In psychophysics, cross-orientation suppression (XOS) and cross-orientation facilitation (XOF) have been measured by investigating mask configuration on the detection threshold of a centrally placed patch of sine-wave grating. Much of the evidence for XOS and XOF comes from studies using low and high spatial frequencies, respectively, where the interactions are thought to arise from within (XOS) and outside (XOF) the footprint of the classical receptive field. We address the relation between these processes here by measuring the effects of various sizes of superimposed and annular cross-oriented masks on detection thresholds at two spatial scales (1 and 7 c/deg) and on contrast increment thresholds at 7 c/deg. A functional model of our results indicates the following (1) XOS and XOF both occur for superimposed and annular masks. (2) XOS declines with spatial frequency but XOF does not. (3) The spatial extent of the interactions does not scale with spatial frequency, meaning that surround-effects are seen primarily at high spatial frequencies. (4) There are two distinct processes involved in XOS: direct divisive suppression and modulation of self-suppression. (5) Whether XOS or XOF wins out depends upon their relative weights and mask contrast. These results prompt enquiry into the effect of spatial frequency at the single-cell level and place new constraints on image-processing models of early visual processing.

^{1}

^{2})]. Contrast is expressed in dB re 1%, given by 20 log

_{10}(

*c*), where

*c*is the Michelson contrast in percent given by

*c*= 100(

*L*

_{max}−

*L*

_{min}) / (

*L*

_{max}+

*L*

_{min}),

*L*is the luminance. (So, for example, −6 dB = 0.5%, 0 dB = 1%; 6 dB = 2%, 12 dB = 4%, and so on). Mean luminance was constant throughout the experiments.

*c*

_{test}and

*c*

_{xcenter}are test and superimposed, cross-oriented mask contrasts (in %), and

*p*,

*q*,

*α*, and

*w*are free parameters. The exponents

*p*and

*q*describe the rate of acceleration of nonlinear contrast responses on the numerator and denominator of the gain control equation, respectively. These parameters were not very well constrained by the data from the experiments here. For the 1 c/deg conditions, where masking was strong and provided some constraint, we allowed

*q*to be free, and set

*p*=

*q*+ 0.4, broadly consistent with numerous observations from pedestal masking studies. For the 7 c/deg conditions, we set

*p*= 2.0 and

*q*= 2.4, also consistent with other work (e.g., Legge & Foley, 1980).

*c*

_{test}term is zero, the saturation constant

*z*does not represent a degree of freedom. As the

*c*

_{test}term is negligible for the low test contrast conditions here,

*z*was set to unity.

*α*and

*w*are the weights of modulatory facilitation and divisive suppression from the mask. But as these can originate from both center and surround in the present study, Equation 1 needs to be extended to accommodate this. From the data we observed little or no systematic difference between the medium surround conditions (*M) and any of the larger surround conditions (*L & *V), and so we treat all of the surround conditions together in the modelling. To do this, we consider a simple additive arrangement of center and surround terms as follows:

*k*are possible, but a convenient one is that it is proportional to the standard deviation of late additive noise in the model. When

*k*is a free parameter, it controls the observer's overall sensitivity (i.e., the vertical position of the masking functions on a log plot). However, our estimates of baseline were based on substantially more data than the thresholds at other mask contrasts, so we constrained the model to intercept the ‘no-mask’ detection threshold in each masking function. We did this by normalizing all of the detection thresholds to the baseline which, from Equations 2 and 3, constrains

*k*= 0.5. Model and data were then de-normalized by the original baseline data before plotting. Thus, for each observer the fitting involved four free parameters for the 7 c/deg conditions (

*α*

_{center},

*α*

_{surround},

*w*

_{center},

*w*

_{surround}) and five free parameters for the 1 c/deg conditions (where the extra parameter was

*q*).

*x*

_{ i}and

_{ i}are the data and model predictions, respectively (test contrasts in %), for the

*i*th of

*n*data points. The model equation was solved numerically for

*c*

_{test}to produce three curves for five masking functions fitted simultaneously to the data (

*n*= 5× number of mask contrasts in each function, excluding 0%).

Condition (test and mask, c/deg) | Observer | RMS error (dB) | p (fixed) | q (free
and fixed) | α _{center} (free) | α _{surround} (free) | w _{center} (free) | w _{surround} (free) | Baseline (no-mask detection threshold, dB) | Parameter scaling factor ( Figure 4) |
---|---|---|---|---|---|---|---|---|---|---|

1 and 1 | DHB | 0.7 | q + 0.4 | 1.56 | 0.588 | 0.049 | 1.2488 | 0.0165 | −1.713 | 0.82 |

1 and 1 | RS | 1.00 | q + 0.4 | 1.56 | 0.268 | 0.061 | 0.5861 | 0.0212 | −2.48 | 0.75 |

1 and 3 | LM | 1.20 | q + 0.4 | 2.16 | 0.251 | 0.005 | 0.1978 | 0.0008 | −5.99 | 0.50 |

1 and 3 | BX | 0.91 | q + 0.4 | 2.77 | 0.200 | 0.080 | 0.0786 | 0.0003 | −4.14 | 0.62 |

7 and 7 | RL | 1.50 | 2.4 | 2.0 | 0.112 | 0.147 | 0.0054 | 0.0039 | 8.71 | 2.73 |

7 and 7 | KS | 1.13 | 2.4 | 2.0 | 0.193 | 0.258 | 0.0131 | 0.0078 | 8.07 | 2.53 |

7 and 7 | DJH | 0.95 | 2.4 | 2.0 | 0.137 | 0.099 | 0.0046 | 0.0023 | 7.89 | 2.48 |

7 and 7 | RJS | 0.84 | 2.4 | 2.0 | 0.038 | 0.061 | 0.0015 | 0.0018 | 7.13 | 2.27 |

*α*) is fairly unaffected by spatial frequency (compare the two green circles), but the suppressive weight is much lower at the higher spatial frequency (compare the two red squares). This is why masking is seen primarily at the low spatial frequency. Now consider how the weights change across mask region. At 7 c/deg, the weights are very similar for the center and the surround (the solid symbols are connected by almost parallel lines), and suppression is relatively weak. This is why the masking functions in Figure 3 (7 c/deg) are all very similar, and dominated by facilitation. In contrast, at 1 c/deg both weights are relatively high in the center, but decline substantially in the surround (the open symbols are connected by lines with negative slope), more so for suppression than facilitation. This is why masking dominates in the center in Figure 2 (1 c/deg), but facilitation becomes apparent when the mask is restricted to the surround.

*c*

_{ped}refers to the contrast of the pedestal plus test increment (if it is present), and

*c*

_{xom}refers to the contrast of a cross-oriented mask. This version of Foley's model has a single divisive term for XOS,

*wc*

_{xom}

^{q}(

*w*is a weight parameter), which adds to a self-suppression term

*c*

_{ped}

^{q}and a saturation constant

*z*on the denominator (Carandini & Heeger, 1994; Foley, 1994; Heeger, 1992; Tolhurst & Heeger, 1997). There is no provision for XOF. The model predicts that a superimposed mask shifts the dip region of the masking function upward, but that the ‘dipper handles’ converge at higher pedestal contrasts (Figure 5A). Thus, the cross-oriented mask produces masking at detection threshold, but facilitation by the pedestal survives this transformation. This model has been very successful at low spatial frequencies where cross-oriented masking is strong (Foley, 1994; Holmes & Meese, 2004). However, it is ruled out here by the results from Experiment 1, which show that when testing at 7 c/deg, superimposed cross-oriented masks produce facilitation, not masking.

^{2}This model can be re-expressed in the terms used here as follows:

*w*= 0, this model has a similar form to the one used by Yu et al. (2002), although they did not express the functional relation between mask contrast and the weight of modulation.

*αc*

_{xom}) from the numerator in Model 1 is now also applied to the pedestal (plus test) contrast on the denominator, replacing the previous divisive term

*αc*

_{xom}

^{ q}. This means that XOS is due to modulation of self-suppression and predicts facilitation at low pedestal contrasts followed by convergence of the ‘dipper handles’ at higher contrasts ( Figure 5C). This pattern is the reverse of that produced by the Foley-type model ( Figure 5A).

*α*) and the denominator (

*b*). With

*b*>

*α*, This model predicts that facilitation occurs at low pedestal contrasts but gives way to further masking as pedestal contrast increases ( Figure 5D). This is the behavior found by Chen and Tyler (2002) for pedestal masking in the presence of cross-oriented flanking masks at a spatial frequency of 4 c/deg. This model has a similar form to that used by Chen and Tyler (2002), although they did not express the functional relation between mask contrast and the weight of modulation.

Observer | Fixed mask | RMS error (dB) | p | q | α | b | z | k | α/b |
---|---|---|---|---|---|---|---|---|---|

RJS | Superimposed | 1.043 | 3.11 | 2.59 | 0.323 | 0.494 | 32.87 | 0.436 | 0.65 |

DHB | Superimposed | 1.027 | 2.98 | 2.42 | 0.321 | 0.417 | 20.99 | 0.591 | 0.77 |

DJH | Superimposed | 0.841 | 2.82 | 2.28 | 0.119 | 0.184 | 22.50 | 0.304 | 0.65 |

RJS | Doughnut | 0.238 | 2.22 | 1.76 | 0.233 | 0.330 | 15.66 | 0.292 | 0.71 |

DHB | Doughnut | 1.017 | 3.35 | 2.75 | 0.22 | 0.17 | 51.58 | 0.690 | 1.29 |

DJH | Doughnut | 0.449 | 2.66 | 2.28 | 0.048 | 0.037 | 30.06 | 0.196 | 1.32 |

*c*

_{ped}term (the pedestal + test contrast) on the denominator of Equation 7. For stimuli with no pedestal (e.g., those in Experiment 1), this term is small, the model equation is dominated by the saturation constant

*z*, and there is little or no masking. This can be overcome by using very large values of

*b*, but this leads to other problems. It means the model does not produce convergence of the dipper handles in pedestal plus fixed mask experiments (see Figure 5A), which is known to happen at low spatial frequencies (Foley, 1994; Holmes & Meese, 2004; Ross & Speed, 1991). Furthermore, the arrangement of modulation on both numerator and denominator means that the model does not describe the decline of facilitation after its onset (see Figure 3).

*w*and represents divisive suppression from the mask (as in Model 1). The other is controlled by

*b*and represents modulation of self-suppression by the mask (as in Model 3). The route to facilitation is the same as for Models 1–3. The hybrid model was fitted to the results from both experiments (not shown) to determine whether the various types of interaction interfere with each other. In doing this, the number of free parameters was constrained to be the same as for Model 1 in Experiment 1 and Model 3 in Experiment 2.

*b*=

*α*/0.65 for the center and

*b*=

*α*/1.32 for the surround, consistent with the extremes of these parameters in Experiment 2 (last column of Table 1). This is probably over-restrictive for the low spatial frequency conditions from Experiment 1, where we have no estimates of

*α*and

*b*. Nevertheless, we found only marginal changes in the quality of the fit for the hybrid model (not shown). For Experiment 2, when

*w*> 0, this adds a constant to the denominator of the model equation (equivalent to increasing the magnitude of

*z*). So long as

*w*was not so large that it overpowered the facilitation produced by

*α*at threshold, we found that it had negligible impact on the fitting (not shown).

*b*terms from the center and the surround. Furthermore, Experiment 1 suggests that the functional relation between mask contrast and facilitatory modulation is a reasonable one at detection threshold, but we do not know whether this relation extends well above threshold. Similarly, we have not tested the functional relation between mask contrast and the modulation of self-suppression.

*α*) fixed across spatiotemporal frequency. They found that the suppressive weight (

*w*) decreased in proportion to the ratio of spatial frequency to temporal frequency (on double log axes).

*z*). This has the effect of depressing the contrast response at lower test contrasts but leaving it intact at higher contrasts. We refer to this as divisive suppression. The second ( Experiment 2) involves modulation of self-suppression by the cross-oriented mask. For a fixed-contrast mask, this has the effect of leaving the contrast response intact at lower contrasts, but depressing it at higher contrasts. We refer to this as modulated self-suppression. The combined effect of divisive suppression and modulated self-suppression is to depress the contrast response across the entire operating characteristic of the detecting mechanism. Of course, this could be achieved more directly by modulating the sum of the saturation constant and self-suppression by a single function of mask contrast. However, the results here and elsewhere indicate that the two processes must act independently. This is because divisive suppression dominates at low spatial frequencies (Foley, 1994; Holmes & Meese, 2004), whereas modulated self-suppression dominates at high spatial frequencies. This need for independence can be seen in the results from Experiment 2. If cross-oriented mask contrast modulated the entire denominator of the model equation with weight

*b*, then the mask would shift the entire dipper functions in Experiment 2 (Figure 6) either upward or downward, depending on the relative values of

*b*and the facilitatory weight

*α*. This is not what we found.

^{1}Elsewhere (Meese & Holmes, 2007) we have found it convenient to use the terms cross-orientation masking (XOM) and cross-orientation facilitation (XOF) to refer to psychophysical phenomena and cross-orientation suppression (XOS) and cross-orientation enhancement (XOE) to refer to the underlying processes. We do not use this terminology to make that distinction here, but use the terms XOS and XOF to refer to two different classes of modulatory interaction (suppressive and facilitatory). More generally, we use the term cross-orientation interactions (XOI) to refer to both or either of these.

^{2}Here we have set

*w*= 0 ( Table 2), but this is not crucial for our point. So long as

*w*is sufficiently small for XOF to remain intact at detection threshold, performance is facilitated across the entire dipper function.