between the Different Axes
In Chapter 7 and 8, we manipulated the number of the colors in the images to investigate the integration of information from color selective symmetry channels at different levels of uncertainty. In these two chapters, all the symmetric image components shared the same symmetry axis. Hence, under these situations, the visual system only needs to monitor color-selective channels tuned to the same axis orientation. However, symmetry perception in a nature scene can be more complicated than this. Sometimes we need to detect two symmetric images or objects in a scene. In this case, these two images or objects do not necessarily share the same symmetry axis. For example, the red apple and the green leaf in the apple tree usually have different symmetry axes. To detect multiple symmetric patterns, the visual system in this case instead monitors channels sensitive to different orientations.
The ecological significance behind these two cases is quite different. The roles of the color in these two cases may also differ. In the latter case, the color information helps object segmentation (Shevell & Kingdom, 2008). That is, the visual system uses color information to distinguish two objects. In the former, however, the visual system instead needs to incorporate different colors into the same pattern to form a coherent percept of a symmetric pattern. Due to such difference, there is a possibility that the mechanisms underlying theses two processes are different. Specifically, maybe the integration between chromatic symmetry channels selective to the same
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orientation axis and to different axes is different. For example, it is possible that the detecting symmetry is more difficult when two symmetric patterns share the same axis than not for the segmentation of two images with different colors helps symmetry detection. In this chapter, we manipulated the orientation of two chromatic symmetric patterns that superimposed on each other, to investigate how these symmetry channels works to achieve these ecological purposes. We also compared the above symmetry detection performance in both chromatic and achromatic images.
9.1. Method
Participant. Three observers attended this experiment: CCW, CPY, and RYT.
RYT was naïve to the purpose of the experiment.
Stimuli. There were four conditions in this experiment, two isoluminance and
two luminance conditions. In each condition, the stimuli in each trial consisted of two symmetric-dot targets or two random-dot noise control superimposed on a random-dot noise mask. The target was a 45° or 135° diagonal symmetric-dot pattern containing only one color. The two isoluminance conditions were RG-S and RG-D, and the two luminance conditions were WK-S, and WK-D, where two capital letters before the hyphens denoted the colors of the two targets respectively (The latter R, G, W, and K were the abbreviation of red, green, white, and black respectively, see Table 4.1 in Chapter 4 for the definition of the color) and the letters after the hyphens denoted the same (S) or the different (D) axis orientations of the two targets. In the same (S) orientation condition, the axis orientations of the two symmetric targets were both 45°
or both 135° from horizontal. In the different (D) orientation conditions, the axis orientation of one symmetric target was 45° and that of another one was 135° from horizontal. The color and the density of the two noise controls were the same as those
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of the two targets. The noise mask was a random-dot noise pattern containing the same color as the two targets. The density of noise mask was from 0 to 5%. The center region of 1.4° * 1.4° visual angle and the grids on and near the two diagonal axes (0.7°) contained no dot to prevent observers from using only the information near the axis to make a judgment.
Procedure. The 2AFC procedure was the same as that in the previous chapters.
The observers’ task was to judge which stimulus interval contained any symmetric pattern. The observers were informed that the axis orientation of the two targets was either the same (both 45° or both 135° from horizontal) or different (45° and 135°
from horizontal). The isoluminance and luminance conditions were run separately, in which the trials of the same and different axis orientation conditions were randomized in the same block to prevent observers from using the top-down strategy.
9.2. Results
Figure 9.1 shows TvD functions for the two isoluminance conditions (RG-S and
RG-D). Each panel represents the TvD functions from one observer. The red and green symbols denote the TvD functions for the RG-S and the RG-D conditions respectively. The smooth curves are fits of the model discussed below.
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Figure 9.1. Target threshold vs. mask density (TvD) functions in the isoluminance condition. Each panel represents the data from one observer. The red and green symbols represent the data points of the RG-S and RG-D conditions respectively. The smooth curves are fits of the model (see text for details).
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For these two isoluminance conditions, the target density threshold increased with noise density. The slope of the increment function was 0.96 in log-log coordinates, averaged across conditions and observers (SE = 0.08). The slope of the TvD functions had no difference between the same- (i.e., RG-S, M = 0.87, SE = 0.04) and the different-orientation conditions (i.e., RG-D, M = 1.05, SE = 0.16) when averaged across three observers (Figure 9.2), t(2) = 0.98, p = .43. The target density threshold was lower in the same-orientation condition than in the different-orientation condition. The red symbols in Figure 9.3 show the threshold difference between the RG-S and the RG-D conditions, averaged across three observers. The orientation effect was from 0.2 to 0.62 log unit (or 1.6 to 4.2 fold change) in threshold measurement.
Condition
Slo pe
RG-S RG-D
0 0.2 0.4 0.6 0.8 1 1.2
RG-S RG-D
0 0.2 0.4 0.6 0.8 1 1.2
Figure 9.2. Slope of the target threshold vs. mask density (TvD) functions in isoluminance conditions. The red and blue bars represent the slopes of the RG-S and the RG-D conditions respectively. The error bar was standard error. There was no significant difference in the slopes of the TvD functions between two conditions.
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The results of the luminance conditions, WK-S and WK-D, were similar to that of the isoluminance conditions. The gray and brown symbols in Figure 9.4 denote the TvD functions for the WK-S and the WK-D conditions respectively. The slope of the TvD functions showed no difference between the same- (i.e., WK-S, M = 0.89, SE = 0.06) and the different- orientation conditions (i.e., WK-D, M = 1.04, SE = 0.13) when averaged across three observers (Figure 9.5), t(2) = 0.83, p = .49. Also, the target density threshold was lower in the same-orientation condition than in the different-orientation condition, as Figure 9.6 shown. The gray symbols in Figure 9.6 represent the threshold difference between the WK-S and the WK-D conditions, averaged across three observers. The threshold reduction in the WK-S condition was from 0.24 to 0.52 log unit, equivalent to 1.7 to 3.3 fold change.
-2.5 -2 -1.5 -1
-0.8 -0.6 -0.4 -0.2 0
-2.5 -2 -1.5 -1
-0.8 -0.6 -0.4 -0.2 0
Noise density (log)
Thr es ho ld di ff er enc e ( lo g)
Figure 9.3. The threshold difference between the same- and the different-orientation conditions in the isoluminance conditions, averaged across three observers. The red symbols represent the threshold difference between the RG-S and the RG-D conditions.
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Figure 9.4. Target threshold vs. mask density (TvD) functions in the luminance conditions. Each panel represents the data from one observer. The gray and brown symbols represent the data points of the WK-S and the WK-D conditions respectively.
The smooth curves are fits of the model (see text for details).
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Figure 9.6. The threshold difference between the same- and the different-orientation conditions in the luminance conditions, averaged across three observers. The gray symbols represent the threshold differences between the WK-S and the WK-D conditions.
Figure 9.5. Slope of the target threshold vs. mask density (TvD) functions in luminance conditions. The red and blue bars represent the slopes of the WK-S and the WK-D conditions respectively. The error bar was standard error. There was no significant difference in the slope of the TvD functions between two conditions.
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Our results also showed the difference in the target density thresholds between the isoluminance and the luminance conditions. Figure 9.7 represents the threshold difference between the two same-orientation conditions (i.e., RG-S and WK-S, pink symbols) and between the two different-orientation conditions (i.e., RG-D and WK-D, green symbols), averaged across three observers. The amount of the threshold difference was from 0.1 to 0.3 log unit, equivalent to 1.2 to 2 fold change, except for the difference between the two different-orientation conditions at the highest noise density.
Noise density (log)
Thr es ho ld di ff er en ce (lo g)
-2.5 -2 -1.5 -1
-0.4 -0.3 -0.2 -0.1 0
0.1 Same orientation Different orientation
-2.5 -2 -1.5 -1
-0.4 -0.3 -0.2 -0.1 0
0.1 Same orientation Different orientation
Figure 9.7. The threshold difference between two same-orientation and between two different-orientation conditions, averaged across three observers. The pink symbols represent the threshold difference between the RG-S and the WK-S conditions. The green symbols represent the threshold difference between the RG-D and the WK-D conditions.
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9.3. Discussion
Our results showed better symmetry detection performance when two symmetric targets shared the same axis than those did not. This suggests different mechanisms mediating these two conditions. Here we applied the model proposed in Chapter 3 to account for our data.
Model implementation. As discussed in Chapter 8, there are occasions that the
visual system needs to monitor more channels than those relevant to the visual task. In this task, the two symmetric targets of different colors either shared the same axis (i.e., both left- or right-diagonal) or had different axes (i.e., left- and right-diagonal). Hence, the system needs to monitor four channels to detect symmetry, two of which are relevant and another two are irrelevant. Take the RG-D condition for example. Since the observer had no prior knowledge of the axis orientation of the target, the visual system needs to monitor all the red diagonal, red right-diagonal, green left-diagonal and green right-left-diagonal symmetry channels to detect symmetry. The two targets were either one red left-diagonal and one green right-diagonal symmetric pattern or one green left-diagonal and one red right-diagonal symmetric pattern.
Hence, only red left-diagonal and green right-diagonal symmetry channels or green left-diagonal and red right-diagonal symmetry channels among the four monitored channels are relevant. Hence, m = 2n in Eq. (6), where n = 2 in this task. The only difference between the same-orientations and the different-orientations is that the two relevant channels are selective to the same orientation in the same-orientation conditions while are selective to the different orientation in the different-orientation conditions.
Again, we set the density of the symmetric component in a random-dot noise pattern as the square of the whole density of that pattern. Hence, the Eq. (2) in the
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interval that contains two targets plus mask can be presented as
for the two relevant color-selective channels while as
( )
2for the two irrelevant channels, where the subscript r and ir indicate the relevant and irrelevant channel respectively, Dt and Db are the target and noise densities respectively, Set is the sensitivity of the symmetry channel to the symmetric pattern, and n is the number of the colors in the image. Here we set n to be 2 for the noise mask contains two colors that are the same as the two targets. Eq. (2) in the interval that contains two noise control plus mask can be presented as
( )
2Again, we used a typical value of 2 for the power for the divisive inhibition input q in Eq. (4) in model implementation (Foley, 1994; Foley & Chen, 1999; Hegger, 1992).
The response of the individual relevant channel in the intervals containing target plus mask in Eq. (3) thus can be expressed by
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while that of irrelevant channel can be expressed by
( ) ( )
1 1 'the response of the individual channel in the intervals containing noise control plus mask in Eq. (3) can be expressed by the response R’ in the interval containing two targets plus mask and noise control plus mask can be expressed by Eq.(23) and (24) in Chapter 8 respectively, where n = 2 in this task as mentioned above. Also, we replaced the v*Db2
in Eq. (5) with (v*Db)2 in practice. Hence, Eq(5) can be expressed as Eq. (25) in this task. The γ in the decision variable in Eq. (25) is set to be 0.71 for all conditions for the standard deviation of the max distribution of four independently and identically distributed samples is 0.71 times the standard deviation of the original distribution (Chen & Tyler, 1999). Hence,
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Eq. (23) – (31) define the whole computation and all the parameters in the model.
Again, we fixed Set to be 1000 and the effect of the internal noise, σa2
to be 1 in all conditions to reduce the mathematical redundancy in the model in practice. We empirically found that fixing all other parameters except Sit.tc and Sit.nc provided a good fit to the data.
Model fits. The model fits are shown as smooth curves in Figure 9.1 and Figure
9.4. Since five data points of CCW in the high noise densities reached the highest
threshold (three data points in the RG-D condition and two data points in the WK-D condition respectively), we excluded these data points in model fitting. Other than that, the model explains 98-99% of the all variability in the thresholds across observers (27 free parameters). The root mean square error (RMSE) is between 0.06 to 0.1 log unit across observers, on par with the average standard error of measurement. Table 9.1 shows the fitted model parameters.
The model fitting results showed that the inhibition from the same-orientation symmetric components of the non-target color (Sit.nc in the same-orientation conditions) was less than 13% of that from the different-orientation ones (Sit.nc in the different-orientation conditions) across three observers. That is, the mutual inhibition between mechanisms responding to the two symmetric patterns sharing the same axis is smaller than that between mechanisms responding to two symmetric patterns about different orientation axes. This suggests that the human visual system integrates the same-axis information by reducing the inhibition between the same-axis channels. On the other hand, when the symmetric components have different axes, the greater inhibition they produced makes the visual system easier to tell them apart. However, the increase of the mutual inhibition also impairs the symmetry detect performance among them. Thus, the difference in the inhibition terms subserves the two ecological
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implications of these two conditions.
CCW CPY RYT
Set* 1000 1000 1000
Sit.tc
Isoluminance conditions 272 1920 693
Luminance conditions 55 62 144
Sit.nc
Same-orientation conditions 8.4 269 158
Different-orientation conditions 680 2708 1228
Sib.tc 701 8227 1208
Sib.nc 342 135 286
z 2.96 1.50 1.00
v 2642 1163 1942
p 2.56 3.12 2.96
Other fixed parameters used in the model fits
σa2* 1 1 1
q* 2 2 2
n* 1 1 1
γ* 0.71 0.71 0.71
*fixed value, not a free parameter
Our results also showed that the inhibition from the target in the luminance conditions was less than 21% of that in the isoluminance conditions across three observers. This corresponds to the better detection performance in the luminance conditions than in the isoluminance conditions. This result is partially consistent with that in Chapter 8. In Chapter 8, we found that both the inhibition term from the symmetric patterns (Sit.tc) and the effect of the external noise (v2) differ between the isoluminance and the luminance conditions. However, our results in this experiment only showed the difference of the inhibition term from the symmetric patterns (Sit.tc)
Table 9.1.
Fitted model parameters.
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between the isoluminance and the luminance conditions. There is no difference in the effect of the external noise between the two conditions. This might be due to the difference in the procedures between these two experiments. As Table 9.1 shown, the mixture of the trials of the same- and the different-orientation conditions in this experiment produced a great deal of increment of the effect of the external noise in both the isoluminance and the luminance conditions. The effect of the external noise (v2) in this experiment reaches 3,000-200,000 times as much as that in the experiment in Chapter 8. The lack of the effect of the external noise between the two conditions might be due to the ceiling effect.
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