Anisotropy for spatial summation of elongated patches of grating:A tale of two tails

Studies of spatial summation often use sinusoidal gratings with blurred edges. When the envelope is elongated (i) along the grating stripes and (ii) at right angles to the grating stripes, we refer to the stimuli as skunk-tails and tiger-tails respectively. Previous work [Polat & Tyler, 1999; Vision Research, 39, 887-895.] has found that sensitivity to skunk-tails is greater than for tiger-tails, but there have been several failures to replicate this result within a subset of the conditions. To address this we measured detection thresholds for skunk-tails, tiger-tails and squares of grating with sides matched to the lengths of the tails. For foveal viewing, we found a contrast sensitivity advantage in the order of 2 dB for skunk-tails over tiger-tails, but only for horizontal gratings. For vertical gratings, sensitivity was very similar for both tail-types. When the stimuli were presented parafoveally (upper right visual field), a small advantage was found for skunk-tails over tiger-tails at both orientations, and spatial summation slopes were close to that of the ideal observer. We did not replicate the findings of Polat & Tyler, but our results are consistent with (i) those of Foley et al. [Foley, J. M., Varadharajan, S., Koh, C. C., & Farias, C. Q. (2007) Vision Research, 47, 85-107.] who used only vertical gratings and (ii) those from modelfest, where only horizontal gratings were used. The small effect of tail-type here suggests an anisotropy in the underlying physiology. © 2007 Elsevier Ltd. All rights reserved.

Grating and plaid masks indicate linear summation in a contrast gain pool

In human vision, the response to luminance contrast at each small region in the image is controlled by a more global process where suppressive signals are pooled over spatial frequency and orientation bands. But what rules govern summation among stimulus components within the suppressive pool? We addressed this question by extending a pedestal plus pattern mask paradigm to use a stimulus with up to three mask components: a vertical 1 c/deg pedestal, plus pattern masks made from either a grating (orientation = -45°) or a plaid (orientation = ±45°), with component spatial frequency of 3 c/deg. The overall contrast of both types of pattern mask was fixed at 20% (i.e., plaid component contrasts were 10%). We found that both of these masks transformed conventional dipper functions (threshold vs. pedestal contrast with no pattern mask) in exactly the same way: The dipper region was raised and shifted to the right, but the dipper handles superimposed. This equivalence of the two pattern masks indicates that contrast summation between the plaid components was perfectly linear prior to the masking stage. Furthermore, the pattern masks did not drive the detecting mechanism above its detection threshold because they did not abolish facilitation by the pedestal (Foley, 1994). Therefore, the pattern masking could not be attributed to within-channel masking, suggesting that linear summation of contrast signals takes place within a suppressive contrast gain pool. We present a quantitative model of the effects and discuss the implications for neurophysiological models of the process. © 2004 ARVO.

Area summation and masking

At detection threshold, sensitivity improves as the area of a test grating increases, but not when the test is placed on a pedestal and the task becomes contrast discrimination (G. E. Legge, & J. M. Foley, 1980). This study asks whether the abolition of area summation is specific to the situation where mask and test stimuli have the same spatial frequency and orientation ("within-channel" masking) or is more general, also occurring when mask and test stimuli are very different ("cross-channel" masking). Threshold versus contrast masking functions were measured where the test and mask were either both small (SS), both large (LL), or small and large, respectively (SL). For within-channel masking, facilitation and area summation were found at low mask contrasts, but the results for SS and LL converged at intermediate contrasts and above, replicating Legge and Foley (1980). For all three observers, less facilitation was found for SL than for SS. For cross-channel masking, area summation occurred across the entire masking function and results for SS and SL were identical. The results for the entire data set were well fit by an extended version of a contrast masking model (J. M. Foley, 1994) in which the weights of excitatory and suppressive surround terms were free parameters. I conclude that (i) there is no empirical abolition of area summation for cross-channel masking, (ii) within-channel area summation can be abolished empirically without being disabled in the model, (iii) observers are able to restrict the area of spatial integration, but not suppression, (iv) extending a cross-channel mask to the surround has no effect on contrast detection, and (v) there is a formal similarity between area summation and contrast adaptation. © 2004 ARVO.

Performance data indicate summation for pictorial depth-cues in slanted surfaces

Over recent years much has been learned about the way in which depth cues are combined (e.g. Landy et al., 1995). The majority of this work has used subjective measures, a rating scale or a point of subjective equality, to deduce the relative contributions of different cues to perception. We have adopted a very different approach by using two interval forced-choice (2IFC) performance measures and a signal processing framework. We performed summation experiments for depth cue increment thresholds between pairs of pictorial depth cues in displays depicting slanted planar surfaces made from arrays of circular 'contrast' elements. Summation was found to be ideal when size-gradient was paired with contrast-gradient for a wide range of depth-gradient magnitudes in the null stimulus. For a pairing of size-gradient and linear perspective, substantial summation (> 1.5 dB) was found only when the null stimulus had intermediate depth gradients; when flat or steeply inclined surfaces were depicted, summation was diminished or abolished. Summation was also abolished when one of the target cues was (i) not a depth cue, or (ii) added in conflict. We conclude that vision has a depth mechanism for the constructive combination of pictorial depth cues and suggest two generic models of summation to describe the results. Using similar psychophysical methods, Bradshaw and Rogers (1996) revealed a mechanism for the depth cues of motion parallax and binocular disparity. Whether this is the same or a different mechanism from the one reported here awaits elaboration.

Spiral mechanisms are required to account for summation of complex motion components

Stimuli from one family of complex motions are defined by their spiral pitch, where cardinal axes represent signed expansion and rotation. Intermediate spirals are represented by intermediate pitches. It is well established that vision contains mechanisms that sum over space and direction to detect these stimuli (Morrone et al., Nature 376 (1995) 507) and one possibility is that four cardinal mechanisms encode the entire family. We extended earlier work (Meese & Harris, Vision Research 41 (2001) 1901) using subthreshold summation of random dot kinematograms and a two-interval forced choice technique to investigate this possibility. In our main experiments, the spiral pitch of one component was fixed and that of another was varied in steps of 15° relative to the first. Regardless of whether the fixed component was aligned with cardinal axes or an intermediate spiral, summation to-coherence-threshold between the two components declined as a function of their difference in spiral pitch. Similar experiments showed that none of the following were critical design features or stimulus parameters for our results: superposition of signal dots, limited life-time dots, the presence of speed gradients, stimulus size or the number of dots. A simplex algorithm was used to fit models containing mechanisms spaced at a pitch of either 90° (cardinal model) or 45° (cardinal+model) and combined using a fourth-root summation rule. For both models, direction half-bandwidth was equated for all mechanisms and was the only free parameter. Only the cardinal+model could account for the full set of results. We conclude that the detection of complex motion in human vision requires both cardinal and spiral mechanisms with a half-bandwidth of approximately 46°. © 2002 Elsevier Science Ltd. All rights reserved.

Adaptation and gain pool summation:Alternative models and masking data

Foley [J. Opt. Soc. Am. A 11 (1994) 1710] has proposed an influential psychophysical model of masking in which mask components in a contrast gain pool are raised to an exponent before summation and divisive inhibition. We tested this summation rule in experiments in which contrast detection thresholds were measured for a vertical 1 c/deg (or 2 c/deg) sine-wave component in the presence of a 3 c/deg (or 6 c/deg) mask that had either a single component oriented at -45° or a pair of components oriented at ±45°. Contrary to the predictions of Foley's model 3, we found that for masks of moderate contrast and above, threshold elevation was predicted by linear summation of the mask components in the inhibitory stage of the contrast gain pool. We built this feature into two new models, referred to as the early adaptation model and the hybrid model. In the early adaptation model, contrast adaptation controls a threshold-like nonlinearity on the output of otherwise linear pathways that provide the excitatory and inhibitory inputs to a gain control stage. The hybrid model involves nonlinear and nonadaptable routes to excitatory and inhibitory stages as well as an adaptable linear route. With only six free parameters, both models provide excellent fits to the masking and adaptation data of Foley and Chen [Vision Res. 37 (1997) 2779] but unlike Foley and Chen's model, are able to do so with only one adaptation parameter. However, only the hybrid model is able to capture the features of Foley's (1994) pedestal plus orthogonal fixed mask data. We conclude that (1) linear summation of inhibitory components is a feature of contrast masking, and (2) that the main aftereffect of spatial adaptation on contrast increment thresholds can be assigned to a single site. © 2002 Elsevier Science Ltd. All rights reserved.

Contrast integration over area is extensive: a three-stage model of spatial summation

Classical studies of area summation measure contrast detection thresholds as a function of grating diameter. Unfortunately, (i) this approach is compromised by retinal inhomogeneity and (ii) it potentially confounds summation of signal with summation of internal noise. The Swiss cheese stimulus of T. S. Meese and R. J. Summers (2007) and the closely related Battenberg stimulus of T. S. Meese (2010) were designed to avoid these problems by keeping target diameter constant and modulating interdigitated checks of first-order carrier contrast within the stimulus region. This approach has revealed a contrast integration process with greater potency than the classical model of spatial probability summation. Here, we used Swiss cheese stimuli to investigate the spatial limits of contrast integration over a range of carrier frequencies (1–16 c/deg) and raised plaid modulator frequencies (0.25–32 cycles/check). Subthreshold summation for interdigitated carrier pairs remained strong (~4 to 6 dB) up to 4 to 8 cycles/check. Our computational analysis of these results implied linear signal combination (following square-law transduction) over either (i) 12 carrier cycles or more or (ii) 1.27 deg or more. Our model has three stages of summation: short-range summation within linear receptive fields, medium-range integration to compute contrast energy for multiple patches of the image, and long-range pooling of the contrast integrators by probability summation. Our analysis legitimizes the inclusion of widespread integration of signal (and noise) within hierarchical image processing models. It also confirms the individual differences in the spatial extent of integration that emerge from our approach.

Contrast summation across eyes and space is revealed along the entire dipper function by a "Swiss cheese" stimulus.

Previous contrast discrimination experiments have shown that luminance contrast is summed across ocular (T. S. Meese, M. A. Georgeson, & D. H. Baker, 2006) and spatial (T. S. Meese & R. J. Summers, 2007) dimensions at threshold and above. However, is this process sufficiently general to operate across the conjunction of eyes and space? Here we used a "Swiss cheese" stimulus where the blurred "holes" in sine-wave carriers were of equal area to the blurred target ("cheese") regions. The locations of the target regions in the monocular image pairs were interdigitated across eyes such that their binocular sum was a uniform grating. When pedestal contrasts were above threshold, the monocular neural images contained strong evidence that the high-contrast regions in the two eyes did not overlap. Nevertheless, sensitivity to dual contrast increments (i.e., to contrast increments in different locations in the two eyes) was a factor of ∼1.7 greater than to single increments (i.e., increments in a single eye), comparable with conventional binocular summation. This provides evidence for a contiguous area summation process that operates at all contrasts and is influenced little, if at all, by eye of origin. A three-stage model of contrast gain control fitted the results and possessed the properties of ocularity invariance and area invariance owing to its cascade of normalization stages. The implications for a population code for pattern size are discussed.

Spatially extensive summation of contrast-energy is revealed by contrast detection of micro-pattern textures

Vision must analyze the retinal image over both small and large areas to represent fine-scale spatial details and extensive textures. The long-range neuronal convergence that this implies might lead us to expect that contrast sensitivity should improve markedly with the contrast area of the image. But this is at odds with the orthodox view that contrast sensitivity is determined merely by probability summation over local independent detectors. To address this puzzle, I aimed to assess the summation of luminance contrast without the confounding influence of area-dependent internal noise. I measured contrast detection thresholds for novel Battenberg stimuli that had identical overall dimensions (to clamp the aggregation of noise) but were constructed from either dense or sparse arrays of micro-patterns. The results unveiled a three-stage visual hierarchy of contrast summation involving (i) spatial filtering, (ii) long-range summation of coherent textures, and (iii) pooling across orthogonal textures. Linear summation over local energy detectors was spatially extensive (as much as 16 cycles) at Stage 2, but the resulting model is also consistent with earlier classical results of contrast summation (J. G. Robson & N. Graham, 1981), where co-aggregation of internal noise has obscured these long-range interactions.

Theory and data for area summation of contrast with and without uncertainty:evidence for a noisy energy model

Contrast sensitivity improves with the area of a sine-wave grating, but why? Here we assess this phenomenon against contemporary models involving spatial summation, probability summation, uncertainty, and stochastic noise. Using a two-interval forced-choice procedure we measured contrast sensitivity for circular patches of sine-wave gratings with various diameters that were blocked or interleaved across trials to produce low and high extrinsic uncertainty, respectively. Summation curves were steep initially, becoming shallower thereafter. For the smaller stimuli, sensitivity was slightly worse for the interleaved design than for the blocked design. Neither area nor blocking affected the slope of the psychometric function. We derived model predictions for noisy mechanisms and extrinsic uncertainty that was either low or high. The contrast transducer was either linear (c1.0) or nonlinear (c2.0), and pooling was either linear or a MAX operation. There was either no intrinsic uncertainty, or it was fixed or proportional to stimulus size. Of these 10 canonical models, only the nonlinear transducer with linear pooling (the noisy energy model) described the main forms of the data for both experimental designs. We also show how a cross-correlator can be modified to fit our results and provide a contemporary presentation of the relation between summation and the slope of the psychometric function.

Area summation of first- and second-order modulations of luminance

To extend our understanding of the early visual hierarchy, we investigated the long-range integration of first- and second-order signals in spatial vision. In our first experiment we performed a conventional area summation experiment where we varied the diameter of (a) luminance-modulated (LM) noise and (b) contrastmodulated (CM) noise. Results from the LM condition replicated previous findings with sine-wave gratings in the absence of noise, consistent with long-range integration of signal contrast over space. For CM, the summation function was much shallower than for LM suggesting, at first glance, that the signal integration process was spatially less extensive than for LM. However, an alternative possibility was that the high spatial frequency noise carrier for the CM signal was attenuated by peripheral retina (or cortex), thereby impeding our ability to observe area summation of CM in the conventional way. To test this, we developed the ''Swiss cheese'' stimulus of Meese and Summers (2007) in which signal area can be varied without changing the stimulus diameter, providing some protection against inhomogeneity of the retinal field. Using this technique and a two-component subthreshold summation paradigm we found that (a) CM is spatially integrated over at least five stimulus cycles (possibly more), (b) spatial integration follows square-law signal transduction for both LM and CM and (c) the summing device integrates over spatially-interdigitated LM and CM signals when they are co-oriented, but not when crossoriented. The spatial pooling mechanism that we have identified would be a good candidate component for amodule involved in representing visual textures, including their spatial extent.

RKHS-Methods at Series Summation for Software Implementation

Reproducing Kernel Hilbert Space (RKHS) and Reproducing Transformation Methods for Series Summation that allow analytically obtaining alternative representations for series in the finite form are developed.

Linear binocular combination of responses to contrast modulation:contrast-weighted summation in first- and second-order vision

Binocular combination for first-order (luminancedefined) stimuli has been widely studied, but we know rather little about this binocular process for spatial modulations of contrast (second-order stimuli). We used phase-matching and amplitude-matching tasks to assess binocular combination of second-order phase and modulation depth simultaneously. With fixed modulation in one eye, we found that binocularly perceived phase was shifted, and perceived amplitude increased almost linearly as modulation depth in the other eye increased. At larger disparities, the phase shift was larger and the amplitude change was smaller. The degree of interocular correlation of the carriers had no influence. These results can be explained by an initial extraction of the contrast envelopes before binocular combination (consistent with the lack of dependence on carrier correlation) followed by a weighted linear summation of second-order modulations in which the weights (gains) for each eye are driven by the first-order carrier contrasts as previously found for first-order binocular combination. Perceived modulation depth fell markedly with increasing phase disparity unlike previous findings that perceived first-order contrast was almost independent of phase disparity. We present a simple revision to a widely used interocular gain-control theory that unifies first- and second-order binocular summation with a single principle-contrast-weighted summation-and we further elaborate the model for first-order combination. Conclusion: Second-order combination is controlled by first-order contrast.

Target/error overlap in jargonaphasia:the case for a one-source model, lexical and non-lexical summation, and the special status of correct responses

We present three jargonaphasic patients who made phonological errors in naming, repetition and reading. We analyse target/response overlap using statistical models to answer three questions: 1) Is there a single phonological source for errors or two sources, one for target-related errors and a separate source for abstruse errors? 2) Can correct responses be predicted by the same distribution used to predict errors or do they show a completion boost (CB)? 3) Is non-lexical and lexical information summed during reading and repetition? The answers were clear. 1) Abstruse errors did not require a separate distribution created by failure to access word forms. Abstruse and target-related errors were the endpoints of a single overlap distribution. 2) Correct responses required a special factor, e.g., a CB or lexical/phonological feedback, to preserve their integrity. 3) Reading and repetition required separate lexical and non-lexical contributions that were combined at output.