In the filled-space (or Oppel-Kundt) illusion, the filled part of the stimulus for most observers appears longer in comparison with the empty one. In the first two experimental series of the present study, we investigated the illusory effect as a function of continuous filling (by a shaft-line segment) of the reference spatial interval of the three-dot stimulus. It was demonstrated that for the fixed length of the reference interval, the magnitude of the illusion increases non-linearly with the shaft length. For the fixed length of the shaft, the illusion magnitude gradually decreases with the lengthening of the reference interval. In the third series, psychophysical examination of the conventional Oppel-Kundt stimulus with different number of equally spaced elements (dots) subdividing its filled part was performed. Based on the analysis of the functional dependencies established, we have proposed a simple computational model that was successfully applied to fit the experimental data obtained in the present study.

In the present study, we have tested the applicability of the computational model of centroid extraction to account for the data collected in experiments with stimuli comprising of closed two-dimensional shapes. The outlined or uniformly filled pie-shaped circular sectors (contextual distractors) were arranged according to the Brentano pattern, and three different stimulus parameters (either the radius or the central angle or the tilt angle of the sectors) were used as independent variables in different series of experiments. It was demonstrated that the model calculations adequately predict the variations of illusion magnitude shown by all the subjects for all independent variables and that there is no significant difference between the data obtained for stimuli with the outlined and uniformly filled distractors. A good correspondence between the computational and experimental data provides convincing evidence in support of the "centroid" explanation of illusions of extent of the Müller-Lyer type.

The Oppel-Kundt illusion was examined in the psychophysical experiments with the classical two-part stimuli and modified three-part figures. The modified versions comprised either one filled medial interval and two empty flanking intervals or one empty space situated in between two fillings. The illusion was measured as a function of the number of filling elements in the referential parts of the figures. The curves obtained by two modified figures and by the original two-part stimulus were quite similar in shape, but the magnitudes of the illusions differed significantly. The figure with two filled intervals yielded about twice-stronger illusory effect than the contrasting figure with a single filled and two empty intervals. The two-part stimulus showed the illusion magnitudes in the midst. Our assumption suggests the illusory effect being related particularly to overestimations of the filled interval when compared with the empty interval displayed side-to-side. The unfilled interval might not contribute to the illusion.

In the present visual psychophysical study, the Oppel-Kundt and and Müller-Lyer illusion magnitudes were measured separately (by single figures) and in combination (by two patterns superposed spatially). Data for 30 subjects revealed extensive variability both for the separate and combined illusion strength. Nevertheless, the effect of addition of the perceived length distortions was established. The combined illusions were significantly stronger than the separate ones. Dynamics of the misperceptions summation was studied by varying length of the Müller-Lyer wings in the superposed stimuli. According to the experimental data obtained, the two misperceptions in length occurred and combined into sensory response varying in dependence on the spatial parameters of the superposed stimuli and on the individual experimental accomplishment. The data supported an explanation for the origin of the filled/unfilled illusion: overestimate of a filled interval length developed due to the spatiotemporal integration along a continuous excitation path elicited by the real or imaginary contours of the filling.

According to the centroid hypothesis, the visual Müller-Lyer-type illusions in which subjects misperceive lines or gaps as longer or shorter depending on surrounding distracters result from the pooling of positioning neural signals such that the perceived object is shifted towards these distracters. However, it is uncertain if this type of pooling is a more general principal that influences perceptions in other sensory modalities, including time perception based on auditory signals. In this study, I show that by applying the principles of the centroid hypothesis, an audial time duration illusion can be constructed. The perception of two sequential time intervals, which were defined by three short tone signals, was distorted by placing distracting white noise sounds near each signal. Misperception magnitude, which peaked at 31%, changed with the time interval between the tone signals and distracters; the relationship between the target-distracter distance and the illusion strength closely paralleled with that of a Müller-Lyer-type illusion, whereby the visual objects were analogically arranged in space rather than time. These results demonstrate that even if signals and distracters are distinguishable, the neural mechanisms for estimating time duration utilize coarser sampling to preserve processing resources at the expense of high accuracy. I hypothesize that systems that are dedicated to visual length and time duration estimations are based on similar perceptual magnitude evaluation algorithms. Moreover, this signal pooling principle may be applicable to other perceptual modalities across different species.