Classical plume theory: 1937-2010 and beyond

Developing a theoretical description of turbulent plumes, the likes of which may be seen rising above industrial chimneys, is a daunting thought. Plumes are ubiquitous on a wide range of scales in both the natural and the man-made environments. Examples that immediately come to mind are the vapour plumes above industrial smoke stacks or the ash plumes forming particle-laden clouds above an erupting volcano. However, plumes also occur where they are less visually apparent, such as the rising stream of warmair above a domestic radiator, of oil from a subsea blowout or, at a larger scale, of air above the so-called urban heat island. In many instances, not only the plume itself is of interest but also its influence on the environment as a whole through the process of entrainment. Zeldovich (1937, The asymptotic laws of freely-ascending convective flows. Zh. Eksp. Teor. Fiz., 7, 1463-1465 (in Russian)), Batchelor (1954, Heat convection and buoyancy effects in fluids. Q. J. R. Meteor. Soc., 80, 339-358) and Morton et al. (1956, Turbulent gravitational convection from maintained and instantaneous sources. Proc. R. Soc. Lond. A, 234, 1-23) laid the foundations for classical plume theory, a theoretical description that is elegant in its simplicity and yet encapsulates the complex turbulent engulfment of ambient fluid into the plume. Testament to the insight and approach developed in these early models of plumes is that the essential theory remains unchanged and is widely applied today. We describe the foundations of plume theory and link the theoretical developments with the measurements made in experiments necessary to close these models before discussing some recent developments in plume theory, including an approach which generalizes results obtained separately for the Boussinesq and the non-Boussinesq plume cases. The theory presented - despite its simplicity - has been very successful at describing and explaining the behaviour of plumes across the wide range of scales they are observed. We present solutions to the coupled set of ordinary differential equations (the plume conservation equations) that Morton et al. (1956) derived from the Navier-Stokes equations which govern fluid motion. In order to describe and contrast the bulk behaviour of rising plumes from general area sources, we present closed-form solutions to the plume conservation equations that were achieved by solving for the variation with height of Morton's non-dimensional flux parameter Γ - this single flux parameter gives a unique representation of the behaviour of steady plumes and enables a characterization of the different types of plume. We discuss advantages of solutions in this form before describing extensions to plume theory and suggesting directions for new research. © 2010 The Author. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Probabilistic computation in human perception under variability in encoding precision.

A key function of the brain is to interpret noisy sensory information. To do so optimally, observers must, in many tasks, take into account knowledge of the precision with which stimuli are encoded. In an orientation change detection task, we find that encoding precision does not only depend on an experimentally controlled reliability parameter (shape), but also exhibits additional variability. In spite of variability in precision, human subjects seem to take into account precision near-optimally on a trial-to-trial and item-to-item basis. Our results offer a new conceptualization of the encoding of sensory information and highlight the brain's remarkable ability to incorporate knowledge of uncertainty during complex perceptual decision-making.

Variability in encoding precision accounts for visual short-term memory limitations.

It is commonly believed that visual short-term memory (VSTM) consists of a fixed number of "slots" in which items can be stored. An alternative theory in which memory resource is a continuous quantity distributed over all items seems to be refuted by the appearance of guessing in human responses. Here, we introduce a model in which resource is not only continuous but also variable across items and trials, causing random fluctuations in encoding precision. We tested this model against previous models using two VSTM paradigms and two feature dimensions. Our model accurately accounts for all aspects of the data, including apparent guessing, and outperforms slot models in formal model comparison. At the neural level, variability in precision might correspond to variability in neural population gain and doubly stochastic stimulus representation. Our results suggest that VSTM resource is continuous and variable rather than discrete and fixed and might explain why subjective experience of VSTM is not all or none.

Does precision decrease with set size?

The brain encodes visual information with limited precision. Contradictory evidence exists as to whether the precision with which an item is encoded depends on the number of stimuli in a display (set size). Some studies have found evidence that precision decreases with set size, but others have reported constant precision. These groups of studies differed in two ways. The studies that reported a decrease used displays with heterogeneous stimuli and tasks with a short-term memory component, while the ones that reported constancy used homogeneous stimuli and tasks that did not require short-term memory. To disentangle the effects of heterogeneity and short-memory involvement, we conducted two main experiments. In Experiment 1, stimuli were heterogeneous, and we compared a condition in which target identity was revealed before the stimulus display with one in which it was revealed afterward. In Experiment 2, target identity was fixed, and we compared heterogeneous and homogeneous distractor conditions. In both experiments, we compared an optimal-observer model in which precision is constant with set size with one in which it depends on set size. We found that precision decreases with set size when the distractors are heterogeneous, regardless of whether short-term memory is involved, but not when it is homogeneous. This suggests that heterogeneity, not short-term memory, is the critical factor. In addition, we found that precision exhibits variability across items and trials, which may partly be caused by attentional fluctuations.

Comparing crowding in human and ideal observers.

A visual target is more difficult to recognize when it is surrounded by other, similar objects. This breakdown in object recognition is known as crowding. Despite a long history of experimental work, computational models of crowding are still sparse. Specifically, few studies have examined crowding using an ideal-observer approach. Here, we compare crowding in ideal observers with crowding in humans. We derived an ideal-observer model for target identification under conditions of position and identity uncertainty. Simulations showed that this model reproduces the hallmark of crowding, namely a critical spacing that scales with viewing eccentricity. To examine how well the model fits quantitatively to human data, we performed three experiments. In Experiments 1 and 2, we measured observers' perceptual uncertainty about stimulus positions and identities, respectively, for a target in isolation. In Experiment 3, observers identified a target that was flanked by two distractors. We found that about half of the errors in Experiment 3 could be accounted for by the perceptual uncertainty measured in Experiments 1 and 2. The remainder of the errors could be accounted for by assuming that uncertainty (i.e., the width of internal noise distribution) about stimulus positions and identities depends on flanker proximity. Our results provide a mathematical restatement of the crowding problem and support the hypothesis that crowding behavior is a sign of optimality rather than a perceptual defect.

Optimal inference of sameness.

Deciding whether a set of objects are the same or different is a cornerstone of perception and cognition. Surprisingly, no principled quantitative model of sameness judgment exists. We tested whether human sameness judgment under sensory noise can be modeled as a form of probabilistically optimal inference. An optimal observer would compare the reliability-weighted variance of the sensory measurements with a set size-dependent criterion. We conducted two experiments, in which we varied set size and individual stimulus reliabilities. We found that the optimal-observer model accurately describes human behavior, outperforms plausible alternatives in a rigorous model comparison, and accounts for three key findings in the animal cognition literature. Our results provide a normative footing for the study of sameness judgment and indicate that the notion of perception as near-optimal inference extends to abstract relations.

A neurophysiologically plausible population code model for feature integration explains visual crowding.

An object in the peripheral visual field is more difficult to recognize when surrounded by other objects. This phenomenon is called "crowding". Crowding places a fundamental constraint on human vision that limits performance on numerous tasks. It has been suggested that crowding results from spatial feature integration necessary for object recognition. However, in the absence of convincing models, this theory has remained controversial. Here, we present a quantitative and physiologically plausible model for spatial integration of orientation signals, based on the principles of population coding. Using simulations, we demonstrate that this model coherently accounts for fundamental properties of crowding, including critical spacing, "compulsory averaging", and a foveal-peripheral anisotropy. Moreover, we show that the model predicts increased responses to correlated visual stimuli. Altogether, these results suggest that crowding has little immediate bearing on object recognition but is a by-product of a general, elementary integration mechanism in early vision aimed at improving signal quality.

A crowding model of visual clutter.

Visual information is difficult to search and interpret when the density of the displayed information is high or the layout is chaotic. Visual information that exhibits such properties is generally referred to as being "cluttered." Clutter should be avoided in information visualizations and interface design in general because it can severely degrade task performance. Although previous studies have identified computable correlates of clutter (such as local feature variance and edge density), understanding of why humans perceive some scenes as being more cluttered than others remains limited. Here, we explore an account of clutter that is inspired by findings from visual perception studies. Specifically, we test the hypothesis that the so-called "crowding" phenomenon is an important constituent of clutter. We constructed an algorithm to predict visual clutter in arbitrary images by estimating the perceptual impairment due to crowding. After verifying that this model can reproduce crowding data we tested whether it can also predict clutter. We found that its predictions correlate well with both subjective clutter assessments and search performance in cluttered scenes. These results suggest that crowding and clutter may indeed be closely related concepts and suggest avenues for further research.

On the generality of crowding: visual crowding in size, saturation, and hue compared to orientation.

Perception of peripherally viewed shapes is impaired when surrounded by similar shapes. This phenomenon is commonly referred to as "crowding". Although studied extensively for perception of characters (mainly letters) and, to a lesser extent, for orientation, little is known about whether and how crowding affects perception of other features. Nevertheless, current crowding models suggest that the effect should be rather general and thus not restricted to letters and orientation. Here, we report on a series of experiments investigating crowding in the following elementary feature dimensions: size, hue, and saturation. Crowding effects in these dimensions were benchmarked against those in the orientation domain. Our primary finding is that all features studied show clear signs of crowding. First, identification thresholds increase with decreasing mask spacing. Second, for all tested features, critical spacing appears to be roughly half the viewing eccentricity and independent of stimulus size, a property previously proposed as the hallmark of crowding. Interestingly, although critical spacings are highly comparable, crowding magnitude differs across features: Size crowding is almost as strong as orientation crowding, whereas the effect is much weaker for saturation and hue. We suggest that future theories and models of crowding should be able to accommodate these differences in crowding effects.

Visual search near threshold: Some features are more equal than others.

While searching for objects, we combine information from multiple visual modalities. Classical theories of visual search assume that features are processed independently prior to an integration stage. Based on this, one would predict that features that are equally discriminable in single feature search should remain so in conjunction search. We test this hypothesis by examining whether search accuracy in feature search predicts accuracy in conjunction search. Subjects searched for objects combining color and orientation or size; eye movements were recorded. Prior to the main experiment, we matched feature discriminability, making sure that in feature search, 70% of saccades were likely to go to the correct target stimulus. In contrast to this symmetric single feature discrimination performance, the conjunction search task showed an asymmetry in feature discrimination performance: In conjunction search, a similar percentage of saccades went to the correct color as in feature search but much less often to correct orientation or size. Therefore, accuracy in feature search is a good predictor of accuracy in conjunction search for color but not for size and orientation. We propose two explanations for the presence of such asymmetries in conjunction search: the use of conjunctively tuned channels and differential crowding effects for different features.

Simultaneous colour search renders other object properties less salient

A common approach to visualise multidimensional data sets is to map every data dimension to a separate visual feature. It is generally assumed that such visual features can be judged independently from each other. However, we have recently shown that interactions between features do exist [Hannus et al. 2004; van den Berg et al. 2005]. In those studies, we first determined individual colour and size contrast or colour and orientation contrast necessary to achieve a fixed level of discrimination performance in single feature search tasks. These contrasts were then used in a conjunction search task in which the target was defined by a combination of a colour and a size or a colour and an orientation. We found that in conjunction search, despite the matched feature discriminability, subjects significantly more often chose an item with the correct colour than one with correct size or orientation. This finding may have consequences for visualisation: the saliency of information coded by objects' size or orientation may change when there is a need to simultaneously search for colour that codes another aspect of the information. In the present experiment, we studied whether a colour bias can also be found in a more complex and continuous task, Subjects had to search for a target in a node-link diagram consisting of SO nodes, while their eye movements were being tracked, Each node was assigned a random colour and size (from a range of 10 possible values with fixed perceptual distances). We found that when we base the distances on the mean threshold contrasts that were determined in our previous experiments, the fixated nodes tend to resemble the target colour more than the target size (Figure 1a). This indicates that despite the perceptual matching, colour is judged with greater precision than size during conjunction search. We also found that when we double the size contrast (i.e. the distances between the 10 possible node sizes), this effect disappears (Figure 1b). Our findings confirm that the previously found decrease in salience of other features during colour conjunction search is also present in more complex (more 'visualisation- realistic') visual search tasks. The asymmetry in visual search behaviour can be compensated for by manipulating step sizes (perceptual distances) within feature dimensions. Our results therefore also imply that feature hierarchies are not completely fixed and may be adapted to the requirements of a particular visualisation. Copyright © 2005 by the Association for Computing Machinery, Inc.

No evidence for an item limit in change detection.

Change detection is a classic paradigm that has been used for decades to argue that working memory can hold no more than a fixed number of items ("item-limit models"). Recent findings force us to consider the alternative view that working memory is limited by the precision in stimulus encoding, with mean precision decreasing with increasing set size ("continuous-resource models"). Most previous studies that used the change detection paradigm have ignored effects of limited encoding precision by using highly discriminable stimuli and only large changes. We conducted two change detection experiments (orientation and color) in which change magnitudes were drawn from a wide range, including small changes. In a rigorous comparison of five models, we found no evidence of an item limit. Instead, human change detection performance was best explained by a continuous-resource model in which encoding precision is variable across items and trials even at a given set size. This model accounts for comparison errors in a principled, probabilistic manner. Our findings sharply challenge the theoretical basis for most neural studies of working memory capacity.