There is no consensus on whether luminance-modulated (LM) and contrast-modulated (CM) stimuli are processed by common or separate mechanisms. To investigate this, the sensitivity variations to these stimuli are generally compared as a function of different parameters (e.g., sensitivity as a function of the spatial or temporal window sizes) and similar properties have been observed. The present study targets the sensitivity difference between LM and CM stimuli processing. Therefore, instead of studying the variation of sensitivity in different conditions, we propose to decompose the sensitivities in internal equivalent noise (IEN) and calculation efficiency (CE) to evaluate at which processing level the two mechanisms differ. For each stimulus type, the IEN and CE of four observers were evaluated using three different carriers (plaid, checkerboard, and binary noise). No significant CE differences were noted in all six conditions (3 carriers × 2 modulation types), but important differences were found between the IEN of the two stimulus types. These data support the hypothesis that the two pathways are initially separate and that the two stimuli may be treated by common mechanisms at a later processing stage. Based on ideal observer analysis, pre-rectification internal noise could explain the difference of IEN between LM and CM stimuli detection when using binary noise as a carrier but not when using a plaid or a checkerboard. We conclude that a suboptimal rectification process causes higher IEN for CM stimuli detection compared with LM stimuli detection and that the intrinsic noise of the binary carrier had a greater impact on the IEN than the suboptimal rectification.

*c*) and effective noise contrast (

*N*

_{eff}) will not affect the observer's performance. The effective noise represents the combination of the internal (observer) and external (stimulus) noise.

*k*

_{ideal}is the

*k*parameter for the ideal observer. An ideal observer is a theoretical observer using all the information available to optimally perform the task. Therefore,

*k*

_{ideal}represents the smallest SNR (

*k*) mathematically possible to detect the signal (

*c*) based on a given threshold criterion.

*N*

_{eff}) will be

*N*and

*N*

_{eq}represent the RMS contrasts of the external noise and IEN, respectively. The IEN models the impact of the internal noise on the sensitivity. Note that if the external noise contrast is equal to the IEN (

*N*=

*N*

_{eq}), the effective noise (

*N*

_{eff}) will be √2 times greater. Therefore, assuming that the calculation is contrast invariant (Equation 1), the IEN will be equal to the external noise contrast that raises the signal contrast threshold (

*c*) by a factor of √2.

*c*) threshold and the external noise contrast (

*N*) can be deduced by combining Equations 1 and 3:

*c*

^{2}) and the external noise variance (

*N*

^{2}) would be linear:

*N*<<

*N*

_{eq}), varying the external noise does not affect significantly the effective noise (Equation 3), and therefore the signal contrast threshold is relatively constant as a function of the external noise. However, in high external noise conditions, varying the external noise has a great impact on the total amount of noise and thereby the signal contrast threshold increases as a function of the external noise.

*M*(

*x,y*)] with a texture [

*T*(

*x,y*)], where the modulation is defined by lower spatial frequencies relative to the texture, and the texture local RMS contrast (

*T*

_{RMS}) is constant throughout the stimulus. Therefore, the modulation represents the global texture contrast variation. The RMS contrast near the position (

*x,y*) is equal to

*T*

_{RMS}

*M*(

*x,y*). The rectification consists in estimating the local (carrier spatial frequency) RMS contrast. This estimation should be applied locally over the entire stimulus (i.e., for all (

*x,y*) positions), which would reconstruct a similar modulation [

*T*

_{RMS}

*M*(

*x,y*)] as the one defining the stimulus [

*M*(

*x,y*)].

*T*

_{RMS}

*M*(

*x,y*)] similar to a LM stimulus without a texture, that is, the modulation

*M*(

*x,y*). In the LM stimulus, each position represents the luminance intensity. For the rectified CM stimulus, each position of the effective stimulus would represent the local contrast modulation of the CM stimulus. Therefore, after a rectification, a CM stimulus would be analogue to a LM stimulus and both could be treated by common mechanisms.

*T*

_{RMS}

*M*(

*x,y*)] by a normal distribution centered at

*βT*

_{RMS}

*M*(

*x,y*) and with a standard deviation of

*N*

_{rect}. The

*β*represents the gain parameter affecting the strength of the rectification output.

*N*

_{rect}represents internal noise that could be added during the rectification process.

*M*(

*x,y*)

*T*(

*x,y*)] with a contrast modulation [

*M*(

*x,y*)] composed of a signal [

*S*(

*x,y*)] with contrast

*S*

_{in}embedded in noise [

*N*(

*x,y*)] with contrast

*N*

_{in}. Using the previously defined rectification, the signal and noise contrast at the output of the rectification would be scaled by a factor of

*βT*

_{RMS}and noise would also be added (

*N*

_{rect}). Consequently, the signal (

*S*

_{in}) and noise (

*N*

_{in}) contrast of the input of the rectification process would result, after the rectification process, as

*S*

_{in}/

*N*

_{in}to

*N*

_{in}<<

*N*′

_{rect}), then the impact of the input noise would not be significant:

*N*

_{in}) the rectification process would affect the observer's performance. However, if the input CM noise is relatively high (

*N*

_{in}>>

*N*′

_{rect}), then the noise added by the rectification (

*N*′

_{rect}) would not be significant:

*x,y*) for the LM stimuli was defined by the addition of a texture to a luminance modulation:

*L*

_{0}is the stimulus luminance average, which was fixed to 59 cd/m

^{2}for the present study.

*M*(

*x,y*) and

*T*(

*x,y*) represent, respectively, the modulation and the texture of the pixel at position (

*x,y*). The texture was added to the LM stimulus to give both stimuli the same contrast average. Therefore, the two stimulus types were similar with the exception that for the CM stimulus, the modulation was applied to the texture instead of the luminance:

*M*(

*x,y*) and

*T*(

*x,y*) over the entire stimulus (all

*x*and

*y*) must be 1 and 0, respectively.

*M*(

*x,y*)] was composed of a signal [

*S*(

*x,y*)], which the subject had to detect, embedded in external noise [

*N*(

*x,y*)]:

*f*) of the Gabor patch was set to a low spatial frequency of 1 cycle per degree (cpd). The phase (

*p*) of the sine wave was randomly set at each stimulus presentation. The standard deviation (

*σ*) of the Gabor patch was set to 1 deg of visual angle,

*c*represented the contrast of the signal, which corresponds to the Michelson contrast once the signal [

*S*(

*x,y*)] is integrated in the modulation (

*M*(

*x,y*), Equation 17). The contrast (

*c*) was the dependent variable.

*T*(

*x,y*)] was −0.5 and 0.5. Therefore, the difference of luminance between the two peeks was equal to the stimulus luminance average (

*L*

_{0}). In other words, for both stimulus types, the contrast average of the whole stimulus was set equal to the luminance average (

*L*

_{0}) of the whole stimulus (Figure 6).

*N*) of the noise was set to 0, 0.0125, 0.050, and 2.00 for LM stimuli and 0, 0.050, 2.00, and 4.00 for CM stimuli. More noise was introduced for CM stimuli because of two reasons. First, because the modulation was multiplied with the texture ranging between −0.5 and 0.5, two times more noise could be used without exceeding the monitor luminance range. Second, as mention above, CM had more IEN so greater external noise was required to derive the IEN and CE.

*k*

_{ideal}) sufficient to perform a task for a given threshold criterion. However, the present section will only show that an ideal observer has the same sensitivities to both LM and CM stimuli. Because the CE (

*k*

_{ideal}/

*k*) is defined relative to the optimal SNR (

*k*

_{ideal}) and that the optimal SNR is the same for both LM and CM stimuli (

*k*

_{ideal LM}

*= k*

_{ideal CM}), deriving the exact optimal SNR is not necessary and is beyond the scope of the present study. The relative difference between CEs of LM and CM stimuli may be compared directly:

_{LM}= CE

_{CM}), then the SNR required for detecting LM stimuli will be equal to the SNR required for detecting CM stimuli (

*k*

_{LM}

*= k*

_{CM}).

*L*

_{0}) is constant in all the testing conditions, it may be abstracted from the stimulus equation and the stimuli may be defined by their contrast function

*C*(

*x,y*) (Linfoot, 1964) instead of their luminance function [

*L*(

*x,y*)]:

*S*(

*x,y*) = 0) is

*T*(

*x,y*) and is known to the ideal observer. Therefore, it may be subtracted from the stimulus equation without affecting the ideal observer's performance:

*C*′

_{LM}(

*x,y*) and

*C*′

_{CM}(

*x,y*) is identical to the one using

*C*

_{LM}(

*x,y*) and

*C*

_{CM}(

*x,y*), respectively.

*S*(

*x,y*)] in LM noise [

*N*(

*x,y*)] or detecting a CM signal [

*S*(

*x,y*)

*T*(

*x,y*)] in CM noise [

*N*(

*x,y*)

*T*(

*x,y*)].

*C*″

_{CM}(

*x,y*) will be identical to

*C*′

_{CM}(

*x,y*), which is the same as using

*C*

_{CM}(

*x,y*).

*C*′

_{LM}(

*x,y*) =

*C*″

_{CM}(

*x,y*) and the ideal observer performance to

*L*

_{LM}(

*x,y*) and

*L*

_{CM}(

*x,y*) is identical to the one using

*C*′

_{LM}(

*x,y*) and

*C*″

_{CM}(

*x,y*), respectively, the performance of an ideal observer using

*L*

_{LM}(

*x,y*) will be equal to the one using

*L*

_{CM}(

*x,y*) as long as it has perfect knowledge of the texture [

*T*(

*x,y*)]. Consequently, for an ideal observer, detecting a LM signal [

*S*(

*x,y*)] in LM noise [

*N*(

*x,y*)] is equivalent as detecting a CM signal [

*S*(

*x,y*)

*T*(

*x,y*)] in CM noise [

*S*(

*x,y*)

*N*(

*x,y*)]. In other words, an ideal observer will require the same SNR (

*k*

_{ideal}) for detecting both stimulus types.

*T*(

*x,y*) at all the positions (

*x,y*) must be known]. For the plaid carrier, the frequency, amplitude, and orientation of the carrier are known except that its phases change randomly at each presentation. However, the phase can easily be deduced precisely because the spatial frequency of the carrier is higher than the rest of the stimulus (signal and noise). Consequently, the local variation only depends on the carrier. For LM stimuli, the signal and noise, which are at lower spatial frequencies, will only change the local mean luminance. For CM stimuli, the signal and the noise will only change the local carrier contrast. Therefore, the phase of the plaid can be detected and the value of the texture [

*T*(

*x,y*)] can be precisely computed at each pixel position (

*x,y*) and abstracted from the equation.

*x,y*) may have two possible values: −0.5 and 0.5. Because it is impossible to have negative luminance pixel values, the luminance range of each pixel [

*L*(

*x,y*)] must be, for a symmetrical reason, between 0 and 2

*L*

_{0}. Based on these constraints and on Equations 3 and 4, the modulation [

*M*(

*x,y*)] can theoretically range between 0.5 and 1.5 for LM stimuli, and between 0 and 2 for CM stimuli. For both LM and CM stimuli, a texture element [

*T*(

*x,y*)] of −0.5 or 0.5 will cause the luminance value at that same position to be bellow or above the luminance average (

*L*

_{0}), respectively. Consequently, an ideal observer can precisely recompute the original texture for all the carriers used in the present study.

*k*and

*N*

_{eq}. The fitting was achieved by minimizing an error function using Excel Solver (Newton method). The error function was the sum over each noise condition of the squared difference in log units between the detection threshold and the predicted threshold by Equation 4.

Plaid | Checkerboard | Binary noise | |||||
---|---|---|---|---|---|---|---|

IEN | CE | IEN | CE | IEN | CE | ||

ela | 1.10 | 0.02 | 0.44 | −0.30 | 0.53 | 0.19 | |

Il | 0.99 | −0.03 | 0.57 | −0.17 | 0.20 | 0.04 | |

mer | 0.98 | 0.12 | 0.71 | 0.12 | 0.25 | −0.03 | |

ra | 1.04 | −0.04 | 0.72 | −0.01 | 0.33 | 0.09 | |

Mean | 1.03 | 0.02 | 0.61 | −0.09 | 0.33 | 0.07 | |

± | 0.03 | 0.04 | 0.08 | 0.11 | 0.08 | 0.05 |

*S*(

*x,y*)

*T*(

*x,y*)] in LM noise [

*N*(

*x,y*)] compared with the LM detection task, which consists in detecting a LM signal [

*S*(

*x,y*)] in LM noise [

*N*(

*x,y*)]. Therefore, the two tasks would only differ by their signal [

*S*(

*x,y*) vs.

*S*(

*x,y*)

*T*(

*x,y*)].

*S*(

*x*,

*y*)], the energy of the LM stimulus [

*S*(

*x*,

*y*)] will be 1/

*T*

_{RMS}

^{2}times greater (

*T*

_{RMS}

^{2}< 1 because −1 <

*T*(

*x*,

*y*) < 1) than the energy of the CM stimulus [

*S*(

*x*,

*y*)

*T*(

*x*,

*y*)]. Because the energy is proportional to the squared contrast, to have the same energy level as the LM stimulus the CM contrast must be 1/

*T*

_{RMS}times greater. Therefore, in the same noise condition [

*N*(

*x*,

*y*)], the LM sensitivity of the ideal observer's with LM internal noise would be 1/

*T*

_{RMS}times greater than the CM sensitivity.

*T*

_{RMS}times more IEN for CM stimuli detection than for LM stimuli detection. Comparing LM and CM IENs (or sensitivities), if we suppose that the significant internal noise occurs prior to the rectification, a factor of

*T*

_{RMS}should be considered. Therefore, pre-rectification IEN (

*N*

_{eq pre-rect}) may be defined as:

*T*

_{RMS}) of the plaid, checkerboard, and binary noise carriers were 0.25, 0.5, and 0.5, respectively. Using binary noise as a carrier, the results (0.33 log) show a difference of IEN (

*N*

_{eq}) near the factor predicted by the ideal observer with LM internal noise (2 or 0.30 log). Therefore, by compensating for the texture contrast (

*N*

_{eq pre-rect}), there was no significant difference between LM and CM IEN (Figure 13). These results suggest that the significant noise occurred prior to the rectification and was common to both tasks.

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