**Abstract**:

**Abstract**
**The human visual system is sensitive to both luminance (first-order) and contrast (second-order) modulations in an image. A linear-nonlinear-linear model is commonly used to explain visual processing of second-order patterns. Here we used a pattern-masking paradigm to compare first-order and second-order visual mechanisms and to characterize the nonlinear properties underlying them. The carriers were either a high-frequency horizontal grating (8 c/°) or a binary random dot pattern; they were either added to a vertical low-frequency (2 c/°) sinusoidal grating (first-order stimuli) or multiplied by it (second-order stimuli). The incremental discrimination threshold of the target was measured with pedestals whose spatial properties matched those of the target, with the exception of contrast (in the first-order pedestal) or modulation depth (in the second-order pedestal). The threshold function showed a typical dipper shape for both first- and second-order stimuli. The results for the first-order stimuli with different types of carrier and the second-order stimuli with a grating carrier were well explained by a divisive inhibition model in which the facilitatory input was divided by the sum of broadband inhibitory inputs. The results for the second-order stimuli with a random-dot carrier were explained by a modified divisive inhibition model that operated on modulation depth. Our results suggest that divisive inhibition is required to explain visual discrimination in both first- and second-order patterns. However, the source and nonlinearity of the divisive inhibition may be different for these two types of patterns and carrier.**

*multiplying*a high-spatial-frequency carrier or a white-noise carrier by a low-spatial-frequency envelope as the second-order stimulus, while luminance-modulated patterns (LM) are constructed by

*adding*the carrier and the envelope. There is no subthreshold summation between the first- and second-order stimuli (Schofield & Georgeson, 1999). The detectability of the envelope improves with carrier contrast for the second-order patterns but degrades for the first-order patterns (Schofield & Georgeson, 1999, 2003). In addition, the shape of the CM contrast-sensitivity function is independent of the spectrum of its noise carrier, but that of the LM contrast-sensitivity function is dependent on the spectrum of its additive noise mask (Schofield & Georgeson, 2003). In addition to psychophysical evidence, there are supporting electrophysiological studies showing that in cats, neurons in areas 17 and 18 have different spatial-frequency and orientation tuning for first- and second-order stimuli (Baker & Mareschal, 2001; Mareschal & Baker, 1998). This also indicates that a neuron's firing rate reflects a combination of inputs from first-order and second-order processing. The temporal property of the second-order system is slightly sluggish compared with that of the first-order system (Schofield & Georgeson, 2000), and it has a sustained temporal response with dynamic carrier noise and a transient temporal response with static carrier noise (Schofield & Georgeson, 2000). A study of visually evoked potential (VEP) has shown that response latency is longer for CM stimuli than for LM stimuli (Calvert, Manahilov, Simpson, & Parker, 2005). This longer latency found in VEP, which is also found in physiological data (Mareschal & Baker, 1998), might be due to (a) the slower first-stage linear filter and (b) the additional processing stage of the second-order pathway.

*C*) and the target plus the pedestal (intensity

*C*+Δ

*C*). In contrast discrimination, a typical result of the experiment plots the detection threshold of the target (Δ

*C*) against the pedestal intensity (

*C*), showing a dipper-shaped target threshold versus the pedestal contrast (T

*v*C) function (Chen & Foley, 2004; Kontsevich & Tyler, 1999; Legge & Foley, 1980). That is, as pedestal contrast increases, the target threshold first decreases (facilitation) and then increases (suppression). The T

*v*C function reflects the response characteristics in the visual system, and the slope of the response function at

*C*is inversely proportional to the contrast discrimination threshold (Δ

*C*) at base contrast

*C*(see Model for the details).

*v*C function for both first- and second-order pattern vision. In the first part of the study, we measured the T

*v*C function by using a 1-D carrier: a horizontal grating. In this experimental design, we could investigate the nonlinear properties of the early and late linear filters. In the second part of the research, we replicated Schofield and Georgeson's (1999) experimental results by using a 2-D noise carrier and adopted the contrast-gain-control model to fit the data. We also compared the function between LM and CM patterns to see how the model parameters could describe the data.

*LM*is the sum of an envelope

*G*(

*x*,

*y*), which is a low-spatial-frequency vertical sinusoidal grating, and a carrier pattern

*N*(

*x*,

*y*), which is either a high-spatial-frequency horizontal sinusoidal grating (Experiment 1) or a 2-D white noise (Experiment 2). That is, where

*I*

_{0}is the mean luminance (32.9 cd/m

^{2}),

*c*is the contrast of the carrier, and

*m*is the contrast of the envelope. Note that

*G*(

*x*,

*y*) is called the “envelope” here to be consistent with the description for the second-order pattern. The function

*w*(

*x*,

*y*) is an isotropic Gaussian window whose scale parameter (“standard deviation”) was 1.23° in Experiment 1 and 0.96° in Experiment 2.

*N*(

*x*,

*y*) in Experiment 1 was a horizontal grating, where

*f*is the spatial frequency of the carrier wave, 8 c/°, which is two octaves higher than the envelope's spatial frequency. The carrier pattern

_{c}*N*(

*x*,

*y*) in Experiment 2 was a 2-D white noise, and the size element was 4 × 4 pixels (3.68′).

*m*for the CM stimuli is called modulation depth, rather than contrast as for the LM stimuli.

*N*(

*x*,

*y*) from its sideband pattern

*N*(

*x,y*) ×

*G*(

*x,y*). This allowed us to present the carrier and sideband on different monitors (Figure 1A) and to manipulate the modulation depth of the CM pattern and the contrast of the LM pattern independently, simply by changing the lookup tables at different monitors. The frequency and orientation information of the sideband image was 8.25 c/

**°**and 23.66

**°**deviated from the orientation of the carrier in Experiment 1.

*c*for LM stimuli (40%, 16%, and 0%) and two carrier contrasts for CM stimuli (40% and 16%) were used in the experiments. In Results, we present contrast values in decibels (dB), or 20 times the logarithm of linear contrast. Thus the carrier contrast of 40% and 16% is equivalent to −8 and −16 dB, respectively.

*m*is the pedestal contrast (for LM) or modulation depth (for CM) and Δ

*m*is the target contrast (for LM) or modulation depth (for CM).

*m*in Equation 1) or modulation depth (

*m*in Equation 5) and seven levels of target contrast (Δ

*m*in Equation 6) or modulation depth (Δ

*m*in Equation 7). There were 10 repetitions of each target contrast or modulation depth in each run. The order of target contrast and modulation depth was randomized within the run. The experimental runs were further blocked by pedestal type (LM or CM) and carrier contrast (−∞, −16, or −8 dB). Each block contained several pedestal contrasts or modulation depths for a specific pedestal type and carrier contrast. That is, each block measured one T

*v*C function. There were four repetitions for each pedestal contrast or modulation depth. Thus the psychometric function for a pedestal had 280 data points. The order of runs in each block was randomized.

*μ*represents the mean and

*σ*the standard deviation of the Gaussian function. The value of

*γ*was set at 0.5, which is the guessing rate for the 2AFC paradigm, and

*λ*was constrained between 0 and 0.05, which indicates the rate of finger error. A maximum-likelihood method was used to estimate

*μ*and

*σ*free parameters (Wichmann & Hill, 2001a, 2001b). The detection threshold was defined as the value of

*μ*, and the standard deviation of the detection threshold was estimated by the bootstrapping method.

*v*C functions for the horizontal grating carrier are shown in Figure 2A. The thresholds for the LM stimuli are represented by circles, while those for the CM stimuli are represented by diamonds. The green circles represent the detection threshold for the no-carrier condition (LM only); the red circles, the low-carrier-contrast (−16 dB) LM condition; and the blue circles, the high-carrier-contrast (−8 dB) LM condition. The T

*v*C functions all show a typical dipper shape. That is, the threshold first decreased (facilitation) and then increased (suppression) with pedestal contrast. All three curves overlap with each other. This is consistent with previous results for these conditions, that the target and pedestal are processed independently in the visual system such that the spatial frequency of the target and the pedestal were three octaves apart or that their orientations were orthogonal (Blakemore & Campbell, 1969; Blakemore, Nachmias, & Sutton, 1970).

*v*C function for the CM stimuli is shown in magenta diamonds for the low-carrier-contrast condition (−16 dB) and cyan diamonds for the high-carrier-contrast condition (−8 dB). The T

*v*C function shows a typical dipper shape, as commonly observed in T

*v*C functions for first-order pattern masking. When there was no pedestal (−∞ pedestal contrast), the detection threshold for lower carrier contrast was about twice (−6 dB) that for high carrier contrast. This result is consistent with previous findings (Schofield & Georgeson, 1999). This difference in threshold decreased as the pedestal modulation depth increased; thus the two T

*v*C functions merged at high modulation depth.

*v*C functions for the 2-D noise carrier. The data for LM stimuli are represented by circles. All LM T

*v*C functions show a typical dipper shape. At low pedestal contrasts, the target threshold is higher when the carrier contrast is higher. At high pedestal contrasts, the carrier contrast has little effect, and the T

*v*C functions for different carrier contrasts merge. This carrier-contrast effect was different from that for the grating carrier, in which the T

*v*C functions were about the same for the three different carrier contrasts, suggesting that contrast of the grating carrier has only trivial influence.

*v*C function for the CM stimuli (represented by diamond symbols) shows that the threshold initially decreased as pedestal modulation depth increased. However, up to the maximum pedestal modulation depth that can be produced by our apparatus, there was no sign of threshold increment at high pedestal modulation depth. This is probably due to the fact that the pedestal modulation depth we used was not high enough to show an inhibitory effect.

*v*C functions merged at high pedestal modulation depth. This effect is similar to that for the grating carrier.

*E*′ of the

*i*-th linear filter, which is centered at position (

*x*,

*y*) and has a sensitivity profile

*f*, to the

_{i}*j*-th image

*L*is given by the cross correlation

_{j}*i*-th linear filter then undergoes a linear transform to produce the

*i*-th early-channel response

*R*, given as Here we drop the notation (

_{i}*x*,

*y*) for simplicity. The numerator in Equation 10 is the half-wave-rectified excitation of the

*i*-th filter,

*E*, raised to a power

_{i,j}*p*. The denominator contains the divisively inhibitory term

*I*and an additive constant

_{j}*z*. The inhibition term is a nonlinear combination of the excitation of a set of relevant channels (Foley, 1994; Heeger, 1992). That is, where

*q*is an exponent parameter and

*w*is the contribution of the

_{i}*i*-th filter excitation to the inhibition inputs.

*T*is then the ratio between the half-wave-rectified second-order linear-filter excitation, raised to a power

*π*, and the late divisive-inhibition input

*K*plus a constant

*ζ*. That is, Again, the divisive inhibition is a nonlinear combination of the linear-filter excitations. That is, where

*θ*is the exponent parameter and

*ω*is the contribution of the

_{k}*k*-th filter excitation to the inhibition inputs.

*R*) and late (

_{i,j}*T*) responses. Therefore, there are two possible decision variables: and where

_{k,j}*D*

_{1}and

*D*

_{2}are the decision variables and

*σ*

_{1}and

*σ*

_{2}are the standard deviations of the noise in the early and late mechanisms, respectively. The subscript

*p*+

*t*denotes the pedestal-plus-target interval, while

*p*denotes the pedestal interval.

*v*C function, variability in the response produced by the noise carrier should also be a constant. That is, the standard deviations

*σ*

_{1}and

*σ*

_{2}are constants.

*(*

_{i}*R*−

_{i,p+t}*R*) or

_{i,p}*(*

_{i}*T*−

_{i,p+t}*T*) is equal to unity.

_{i,p}*D*

_{1}), we would not compute

*D*

_{2}, for reasons of parsimony. That is,

*D*

_{2}was implemented in the model only in the conditions where it was impossible for

*D*

_{1}to reach the criterion.

*Se*is a constant called the excitatory sensitivity of the mechanism,

_{j}*c*

_{carrier}is the contrast of the carrier, and

*c*

_{env}is the contrast of the envelope. The derivation of Equation 16 is shown in the Appendix. The inhibition of the linear output is the same as in Equation 16 but with inhibitory sensitivity (

*Si*

_{env},

*Si*

_{carrier}) of the mechanism. In addition, we assume that the observer can see the difference between the pedestal-plus-target and the pedestal alone if the responses to these two intervals are sufficiently different in at least one channel. That is, we only need to consider the channel with the largest response difference between the two intervals. Since

*c*

_{carrier}is the same in both intervals, we only need to consider the mechanism that is most sensitive to the envelope, even though it has response to the carrier. In addition, since the modulation of the envelope is greatest at around the center of the image, the receptive field of this mechanism should be centered on the center of the image. Hence (

*x,y*) = (0,0) in Equation 16. In practice,

*Se*

_{carrier}was fixed to 0 for the grating carrier and a free parameter for the noise carrier. The reason for this is that the peak spatial frequency of the grating carrier is two octaves apart from that of the envelope and that the orientation of carrier and envelope is orthogonal, and hence the target should produce little response in the mechanism responding to the envelope. On the other hand, the noise carrier is a very broadband stimulus, whose spectrum overlaps with that of the envelope and thus may produce a response in the envelope mechanism. Hence there were five parameters (

*Se*

_{env},

*Si*

_{env},

*p*,

*q*, and

*z*) for fitting data with LM patterns with a sine-wave carrier, and seven parameters (

*Se*

_{env},

*Si*

_{env},

*Se*

_{carrier},

*Si*

_{carrier},

*p*,

*q*, and

*z*) for LM patterns with a noise carrier. Based on the pilot fitting results, the value of

*Se*

_{carrier}is close to 0; therefore we fixed the value at 0 as well for both carrier types. The best-fit parameters for the grating carrier are shown in Table 1, and for the noise carrier in Table 2. The fits are shown in Figure 2. The model fit the LM-pattern results very well. For the LM pattern with sine-wave carrier, the root-mean-square error (RMSE) was 1.86 for CWC and 2.48 for PCH. This is similar to the mean standard error of measurement: 2.21 for CWC and 1.34 for PCH. For the LM pattern with noise carrier, the RMSE was 2.36 for CWC and 2.36 for PCH. This is at a reasonable range for the mean standard error of measurement: 1.52 for CWC and 1.37 for PCH. Both LM patterns showed that the values of

*p*and

*q*are significantly deviated from 1 and the value of

*p*is larger than that of

*q*, suggesting that the inhibitory pooling is weaker than the facilitation and a simple rectification cannot explain our results. In addition, the value of

*Se*is larger than that of

*Si*, also suggesting that inhibition is weaker than facilitation, which is consistent with previous findings for first-order pattern vision (Foley, 1994; Foley & Chen, 1999).

CWC | PCH | |

LM | ||

Se_{env}* | 100 | 100 |

Si_{env} | 57.90 | 59.92 |

Se_{carrier}* | 0 | 0 |

Si_{carrier}* | 0 | 0 |

p | 2.50 | 2.31 |

q | 2.08 | 1.93 |

z | 12.97 | 9.40 |

SSE | 55.70 | 89.23 |

# datum | 30 | 36 |

MSE | 1.86 | 2.48 |

SE | 2.21 | 1.34 |

CM | ||

Se_{side}* | 100 | 100 |

Si_{side} | 89.00 | 86.07 |

Se_{carrier}* | 0 | 0 |

Si_{carrier} | 3.76 | 5.90 |

p | 4.13 | 4.20 |

q | 3.67 | 3.73 |

z | 323.70 | 405.68 |

SSE | 65.58 | 22.29 |

# datum | 23 | 23 |

MSE | 2.85 | 0.97 |

SE | 1.50 | 1.24 |

CWC | PCH | |

LM | ||

Se_{env}* | 100 | 100 |

Si_{env} | 62.54 | 60.27 |

Se_{carrier}* | 0 | 0 |

Si_{carrier} | 10.63 | 6.06 |

p | 2.70 | 3.22 |

q | 2.23 | 2.92 |

z | 22.31 | 31.78 |

SSE | 54.07 | 99.05 |

# datum | 23 | 42 |

MSE | 2.36 | 2.36 |

SE | 1.52 | 1.37 |

CM | ||

Se_{env}* | 100 | 100 |

Si_{env} | 145.38 | 58.46 |

R | 0.13 | 0.10 |

k | 1.25 | 1.60 |

π | 2.85 | 2.63 |

θ | 1.92 | 2.30 |

ζ | 51.56 | 17.27 |

SSE | 65.43 | 41.25 |

# datum | 23 | 23 |

MSE | 2.85 | 1.79 |

SE | 1.33 | 1.32 |

*CM*(

*x*,

*y*) and a linear filter whose sensitivity profile

*f*(

*x*,

*y*) is defined by a Gabor function and whose spatial frequency and orientation are the same as those of the carrier (see Appendix for the derivation). The result can be written as where

*Se*

_{carrier}and

*Se*

_{side}are constants which are related to the sensitivity to carrier and sideband images, respectively;

*c*is the carrier contrast; and

*m*is the envelope-modulation depth. For a CM pattern with a sine-wave carrier, it may be possible to find a mechanism whose

*Se*

_{side}is large enough that the difference in envelope modulation between the two intervals in a trial would allow the early decision variable

*D*

_{1}to surpass the noise level. Hence, it is not necessary to have the late mechanism involved. As a result, the model for CM patterns with sine-wave carrier is the same as the model for LM patterns. The parameters are shown in Table 1 and the results are shown in Figure 2. The model fit the CM-pattern results well: The RMSE was 2.85 for CWC and 0.97 for PCH. This is a reasonable range compared to the mean standard error of measurement: 1.50 for CWC and 1.24 for PCH. The values of

*Se*

_{carrier}and

*Si*

_{carrier}are small, suggesting that the carrier itself did not contribute much to the response of the detecting mechanism. The model is well fit to the data, suggesting that perceiving a second-order pattern does not necessarily require a second-order mechanism.

*de*is a fixed value and

*c*

_{noise}is the noise-carrier contrast. In our experiment, two levels of carrier contrast were used, and their relative output was constant under different modulation depths, no matter what kind of nonlinear transform was adapted. Therefore, the output ratio between high and low carrier contrasts can be set to be a constant value, and thus in which

*R*represents the output after the nonlinear transform and also the input to the second-stage linear filter. Thus we assumed the output for the second linear stage of the linear filter,

*R*,

_{j}*j*=

*l*or

*h*, is the excitatory input from the first stage of the process. We used seven parameters to fit the data (

*R*,

_{l}*k*,

*Se*

_{env},

*Si*

_{env},

*π*,

*θ*, and

*ζ*). The RMSE was 2.85 for CWC and 1.79 for PCH. This is at the reasonable range of mean standard error of measurement: 1.33 for CWC and 1.32 for PCH. The fitted results showed that the values of

*π*and

*θ*are significantly deviated from 1 and the value of

*π*is larger than that of

*θ*, suggesting that the inhibitory pooling is weaker than the facilitation and a simple rectification between early and late linear filters cannot explain our results.

*c*is the carrier contrast and

*m*is the modulation depth (Schofield & Georgeson, 1999). We measured the discrimination threshold for the noise carriers at −16 dB and −8dB and found the contrast needed to be raised to the value Δ

*c*+

*c*to 1.41

*c*and 1.22

*c*for −16 and −8 dB, respectively. Thus a detectable RMS contrast by modulation depth

*m*> 1 or 0.98 was needed for low and high carrier contrast, respectively, by setting

*C*

_{r.m.s.}to 1.41

*c*and 1.22

*c*in Equation 21. For the grating-based carrier, the discrimination threshold was also measured, and c > 1 and 0.86 were derived for low and high carrier contrast, respectively. Our experimental results did not use such high modulation depth; therefore, our results cannot be explained by RMS contrast changes. In conclusion, our experimental stimuli were not contaminated by a luminance artifact.

*v*C function for the second-order stimuli?

*v*C functions for grating-based LM and CM stimuli can be well described by a divisive-inhibition model. This indicates that the CM discrimination task might be achieved by off-orientation (sideband information) discrimination instead of second-order pattern discrimination, and suggests that a contrast-gain-control type of nonlinear process is involved in the pattern discrimination. Even though the contrast-gain control right after an early linear filter can explain the CM results, it does not mean the late linear filter is not necessary. This is because adding the late linear filter is equal to multiplying a constant value by the output of gain-control processing, and this would not influence the shape of the response function.

*v*C functions between the first- and second-order stimuli

*v*C function, because the carrier information we used was orthogonal and two octaves higher than the envelope's spatial frequency. Therefore, the influence of the carrier was small in LM. Again, the difference in performance between LM and CM demonstrates that the carrier plays a different role in LM and CM patterns. In our model, the output of the linear filters for CM patterns contains two pieces of information: carrier and sideband. The influential factors from the sideband (

*Se*

_{side}) were larger than those from the carrier (

*Se*

_{carrier}), indicating the importance of sideband information. Our fitting results were also consistent with image-classification findings, which have shown that human observers may detect contrast modulation of a sinusoidal carrier by using sideband information (Manahilov, Simpson, & Calvert, 2005).

*v*C functions for LM and CM stimuli were similar and can be explained by divisive inhibition, which occurs at different stages of processing. Both LM and CM stimuli were influenced by noise contrast, but in opposite ways: An increase in carrier contrast decreased the detectability of the LM target but increased the detectability of the CM target. This is consistent with previous findings (Schofield & Georgeson, 1999).

*v*C function for both the first-order and second-order stimuli, and our proposed divisive-inhibition model was well able to account for the T

*v*C function for both the first- and second-order stimuli with grating and noise carriers. Combining noise-based and grating-based CM-pattern results, we found that (a) a divisive inhibition is needed after the second linear filter to describe the CM pattern with noise carrier; (b) the exponent for the excitation signal was significantly different from that for the inhibition signal in the late mechanism, accounting for the result for the CM pattern with noise carrier; and (c) the first stage of divisive inhibition is necessary to account for the CM pattern with grating carrier. Thus, rectification between early and late linear filters alone cannot explain the discrimination threshold for the second-order stimuli, and divisive inhibition is necessary to explain the performance. Furthermore, the divisive inhibition is also determined by the first-order information in the stimuli.

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*I*

_{0}: mean luminance

*c*: contrast of the carrier

*m*: contrast of the LM stimuli or modulation depth of the CM stimuli

*E*: excitatory output for early linear filter

*I*: inhibitory output for early linear filter

*p*: exponent of excitation for early linear filter

*q*: exponent of inhibition for early linear filter

*z*: semisaturation constant for early contrast- gain-control processing

*R*: early response

*H*: excitation of a later linear filter

*K*: inhibition of a later linear filter

*π*: exponent of excitation for late linear filter

*θ*: exponent of inhibition for late linear filter

*ζ*: semisaturation constant for late contrast-gain control

*T*: late response

*D*

_{1}: decision variable for the early response

*D*

_{2}: decision variable for the late response

*Se*: excitatory sensitivity parameters of spatial filters

*Si*: inhibitory sensitivity parameters of spatial filters

*v*C function for LM and CM patterns with periodical carriers