Tiny Squares at the Hermann Grid Corners Can Completely Remove the Illusion

Introduction

The most well-known version of the Hermann grid illusion (1870) consists of a two-dimensional array of equally spaced black squares. Illusory dark spots/blobs are seen on the white at the junction of the grid lines. This is depicted below in Figure 1(a), (c), (e), and (g). The phenomenon of appearance of these spots remains tolerant across a wide range of variations in contrast, spacing, angular rotation, and so on (Spillmann, 1994). Ever since its inception, the Hermann grid illusion has remained a constant subject of study in visual psychophysics through the works of Hering (1920), Baumgartner (1960), Spillmann and Levine (1971), Spillmann (1994), De Lafuente and Ruiz (2004), Geier, Bernáth, Hudák, and Séra (2008), Hamburger, Baier, and Spillmann (2012), and many others. The effect was generally explained by Baumgartner’s (1960) hypothesis that the illusory blobs at the junction of the grids are generated by the excitatory-inhibitory response of the receptive field of the retinal ganglion cells, till there came several challenges to this explanation during the past three decades (Geier et al., 2008; Lingelbach, Block, Hatzky, & Reisinger, 1985; Schiller & Carvey, 2005; Spillmann, 1994). These works tried to propose alternative theories that mostly involve the role of higher level vision to explain the mechanism of the formation of the illusory dark spots, though admittedly the neural mechanism behind the illusory perception of the dark spots is still not very well understood. Some of the stimuli demonstrated in their works by the abovementioned researchers that challenges Baumgartner’s (1960) low-level vision-based explanation are displayed in Figure 1(b), (d), and (f) which may be seen in comparison to their corresponding classical versions shown in Figure 1(a), (c), and (e), respectively. Other spatial filtering models based on receptive field modeling have also not been able to make much dent in revealing the neural correlates of Hermann grid illusion especially in the light of the abovementioned challenging stimuli (Ghosh, Sarkar, & Bhaumik, 2006; Yu, Yamauchi, & Choe, 2004). This work too does not as such provide any new insight with respect to understanding the visual signal processing mechanism that results in this illusory perception, but like the other stimuli shown in Figure 1, it demonstrates a novel variation of the Hermann grid stimulus (Figure 1(h)) that can completely wipe out the illusory effect, in the same sense as the one shown by Geier et al. (2008; Figure 1(b)). The experiments concerning the variations or distortions in the classical Hermann grid illusion prior to Geier et al. (2008) were all based on asking the subjects how intensively they perceived the illusory spots, and several conditions, that lead to weakening of the effect, were discovered from these experiments. Compared to this, for both Geier et al. (2008) and the present authors, the attempt instead was to determine the limiting condition for the illusion itself to completely disappear, beyond mere weakening of the effect. Geier et al. (2008) defined this limit as the distortion tolerance that refers to the deviation from the straightness of the grids. In this work, we demonstrate that the illusory spots completely disappear not just by curving the middle portion of the grid block edges, as shown in Figure 5 of Geier et al. (2008), but even when tiny perturbing squares are introduced only at the corners without directly interfering with the rest of the grid edge (Figure 2(c)–(f)). Furthermore, this happens even as the grid square edges (except at corners) remain right-angled in contrast to several of the stimuli demonstrated by Schiller & Carvey (2005), where they observed weakening of the effect with orientation changes of the bars (e.g., Figure 1(d)). Moreover, in correspondence to the distortion tolerance defined by Geier, Sera, and Bernath (2005), this work determines the gray-level intensity threshold for the perturbing corner stimuli in order to make the illusory blobs disappear completely.

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Figure 1.

(a, c, e, and g) The classical Hermann grid. (b) The distorted (sinusoid) grid reproduced from Geier et al. (2008). (d) The Schiller and Carvey (2005) modification with the vertical lanes distorted into a zigzag path. (f) The Spillmann (1994) modification where the relative positions of some of the grid squares have changed (h) The proposed Hermann grid modification where the tiny perturbing squares are placed overlapping with each grid square corner. While modifications in (d) and (f) succeed in reducing the illusory strength, those in (b) and (h) completely wipe out the illusory spots.

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Figure 2.

Modified Hermann grid stimulus with tiny gray squares added to the corners. The columns correspond to the size of the grid squares. The rows correspond to the intensity values of the tiny squares added to the corners of the large squares. The images on left column (a, c, e, and g) correspond to small square sizes, whereas the images on the right column (b, d, f, and h) correspond to large square sizes. The gap between the grid squares is kept fixed. The first row (a and b) has tiny squares added to the corners with a grayscale value identical to the white background. The second row (c and d) has tiny squares with a high grayscale value (but less than in a and b) added to the corners. The tiny squares added to the third row (e and f) are darker than those on the second row. The tiny squares on the fourth row (g and h) have grayscale values same as the large squares of the Hermann grid.

Materials and Method

The experiments involved showing the subjects the modified Hermann grid stimuli and recording their responses as explained next. The classical Hermann grid illusion was modified by introducing some perturbing stimuli. These perturbing stimuli are simply tiny squares that are added to the four corners of each grid square. The centers of these tiny squares coincide with the corresponding corners of the grid squares. The resultant stimuli are shown in Figure 2, which also depicts a series of stimuli that were shown to the subjects with increasing width of the original grid squares, keeping the gap between the squares fixed. The width of the grid squares used in the experiment varied from 60 pixels to 80 pixels in steps of 5 pixels. The gap between the grid squares was kept fixed at 10 pixels for all stimuli. The perturbing tiny squares added to the four corners of the larger grid squares were 4 pixels by 4 pixels in size, out of which 12 pixels fell outside the boundaries of the larger grid squares and 4 pixels fell within. The columns in Figure 2 correspond to the size of the grid squares. The rows correspond to the intensity values of the tiny squares added to the corners of the large squares. The images on left column (Figure 2(a), (c), (e), and (g)) correspond to a smaller grid square size, whereas the images on the right column (Figure 2(b), (d), (f), and (h)) correspond to a larger grid square size. The first row (a and b) has tiny squares added to the corners with a grayscale value identical to the white background. The second row (c and d) has tiny squares with lower grayscale value added to the corners. The tiny squares added to the third row (e and f) are darker than those on the second row. Finally, the tiny squares on the fourth row (g and h) have grayscale values same as the large squares of the Hermann grid. The subjects were shown a particular stimulus image and then asked whether the illusory effect was perceivable or not. For every stimulus shown, the subjects responded with either a YES or a NO depending upon whether the illusory effect was visible to them or not. Then, the gray-level values of the added squares were changed in steps starting from white to black and at each step the perceptibility of the illusory effect was recorded. The value of the gray level at which the transition from YES to NO (or from NO to YES) occurred was marked, and from the transition value, the gray-level threshold was obtained. Then, the square side length was changed, keeping the gap between the grid squares fixed, and the trial was repeated. This was done for a total of five different values of square side length. For each subject, the whole procedure described earlier was repeated 10 times, and for each of the square side lengths, the average value of the thresholds was taken as the final value of the gray-level thresholds of the tiny added squares. The reason for keeping the gap width fixed is to obtain different values for the ratio of the gap to the square-size. The aim was to measure the threshold for different ratios of the gap size to the square size, since the illusion strength depends upon this ratio.

All the stimuli were generated on a PC with a 17 in. LCD display at 1,280 × 1,024 resolution. Frame refresh rate was 60 Hz. Color depth was 24 bits (8 bits each for R, G, and B channels). The display unit was calibrated using a photometer, and the calibration data were used to linearize the display. Distance between screen and observer was fixed at 70 cm. Three subjects participated in the experiments. These include one of the authors. The stimuli were created and shown using the Psychophysics toolbox for MATLAB.

Results and Discussion

In our experiments, we have made a very small modification to the classical Hermann grid stimulus by adding a few perturbing pixels to the four corners of the grid squares. This is shown in Figure 1 along with some of the similar modifications reported in earlier studies (Geier et al., 2008; Lingelbach et al., 1985; Schiller & Carvey, 2005; Spillmann, 1994) that demonstrate a reduction/removal of the illusory effect. In Figure 1(b), the distorted (sinusoid) grid is reproduced from Geier et al. (2008). In Figure 1(d), the Schiller and Carvey (2005) modification is shown with the vertical lanes distorted into a zigzag path. Figure 1(f) displays the Spillmann (1994) modification where the relative positions of some of the grid squares have shifted. Finally, in Figure 1(h), the proposed Hermann grid modification is demonstrated where the tiny perturbing squares are placed overlapping with each grid square corner. While modifications in Figure 1(d) and (f) succeed in reducing the illusory strength, those in Figure 1(b) and (h) completely wipe out the illusory spots. The present novel modification also does not involve any change in relative position or orientation of the original grid squares.

The gray-level intensity threshold versus square side-length plots for the three subjects is illustrated in Figure 3. From these graphs, it can be clearly observed that, for a particular subject, and size of the original squares, there is an intensity threshold of the perturbing small squares, below which the illusion becomes imperceptible. Moreover, it can be seen that there is very little variation in the threshold value over a range of the grid square sizes. More subjects, however, are possibly needed to be involved to infer about the dynamics of the intensity thresholds and get an idea about the distribution of the intensity threshold for the disappearance of the illusory blobs. It may further be seen from the graph (Figure 3) that while for two of the subjects, the relatively tiny squares, with quite low threshold gray value (about 20% of the maximum in the stimulus) is enough to wipe out the Hermann grid illusion across the range of the grid squares, for the third subject the threshold was higher (about 60%), though the perturbing squares remain as tiny as in the case of the other two subjects.

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Figure 3.

Graph of Intensity threshold versus side length of Hermann grid squares depicts that the gray-level intensity threshold remains almost the same for each subject over a range of grid square length, in presence of the perturbing squares as in Figure 2.

Conclusion

From the experiments presented in this work it was demonstrated that extremely tiny changes at the corners of the original stimulus can bring about drastic effects in the Hermann grid illusion. Currently, we find no suitable explanation to this interesting phenomenon. However, there are some clues based on some previous works in the relevant domain. Bakshi and Ghosh (2012), for instance, while discussing on the abrupt disappearance of the Mach bands at step edges, and the scaling property of the Mach band widths, had concluded that linear models are not suitable candidates for explaining why Mach bands are strong at any ramp edge, but vanish abruptly as soon as the stimulus changes from a ramp to a step. A similar phenomenon is being found here; it is also an abrupt disappearance, occurring at an extremely small-scale perturbation. A common neural mechanism may underlie both these phenomena of Mach band and Hermann grating. The perturbation proposed in this work also has some similarity with that proposed by Geier et al. (2008) where the grids become wavy and the visual system tolerates some tilt/curviness as Hamburger et al. (2012) have shown. We have also proposed that in correspondence to the distortion tolerance defined by Geier et al. (2005) for Hermann grid, there exists a gray-level intensity threshold for the present perturbing stimuli in order to make the illusory blobs disappear completely. From the perspective of spatial filtering algorithms, such effects present a significant challenge. Spatial filtering algorithms apply one or more filtering operations, such as linear convolutions, on an input stimulus and interpret the output as the perceived image which also contains the illusory effect. Spatial filtering algorithms have a continuous nature, by which we mean that a small change to the input stimulus will produce a small change in its output. Thus, it will be difficult for a spatial filtering algorithm to produce an abrupt disappearance of the illusory effect just by the addition of tiny squares to the input stimulus that change only a tiny fraction of the input pixels. One possibility is that this could be a masking effect of illusion suppression (Meese & Hess, 2004; Summers & Meese, 2009). Betz, Shapley, Wichmann, and Maertens (2015), for instance, have shown that White’s illusion can be masked by adding narrowband noise, an effect which they show to be difficult to explain using spatial filtering models. However, unlike these White’s illusion-masking experiments, it is difficult to perform brightness matching experiments on the Hermann grid illusion using the tiny squares as a mask because the illusory blobs disappear as soon as one focuses on one of them. An observer who concentrates upon the blobs is immediately aware of their illusoriness, which is not the case in White’s illusion or Simultaneous Brightness Contrast illusion where the brightness shift remains visible even when the viewer focuses upon it. The Hermann grid illusion, as stated by Lothar Spillmann (1994), thus continues to remain an intriguing and captivating tool for studying the human perceptive field organization even after the lapse of another quarter of a century since then.