaces. In addition, the Garden-of-Eden problem is discussed, and a simple self-reproduction property is explained.     

What does the occluding contour tell us about solid shape?

A new theorem is discussed that relates the apparent curvature of the occluding contour of a visual shape to the intrinsic curvature of the surface and the radial curvature. This theorem allows the formulation of general laws for the apparent curvature, independent of viewing distance and regardless of the fact that the rim (the boundary between the visible and invisible parts of the object) is a general, thus twisted, space curve. Consequently convexities, concavities, or inflextions of contours in the retinal image allow the observer to draw inferences about local surface geometry with certainty. These results appear to be counterintuitive, witness to the treatment of the problem by recent authors. It is demonstrated how well-known examples, used to show how concavities and convexities of the contour have no obvious relation to solid shape, are actually good illustrations of the fact that convexities are due to local ovoid shapes, concavities to local saddle shapes.