A Neuro-Mathematical Model for Geometrical Optical Illusions

J Math Imaging Vis manuscript No. (will be inserted by the editor)

A neuro-mathematical model for geometrical optical illusions

B. Franceschiello · A. Sarti · G. Citti

Received: date / Accepted: date

Abstract Geometrical optical illusions have been ob- late to misjudgements of geometrical properties of con- ject of many studies due to the possibility they offer to tours and they show up equally for dark configura- understand the behaviour of low-level visual process- tions on a bright background and viceversa. For the ing. They consist in situations in which the perceived interested reader, a historical survey of the discovery geometrical properties of an object differ from those of of geometrical-optical illusions is included in Appendix the object in the visual stimulus. Starting from the ge- I of [41]. Our intention here is not to make a classi- ometrical model introduced by Citti and Sarti in [3], we fication of these phenomena, which is already widely provide a mathematical model and a computational al- present in literature (Coren e Girgus, 1978, [5]; Robin- gorithm which allows to interpret these phenomena and son, 1998, [31]; Wade, 1982, [38]). The aim of this paper to qualitatively reproduce the perceived misperception. is to propose a mathematical model for GOIs based on the functional architecture of low level visual cortex Keywords Geometrical optical illusions · Neuro- (V1/V2). This neuro-mathematical model will allow us mathematical model · Visual cortex · Infinitesimal to interpret at a neural level the origin of GOIs and to strain theory · Hering illusion reproduce the arised percept for this class of phenomena. The main idea is to adopt the model of the functional geometry of V1 provided in [3] and to consider 1 Introduction that the image stimulus will modulate the connectivity. When projected onto the visual space, the modulated Geometrical-optical illusions (GOIs) have been discov- connectivity gives rise to a Riemannian metric which ered in the XIX century by German psychologists (Op- is at the origin of the visual space deformation. The pel 1854 [29], Hering, 1878, [15]) and have been de- displacement vector field at every point of the stimu- fined as situations in which there is an awareness of a lus is mathematically computed by solving a Poisson mismatch of geometrical properties between an item in problem and the perceived image is finally reproduced. object space and its associated percept [41]. The dis- The considered phenomena consist, as shown in figure arXiv:1611.08844v1 [cs.CV] 27 Nov 2016 tinguishing feature of these illusions is that they re- 1, in straight lines over different backgrounds (radial B. Franceschiello lines, concentric circles, etc). The interaction betwen Centre d’analyse et mathématiquessociales, CNRS - EHESS target and context either induces an effect of curva- Paris, France ture of the straight lines (fig. 1a, 1b, 1c), eliminates the E-mail: [email protected] bending effect (fig. 1d), or induces an effect of unpar- A. Sarti allelism (fig. 1e). The paper is organised as follows: in Centre d’analyse et mathématiquessociales, CNRS - EHESS section 2 we review the state of the art concerning the Paris, France E-mail: [email protected] previous mathematical models proposed. In section 3 we will briefly recall the functional achitecture of the G. Citti Dipartimento di Matematica, Universitàdi Bologna visual cortex and the cortical based model introduced Bologna, Italy by Citti and Sarti in [3]. In section 4 we will introduce E-mail: [email protected] the neuro-mathematical model proposed for GOIs, tak- 2 B. Franceschiello et al. ing into account the modulation of the functional ar- of angular expansion, which is the tendence to perceive chitecture induced by the stimulus. In 5 the numerical under certain conditions acute angles as larger and ob- implementation of the mathematical model will be ex- tuse ones as smaller) modeling the generated perceived plained and applied to a number of examples. Results curves as orbits of a Lie transformation group acting are finally discussed. on the plane. The proposed model allows to classify the perceptual invariance of the considered phenomena in terms of Lie Derivatives, and to predict the slope. An- 2 Geometrical optical illusions other model mathematically equivalent to the one proposed by Hoffman has been proposed by Smith, [35], 2.1 Role of GOIs who stated that the apparent curve of geometrical optical illusions of angle can be modeled by a first-order In psychology the distal stimulus is defined as the light differential equation depending on a single parameter. reflected off a physical object in the external world; when By computing this value an apparent curve can be cor- we look at an image (distal stimulus) we cannot actu- rected and plotted in a way that make the illusion be- ally experience the image physically with vision, we can ing not perceived anymore (see for example fig. 8 of only experience it in our mind as proximal stimulus [23], [35]). This permits to introduce a quantitative analysis [13]. Geometrical optical illusions arise when the distal of the perceived distortion. Ehm and Wackerman in [9], stimulus and its percept differ in a perceivable way. As started from the assumption that GOIs depend on the explained by Westheimer in [41], we can conveniently context of the image which plays an active role in al- divide illusions into those in which spatial deformations tering components of the figure. On this basis they pro- are a consequence of the exigencies of the processing vided a variational approach computing the deformed in the domain of brightness and the true geometrical- lines as minima of a functional depending on length of optical illusions, which are misperceptions of geomet- the curve and the deflection from orthogonality along rical properties of contours in simple figures. Some of the curve. This last request is in accordance to the phe- the most famous geometric illusions of this last type are nomenological property of regression to right angle. One shown in figure 1. The importance of this study lies in of the problems pointed out by the authors is that the the possibility, through the analysis of these phenom- approach doesn’t take into account the underlying neu- ena combined with physiological recordings, to help to rophysiological mechanisms. An entire branch for mod- guide neuroscientific research (Eagleman, [8]) in under- eling neural activity, the Bayesian framework, had its standing the role of lateral inhibition, feedback mech- basis in Helmholtzs theory [14]: our percepts are our anisms between different layers of the visual process best guess as to what is in the world, given both sensory and to lead new experiments and hypothesis on recep- data and prior experience. The described idea of uncon- tive fields of V1 and V2. Many studies, which relies scious inference is at the basis of the Bayesian statisti- on neuro-physiological and imaging data, show the evi- cal decision theory, a principled method for determining dence that neurons in at least two visual areas, V1 and optimal performance in a given perceptual task ([12]). V2, carry signals related to illusory contours, and that These methods consists in attributing a probability to signals in V2 are more robust than in V1 ([37], [28], each possible true state of the environment given the reviews [8], [27]). A more recent study on the tilt il- stimulus on the retina and then to establish the way lusion, in which the perceived orientation of a grating prior experience influences the final guess, the built differs from its physical orientation when surrounded by proximal stimulus (see [21] for examples of Bayesian a tilted context, measured the activated connectivity in models in perception). An application of this theory to and between areas of early visual cortices ([36]). These motion illusions has been provided by Weiss et al in findings suggest that for GOIs these areas may be in- [40], and a review in [12]. Fermüllerand Malm in [11] volved as well. Neurophysiology can help to provide a attributed the perception of geometric optical illusions physical basis to phenomenological experience of GOIs to the statistics of visual computations. Noise (uncer- opening to the possibility of mathematically modeling tainty of measurements) is the reason why systematic them and to integrate subjective and objective experi- errors occur in the estimation of the features (intensity ences. of the image points, of positions of points and orientations of edge elements) and illusions arise as results of errors due to quantization. Walker ([39]) tried to com- 2.2 Mathematical models proposed in literature bine neural theory of receptive field excitation together with mathematical tools to provide an equation able to The pioneering work of Hoffman [16] dealt with illu- determine the disparity between the apparent line of an sions of angle (i.e. the ones involving the phenomenon A neuro-mathematical model for geometrical optical illusions 3

(a) Hering illusion (b) Wundt Illusion (c) Square in Ehrenstein context

(d) Wundt-Hering illusion (e) Zollner illusion

Fig. 1: a Hering illusion: the two vertical lines are straight and parallel, but since they are presented in front of a radial background the lines appear as if they were bowed outwards. b Wundt-Illusion: the two horizontal lines are both straight, but they look as if they were bowed inwards. c Square shape over Ehrenstein context: the context of concentric circles bends the edges of the square toward the center of the image. d Wundt-Hering illusions merged together: the horizontal lines are straight and parallel and the presence of inducers which bow them outwards and inwards at the same time inhibits the bending eﬀect. e Zollner illusion: a pattern of oblique inducers surrounding parallel lines creates the illusion they are unparallel

illusion and its corresponding actual line, in order to re- when a stimulus is applied to a point ξ = (ξ1, ξ2) of the produce the perceptual errors that occur in GOIs (the retinal plane. ones involving straight lines). In our model we aim to combine psycho-physical evidence and neurophysiolog- 3.1.1 The set of simple cells receptive profiles ical findings, in order to provide a neuro-mathematical model able to interpret and simulate GOIs. Simple cells of visual cortices V1 and V2 are sensitive to position and orientation of the contrast gradient of an image. Their properties have been experimentally 3 The classical neuromathematical model of described by De Angelis in [7], see figure 2. From the V1/V2 neurophysiological point of view the orientation selectivity, the spatial and temporal frequency of cells in V2 3.1 Neurogeometry of the primary visual cortex differs little from the one in V1 ([24]). Receptive fields in V2 are larger from those in V1 ( [20], [24]). Considering The visual process is the result of several retinic and a basic geometric model, the set of simple cells RPs can cortical mechanisms which act on the visual signal. The be obtained via translations of vector x = (x1, x2) and retina is the first part of the visual system responsible rotation of angle θ from a unique mother profile ψ0(ξ). for the trasmission of the signal, which passes through Daugman [6], Jones and Palmer [18] showed that Gabor the Lateral Geniculate Nucleus, where a pre-processing filters were a good approximation for receptive profiles is performed and arrives in the visual cortex, where it is of simple cells in the primary visual cortices V1 and further processed. The receptive field (RF) of a cortical V2. Another approach is to model receptive profiles as neuron is the portion of the retina which the neuron Gaussian derivatives, as introduced by Young in [43] reacts to, and the receptive profile (RP) ψ(ξ) is the and Koenderink in [22], but for our purposes the two function that models the activation of a cortical neuron approaches are equivalent. A good expression for the 4 B. Franceschiello et al.

R2 → R+ activates the retinal layer of photoreceptors, the neurons whose RFs intersect M spike and their spike frequencies O(x1, x2, θ) can be modeled (taking into account just linear contributions) as the integral of the signal I(x1, x2) with the set of Gabor ﬁlters. The expression for this output is: Z

O(x1, x2, θ) = I(ξ1, ξ2) ψ(x1,x2,θ) (ξ1, ξ2) dξ1dξ2. (3) M Fig. 3: In each image: (top) even part of Gabor filters In the right hand side of the equation the integral of (real part), (bottom) odd one (imaginary part). Corre- the signal with the real and imaginary part of the Ga- sponding orientation from left to right: θ = 0, θ = π/6, bor filter is expressed. The two families of cells have θ = 2π/3, θ = 5π/6, with σ = 4.48 pixels different shapes, hence they detect different features. In particular odd cells will be responsible for boundary mother Gabor filter is: detection.

ξ2 −(ξ2+ 2 ) ¯ 1 1 4 2ibξ2 ψ0(ξ) = ψ0(ξ1, ξ2) = e 2σ2 e σ , (1) 3.1.3 Hypercolumnar structure 4πσ2 where ¯b = 0.56 is the ratio between σ and the spatial The term functional architecture refers to the organ- wavelength of the cosine factor. isation of cells in the primary visual cortex in structures. The hypercolumnar structure, discovered by the neuro-physiologists Hubel and Wiesel in the 60s ([17]), organizes the cells of V1/V2 in columns (called hyper- colums) covering a small part of the visual ﬁeld M ⊂ R2 and corresponding to parameters such as orientation, scale, direction of movement, color, for a ﬁxed retinal position (x1, x2).

Fig. 2: In vivo registered odd receptive ﬁeld (left, from (De Angelis et al., 1995) [7]) and a schematic representation of it as a Gabor ﬁlter (right), see (1)

Translations and rotations can be expressed as: x1 cos θ − sin θ ξ1 A(x1,x2,θ)(ξ) = + . (2) x2 sin θ cos θ ξ2 Hence a general RP can be expressed as:

−1 ψ (ξ1, ξ2) = ψ0(A (ξ1, ξ2)). (x1,x2,θ) (x1,x2,θ) A set of RPs generated with equation 2 is shown in ﬁgure 3.

Fig. 4: Top: representation of hypercolumnar structure, 3.1.2 Output of receptive proﬁles for the orientation parameter, where L and R represent 2 the ocular dominance columns (Petitot [30]). Bottom: The retinal plane is identiﬁed with the R -plane, whose for each position of the retina (x1, x2) we have the set local coordinates will be denoted with x = (x1, x2). of all possible orientations When a visual stimulus I of intensity I(x1, x2): M ⊂ A neuro-mathematical model for geometrical optical illusions 5

In our framework over each retinal point we will with i, j = 1, 2, 3. Cortical curves in V1 will be a linear consider a whole hypercolumn of cells, each one sensi- combination of vector fields X1 and X2, the generators tive to a specific instance of orientation. Hence for each of the 2-dimensional horizontal space, while they will 2 position (x1, x2) of the retina M ⊂ R we associate a have a vanishing component along X3 = [X1,X2]. The whole set of filters functional architectures built in R2 ×S1 correspond to the neural connectivity measured by Angelucci et al. RP = {ψ : θ ∈ S 1}. (4) (x1,x2) (x1,x2,θ) in [1] and Bosking et al. in [2]. For a qualitatively and This expression associates to each point of the proximal quantitative comparison between the kernels and the stimulus in R2 all possible feature orientations into the connectivity patterns see Favali et al. in [10]. In our space of features S 1, and defines a fiber over each point work a local formulation of the kernel presented in [10]

1 will be used. Furthermore, it has been shown by San- {θ ∈ S }. guinetti et al. in [32] that the geometry of fuctional ar- In this way the hypercolumnar structure is described in chitecture formally introduced in 3 is naturally encoded terms of diﬀerential geometry, but we need to explain in the statistics of natural images. Hence these geomet- how the orientation selectivity is performed by the cor- rical structures are compatible with Bayesian learning tical areas in the space of feature S 1 ([3]). methods.

3.1.4 Cortical connectivity 4 The neuro-mathematical model for GOIs Physiologically the orientation selectivity is the action of short range connections between simple cells belong- 4.1 Output of Simple Cells and connectivity metric ing to the same hypercolumn to select the most proba- ble response from the energy of receptive profiles. Hori- We consider simple cells at fixed value of σ depending zontal connections are long ranged and connect cells of on position and orientation. For fixed value of (x1, x2), ij approximately the same orientation. Since the connec- we restrict the connectivity tensor (g )i,j=1,2,3 to the tivity between cells is defined on the tangent bundle, we R2 plane, subset of the tangent plane to R2×S1 at the define now the generator of this space. The change of point (x1, x2, θ), and obtain the tensor variable defined through A in (2) acts on the basis for 2 the tangent bundle ( ∂ , ∂ ) giving as frame in polar cos θ sin θ cos θ ∂x1 ∂x2 2 . coordinates: sin θ cos θ sin θ ∂ ∂ ∂ ∂ For every value of θ this tensor has only one non zero X1 = cos θ +sin θ ,X3 = − sin θ +cos θ . ∂x1 ∂x2 ∂x1 ∂x2 eigenvalue. The corresponding eigenvector has direction θ. We will assign to the norm of the output the usual As presented in [3], the whole space of features (x1, x2, θ) is described in terms of a 3-dimensional fiber bundle, meaning of energy whose generators are X1, X3 for the base and E(x1, x2, θ) = kO(x1, x2, θ)k, ∂ X = , 2 ∂θ where the output is defined in (3) and is evaluated at for the fiber. These vector fields generate the tangent the fixed value of σ. We will discuss in section 5 the bundle of R2 ×S1. choice of σ for our experiments. Each point of the hy- Since horizontal connectivity is very anysotropic, percolumn is weighted by the energy of simple cells nor- the three generators are weighted by a strongly anysotropic malized over the whole set of hypercolumn responses: metric. We introduce now the sub-Riemannian metric with whom Citti and Sarti in [3] proposed to endow E(x , x , θ) 2 1 1 2 the ×S group to model the long range connectiv- π . (6) R R E(x , x , θ)dθ ity of the primary visual cortex V1. Starting from the 0 1 2 vector fields X1, X2 and X3 we define a metric gij for The normalization of the output expresses the proba- which the inverse (responsible for the connectivity in bility that a specific cell sensitive to θ within the hy- the cortex) is: percolumn over (x1, x2) is selected. The mechanism of intracortical selection attributing a probability to each  cos2 θ sin θ cos θ 0  possible orientation (state) given the initial stimulus is ij 2 g (x1, x2, θ) =  sin θ cos θ sin θ 0  , (5) connected to the long-range activity: simple cells be- 0 0 1 longing to different hypercolumns in a neighbourhood 6 B. Franceschiello et al. of a point (x1, x2) sensitive to the same orientation will have a high probability. The connectivity tensor restricted to the R2 plane and modulated by the output of simple cells will become: E(x , x , θ) cos2 θ sin θ cos θ 1 2 . (7) R π sin θ cos θ sin2 θ 0 E(x1, x2, θ)dθ This last expression corresponds to a connectivity polarized by the normalized energy of simple cells shown in (6) at points (x, y, θ). The overall cometric (inverse (a) Hering illusion, distal stimu- of the metric tensor) arising from the action within the lus hypercolumn over each retinal point (x1, x2) is obtained summing up along θ the previous modulated metric in (7) and will have the following expression:

2 R π cos θ sin θ cos θ 0 E(x1, x2, θ) 2 dθ −1 −1 sin θ cos θ sin θ p (x1, x2) = γ R π , 0 E(x1, x2, θ)dθ (8)

where γ−1 is a constant factor. This tensor will have principal eigenvector in the direction θ¯, the orientation corresponding to the maximum energy within the hy- (b) Tensor representation percolumn. A visualization of p−1 is given in figure 5. Hence this process describes the selection at every point (x1, x2) of the most likely direction of propagation of the connectivity, expressed by the values attained by (c) Detail the energy. Fig. 5: a Proximal stimulus (Hering illusion). b Rep- resentation of p−1 (blue). Principal and second eigen- 4.2 From metric tensor field to image distortion vectors correspond to first and second semi-axes of the ellipses. Lengths of the semi-axes is given by the magni- In the previous section we described the response of the tude of the corresponding eigenvalues. Principal eigen- cortex in the presence of a visual stimulus. vectors of ellipses are oriented along the maximum ac- ¯ (1) The distal stimulus is projected onto the cortex by tivity registred at θ over each point (x1, x2), marked in means of activity of simple cells. cyan vector. c Here we show a detail of the tensor field representation: we notice that along parts of the stim- (2) The joint action of the short and long range con- ulus strongly oriented, ellipses are elongated. As far as −1 nectivity induces a Riemannian tensor p on the we move further from the level-lines, ellipses lost their 2 R retinal plane. elongated form and become rounded, since the stimulus Even though it is not completely clear in which corti- does not respond to a preferred orientation anymore cal area the perceived image is reconstructed, from a phenomenological point of view it is evident that our visual system recostructs the perceived image. a distorted image which models the proximal one. In Hence a third mechanism takes place, able to con- this way we justify the mechanism at the basis of geo- struct the perceived stimulus from the cortical activa- metrical optical illusions. tion. With this mechanism the image distortion which induces the metric tensor p (inverse of p−1) is esti- In this approach we consider the medium to be sub- mated. Here we propose to apply infinitesimal strain jected only to small displacements, i.e. the geometry theory and to identify its inverse p with the strain ten- of the medium and its constitutive properties at each sor to compute the deformation. Once the displacement point of the space are assumed to be unchanged by de- vector field is applied to the distal stimulus, we obtain formation. A neuro-mathematical model for geometrical optical illusions 7

(a) Hering with red lines superimposed (b) Displacement vector ﬁelds (c) Proximal stimulus

Fig. 6: a Here we superimpose two red lines to the original distal stimulus (Hering illusion) to remark that vertical lines present in the stimulus are straight. b Representation of the displacement ﬁeld {u¯(x , x )} 2 . 1 2 (x1,x2)∈R c Perceived deformation

4.2.1 Strain tensor - displacement vector field We can now express the right Cauchy-Green tensor in terms of displacement: The mathematical question is how to reconstruct the p = p (x) = (∇Φ)T (∇Φ) = (∇u¯ + Id)T (∇u¯ + Id) displacement starting from the strain tensor p. We think ij at the deformation induced by a geometrical optical il- = (∇u¯)T (∇u¯) + (∇u¯) + (∇u¯)T + Id. 2 lusion as a map between the R plane equipped with the The concept of strain is used to evaluate how much 2 metric p and the R plane with the Euclidean metric a given displacement differs locally from a rigid body Id: displacement. One of the strain tensors for large defor- 2 2 Φ :(R , p) → (R , Id). mations is the so called Green-Lagrangian strain tensor or Green-Saint Venant strain tensor defined as: From the mathematical point of view this means that 1 1 we look for the change of variable which induces the E = (p − Id) = ((∇Φ)T (∇Φ) − Id), new metric, i.e. 2 2 which can be written in terms of the displacement as ∂Φk ∂Φl before: Id = p (x), kl ij 1 ∂xi ∂xj E(¯u) = ((∇u¯) + (∇u¯)T + (∇u¯)T (∇u¯)). 2 2 with x = (x1, x2) ∈ R (see Jost [19]), obtaining the For infinitesimal deformations of a continuum body, in relation: which the displacement gradient is small (k∇u¯k 1), it is possible to perform a geometric linearization of T p(x) = (∇Φ) (∇Φ). (9) strain tensors introduced before, in which the non-linear second order terms are neglected. The linearized Green- −1 −1 Let us notice p corresponds to Φ , the map rep- Saint Venant tensor has the following form: resenting the process which builds the modulated con- 1 nectivity we discussed before. In strain theory p sat- E(¯u) ≈ (¯u) = ((∇u¯) + (∇u¯)T ), (10) 2 isfying (9) is called right Cauchy-Green tensor associated to the deformation Φ, which from the physi- which is used in the study of linearized elasticity, i.e. 2 the study of such situations in which the displacements cal point of view is a map Φ : Ω¯ → R associat- ing the points of the closure of a bounded open set of the material particles of a body are assumed to be 2 2 small (as stated at the beginning, infinitesimal strain Ω ⊂ R (initial configuration of a body) to Φ(Ω) ⊂ R (deformed configuration). For references see [25], [26]. theory.) It is possible to introduce the displacement as a map Here we give the expression in components of (¯u): 2 u¯(x1, x2) = Φ(x1, x2) − (x1, x2), where (x1, x2) ∈ R . It ! follows ∂u1 1 ∂u1 + ∂u2 (¯u) = ∂x1 2 ∂x2 ∂x1 , (11) ij 1 ∂u2 + ∂u1 ∂u2 ∇u¯ = ∇Φ − Id. 2 ∂x1 ∂x2 ∂x2 8 B. Franceschiello et al. whereu ¯ = (u1, u2). Expressing ij in terms of the metric (pij)i,j with whom the initial configuration of the considered body was equipped we obtain: 1 E = ((p ) − Id) ≈ (¯u), (12) 2 ij ij ij and in its matrix form: ! p p 1 0 ∂u1 1 ∂u1 + ∂u2 11 12 ∂x1 2 ∂x2 ∂x1 − = 1 ∂u2 ∂u1 ∂u2 . p21 p22 0 1 + 2 ∂x1 ∂x2 ∂x2 (a) (b) (13)

4.2.2 Poisson problems - displacement

Starting from (13) we obtain a system of equations with this form:

 ∂u1 p11 − 1 =  ∂x1 ∂u2 p22 − 1 = (14) ∂x2  1 ∂ ∂  p12 = p21 = 2 ( ∂x u1 + ∂x u2) 2 1 (c) (d) Differentiating, substituting and imposing Neumann boundary conditions to system (14) we end up with the fol- Fig. 7: a We superimpose two red vertical lines to the lowing differential system: Hering illusion, represented in figure 1a, in order to remark that vertical lines present in the stimulus are  ∂ ∂ ∂ ∆u1 = p11 + 2 p12 − p22 in M −1 ∂x1 ∂x2 ∂x1 straight. b Representation of p , projection onto the  ∂ ∂ ∂  ∆u2 = p22 + 2 p12 − p11  ∂x2 ∂x1 ∂x2 retinal plane of the polarized connectivity in 7. The (15) first eigenvalue has direction tangent to the level lines  ∂  ∂n u1 = 0 in ∂M of the distal stimulus. In blue the tensor field, in cyan  ∂ ∂n u2 = 0 the eigenvector related to the first eigenvalue. c Com- Let us explicitly note that tensor p is obtained after puted displacement fieldu ¯ : R2 → R2. d Displacement convolution of Gabor filters, so that it is differentiable, applied to the image. In black we represent the proxi- allowing to write the system. Hence we solve (15), re- mal stimulus as displaced points of the distal stimulus: covering the displacement fieldu ¯(x1, x2). (x1, x2) +u ¯(x1, x2). In red we give two straight lines as reference, in order to better clarify the curvature of the target lines 5 Numerical implementation and results

The inverse of tensor expressed in formula (8) is com- 5.1 Hering illusion puted discretizing θ as a vector of 32 values equally spaced in the interval [0, π]. The scale parameter σ The Hering illusion, introduced by Hering, a German varies in dependence of the image resolution and is set physiologist, in 1861 [15] is presented in figure 1a. In in concordance with the stimulus processed. It is taken this illusion two vertical straight lines are presented quite large in all examples in such a way to obtain a in front of radial background, so that the lines appear smooth tensor field covering all points of the image. as if they were bowed outwards. In order to help the This is in accordance with the hypothesis previously reader, in figure 7a we superpose to the initial illusion introduced that mechanisms in V2, where the receptive two red vertical lines, which indeed coincide with the field of simple cells is larger than in V1, play a role ones present in the stimulus. As described in the previ- in such phenomena. The constant γ has been chosen ous sections, we first convolve the distal stimulus with for all the examples as γ = 2 · 10−2. The differential the entire bank of Gabor filters: we take 32 orientations problem in (15) is approximated with a central finite selected in [0, π), σ = 6.72 pixels. Following the process, difference scheme and it is solved with a classical PDE we compute p−1 using equation (8), we solve equation linear solver. We now start discussing all results ob- (15) obtaining the perceived displacementu ¯ : R2 → R2. tained through the presented algorithm. Once it is applied to the initial stimulus, the proxi- A neuro-mathematical model for geometrical optical illusions 9 mal stimulus is recovered. The result of computation is pre/post processing. In figure 10 details of the distances shown in figure 7. The distorted image folds the parallel between the bent curves and the original straight lines lines (in black) against the straight lines (in red) of the are shown. original stimulus (figure 7d).

(a) (b)

Fig. 9: a Here we superimpose two red lines to the Wundt illusion, presented in figure 1b, in order to clarify that the horizontal lines present in the image are indeed straight. b Representation of p−1, projection onto the retinal plane of the polarized connectivity in 7. The first eigenvalue has direction tangent to the level lines of the distal stimulus. In blue the tensor field, in cyan the eigenvector related to the first eigenvalue. c Computed Fig. 8: Details of the perceived distortion in the com- displacement fieldu ¯. d Displacement applied to the im- puted proximal stimulus age. In black we represent the proximal stimulus as displaced points of the distal stimulus: (x1, x2)+u ¯(x1, x2). In red we give two straight lines as reference, in order to put in evidence the curvature of the target lines 5.2 Wundt Illusion

A variant of the Hering illusion, introduced by Wundt in the 19th century, [42] is presented in figure 1b. In this illusion two straight horizonal lines look as if they were bowed inwards, due to the distortion induced by the crooked lines on the background. For the convolution of the distal stimulus with Gabor filters we select 32 orientations in [0, π), σ = 11.2 pixels. Then we apply the previous model, and obtain the result presented in figure 9. Computed vector fields are concentrated in the central part of the image and point toward the center. They indicate the direction of the displacement, which bends the parallel lines inwards. In figure 9d the proximal stimulus is computed through the expression: (x1, x2)+u ¯(x1, x2). In black we indicate displaced dots of the initial image: the straight lines of the distal stimulus are bent by the described mechanism (black). In red we put the straight lines of the original distal Fig. 10: Details of the perceived distortion in the com- stimulus. This provide a comparison between the lines puted proximal stimulus 10 B. Franceschiello et al.

5.3 Square shape over Ehrenstein context nearer the center than in the reference Hering illusion. For coherence with the Hering example, orientations This illusion, introduced by Ehm and Wackermann in selected are 32 in [0, π) and σ = 6.72 pixels. All other [9], consists in presenting a square over a background of parameters are fixed during these three experiments. concentric circles, figure 1c. This context, the same we find in Ehrenstein illusion, bends the edges of the square (red lines in 11a) toward the center of the image. Here we take the same number of orientations, 32, selected in [0, π) and σ = 13.44 pixels. The resulting distortion is shown in figure 11d.

(a) (b)

Fig. 12: a Hering illusion, distal stimulus. b Modified Hering illusion, in this example straight lines are further (c) (d) from the center with respect to the classical example of Hering illusion. c Modified Hering illusion, in this Fig. 11: a Here we superimpose red edges to the original example straight lines are placed nearer the center with illusion shown in 1c. b Representation of p−1, projec- respect to the classical example of the Hering illusion. d tion onto the retinal plane of the polarized connectiv- Modified Hering illusion with a incoherent background, ity in 7. The first eigenvalue has direction tangent to composed by random-oriented segments the level lines of the distal stimulus. In blue the tensor field, in cyan the eigenvector related to the first eigenvalue. c Computed displacement fieldu ¯. d Displacement applied to the image. In black we represent the proximal stimulus as displaced points of the distal stimulus: In the proposed modified Hering illusions the verti- (x1, x2)+u ¯(x1, x2). In red we give a square as reference, cal lines are straight and parallel as in the Hering, but in order to put in evidence the curvature of the target since they are located further/nearer the center of the lines image the perceived bending results to be less/more in- tense. In accordance with the displacement vector fields shown in figure 7c, as far as we outstrip/approach the 5.4 Modified Hering illusion center the magnitude of the computed displacement de- creases/increases. In figure 12d two straight lines are Here we present three modified Hering illusions (see fig- put over an incoherent background, composed by ran- ure 12): in the first one straight lines are positioned dom oriented segments. As we can see from 13d, any further from the center than in the classical Hering il- displacement is perceived nor computed by the present lusion. In the second one straight lines are positioned algorithm. A neuro-mathematical model for geometrical optical illusions 11

(a) (b)

Fig. 14: a Here we superimpose two red horizontal lines to the original Wundt-Hering illusion shown in 1d. b Representation of p−1, projection onto the retinal plane of the polarized connectivity in 7. The first eigenvalue has direction tangent to the level lines of the distal stim- (d) (c) ulus. In blue the tensor field, in cyan the eigenvector related to the first eigenvalue. c Computed displacement Fig. 13: a Displacement applied to the Hering illu- fieldu ¯. d Displacement applied to the image. In black sion. b Displacement applied to the first modified Her- we represent the proximal stimulus as displaced points ing illusion, in which the distance from the center is of the distal stimulus: (x , x ) +u ¯(x , x ). In red we increased. c Displacement applied to second modified 1 2 1 2 give two straight lines as reference, in order to put in Hering illusion, in which the distance from the center evidence the curvature of the target lines is decreased. d Displacement applied to the third modified Hering illusion, with an incoherent background of random-oriented segments. In this last example no deformation is perceived. In black we represent the proximal stimulus as displaced points of the distal stimulus: (x1, x2) +u ¯(x1, x2). In red we give two straight lines as reference, in order to put in evidence how much target lines are bent a, b, c or not bent d

5.5 Wundt-Hering illusion

The Wundt-Hering illusion (figure 1d) combines the effect of the background of the Hering and Wundt illusions. In this illusion two straight horizontal lines are 5.6 Zöllnerillusion presented in front of inducers which bow them outwards and inwards at the same time, inhibiting the bending effect. As a consequence the horizontal lines are indeed perceived as straight. As previously explained for the The Zöllnerillusion (figure 1e) consists in a pattern of modified Hering illusion, also this phenomenon can be oblique segments surrounding parallel lines, which cre- interpreted in terms of lateral interaction between cells ates the effect of unparallelism. As in the previous ex- belonging to the same neighborhood. Here we take 32 periments, in figure 15a we superimpose two red lines to orientations selected in the interval [0, π), σ = 6.72 pix- identify the straight lines. Here we take 32 orientations els. selected in the interval [0, π), σ = 10.08 pixels. 12 B. Franceschiello et al.

involving the feature of scale, starting from models provided by Sarti, Citti and Petitot in [33], [34]. This will allow to provide a model for scale illusions, such as the Delbouf, see [4]. Indeed, another direction for future works will be to provide a quantitative analysis for the described phenomena, such as the one proposed by Smith [35] and to direct compare the developed theory with observations of GOIs through neuro-imaging techniques.

Acknowledgements This project has received funding from (a) (b) the European Unions Seventh Framework Programme, Marie Curie Actions- Initial Training Network, under grant agreement n. 607643, “Metric Analysis For Emergent Technologies (MAnET)”. We would like to thank B. ter Haar Romeny, University of Technology Eindhoven, and M. Ursino, Univer- sity of Bologna, for their important comments and remarks to the present work.

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