Gunnar Schmidtmann

Modeling probability and additive summation for detection across multiple mechanisms under the assumptions of signal detection theory.

Many studies have investigated how multiple stimuli combine to reach threshold. There are broadly speaking two ways this can occur: additive summation (AS) where inputs from the different stimuli add together in a single mechanism, or probability summation (PS) where different stimuli are detected independently by separate mechanisms. PS is traditionally modeled under high threshold theory (HTT); however, tests have shown that HTT is incorrect and that signal detection theory (SDT) is the better framework for modeling summation. Modeling the equivalent of PS under SDT is, however, relatively complicated, leading many investigators to use Monte Carlo simulations for the predictions. We derive formulas that employ numerical integration to predict the proportion correct for detecting multiple stimuli assuming PS under SDT, for the situations in which stimuli are either equal or unequal in strength. Both formulas are general purpose, calculating performance for forced-choice tasks with M alternatives, n stimuli, in Q monitored mechanisms, each subject to a non-linear transducer with exponent τ. We show how the probability (and additive) summation formulas can be used to simulate psychometric functions, which when fitted with Weibull functions make signature predictions for how thresholds and psychometric function slopes vary as a function of τ, n, and Q. We also show how one can fit the formulas directly to real psychometric functions using data from a binocular summation experiment, and show how one can obtain estimates of τ and test whether binocular summation conforms more to PS or AS. The methods described here can be readily applied using software functions newly added to the Palamedes toolbox.

Detecting shapes in noise: tuning characteristics of global shape mechanisms

The proportion of signal elements embedded in noise needed to detect a signal is a standard tool for investigating motion perception. This paradigm was applied to the shape domain to determine how local information is pooled into a global percept. Stimulus arrays consisted of oriented Gabor elements that sampled the circumference of concentric radial frequency (RF) patterns. Individual Gabors were oriented tangentially to the shape (signal) or randomly (noise). In different conditions, signal elements were located randomly within the entire array or constrained to fall along one of the concentric contours. Coherence thresholds were measured for RF patterns with various frequencies (number of corners) and amplitudes (“sharpness” of corners). Coherence thresholds (about 10% = 15 elements) were lowest for circular shapes. Manipulating shape frequency or amplitude showed a range where thresholds remain unaffected (frequency ≤ RF4; amplitude ≤ 0.05). Increasing either parameter caused thresholds to rise. Compared to circles, thresholds increased by approximately four times for RF13 and five times for amplitudes of 0.3. Confining the signals to individual contours significantly reduced the number of elements needed to reach threshold (between 4 and 6), independent of the total number of elements on the contour or contour shape. Finally, adding external noise to the orientation of the elements had a greater effect on detection thresholds than adding noise to their position. These results provide evidence for a series of highly sensitive, shape-specific analysers which sum information globally but only from within specific annuli. These global mechanisms are tuned to position and orientation of local elements from which they pool information. The overall performance for arrays of elements can be explained by the sensitivity of multiple, independent concentric shape detectors rather than a single detector integrating information widely across space (e.g. Glass pattern detector).

Rejecting probability summation for radial frequency patterns, not so Quick!

Radial frequency (RF) patterns are used to assess how the visual system processes shape. They are thought to be detected globally. This is supported by studies that have found summation for RF patterns to be greater than what is possible if the parts were being independently detected and performance only then improved with an increasing number of cycles by probability summation between them. However, the model of probability summation employed in these previous studies was based on High Threshold Theory (HTT), rather than Signal Detection Theory (SDT). We conducted rating scale experiments to investigate the receiver operating characteristics. We find these are of the curved form predicted by SDT, rather than the straight lines predicted by HTT. This means that to test probability summation we must use a model based on SDT. We conducted a set of summation experiments finding that thresholds decrease as the number of modulated cycles increases at approximately the same rate as previously found. As this could be consistent with either additive or probability summation, we performed maximum-likelihood fitting of a set of summation models (Matlab code provided in our Supplementary material) and assessed the fits using cross validation. We find we are not able to distinguish whether the responses to the parts of an RF pattern are combined by additive or probability summation, because the predictions are too similar. We present similar results for summation between separate RF patterns, suggesting that the summation process there may be the same as that within a single RF.

Distinct lower visual field preference for object shape

Humans manipulate objects chiefly within their lower visual field, a consequence of upright posture and the anatomical position of hands and arms.This study tested the hypothesis of enhanced sensitivity to a range of stimuli within the lower visual field. Following current models of hierarchical processing within the ventral steam, discrimination sensitivity was measured for orientation, curvature, shape (radial frequency patterns), and faces at various para-central locations (horizontal, vertical, and main diagonal meridians) and eccentricities (5° and 10°). Peripheral sensitivity was isotropic for orientation and curvature. By contrast, observers were significantly better at discriminating shapes throughout the lower visual field compared to elsewhere. For faces, however, peak sensitivity was found in the left visual field, corresponding to the right hemispheric localization of human face processing. Presenting head outlines without any internal features (e.g., eyes, mouth) recovered the lower visual field advantage found for simple shapes. A lower visual field preference for the shape of an object, which is absent for more localized information (orientation and curvature) but also for more complex objects (faces), is inconsistent with a strictly feed-forward model and poses a challenge for multistage models of object perception. The distinct lower visual field preference for contour shapes is, however, consistent with an asymmetry at intermediate stages of visual processing, which may play a key role in representing object characteristics that are particularly relevant to visually guided actions.

Shape recognition: convexities, concavities and things in between.

Visual objects are effortlessly recognized from their outlines, largely irrespective of viewpoint. Previous studies have drawn different conclusions regarding the importance to shape recognition of specific shape features such as convexities and concavities. However, most studies employed familiar objects, or shapes without curves, and did not measure shape recognition across changes in scale and position. We present a novel set of random shapes with well-defined convexities, concavities and inflections (intermediate points), segmented to isolate each feature type. Observers matched the segmented reference shapes to one of two subsequently presented whole-contour shapes (target or distractor) that were re-scaled and re-positioned. For very short segment lengths, performance was significantly higher for convexities than for concavities or intermediate points and for convexities remained constant with increasing segment length. For concavities and intermediate points, performance improved with increasing segment length, reaching convexity performance only for long segments. No significant differences between concavities and intermediates were found. These results show for the first time that closed curvilinear shapes are encoded using the positions of convexities, rather than concavities or intermediate regions. A shape-template model with no free parameters gave an excellent account of the data.

Probability, not linear summation, mediates the detection of concentric orientation-defined textures

Previous studies investigating signal integration in circular Glass patterns have concluded that the information in these patterns is linearly summed across the entire display for detection. Here we test whether an alternative form of summation, probability summation (PS), modeled under the assumptions of Signal Detection Theory (SDT), can be rejected as a model of Glass pattern detection. PS under SDT alone predicts that the exponent β of the Quick- (or Weibull-) fitted psychometric function should decrease with increasing signal area. We measured spatial integration in circular, radial, spiral, and parallel Glass patterns, as well as comparable patterns composed of Gabors instead of dot pairs. We measured the signal-to-noise ratio required for detection as a function of the size of the area containing signal, with the remaining area containing dot-pair or Gabor-orientation noise. Contrary to some previous studies, we found that the strength of summation never reached values close to linear summation for any stimuli. More importantly, the exponent β systematically decreased with signal area, as predicted by PS under SDT. We applied a model for PS under SDT and found that it gave a good account of the data. We conclude that probability summation is the most likely basis for the detection of circular, radial, spiral, and parallel orientation-defined textures.

Nothing more than a pair of curvatures: A common mechanism for the detection of both radial and non-radial frequency patterns

ABSTRACT Radial frequency (RF) patterns, which are sinusoidal modulations of a radius in polar coordinates, are commonly used to study shape perception. Previous studies have argued that the detection of RF patterns is either achieved globally by a specialized global shape mechanism, or locally using as cue the maximum tangent orientation difference between the RF pattern and the circle. Here we challenge both ideas and suggest instead a model that accounts not only for the detection of RF patterns but also for line frequency patterns (LF), i.e. contours sinusoidally modulated around a straight line. The model has two features. The first is that the detection of both RF and LF patterns is based on curvature differences along the contour. The second is that this curvature metric is subject to what we term the Curve Frequency Sensitivity Function, or CFSF, which is characterized by a flat followed by declining response to curvature as a function of modulation frequency, analogous to the modulation transfer function of the eye. The evidence that curvature forms the basis for detection is that at very low modulation frequencies (1–3 cycles for the RF pattern) there is a dramatic difference in thresholds between the RF and LF patterns, a difference however that disappears at medium and high modulation frequencies. The CFSF feature on the other hand explains why thresholds, rather than continuously declining with modulation frequency, asymptote at medium and high modulation frequencies. In summary, our analysis suggests that the detection of shape modulations is processed by a common curvature‐sensitive mechanism that is subject to a shape‐frequency‐dependent transfer function. This mechanism is independent of whether the modulation is applied to a circle or a straight line.

RF shape channels: The processing of compound Radial Frequency patterns.

• Radial Frequency patterns (RF) have been extensively used to study intermediate stages of shape perception (Wilkinson et al., 1998; Loffler, 2008; Schmidtmann et al., 2012). • Combinations of RF patterns can be used to investigate natural shapes like faces or fruits/vegetables. • Previous studies showed evidence that different RF patterns are processed by different independent narrowly-tuned RF shape channels (Bell & Badcock, 2009; Bell et al., 2007; 2009, cf. Dickinson et al, 2013).

Sensitivity to binocular disparity is reduced by mild traumatic brain injury.

Purpose The impairment of visual functions is one of the most common complaints following mild traumatic brain injury (mTBI). Traumatic brain injury-associated visual deficits include blurred vision, reading problems, and eye strain. In addition, previous studies have found evidence that TBI can diminish early cortical visual processing, particularly for second-order stimuli. We investigated whether cortical processing of binocular disparity is also affected by mTBI. Methods In order to investigate the influence of mTBI on global stereopsis, we measured the quick Disparity Sensitivity Function (qDSF) in 22 patients with mTBI. Patients with manifest strabismus and double vision were excluded. Compared with standard clinical tests, the qDSF is unique in that it offers a quick and accurate estimate of thresholds across the whole spatial frequency range. Results Results show that disparity sensitivity in the mTBI patients were significantly reduced compared with the normative dataset (n = 61). The peak spatial frequency was not affected. Conclusions Our results suggest that the reduced disparity sensitivity in patients with mTBI is more likely caused by cortical changes (e.g., axonal shearing, or reduced interhemispheric communication) rather than oculomotor dysfunction.

Imagining circles--Empirical data and a perceptual model for the arc-size illusion.

An essential part of visual object recognition is the evaluation of the curvature of both an objects outline as well as the contours on its surface. We studied a striking illusion of visual curvature--the arc-size illusion (ASI)--to gain insight into the visual coding of curvature. In the ASI, short arcs are perceived as flatter (less curved) compared to longer arcs of the same radius. We investigated if and how the ASI depends on (i) the physical size of the stimulus and (ii) on the length of the arc. Our results show that perceived curvature monotonically increases with arc length up to an arc angle of about 60°, thereafter remaining constant and equal to the perceived curvature of a full circle. We investigated if the misjudgment of curvature in the ASI translates into predictable biases for three other perceptual tasks: (i) judging the position of the centre of circular arcs; (ii) judging if two circular arcs fall on the circumference of the same (invisible) circle and (iii) interpolating the position of a point on the circumference of a circle defined by two circular arcs. We found that the biases in all the above tasks were reliably predicted by the same bias mediating the ASI. We present a simple model, based on the central angle subtended by an arc, that captures the data for all tasks. Importantly, we argue that the ASI and related biases are a consequence of the fact that an objects curvature is perceived as constant with viewing distance, in other words is perceptually scale invariant.