Fig 1.
(A) Simultaneous brightness contrast: The two gray patches with identical luminance increase their brightness difference with their respective backgrounds. (B) White’s effect is consistent with contrast at the horizontal contours of the bars, and with assimilation along the vertical contours. (C, D) Examples of brightness assimilation: The gray structures with identical luminance decrease their brightness difference with their respective background. (E) Sensitivity to context: The gray triangles have identical luminance and selectively contrast with the cross (left) or the background (right). (F) White’s effect is still intact in the presence of noise.
Fig 2.
Each of the three stages is mathematically specified in the Methods Section. (A) Stage 1: The Contrast-only channel and Contrast-Luminance channel are instantiated by filtering an input image with a corresponding set of Gabor filters with high spatial resolution (0.25 cycles per pixel) and coarse resolution (0.125 cycles per pixels), respectively. The local energy map is computed from the Contrast-only channel. (B) Stage 2: The kernel of the dynamic filter is estimated from the local energy map. Dynamic filtering equalizes the amplitude spectrum of the energy map, reducing redundancy. The decorrelated energy map serves as gain control for both contrast channels. (C) Stage 3: The output of the model is a brightness map that is obtained by solving an inverse problem, that is recovering the image from both contrast channels. Note that the two contrast channels do not interact with each other before Stage 3.
Fig 3.
Sensitivity to contrast and luminance.
(A) (left) Fruits filtered by a Contrast-Luminance filter g (Eq 1). (Middle) Fruits filtered by a Contrast-only filter g with the same orientation (Eq 1). (Right) Fruits filtered by a low-pass filter. (B) Example luminance step (values -0.3 to 0.5) and an example for a homogeneous region with luminance 0.3. (C) Responses (maximum) of Contrast-Luminance filters (solid lines) and Contrast-only filters (dashed lines) to the homogeneous region (red color) and the luminance step (blue color). For the luminance step, the lower luminance was increased from -0.5 to 0.5. Negative filter responses result from the use of negative input values.
Fig 4.
(A) The arrows indicate the steps in order to obtain the gain control map: (i) A local energy map is computed; (ii) a dynamical filter is constructed with a customized zero-phase whitening procedure (ZCA, see section C in S1 Text); (iii) a gain control map is obtained by filtering the energy map with the dynamical filter (see subsection Gain Control Map). (B) The power spectrum (= square of amplitude spectrum) before and after of applying the dynamic filter on the local energy map.
Fig 5.
Filter responses before and after applying Eq 4.
Each curve represents a different scalar value for the gain control G (notice that in Eq 4, G is two-dimensional). The values of G corresponding to each curve are indicated in the figure legend.
Fig 6.
Contribution of each channel to brightness estimation.
(A) A luminance staircase (giving rise to Chevreul’s Illusion) served as input. (B) The resulting brightness profiles are estimated at 10 iterations. The blue curve (legend label “CL”) uses only the Contrast-Luminance channel in Eq 5. The red curve (“CO”) uses only the Contrast-Only channel without responses to luminance. (C) Resulting brightness profile at the stop criterion for the Contrast-Luminance channel. Aliasing artifacts appear due to undersampling. (D) Brightness estimation with both channels. The Contrast-only channel eliminates undersampling.
Fig 7.
(A) Scenario 1: A disk embedded in a redundant pattern of eight squares served as input (first column). The middle column depicts the corresponding gain control map G, and the right column the profiles of input (black line) and brightness estimation (red line). The brightness of the center disk ts enhanced with respect to luminance, meaning that a brightness contrast effect is predicted merely based on redundancy (but not on grounds of luminance—note that all features have the same luminance). (B) Scenario 2. (Top) The input consists of a series of nine squares arranged in a spatially redundant pattern, where the middle square has a different luminance. The profile plot suggests an overall increase in brightness contrast: Brightness of the middle square is further reduced, while the brightness of the surrounding squares is enhanced. (Bottom) While the brightness contrast also increased in the display with the bright middle square, this increase in contrast is caused nearly exclusively by the middle square.
Fig 8.
A luminance step and a luminance staircase, respectively, served as input images. Activity in response to the luminance step is close to the control parameter of Eq 4 (τ = 0.5), producing barely changes in the corresponding brightness estimation at the edges. In contrast, for the luminance staircase, the activity at the edges is relatively far from the control parameter, inducing a boost (an increment of brightness contrast) in the corresponding brightness estimation.
Fig 9.
Model prediction for simultaneous brightness contrast (SBC).
(A) Simultaneous brightness contrast display (model input). (B) The corresponding Gain Control Map. (C) Profile plot of the estimated brightness map (red line) and the input (black line). (D) Mean absolute brightness difference between the left and the right patch as predicted by our model (filled circles). The continuous (red) lines show the fit of y = a + blog(x) to the model data. The fit was carried out by linear regression with fitting parameters: intercept a = −0.3851, slope b = −0.0823, R2 = 0.9831, and RMSE = 0.0086.
Fig 10.
Prediction for Benary Cross illusion.
(A) Benary Cross (input). Both triangles have the same intensity, but the triangle embedded in the cross is perceived as brighter. (B) In the gain control map, redundant edges (=aligned with the cross) of the triangle are weakened. (C) Profile plots of predicted brightness (red line) versus luminance (black line). The left profile plot shows the left triangle.
Fig 11.
Model prediction for reverse contrast effect.
(A) Simultaneous Brightness Contrast (SBC, top) and Reverse Contrast (bottom) which is constructed by adding flanking bars to the SBC configuration. All gray patches have the same luminance. Reverse contrast can be explained either by assimilation with the in-between bars that have the same intensity as the background, or as contrast with the flanking bars that have the opposite luminance to the background. (B) Gain Control Maps obtained by dynamic filtering. Notice the suppression of parallel edges corresponding to flanking patches with the same intensity. (C) Profile plots of predicted brightness (red line) versus luminance (black line).
Fig 12.
Model prediction for reverse contrast effect for displays with different configurations.
(A) Reverse contrast with a varying number of adjacent bars to the gray patch. The bar plots show the predicted brightness difference between the gray patches for the corresponding display (a positive value indicates contrast, while negative values “reverse contrast”). (B) Reverse contrast where the good continuation of the end points is varied. This in turn affect the suppression of redundant edges, which increases with the alignment of the flanking bars.
Fig 13.
Model prediction for White’s illusion.
(A) Top: White’s illusion; middle: the corresponding gain control map; bottom: profile plot of estimated brightness (red line) and luminance (black line). (B) With smaller bar height, the brightness difference between the bars increases. (C) Modification of White’s illusion which produces a strong contrast effect. (D) Surface plot of the estimated brightness difference (effect strength) between the bars as a function of bar height (in units of pixels) and spatial frequency of the background stripes (in units of cycles/image). Image size was 256 x 256 pixels.
Fig 14.
Model prediction for Todorovic’s illusion.
(A) Top: A variation of Todorovic’s original illusion with the gray disks in the foreground (Context A). An effect is hardly perceivable. Middle: The corresponding Gain Control Map. Bottom: Profile plots of estimated brightness (red line) and luminance (black line). The model predicted at most a very weak effect. (B) Original Todorovic Illusion (Context B), where the occluded left disk is perceived as being brighter. (C) Reversed Todorovic Illusion (Context C). Now it looks like viewing the disks on a single square background through a window cross, and the left disk is perceived as being darker.
Fig 15.
Brightness dependence on target size for Todorovic’s display.
The Scheme shows the smallest and biggest disk size that was used with respect to the squares in order to generate the plots. Each plot indicates the brightness effect for each of the three Contexts shown in Fig 14. The empty circles indicate the predicted brightness of the disk with the white squares. The filled symbols show disk brightness with the black squares. The continuous lines show the estimated brightness difference between both disks and indicate the predicted strength of the illusion.
Fig 16.
Model predictions for Dungeon, Checkerboard and Shevell.
In each display, the gray areas have the same luminance, yet they are perceived differently because of assimilation with the adjacent structures. (A) Top: Dungeon illusion. Middle, corresponding gain control map. Bottom, profile plot of the estimated brightness (red line) compared with the input (black line). (B) Checkerboard illusion. (C) Shevell’s Rings. Notice that this illusion cannot be explained with T-junctions.
Fig 17.
Prediction for Craik-O’Brien-Cornsweet Effect (COCE).
(A) Top: COCE along with the Gain Control Map. Notice that black lines (adjacent to the edges) in the Gain Control Map, which indicate negative values. Bottom: The first profile plot shows the predicted brightness (red line) along with input luminance (black line). The second profile plot shows the predicted brightness without low-pass filtering after the gain control mechanism (Eq 4). (B) The cow-skin illusion is a variant of the COCE without luminance gradients. It consists only of adjacent black and white lines. Our model consistently predicted this illusion: The brightness map generated by our model is shown at the bottom right.
Fig 18.
Model predictions for Hermann/Hering grid and corrugated grid.
(A) Hermann/Hering (HG) illusion and a corrugated version of it. At the intersections of the white grid lines, illusory gray spots are perceived in the HG, but not in the corrugated grid. (B) The corresponding gain control maps of the input images of A. (C) The brightness estimation from our model. The surfaces plots (insets) illustrate the 3D profile of the brightness estimation corresponding to regions highlighted with red. (D) The predicted brightness magnitude at the intersections as a function of the ratio , where the red curve corresponds to the corrugated grid, and the blue curve to the HG.
Fig 19.
Model predictions for luminance staircase and pyramid (Chevreul’s illusion).
(A) Top: Luminance staircase. Bottom: luminance pyramid. (B) The corresponding gain control maps as a result of dynamic filtering. (C) Top: Profile plot of the estimated brightness (red line) of the luminance staircase (black line). Bottom: The induced brightness, which consists of the difference map between estimated brightness and input luminance (i.e., brighter gray level mean positive values, and darker gray levels mean negative values).
Fig 20.
Model prediction for mach bands.
(A) Top, A luminance ramp that leads to the perception of Mach bands close to the knee points of the ramp. Bottom, a luminance step (no Mach bands are perceived). (B) Profiles plots of estimated brightness (red line) compared with the corresponding input (black line) of A. (C) Brightness magnitude (at the inflection point of the ramp) as a function of ramp width. The plots show the predictions of our model on the perceived strength of the bright Mach band. The colored curves (left axis label: response amplitude) represent model predictions for different dynamic ranges (i.e., differences between luminance values of the upper and the lower plateau, see legend). The gray curves are the threshold contrasts (axis label on the right) for seeing the bright Mach bands at trapezoidal waveforms according to [104]. The trapezoidal waves are characterized by a shape parameter t (see legend; t = 0.5 corresponds to a triangular wave).
Fig 21.
Model prediction for grating induction.
(A) The Grating Induction refers to the illusory perception of a brightness modulation across the gap (=test field) between the inducer gratings. The brightness modulation is perceived in opposite phase to the inducer gratings. (B) When the inducer gratings stand in opposite phase to each other, then brightness modulation is considerably reduced. The corresponding gain control maps are shown in the middle row, and the last row shows the induced brightness, which consists of the difference map between estimated brightness and input luminance (i.e., brighter gray levels mean positive values, and darker gray levels mean negative values). (C) Top: Profile plot of brightness estimation for display A (black line) and display B (blue line). The dashed blue line shows the luminance profile of the inducer grating A. Middle, modulation depth as a function of the phase difference between the two inducer gratings. Bottom: Surface plot that shows how modulation depth depends on test field width and spatial frequency of the inducer grating.
Fig 22.
(A) Top: Fruits image with additive white noise (SNR = 2.6266dB and PSNR = 8.9813dB) along with the corresponding model output (SNR = 5.4682db and PSNR = 11.8228dB); The models capacity for noise removal is worse than an algorithm based on Tikhonov regularization (SNR = 8.30dB and PSNR = 14.65dB; [111]. Middle: A high-dynamic-range version of a real image (where dynamic range of each quadrant decreases clockwise by one order of magnitude) and model output; the dynamic range of the input is 1, and that of the output is 0.9596. Bottom: Bridge image alongside with corresponding model output (SNR = 6.1084dB and PSNR = 13.6201dB). (B) Top: Simultaneous Brightness Contrast display with additive white noise and corresponding brightness profile (red line) as predicted by the model. The dashed line indicates the gray level of the gray squares. Middle: White Effect. Bottom: Benary Cross.