Figure 1.
Synthetic “Dead-Leaves” Stimuli
(A) “Dead-leaves” example, composed of occluding circular disks with radius r and distribution 1/r3. The intensity of each “leaf” is independently drawn from a uniform distribution.
(B) Reflectance matrix (R), which represents a 20 × 20 subsection randomly chosen from the larger “dead-leaves” stimulus. Typically, between 40 and 60 “leaves” were at least partially visible in each reflectance map.
(C) The light falling on a typical surface will come from many sources, so we model illumination with a more gradual change across space than for reflection (see Methods for details). The example illumination matrix (I) shown here is a 20 × 20 section chosen from a similar map as R but with larger disks than with reflection maps, typically containing 10–15 leaves. These were then heavily blurred producing maps of typically 200–400 distinct levels of intensity, but with a high level of spatial correlation.
(D) Stimulus intensity matrix (S), which is the pixel-wise product of B and C: S = I × R. All the values are in the range 0…1.
Figure 2.
(A) Simultaneous brightness contrast illusion. See text for explanation.
(B) Articulated surrounds with mean S = 0.25 and S = 0.75 with same target intensity S = 0.4. See text for explanation.
(C) Concentric rings, both with an average intensity of S = 0.5 and a target intensity of S = 0.5.
In (A–C), ANNs predict a higher reflectance for the stimulus on the left compared to the stimulus on the right.
Figure 3.
Artificial Neural Network Responses to Optical Illusions
(A) Vasarely illusion stimulus on left. Note the illusory regions of lightness along the diagonals formed by the corners of the concentric squares. Blue line on right indicates the intensity profile at the corresponding points of the blue line on the left. Red line shows the relative reflectance predicted by the networks. Units are arbitrary and so are not plotted throughout this figure.
(B) Mach band stimulus consisting of a dark bar (left) and a light bar (right) with a linear gradient between. Blue line on right indicates the intensity profile at the corresponding points of the blue line on the left. Red line shows the relative reflectance predicted by the networks.
(C) Chevreul illusion stimulus, with five bars of uniform reflectance. Blue line on right indicates the intensity profile at the corresponding points of the blue line on the left. Red line shows the relative reflectance predicted by the networks.
(D) Hermann grid illusion stimulus. Blue line on right indicates the intensity profile at the corresponding points of the blue line on the left. Red line shows the relative reflectance predicted by the networks.
(E) White's illusion on left. The grey areas indicated have the same physical reflectance, although the left-hand one appears darker than the other. The ANN response on the right corresponds to this human experience, with the first perceived reflectance (1) being darker than the second (2).
Figure 4.
White's Stimuli and ANN Responses
(A) Three White's stimuli of varying spatial frequency and (B) three White's stimuli with different target patch heights. In all cases, the left-hand target patch has the same intensity as the right-hand patch, but generally appears darker to humans. The stimuli seen by the ANNs are 20 × 20 pixels.
(C) Mean ANN responses to White's stimuli of varying frequencies with varying test patch heights. Each value is the difference in predicted reflectance for the two test patches. A positive difference means that the test patch on the light bar appears darker than the test patch on the dark bar; a negative difference means the test patch on the light bar appears lighter than the patch on the dark bar. The former is consistent with White's illusion, the latter with brightness contrast. The results show i) that decreasing the frequency of the background stripes (i.e., making them wider) also decreases the strength of White's illusion; and also ii) that increasing the height of the test patch decreases the strength of White's illusion. Both results correspond to human psychophysical responses [5].
Figure 5.
Conditional Probability Distribution of Reflectance Given Past Experience and a Particular Stimulus as Context
A maximum-likelihood estimation allows the observer to predict the target reflectance and will be correct (approximately) most of the time. If the true reflectance actually lies in a low-likelihood tail of the distribution, then the resulting percept is an illusion.