The perception of number from the separability of the stimulus: The Stroop effect revisited

The literature on numerical perception is reviewed from the standpoint of research on selective attention, and predictions are made concerning the dimensional interaction between physical and numerical size of numerals. We manipulated stimulus differences to make the classification of numerical value slightly better (Experiment 1), substantially better (Experiment 2), or worse (Experiments 3–4) than classification of physical size. Garner, Stroop, and redundancy effects were used to gauge the degree of interactive processing. For nearly matched discriminability, both number and size appeared separable when the dimensions were varied orthogonally, but showed Stroop interference and redundancy gain when the dimensions were varied in a correlative fashion. When mismatched, asymmetric Garner and Stroop effects emerged in orthogonal contexts along with Stroop and redundancy gains in correlative contexts. These findings define a unique relation: Numerical value and physical size were optionally separable dimensions. We conclude that a magnitude representation is not mandatory for the perception of numerals. Our conclusions offer a new perspective for understanding both numerical perception and the Stroop phenomenon itself.     

Spatial metrics of integral and separable dimensions

Two experiments were conducted to investigate the relationship between dimensional integrality and the form of the combination rule or spatial metric used in a similarity judgment task. In Experiment 1, two groups of eight subjects judged the dissimilarity of all pairwise combinations of 12 rectangles differing in height and width (integral dimensions) or 12 circles differing in size and diameter orientation (separable dimensions). The dimensional organization and spatial metric of both sets of dimensions were determined. The results showed that differences in height and width contributed independently to judgments of the overall dissimilarity of rectangles and that these dimensions were combined using a Euclidean metric. In contrast, substantial interactions between circle size and diameter orientation were found. Combinations of these dimensions also appeared to violate the triangle inequality implying that no spatial metric was appropriate. In Experiment 2, parallelogram size and tilt were similarly analyzed. Although some degree of dimensional interaction was observed, it was found that on the average these dimensions were combined using a city-block metric. In a subsequent speeded classification task, orthogonal interference was observed, which suggested that size and tilt are integral dimensions. The implications of both experiments for the supposed association between the Euclidean metric and dimensional integrality are discussed.     

Dimensional separability or flexibility of attention? Age trends in perceiving configural stimuli

The hypothesis that perceptual development proceeds from less to greater dimensional separability was tested by giving a speeded classification task to first and fouth graders. In Experiment 1, children sorted line drawings of a cartoon figure into two piles according to his eyes (open or closed) or his mouth (smiling or frowning). First graders exhibited the pattern characteristic of integral dimensions: An interference effect was found when dimensions were combined orthogonally and a facilitation effect when dimensions were correlated, relative to a condition in which only one dimension varied. Fourth graders exhibited an optional separable pattern: There was facilitation in the correlated condition but no interference in the orthogonal condition. These results support the hypothesis that development proceeds toward greater flexibility of attention rather than simply toward increasing separability. A flexible field hypothesis of visual attention was suggested to explain how older children can exercise such optional separability. This hypothesis correctly predicted the asymmetrical relation between dimensions found in Experiments 2 and 3: For both first and fourth graders, the eyes of a cartoon figure (open or closed) and his location (right or left) were nonseparable when the eyes were relevant and separable (or optional separable) when location was relevant.     

Is stimulus structure in the Mind's eye? An examination of dimensional structure in iconic memory

Three experiments examined integral and separable dimensions as attentional selective cues from iconic memory. Sperling's partial-report task was employed with physically separable and physically integral stimuli. Stimulus displays were varied in terms of the redundancy of the irrelevant non-cued dimensions—i.e. control, correlated, and orthogonal conditions within Garner's (1974a) framework—and selective readout performance was measured. The results demonstrate that, in initial processing, physically separable dimensions (e.g., circle size vs. angle of radial line) can be selectively attended to as independent dimensions, but physically integral dimensions (e.g., size vs. brightness and height vs. width) can only be represented as wholistic integral objects. Implications of these findings for current models of feature integration and perceptual development are discussed.     

Dimensional interaction between color and pitch

In three experiments I investigated the nature of cross-modal dimensional interaction by testing speeded classification of the synesthetically corresponding dimensions of color (white-black) and pitch (high-low). Experiment 1 showed significant Garner interference when these dimensions were varied orthogonally--redundancy gain for positively correlated dimensions and redundancy loss for negatively correlated dimensions. Attributes from synesthetically congruent stimuli were classified faster than attributes from incongruent stimuli (a congruity effect). Experiment 2 tested a perceptual explanation of this interaction (i.e., that color and pitch are configural dimensions) by using Pomerantz's (1986) diagnostic (comparison of selective and divided attention performance). The configurality hypothesis received little support. Experiment 3 examined the effect of optional processes on color and pitch classification. The results suggest that partly strategic and partly mandatory components may constitute overall performance. Three alternative explanations of the color-pitch interaction--perceptual, semantic, and response based--are evaluated in the context of the present results.     

Linear separability in superordinate natural language concepts

Two experiments are reported in which linear separability was investigated in superordinate natural language concept pairs (e.g., toiletry-sewing gear). Representations of the exemplars of semantically related concept pairs were derived in two to five dimensions using multidimensional scaling (MDS) of similarities based on possession of the concept features. Next, category membership, obtained from an exemplar generation study (in Experiment 1) and from a forced-choice classification task (in Experiment 2) was predicted from the coordinates of the MDS representation using log linear analysis. The results showed that all natural kind concept pairs were perfectly linearly separable, whereas artifact concept pairs showed several violations. Clear linear separability of natural language concept pairs is in line with independent cue models. The violations in the artifact pairs, however, yield clear evidence against the independent cue models.     

Flexible cue combination in the guidance of attention in visual search

Hodsoll and Humphreys (2001) have assessed the relative contributions of stimulus-driven and user-driven knowledge on linearly- and nonlinearly separable searches. However, the target feature used to determine linear separability in their task (i.e., target size) was required to locate the target. In the present work, we investigated the contributions of stimulus-driven and user-driven knowledge when a linearly- or a nonlinearly-separable feature is available but not required for target identification. We asked observers to complete a series of standard color × orientation conjunction searches in which target size was either linearly- or nonlinearly separable from the size of the distractors. When guidance by color × orientation and guidance by size information are both available, observers rely on whichever information results in the best search efficiency. This is the case irrespective of whether we provide target foreknowledge by blocking stimulus conditions, suggesting that feature information is used in both a stimulus-driven and a user-driven fashion.     

Driving attention with the top down: The relative contribution of target templates to the linear separability effect in the size dimension

Bauer, Jolicoeur, and Cowan (1996a, 1996b, 1998) have shown that visual search for a target among distractors is apparently serial if the target is nonlinearly separable from the distractors in a particular feature space (e.g., color or size). In contrast, if the target is linearly separable from the distractors, search is relatively easy and seemingly spatially parallel. We examined the contribution of top-down knowledge of the target to the linear separability effect on search. Two visual search experiments were conducted using small, medium, or large circles as targets. In the first experiment, participants could use knowledge of the target to guide search, whereas, in the second, the target was unknown on each trial. Search for a medium (nonlinearly separable) target among small or large distractors benefited least from knowledge of the target as compared with search for a small or large target. Thus, the linear separability effect can be determined in part by use of top-down knowledge to facilitate the detection of targets at the ends of a continuum defining the stimuli.     

Driving attention with the top down: the relative contribution of target templates to the linear separability effect in the size dimension

Bauer, Jolicoeur, and Cowan (1996a, 1996b, 1998) have shown that visual search for a target among distractors is apparently serial if the target is nonlinearly separable from the distractors in a particular feature space (e.g., color or size). In contrast, if the target is linearly separable from the distractors, search is relatively easy and seemingly spatially parallel. We examined the contribution of top-down knowledge of the target to the linear separability effect on search. Two visual search experiments were conducted using small, medium, or large circles as targets. In the first experiment, participants could use knowledge of the target to guide search, whereas, in the second, the target was unknown on each trial. Search for a medium (nonlinearly separable) target among small or large distractors benefited least from knowledge of the target as compared with search for a small or large target. Thus, the linear separability effect can be determined in part by use of top-down knowledge to facilitate the detection of targets at the ends of a continuum defining the stimuli.     

Development and the continuum of dimensional separability

Recent studies of adult perception may be interpreted in terms of a continuum from the integral to the separable perception of multidimensional variation. Three experiments examined the existence of such a continuum across developmental levels. These experiments are important because several developmental studies suggest that the separability of dimensions increases with age. Despite this assumed developmental trend, it was hypothesized that therelative separability of dimensions does not change with age. The results of the three experiments did not support the hypothesis. The ordering of dimensions by their relative separability changes with development. These results suggest that the relative separability and potential separability of dimensions are not fixed. In light of the results, it is hypothesized that the integrality or separability of dimensions in adult perception is primarily determined by perceptual learning and not by the inherent characteristics of combinations of stimulus dimensions.     

Ebbinghaus illusion: context, contour, and age influence the judged size of a circle amidst circles

Subjects (groups of age 6, 8, 10, 12, and 21 years, with 60 or more persons per group) made comparative size judgments between a single circle and a standard circle that was flanked by four context circles. The context circles varied in diameter, proximity to the standard circle, and amount of circumference displayed. (When circumferences are complete, the display is the usual Ebbinghaus configuration. A variant consisted of the Ebbinghaus display with the outer three quarters of each context-circle circumference removed.) A contour-plus-context theory accounts for the data: Contours attract at every age level, and context (size of flanking circles relative to the standard circle) leads to constrast in judged size beyond age 6. Contour and context effects decreased with decreasing proximity between context circles and the standard circle.     

On Separability of and Interference Between Tracking Dimensions in Dual-Axis Tracking

Time-shared tasks may conceivably be separable or integral. A case in which the question of separability seems quite relevant is dual-axis tracking. To test the interaction between tracking dimensions, we first studied whether they interfere with each other. Practiced subjects performed tracking on one or two axes, with or without feedback indicators and with or without a requirement to allocate resources unevenly between axes. They also performed with or without a concurrent binary classification of visually presented digits which were presented within a moving square that served as the target for tracking. Small deficits were found in the performance of both tracking and digit classification when performed together. However, the conditions of tracking did not have a discernible effect on either tracking or digit classification. Hence, the introduction of a second tracking axis probably does not have harmful consequences either on tracking itself or on any other task time-shared with tracking.

Further studies were conducted to examine whether the absence of an effect of number of tracking axes is due to their integrality. Ordinary position tracking was paired either with another similar task on the other axis or with a novel sort of tracking in which subjects had to continually match sizes of moving rectangles. Tasks were paired under both divided-attention and focused-attention instructions. No interference on position tracking was observed even when the types of task on the two axes differed, and no other evidence for integrality of the homogeneous task pairs was found.     

Attentional Modulation of Size Contrast

A test circle surrounded by smaller context circles appears larger if presented in isolation, whereas a test circle surrounded by large context circles is seen as smaller than in isolation. Two experiments are reported indicating that this phenomenon, the Ebbinghaus illusion, depends on whether subjects are attending to the context circles. Subjects first saw a reference circle and then a briefly presented (150 msec) test circle. Their task was to determine whether the test circle was larger or smaller than the reference. The test circle was surrounded by smaller context circles of one colour arrayed along a horizontal axis centred on the test, and larger context circles of a different colour arrayed along a vertical axis centred on the test. Subjects judged both the size of the test and the colours of either the small or large context circles. Perceived test size changed systematically, depending on which context circles were task-relevant.     

Visual search for size-defined target objects is modulated by the Ebbinghaus apparent-size illusion: facilitatory and inhibitory effects of the context objects

Five experiments were carried out to investigate under which conditions the apparent size of objects is computed and exploited optimally in visual search for size-defined targets. Observers searched for a target test circle that was retinally larger than the distractor test circles, with both types of circles surrounded by context circles modulating the apparent size of the test circles (Ebbinghaus illusion). RTs were the faster the better test circles could be differentiated from the context circles, ie with smaller numbers of context circles, larger distances, and higher lightness (or colour) contrast between test and context circles. Apparent-size modulation had a strong influence on search RTs, resulting in faster RTs with smaller, and slower RTs with larger, context circles. A model assuming overall facilitatory effects of the apparent-size modulation and interference effects arising from decreasing test-context circle discriminability can explain the present results.     

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.     

The linear separability effect in color visual search: Ruling out the additive color hypothesis

Bauer, Jolicoeur, and Cowan (1996b, 1996c) demonstrated difficult visual search for color targets that were not linearly separable (in color space) from two distractor colors and easier search for linearly separable targets. This suggested that search is mediated by a chromatically linear discrimination mechanism (see D’Zmura, 1991). However, in those experiments, the targets that were not linearly separable fell midway between the distractor colors and thus corresponded to the admix of the distractor colors. An alternate interpretation of the results of Bauer et al. is that search was more difficult when the target corresponded to the distractor admix than when it did not. We tested this hypothesis in three experiments by contrasting conditions in which a target that was not linearly separable did or did not correspond to the admix of the distractor colors. In all cases, a target that was not linearly separable produced difficult search, demonstrating that linear separability determines search performance.     

Perceptual and decisional factors influencing the discrimination of inversion in the Thatcher illusion

The Thatcher illusion (Thompson, 1980) is considered to be a prototypical illustration of the notion that face perception is dependent on configural processes and representations. We explored this idea by examining the relative contributions of perceptual and decisional processes to the ability of observers to identify the orientation of two classes of forms-faces and churches-and a set of their component features. Observers were presented with upright and inverted images of faces and churches in which the components (eyes, mouth, windows, doors) were presented either upright or inverted. Observers first rated the subjective grotesqueness of all of the images and then performed a complete identification task in which they had to identify the orientation of the overall form and the orientation of each of the interior features. Grotesqueness ratings for both classes of image showed the standard modulation of rated grotesqueness as a function of orientation. The complete identification results revealed violations of both perceptual and decisional separability but failed to reveal any violations of within-stimulus (perceptual) independence. In addition, exploration of a simple bivariate Gaussian signal detection model of the relationship between identification performance and judged grotesqueness suggests that within-stimulus violations of perceptual independence on their own are insufficient for producing the illusion. This lack of evidence for within-stimulus configurality suggests the need for a critical reevaluation of the role of configural processing in the Thatcher illusion. (PsycINFO Database Record (c) 2011 APA, all rights reserved).     

Categorization response time with multidimensional stimuli

Categorization response time (RT) was examined in three separate experiments, in each of which exemplars varied on two physical dimensions. Three different types of stimuli were used: (1) horizontal and vertical line segments of varying length that were joined at an upper left corner, (2) rectangles of varying width and height, and (3) circles or semicircles of varying size with a radial arm of varying orientation. No evidence was found that stimulus familiarity or the category prototypes played any special role in determining categorization RT. Instead, RT decreased with distance from the stimulus to the categorization decision bound.     

Interference and dominance in texture segregation: hue, geometric form, and line orientation

Five experiments were designed to test whether (1) lowering the similarity of elements within a region of texture (low-similarity arrays) would interfere with texture segregation, and (2) there would be dominance of one type of property difference over another in determining an observer's choice of boundary in two-boundary (ambiguous) displays. In Experiments 1 and 2, the interference question was assessed using stimuli formed from the dimensions hue and geometric form (circle/square or straight/curved novel shapes). The results indicated that when boundary judgments were based on form differences, segregation was significantly impaired by hue variation. However, hue segregations were not affected by form variation. In Experiments 3-5, the dominance question was examined using stimuli formed from hue and geometric form, as well as those formed from hue and line orientation (horizontal/vertical). Analyses revealed that there was no dominance of one type of property difference over another. Rather, observers' performance was completely predicted by the relative discriminabilities of the two boundaries. These findings support Beck's (1982) model of textural segmentation and call into question traditional notions of the preattentive stage of perceptual processing.     

Dark adaptation and short-wavelength backgrounds decrease perceived size.

The effects of background luminance, contrast, and background wavelength on the perceived size of small line figures were studied at mesopic levels of light adaptation. Perceived size diminished at low levels of background luminance. The effect disappeared at high levels of luminance. Perceived size of luminous circles increased as a logarithmic function of background luminance when the background intensity did not exceed 25 td(1). The strength of the size effect decreased as a function of circle diameter from 0-125 to 2 deg of visual angle(2). Perceived size of small luminous circles, subtending less than 0-5 deg, also increased as a function of contrast at low values of contrast but at very high values of contrast there was a decrease in perceived size. Background luminance had the same effect on the perceived size of circles as on the perceived size of spatial cycles in gratings. Control experiments led to the conclusion that dark adaptation is the primary source of the size effects. The main evidence for this conclusion was obtained from a demonstration that the same background luminance produced either an increase or a decrease in perceived size, depending on the adaptational state of the eye. It was also found that a shift from cone vision to rod vision contributes to the effects, for a stimulus looked smaller on a short-wavelength background than on a long-wavelength background. The size effects can be predicted from the changes of receptive-field properties of single neurones under corresponding conditions of stimulation, if it is assumed that the perception of size is mediated by size-specific channels formed of single neurones. Stimulation that leads to an activation of small receptive fields appears to indicate to the brain the presence of small retinal images. If small receptive fields are experimentally made responsive to larger retinal images, an underestimation of size results.