Geometrical Optical Illusions Research Articles

Effects of surrounding frames and inner targeted line segments on event-related potentials in geometric-optical illusions.

The Sander illusion and the horizontal-vertical (H-V) illusion are both size and orientation geometric-optical illusions. The Sander geometric figures can be simply regarded as being made up of surrounding frames and inner targeted line segments. Similarly, H-V illusory geometric figures are made up of the targeted line segments. The role of surrounding frames and inner targeted line segments in the perception and cognition of geometric-optical illusions is not well understood. The time course of event-related potentials (ERP) and the ERP-based standardized low-resolution electromagnetic tomography (sLORETA) source localization were investigated in the Sander illusion and the H-V illusion, which had the same length as the targeted line segments, respectively. The P1, N1, P2, N2 and P3 components of the ERP were focused and measured. The ERP results demonstrated that the existence of surrounding frames in the Sander illusions-induced significant alterations in the P1, N1, P2, N2 and P3 components, compared with the H-V illusion without surrounding frames. In the Sander illusion, different tilted line segments and surrounding frames resulted in significant differences in the P2, N2 and P3 components. The sLORETA results also demonstrated brain activities of source localization as a function of the surrounding frames and the tilted inner line segments. These findings implicate that the perceptual and cognitive processes of the geometric-optical illusions are correlated to the surrounding frames/background, as well as the orientation/direction of inner targeted line segments in geometric figures.

The Müller-Lyer line-length task interpreted as a conflict paradigm: A chronometric study and a diffusion account

We propose to interpret tasks evoking the classical Müller-Lyer illusion as one form of a conflict paradigm involving relevant (line length) and irrelevant (arrow orientation) stimulus attributes. Eight practiced observers compared the lengths of two line-arrow combinations; the length of the lines and the orientation of their arrows was varied unpredictably across trials so as to obtain psychometric and chronometric functions for congruent and incongruent line-arrow combinations. To account for decision speed and accuracy in this parametric data set, we present a diffusion model based on two assumptions: inward (outward)-pointing arrows added to a line (i) add (subtract) a separate, task-irrelevant drift component, and (ii) they reduce (increase) the distance to the barrier associated with the response identifying this line as being longer. The model was fitted to the data of each observer separately, and accounted in considerable quantitative detail for many aspects of the data obtained, including the fact that arrow-congruent responses were most prominent in the earliest RT quartile-bin. Our model gives a specific, process-related meaning to traditional static interpretations of the Müller-Lyer illusion, and combines within a single coherent framework structural and strategic mechanisms contributing to the illusion. Its central assumptions correspond to the general interpretation of geometrical-optical illusions as a manifestation of the resolution of a perceptual conflict (Day & Smith, 1989; Westheimer, 2008).

Geometrical optical illusion via sub-Riemannian geodesics in the roto-translation group

We present a neuro-mathematical model for geometrical optical illusions (GOIs), a class of illusory phenomena that consists in a mismatch of geometrical properties of the visual stimulus and its associated percept. They take place in the visual areas V1/V2 whose functional architecture have been modeled in previous works by Citti and Sarti as a Lie group equipped with a sub-Riemannian (SR) metric. Here we extend their model proposing that the metric responsible for the cortical connectivity is modulated by the modeled neuro-physiological response of simple cells to the visual stimulus, hence providing a more biologically plausible model that takes into account a presence of visual stimulus. Illusory contours in our model are described as geodesics in the new metric. The model is confirmed by numerical simulations, where we compute the geodesics via SR-Fast Marching.

The Ebbinghaus illusion in contrast-defined and orientation-defined stimuli

In the retinal image of the natural world, edges and shapes can be defined by first-order attributes, such as luminance, and second-order attributes, such as contrast and texture. Previous studies have suggested that, in the human visual system, these attributes are initially detected separately and integrated later. Thus, comparing the strength of different geometrical optical illusions in stimuli, in which different elements are defined by the same or different attributes, is helpful to investigate at which stage the underlying mechanism of the illusion is located. We investigated whether there is a single common mechanism underlying the Ebbinghaus illusion in stimuli defined by different attributes. We used the traditional Ebbinghaus (Titchener) illusion figure: a target disk surrounded by smaller or larger inducer disks. The background and stimuli consisted of sine-wave gratings. We manipulated the luminance, contrast, and grating orientations of the target disk and inducer disks to create stimuli defined by each of these attributes. We then examined whether the illusion occurred in stimuli defined by each single attribute and in compound stimuli, in which the target and inducers were defined by different attributes. We found that the Ebbinghaus illusion occurred with the same strength in stimuli defined by all three attributes. We also found an asymmetry, such as the second-order inducers affected the first-order target less than they affected the second-order targets, but the first-order inducers affected all targets similarly. Our findings suggest that different attributes are likely to be integrated into a cue-invariant shape representation prone to the Ebbinghaus illusion. However, first-order and second-order stimuli may differently contribute to the quantitative aspect of the illusion, resulting in the asymmetric illusion strength.

A Neuromathematical Model for Geometrical Optical Illusions

Geometrical optical illusions have been object of many studies due to the possibility they offer to understand the behaviour of low-level visual processing. They consist in situations in which the perceived geometrical properties of an object differ from those of the object in the visual stimulus. Starting from the geometrical model introduced by Citti and Sarti in [3], we provide a mathematical model and a computational algorithm which allows to interpret these phenomena and to qualitatively reproduce the perceived misperception.

Johann Joseph on Geometrical-Optical Illusions: A Translation and Commentary.

The term geometrical–optical illusions was coined by Johann Joseph Oppel (1815–1894) in 1855 in order to distinguish spatial distortions of size and orientation from the broader illusions of the senses. We present a translation of Oppel’s article and a commentary on the material described in it. Oppel did much more than give a name to a class of visual spatial distortions. He examined a variety of figures and phenomena that were precursors of later, named illusions, and attempted to quantify and interpret them.

Top-down Approach to the Investigation of the Neural Basis of Geometric-optical Illusions: Understanding the Brain as a Theoretical Entity

: Geometric-optical illusions have been the subjects of research interest in a number of disciplines in science. Moreover, investigation of the patients’ reactions to illusory configurations has been somewhat instrumental in the understanding of impaired neuro-cognitive processes underlying some of the neurological and/or psychiatric disorders. Recently, neuroscientists have made some progress in understanding the neural underpinning of the geometric-optical illusions. However, a closer collaboration between psychology and neuroscience may lead to a better understanding of not only the neural basis of the illusions but the function of the brain in general. The purpose of the present analysis is to outline a sound epistemological ground for such a relationship and to demonstrate how psychological theories may potentially play a guiding role in the context of scientific discoveries in the neuroscience of illusory phenomena. In order to do so, two concepts of the “many-one” relationship between the mental and the neural states and “context-sensitivity” will be described with regard to the possible relationships between perception and brain in the context of research on illusions. In addition, the implications of the top-down strategy for research in Psychiatry will be explained and the strategy will be discussed as a path towards the unification of scientific explanations.

Geometric–optical illusions and Riemannian geometry

Geometric–optical illusions (goi) are a subclass of a vast variety of visual illusions. A special class of gois originates from the superposition of a simple geometric figure (“target”) with an array of non-intersecting curvilinear elements (“context”) that elicits a perceptual distortion of the target element. Here we specifically deal with the case of circular targets. Starting from the fact that (half)circles are geodesics in a model of hyperbolic geometry, we conceive of the deformations of the target as resulting from a context-induced perturbation of that “base” geometry. We present computational methods for predicting distorted shapes of the target in different contexts, and we report the results of a psychophysical pilot experiment with eight subjects and four contexts to test the predictions. Finally, we propose a common scheme for modeling gois associated with more general types of target curves, subsuming those studied previously.

The elusive Johann Joseph Oppel (1815-1894).

Oppel is frequently cited for coining the term 'geometrical optical illusions' in 1855, but portrayals of him have rarely been sighted. We present a portrait of him and a brief description of his contributions to the study of illusions as well as other areas of sound and vision, particularly stereoscopy, colour, and motion. Despite the citations in the context of illusions, Oppel remains an elusive figure in the history of science.

Geometrical optical illusionists.

Geometrical optical illusions were given this title by Oppel in 1855. Variants on such small distortions of visual space were illustrated thereafter, many of which bear the names of those who first described them. Some original forms of the geometrical optical illusions are shown together with 'perceptual portraits' of those who described them. These include: Roget, Chevreul, Fick, Zöllner, Poggendorff, Hering, Kundt, Delboeuf Mach, Helmholtz, Hermann, von Bezold, Müller-Lyer, Lipps, Thiéry, Wundt, Münsterberg, Ebbinghaus, Titchener, Ponzo, Luckiesh, Sander, Ehrenstein, Gregory, Heard, White, Shepard, and. Lingelbach. The illusions are grouped under the headings of orientation, size, the combination of size and orientation, and contrast. Early theories of illusions, before geometrical optical illusions were so named, are mentioned briefly.

Modeling geometric–optical illusions: A variational approach

Visual distortions of perceived lengths, angles, or forms, are generally known as “geometric–optical illusions” (goi). In the present paper we focus on a class of gois where the distortion of a straight line segment (the “target” stimulus) is induced by an array of non-intersecting curvilinear elements (“context” stimulus). Assuming local target–context interactions in a vector field representation of the context, we propose to model the perceptual distortion of the target as the solution to a minimization problem in the calculus of variations. We discuss properties of the solutions and reproduction of the respective form of the perceptual distortion for several types of contexts. Moreover, we draw a connection between the interactionist model of gois and Riemannian geometry: the context stimulus is understood as perturbing the geometry of the visual field from which the illusory distortion naturally arises. The approach is illustrated by data from a psychophysical experiment with nine subjects and six different contexts.

Relational psychophysics: Messages from Ebbinghaus’ and Wertheimer's work

In past and modern psychophysics there are several unresolved methodological and philosophical problems of human and animal perception, including the outstanding question of the relational basis of whole psychophysics. Here the main issue is discussed: if, and to what extent, there are viable bridges between the traditional “gestalt” oriented approaches and the modern perceptual-cognitive perspectives in psychophysics. Thereby the key concept of psychological “frame of reference” is presented by pointing to Hermann Ebbinghaus’ geometric-optical illusions, on the one hand, and Max Wertheimer's treatment of the traditional transposition phenomenon, on the other hand. A much-needed theoretical reorientation of future research may help to overcome the philosophical narrowness of present-day human and comparative psychophysics.

Isolusions: Evidence for strong geometric-optical illusions under isoluminance

It has long been known that color and luminance are processed together by the same parvo retinal ganglion cells (De Valois & De Valois, 1988). Recently it has been shown that in the primary visual cortex cells respond to oriented chromatic contrast (Johnson et al., Nature Neurosci. 4, 2001), and that the vast majority of color-selective neurons in V1 and V2 are also selective for orientation (Friedman et al., J. Physiol., 548, 2003). On the contrary, a largely segregated processing of unoriented chromatic contrast and oriented luminance contrast has been hypothesized. This idea was bolstered by observations that geometrical-optical illusions vanish under isoluminance (Livingstone & Hubel, J. Neurosci. 7, 1987). Here we examine the strength of various geometric-optical illusions under isoluminance (Delboeuf, Ebbinghaus, Hering, Judd, Mueller-Lyer, Poggendorff, Ponzo, Vertical, Zoellner.) Subjects interactively manipulated computer-generated line drawings to counteract the illusion. In all cases, illusions presented under isoluminance (both for colors drawn from the cardinal L-M or S-(L+M) directions) were as high as the luminance versions (both for high and low contrast). In two control experiments we tested whether subjective deviations from the nominal isoluminance or chromatic aberrations may have caused the observed strong illusions under isoluminance. Again, no significant differences between the illusions under luminance or isoluminance occurred. Our findings show that geometric-optical illusions under isoluminance are as strong as when presented in luminance, supporting the joint processing of oriented color and luminance contrast in early visual areas. Furthermore, the results show that our conscious percept is deceived similarly for both isoluminance and luminance conditions, suggesting that the joint processing for chromatic and luminance defined contours extends well beyond early visual areas.

Application of Riesz transforms to the isotropic AM-PM decomposition of geometrical-optical illusion images

The existence of a special second-order mechanism in the human visual system, able to demodulate the envelope of visual stimuli, suggests that spatial information contained in the image envelope may be perceptually relevant. The Riesz transform, a natural isotropic extension of the Hilbert transform to multidimensional signals, was used here to demodulate band-pass filtered images of well-known visual illusions of length, size, direction, and shape. We show that the local amplitude of the monogenic signal or envelope of each illusion image conveys second-order information related to image holistic spatial structure, whereas the local phase component conveys information about the spatial features. Further low-pass filtering of the illusion image envelopes creates physical distortions that correspond to the subjective distortions perceived in the illusory images. Therefore the envelope seems to be the image component that physically carries the spatial information about these illusions. This result contradicts the popular belief that the relevant spatial information to perceive geometrical-optical illusions is conveyed only by the lower spatial frequencies present in their Fourier spectrum.

Paradoxical form of filled/empty optical illusion.

The filled/empty illusion (Oppel-Kundt) is one of the oldest geometrical-optical illusions, but the determinants of the illusion are not yet sufficiently understood. We studied magnitude of the illusory effect as a function of the height of vertical strokes subdividing a spatial extension of fixed length, using the psychophysical standard-variable matching paradigm. For vertical strokes shorter than, or of the same height as strokes delimiting the standard, the length was over-reproduced consistently with earlier studies of the illusion. However, for vertical strokes three times longer than the delimiters, the illusory effect paradoxically decreased, and attained negative values in two of six subjects. The magnitude of the effect thus depends on the patterning of the space between the delimiters, not merely on the number of subdividing elements.

Size and Direction of Distortion in Geometric-Optical Illusions: Conciliation between the Müller-Lyer and Titchener Configurations

Over the past few decades, different theories have been advanced to explain geometric-optical illusions based on various perceptual processes such as assimilation and/or contrast. Consistent with the contradictory effects of assimilation and contrast, Pressey's assimilation theory provided an explanation for the Müller-Lyer illusion, but failed to account for the Titchener (Ebbinghaus) illusion. A model that explains both Müller-Lyer and Titchener illusions according to a common underlying process may outline a unified explanation for a variety of geometric-optical illusions. In order to develop such a model, the concept of empty space is introduced as an area of the illusory figure that is not filled by line drawings. It was predicted that the magnitude of illusion would increase with the area of the empty space around the illusory figures. The effect of empty space on the magnitude of perceptual distortion was measured in Müller-Lyer figures, with outward arrowheads of different length. The results indicated an overestimation of the target stimulus in all of the figures. Nevertheless, consistent with the prediction of the present model, the horizontal line in the Müller-Lyer figure with the longest arrowheads appeared shorter than that with the shortest arrowheads, although the size contrast of these figures was the same. According to the analysis proposed in the present study, the area of empty space not only affects the magnitude of illusion but also serves as a contextual cue for the perceptual system to determine the direction of illusion (orientation). The functional relationships between the size contrast and empty space provide a common explanation for the Müller-Lyer, Titchener, and a variety of other geometric-optical illusions.

Illusions in the spatial sense of the eye: Geometrical–optical illusions and the neural representation of space

Differences between the geometrical properties of simple configurations and their visual percept are called geometrical–optical illusions. They can be differentiated from illusions in the brightness or color domains, from ambiguous figures and impossible objects, from trompe l’oeil and perspective drawing with perfectly valid views, and from illusory contours. They were discovered independently by several scientists in a short time span in the 1850’s. The clear distinction between object and visual space that they imply allows the question to be raised whether the transformation between the two spaces can be productively investigated in terms of differential geometry and metrical properties. Perceptual insight and psychophysical research prepares the ground for investigation of the neural representation of space but, because visual attributes are processed separately in parallel, one looks in vain for a neural map that is isomorphic with object space or even with individual forms it contains. Geometrical–optical illusions help reveal parsing rules for sensory signals by showing how conflicts are resolved when there is mismatch in the output of the processing modules for various primitives as a perceptual pattern’s unitary structure is assembled. They point to a hierarchical ordering of spatial primitives: cardinal directions and explicit contours predominate over oblique orientation and implicit contours (Poggendorff illusion); rectilinearity yields to continuity (Hering illusion), point position and line length to contour orientation (Ponzo). Hence the geometrical–optical illusions show promise as analytical tools in unraveling neural processing in vision.

Emmert's Law and the moon illusion

A cognitive account is offered of puzzling, though well known phenomena, including increased size of afterimages with greater distance (Emmert's Law) and increased size of the moon near the horizon (the Moon Illusion). Various classical distortion illusions are explained by Size Scaling when inappropriate to distance, 'flipping' depth ambiguity being used to separate botton-up and top-down visual scaling. Helmholtz's general Principle is discussed with simpler wording - that retinal images are attributed to objects - for object recognition and spatial vision.

Geometric-optical illusions at isoluminance

The idea of a largely segregated processing of color and form was initially supported by observations that geometric-optical illusions vanish under isoluminance. However, this finding is inconsistent with some psychophysical studies and also with physiological evidence showing that color and luminance are processed together by largely overlapping sets of neurons in the LGN, in V1, and in extrastriate areas. Here we examined the strength of nine geometric-optical illusions under isoluminance (Delboeuf, Ebbinghaus, Hering, Judd, Müller-Lyer, Poggendorff, Ponzo, Vertical, Zöllner). Subjects interactively manipulated computer-generated line drawings to counteract the illusory effect. In all cases, illusions presented under isoluminance (both for colors drawn from the cardinal L−M or S−(L+M) directions of DKL color space) were as effective as the luminance versions (both for high and low contrast). The magnitudes of the illusion effects were highly correlated across subjects for the different conditions. In two additional experiments we determined that the strong illusions observed under isoluminance were not due to individual deviations from the photometric point of isoluminance or due to chromatic aberrations. Our findings show that our conscious percept is affected similarly for both isoluminance and luminance conditions, suggesting that the joint processing for chromatic and luminance defined contours may extend well beyond early visual areas.

Uncertainty in visual processes predicts geometrical optical illusions

It is proposed in this paper that many geometrical optical illusions, as well as illusory patterns due to motion signals in line drawings, are due to the statistics of visual computations. The interpretation of image patterns is preceded by a step where image features such as lines, intersections of lines, or local image movement must be derived. However, there are many sources of noise or uncertainty in the formation and processing of images, and they cause problems in the estimation of these features; in particular, they cause bias. As a result, the locations of features are perceived erroneously and the appearance of the patterns is altered. The bias occurs with any visual processing of line features; under average conditions it is not large enough to be noticeable, but illusory patterns are such that the bias is highly pronounced. Thus, the broader message of this paper is that there is a general uncertainty principle which governs the workings of vision systems, and optical illusions are an artifact of this principle.