The aesthetic appeal of minimal structures: judging the attractiveness of solutions to traveling salesperson problems

Ormerod and Chronicle (1999) reported that optimal solutions to traveling salesperson problems were judged to be aesthetically more pleasing than poorer solutions and that solutions with more convex hull nodes were rated as better figures. To test these conclusions, solution regularity and the number of potential intersections were held constant, whereas solution optimality, the number of internal nodes, and the number of nearest neighbors in each solution were varied factorially. The results did not support the view that the convex hull is an important determinant of figural attractiveness. Also, in contrast to the findings of Ormerod and Chronicle, there were consistent individual differences. Participants appeared to be divided as to whether the most attractive figure enclosed a given area within a perimeter of minimum or maximum length. It is concluded that future research in this area cannot afford to focus exclusively on group performance measures.     

The perception of minimal structures: performance on open and closed versions of visually presented Euclidean travelling salesperson problems

The planar Euclidean version of the travelling salesperson problem (TSP) requires finding a tour of minimal length through a two-dimensional set of nodes. Despite the computational intractability of the TSP, people can produce rapid, near-optimal solutions to visually presented versions of such problems. To explain this, MacGregor et al (1999, Perception 28 1417-1428) have suggested that people use a global-to-local process, based on a perceptual tendency to organise stimuli into convex figures. We review the evidence for this idea and propose an alternative, local-to-global hypothesis, based on the detection of least distances between the nodes in an array. We present the results of an experiment in which we examined the relationships between three objective measures and performance measures of optimality and response uncertainty in tasks requiring participants to construct a closed tour or an open path. The data are not well accounted for by a process based on the convex hull. In contrast, results are generally consistent with a locally focused process based initially on the detection of nearest-neighbour clusters. Individual differences are interpreted in terms of a hierarchical process of constructing solutions, and the findings are related to a more general analysis of the role of nearest neighbours in the perception of structure and motion.     

Human performance on visually presented Traveling Salesman problems

Little research has been carried out on human performance in optimization problems, such as the Traveling Salesman problem (TSP). Studies by Polivanova (1974, Voprosy Psikhologii, 4, 41–51) and by MacGregor and Ormerod (1996, Perception & Psychophysics, 58, 527–539) suggest that: (1) the complexity of solutions to visually presented TSPs depends on the number of points on the convex hull; and (2) the perception of optimal structure is an innate tendency of the visual system, not subject to individual differences. Results are reported from two experiments. In the first, measures of the total length and completion speed of pathways, and a measure of path uncertainty were compared with optimal solutions produced by an elastic net algorithm and by several heuristic methods. Performance was also compared under instructions to draw the shortest or the most attractive pathway. In the second, various measures of performance were compared with scores on Raven's advanced progressive matrices (APM). The number of points on the convex hull did not determine the relative optimality of solutions, although both this factor and the total number of points influenced solution speed and path uncertainty. Subjects' solutions showed appreciable individual differences, which had a strong correlation with APM scores. The relation between perceptual organization and the process of solving visually presented TSPs is briefly discussed, as is the potential of optimization for providing a conceptual framework for the study of intelligence. Received: 28 December 1998 / Accepted: 20 January 2000     

The importance of the convex hull for human performance on the traveling salesman problem: a comment on MacGregor and Ormerod (1996)

MacGregor and Ormerod (1996) have presented results purporting to show that human performance on visually presented traveling salesman problems, as indexed by a measure of response uncertainty, is strongly determined by the number of points in the stimulus array falling inside the convex hull, as distinct from the total number of points. It is argued that this conclusion is artifactually determined by their constrained procedure for stimulus construction, and, even if true, would be limited to arrays with fewer than around 50 points.     

Convex hull or crossing avoidance? Solution heuristics in the traveling salesperson problem

Untrained adults appear to have access to cognitive processes that allow them to perform well in the Euclidean version of the traveling salesperson problem (E-TSP). They do so despite the famous computational intractability of the problem, which stems from its combinatorial complexity. A current hypothesis is the humans' good performance is based on following a strategy of connecting boundary points in order (the convex hull hypothesis). Recently, an alternative has been proposed, that performance is governed by a strategy of avoiding crossings. We examined the crossing avoidance hypothesis from the perspectives of its capacity to explain existing data, its theoretical adequacy, and its ability to explain the results of three new experiments. In Experiment 1, effects on the solution quality of number of points versus number of interior points were compared. In Experiment 2, the distributions of observed paths were compared with those predicted from the two hypotheses. In Experiment 3, figural effects were varied to induce crossings. The results of the experiments were more consistent with the convex hull than with the crossing avoidance hypothesis. Despite its simplicity and intuitive appeal, crossing avoidance does not provide a complete alternative to the convex hull hypothesis. Further elucidation of human strategies and heuristics for optimization problems such as the E-TSP will aid our understanding of how cognitive processes have adapted to the demands of combinatorial difficulty.     

Human behaviour in the Euclidean Travelling Salesperson Problem: Computational modelling of heuristics and figural effects

The Travelling Salesperson Problem (TSP) describes a situation where an imaginary individual wishes to visit multiple cities once before returning to his/her own city. This type of problem is known as a nondeterministic polynomial (NP) hard problem, since the factorial number of solutions results in it being impractical to solve using exhaustive processing. Interestingly, when presented as a Euclidean graph (i.e., ETSP), humans identify near optimal solutions almost effortlessly, despite billions of possible tours. In this study, we consider human processing of the ETSP, and introduce the reader to a number of factors that literature proposes as impacting human performance. We hypothesise that: (i) human ETSP activity may be modelled by considering the quotient relationship between node-to-node and node-to-centroid distances; and (ii) consideration of figural effects can optimise automated TSP solution generation. In this paper human processing based heuristics are developed, i.e. replacing the cost function within the nearest neighbour algorithm, to guide node selection. Results showed that the quotient relationship between node-to-node and node-to-centroid distances can be used to closely model average human performance, across a range of ETSP graphs. Interestingly, however, additional consideration of graph figural effects (e.g. distance between boundary points in the convex hull, standard deviation of distances between nodes that make up the convex hull, and number of nodes in the convex hull) results in significantly improved tour costs.     

Congruency effects in dot comparison tasks: convex hull is more important than dot area

The dot comparison task, in which participants select the more numerous of two dot arrays, has become the predominant method of assessing Approximate Number System (ANS) acuity. Creation of the dot arrays requires the manipulation of visual characteristics, such as dot size and convex hull. For the task to provide a valid measure of ANS acuity, participants must ignore these characteristics and respond on the basis of number. Here, we report two experiments that explore the influence of dot area and convex hull on participants’ accuracy on dot comparison tasks. We found that individuals’ ability to ignore dot area information increases with age and display time. However, the influence of convex hull information remains stable across development and with additional time. This suggests that convex hull information is more difficult to inhibit when making judgements about numerosity and therefore it is crucial to control this when creating dot comparison tasks.     

Spatial and contextual factors in human performance on the travelling salesperson problem

The travelling salesperson problem (TSP) provides a realistic and practical example of a visuo-spatial problem-solving task. In previous research, we have found that the quality of solutions produced by human participants for small TSPs compares well with solutions from a range of computer algorithms. We have proposed that the ability of participants to find solutions reflects the natural properties of human perception, solutions being found through global perceptual processing of the problem array to extract a best figure from the TSP points. In this paper, we extend the study of human performance on the task in order to understand further how human abilities are utilised in solving real-world TSPs. The results of experiment 1 show that high levels of solution quality are maintained in solving larger TSPs than had been investigated previously with human participants, and that the presence of an implied real-world context in the problems has no effect upon performance. Experiment 2 demonstrated that the presence of regularity in the point layout of a TSP can facilitate performance. This was confirmed in experiment 3, where effects of the internality of point clusters were also found. All three experiments were consistent with a global, perceptually based approach to the problem by participants. We suggest that the role of perceptual processing in spatial problem-solving is an important area for further research in both theoretical and applied domains.     

On the perception of shape from shading

The extraction of three-dimensional shape from shading is one of the most perceptually compelling, yet poorly understood, aspects of visual perception. In this paper, we report several new experiments on the manner in which the perception of shape from shading interacts with other visual processes such as perceptual grouping, preattentive search (“pop-out”), and motion perception. Our specific findings are as follows: (1) The extraction of shape from shading information incorporates at least two “assumptions” or constraints—first,that there is a single light source illuminating the whole scene, and second, that the light is shining from “above” in relation to retinal coordinates. (2) Tokens defined by shading can serve as a basis for perceptual grouping and segregation. (3) Reaction time for detecting a single convex shape does not increase with the number of items in the display. This “pop-out” effect must be based on shading rather than on differences in luminance polarity, since neither left-right differences nor step changes in luminance resulted in pop-out. (4) When the subjects were experienced, there were no search asymmetries for convex as opposed to concave tokens, but when the subjects were naive, cavities were much easier to detect than convex shapes. (5) The extraction of shape from shading can also provide an input to motion perception. And finally, (6) the assumption of “overhead illumination” that leads to perceptual grouping depends primarily on retinal rather than on “phenomenal” or gravitational coordinates. Taken collectively, these findings imply that the extraction of shape from shading is an “early” visual process that occurs prior to perceptual grouping, motion perception, and vestibular (as well as “cognitive”) correction for head tilt. Hence, there may be neural elements very early in visual processing that are specialized for the extraction of shape from shading.     

On the perception of shape from shading.

The extraction of three-dimensional shape from shading is one of the most perceptually compelling, yet poorly understood, aspects of visual perception. In this paper, we report several new experiments on the manner in which the perception of shape from shading interacts with other visual processes such as perceptual grouping, preattentive search ("pop-out"), and motion perception. Our specific findings are as follows: (1) The extraction of shape from shading information incorporates at least two "assumptions" or constraints--first, that there is a single light source illuminating the whole scene, and second, that the light is shining from "above" in relation to retinal coordinates. (2) Tokens defined by shading can serve as a basis for perceptual grouping and segregation. (3) Reaction time for detecting a single convex shape does not increase with the number of items in the display. This "pop-out" effect must be based on shading rather than on differences in luminance polarity, since neither left-right differences nor step changes in luminance resulted in pop-out. (4) When the subjects were experienced, there were no search asymmetries for convex as opposed to concave tokens, but when the subjects were naive, cavities were much easier to detect than convex shapes. (5) The extraction of shape from shading can also provide an input to motion perception. And finally, (6) the assumption of "overhead illumination" that leads to perceptual grouping depends primarily on retinal rather than on "phenomenal" or gravitational coordinates. Taken collectively, these findings imply that the extraction of shape from shading is an "early" visual process that occurs prior to perceptual grouping, motion perception, and vestibular (as well as "cognitive") correction for head tilt. Hence, there may be neural elements very early in visual processing that are specialized for the extraction of shape from shading.     

Electrophysiological and phenomenological effects of short-term immersion in an altered sensory environment

We examined the spontaneous cerebral electrophysiology and phenomenology during short-term perceptual deprivation consisting of an edgeless visual field combined with monotonous auditory input that eliminated potential grounding cues (multimodal Ganzfeld). Subjects (N = 22) were instructed to self-report perceptual fading using a button press. Relaxed wakefulness with closed eyes and viewing of a time-varying stimulus array served as control conditions. The power of parieto-occipital alpha rhythms during perceptual deprivation was midway between the eyes-closed and eyes-open conditions, with a state-specific frequency acceleration. Oscillatory alpha power remained enhanced in the multimodal Ganzfeld relative to viewing time-varying signals, despite no indication of diminished brain arousal. Subjects experienced a range of perceptual phenomena while in the altered sensory environment and individuals with faster alpha oscillations self-reported a greater number of fading episodes. We suggest that alpha-band electroencephalogram (EEG) dynamics signal internally oriented mentation in response to brief perceptual deprivation.     

Discriminating between strong and weak structures in three‐mode principal component analysis

Recently, a number of model selection heuristics (i.e. DIFFIT, CORCONDIA, the numerical convex hull based heuristic) have been proposed for choosing among Parafac and/or Tucker3 solutions of different complexity for a given three‐way three‐mode data set. Such heuristics are often validated by means of extensive simulation studies. However, these simulation studies are unrealistic in that it is assumed that the variance in real three‐way data can be split into two parts: structural variance, due to a true underlying Parafac or Tucker3 model of low complexity, and random noise. In this paper, we start from the much more reasonable assumption that the variance in any real three‐way data set is due to three different sources: (1) a strong Parafac or Tucker3 structure of low complexity, accounting for a considerable amount of variance, (2) a weak Tucker3 structure, capturing less prominent data aspects, and (3) random noise. As such, Parafac and Tucker3 simulation studies are run in which the data are generated by adding a weak Tucker3 structure to a strong Parafac or Tucker3 one and perturbing the resulting data with random noise. The design of these studies is based on the reanalysis of real data sets. In these studies, the performance of the numerical convex hull based model selection method is evaluated with respect to its capability of discriminating strong from weak underlying structures. The results show that in about two‐thirds of the simulated cases, the hull heuristic yields a model of the same complexity as the strong underlying structure and thus succeeds in disentangling strong and weak underlying structures. In the vast majority of the remaining third, this heuristic selects a solution that combines the strong structure and (part of) the weak structure.     

Investigating the effectiveness of perceptual treatments on a crest vertical curve: A driving simulator study

Rural roads are characterized by a high percentage of run-off-the-road accidents and head-on collisions, mainly caused by inappropriate speeds and failure to maintain a proper lateral position along the roadway alignment. Among several road safety treatments, low-cost perceptual measures are considered an effective tool, as they generally increase the risk perceived by drivers, or even alter the drivers’ speed perception, and consequently tempting them to decrease their speeds. Their effectiveness has been widely recognized in a number of studies, especially with respect to road intersections and curves.The overall aim of this study is to investigate the effects of different perceptual treatments on driving speed, along a crest vertical curve of an existing two-lane rural road, in order to identify the most effective measure to reduce speed and define its subsequent implementation in the field. Three perceptual treatments were tested using a driving simulator: white peripheral transverse bars, red peripheral transverse bars and optical speed bars, with each one being painted along the approaching tangent to the crest vertical curve. The effects of these speed-reducing measures were investigated using a sample of forty-four participants, by comparing the driving speeds with those recorded under a baseline condition (without a treatment); these were also used to validate the driving simulator’s speed measurements with those found in the field. Moreover, subjective measures were collected, consisting of the driver’s static evaluation of the desired speed, risk perception and markings comprehension, based on screen shot pictures that represented the simulated configurations of the treatments.The findings demonstrated an overall effectiveness of the perceptual treatments, although only the red peripheral transverse bars were found to significantly reduce the driving speeds (−6 km/h). The analysis of the questionnaire yielded interesting information and demonstrated the importance of performing driving simulation tests for evaluating the effectiveness of perceptual treatments.Finally, the results confirmed the enormous potential of using driving simulators to pinpoint a number of speed-reducing measures, and consequently select the most effective one that reduces cost and promotes safety before its actual implementation in the field.     

Watching for gains and losses: The effects of motivational challenge and threat on attention allocation during a visual search task

This experiment tests predictions based on research and evidence around the biopsychosocial model (BPSM) that people in a challenge state have faster, more gain orientated search patterns than those in a threat state. Participants (n = 44) completed a motivated performance task involving the location of a target appearing in one of two search arrays: one associated with gaining points and the other associated with avoiding the loss of points. Midway through the task, participants received a false feedback prime about their performance invoking either challenge or threat. We found that participants receiving a challenge prime (high performance feedback) spent longer searching the gain array and made fewer fixations on the loss array. Those receiving a threat prime (low performance feedback) made fewer fixations on the gain array. These findings are in line with the BPSM and provide evidence that allocation of attention (measured using eye movement data) is related to challenge and threat.     

Classical geometric illusion effects with nonclassical stimuli: Angular induction from decomposing lines into point arrays

Angular induction is the process by which one line segment can bias judgment of orientation and/or collinearity of another segment, and it has been established that the magnitude of error is a determinate function of the relative angle between the two. We examined how these known relationships are affected by decomposing the induction segment into an array of scattered points. The bias that was produced by such arrays was found to be consistent with a formal model of angular induction, with the strength of the effect decreasing as the scatter among the points was increased. This decline in strength was almost linear with a logarithmic transform of the dimensions of the stimulus array. We also evaluated the hypothesis that the induction stimulus is detected by one or more channels—for example, neurons—for which the sensitivity profiles are modeled as Gabor wavelets. The change in induction strength with increasing point scatter was not predicted by a single width of channel. However, the combined activity of an ensemble of channels that differed in width did match the perceptual effects if one also stipulated that each channel would respond maximally to a fine-line stimulus.     

Aging and the use of an in-vehicle intersection crossing assist system: An on-road study

One of the principal facets of age-related decline–diminished perceptual ability, can also be viewed as a prominent factor when crossing intersections, particularly rural intersections that have disproportionately high fatality rate and where vehicles travel at higher velocities. Providing information through in-vehicle technology may aid drivers in improving crossing of such intersections. The current study examines the efficacy of an in-vehicle intersection crossing assist system in a real-world rural setting across age groups. Thirty-two, older and younger drivers completed several crossings of a busy rural intersection. Drivers completed two blocks of trials in which the presence/absence of the in-vehicle system was counterbalanced. The results showed a limited impact of the system on driving performance, exhibited in longer wait time before crossing and rising trend towards reduced probability of accepting small crossing gaps. Older drivers performed similarly to younger, although they showed a greater tendency towards conservative driving behaviour. The current study represents an initial effort to examine an in-vehicle intersection crossing assist system in a real-world rural environment, generating results that reveal a potential for these types of systems to be assistive to drivers across age groups and increase the safety at rural intersections.     

Contraries as an effective strategy in geometrical problem solving

A focused review of the literature on reasoning suggests that mechanisms based upon contraries are of fundamental importance in various abilities. At the same time, the importance of contraries in the human perceptual experience of space has been recently demonstrated in experimental studies. Solving geometry problems represents an interesting case as both reasoning abilities and the manipulation of perceptual–figural aspects are involved.

In this study we focus on perceptual changes in geometrical problem solving processes in order to understand whether a mental manipulation in terms of opposites might help. Four conditions were studied, two of which concerned the search for contraries as an implicit or explicit strategy.

Results demonstrated that contraries, when used explicitly in solution processes, constitute an effective heuristic: The number of correct solutions increased, less time was needed to find a solution and participants were oriented towards the use of perception-based solutions—not only were perceptual solutions more frequent, but also, more specifically, the number of correct perceptual solutions increased. These last results concerning perception-based solutions were found both when participants were advised about the usefulness of the strategy and when they were not advised. Differences concerning which aspects of a problem were focused on during the solution process also emerged.     

An ordinal evaluation of categorical judgement data by random utilities and a corresponding correlation analysis

Categorical judgement data are analyzed along the lines of random utility theory. A class of orders is introduced (categorical weak orders); their characteristic vectors are regarded as points in a Euclidean space; their convex hull forms a polytope whose facets are fully characterized. This polytope is shown to correspond to an order polytope. Furthermore, its relation to the biorder polytope is pointed out. The convex representations of a given point of the polytope are discussed. The impact of these results on the methods of analyzing data arising from a categorical judgement procedure is outlined. In particular, some consequences are drawn with respect to the usual evaluation of correlations of such data.     

A computer art assessment system for the evaluation of space usage in drawings with application to the analysis of its relationship to level of dementia

Space usage in drawings is an important element that provides useful information about the drawer’s level of energy and psychological status. In this paper, we delineate the development of a computer system to objectively and quantitatively evaluate the space usage using regression models and computer analysis. We formulate regression models whose dependent variables are the grades and the ranks of space usages. We emphasize the contribution of the area of the painted convex hull in explaining the grades and ranks, which we propose as an independent variable. All possible independent variables including the area of painted regions and the number of colors used are evaluated by the computer algorithms developed in this study. We measure the validity of the system for evaluating grades and ranks. As an application to a real case, the system has been used to investigate the relationship between the space usage and the level of dementia.