∎

11institutetext: N. Nematzadeh 22institutetext: College of Science and
Engineering, Flinders University, Adelaide, Australia
Faculty of Mechatronics, Dept. of Science and Engineering, Karaj Branch,
Islamic Azad University (KIAU), Karaj-Alborz, Iran
Tel.: +61-08-74219555
Fax: +61-08-82013399
22email<EMAIL_ADDRESS>33institutetext: D. M. W. Powers
44institutetext: College of Science and Engineering, Flinders University,
Adelaide, Australia

# A Predictive Account of Café Wall Illusions
Using a Quantitative Model

Nasim Nematzadeh  David M.W. Powers

###### Abstract

This paper explores the tilt illusion effect in the Café Wall pattern using a
classical Gaussian Receptive Field model. In this illusion, the mortar lines
are misperceived as diverging or converging rather than horizontal. We examine
the capability of a simple bioplausible filtering model to recognize different
degrees of tilt in the Café Wall illusion based on different characteristics
of the pattern. Our study employed a Difference of Gaussians model of retinal
to cortical “ON” center and/or “OFF” center receptive fields. A wide range of
parameters of the stimulus, for example mortar width, luminance, tiles
contrast, phase of the tile displacement, have been studied for their effects
on the induced tilt in the Café Wall illusion. Our model constructs an edge
map representation at multiple scales that reveals tilt cues and clues
involved in the illusory perception of the Café Wall pattern. We show here
that our model can not only detect the tilt in this pattern, but also allows
us to predict the strength of the illusion and quantify the degree of tilt.
For the first time, quantitative predictions of a model are reported for this
stimulus considering different characteristics of the pattern. The results of
our simulations are consistent with previous psychophysical findings across
the full range of Café Wall variations tested. Our results also suggest that
the Difference of Gaussians mechanism is at the heart of the effects explained
by, and the mechanisms proposed for, the Irradiation, Brightness Induction,
and Bandpass Filtering models.

###### Keywords:

Visual perception Biological neural networks Geometrical illusions Tilt
effects Café Wall illusion Difference of Gaussians Perceptual grouping
Classical Receptive Fields Retinal processing models Retinal Ganglion Cells
Lateral Inhibition

##  1 Introduction

###  1.1 Illusions and their explanations

Visual illusions have the potential to give insight into the biology of vision
Changizi08 ; Eagleman01 ; Grossberg88 . They further open windows for solving
the engineering and application specific problems relevant to image processing
tasks including edge detection and feature selection, as we seek to attain
human level performance. Many illusions have been used to evaluate such
effects, notably the Café Wall Gregory79 ; McCourt83 ; Morgan86  and Mach
Bands Kingdom92 ; Pessoa96 . The Café Wall is a member of the Twisted Cord
family of Geometrical Illusions, for which a number of explanations have been
proposed Gregory79 ; Earle93 ; Woodhouse87 . The Munsterberg version of this
pattern is a chessboard with dark very thin separators between shifted rows of
black and white tiles, giving the illusion of tilted contours dividing the
rows.

The Café Wall pattern (Fig. 1-center) has grey stripes interpreted as mortar
lines dividing shifted rows of black and white tiles, inducing a perception of
diverging and converging of the mortar lines. Morgan and Moulden Morgan86
suggest the mortar lines are critical for the strength of illusion and that
the illusion has its highest strength when the luminance of the mortar is
intermediate relative to the luminance of the tiles. We consider the whole
range of luminance for mortar lines from Black to White as different
variations of the Café Wall illusion. Other significant variations include the
Hollow Square Woodhouse87 ; Bressan85 , Fraser Pattern Fraser08 , Spiral Café
Wall (Fig. 1-left), Spiral Fraser; and even variations of the Zollner Illusion
(Fig. 1-right) McCourt83 ; Bressan85 ; Westheimer07  where small parallel line
segments (inducing horizontal/vertical lines) on parallel lines result in
perceiving these lines as being convergent or divergent. It has been suggested
that the Café Wall illusion originates from the inducing effect of Twisted
Cord Fraser08  elements and then integration of these to an extended
continuous contour along the whole mortar line Grossberg85 ; Moulden79 .

Over the last few decades, many low-level and high-level models have been
proposed with the aim of explaining various Geometrical/Tilt Illusions.
However, there are many issues unaddressed across the range of the Geometrical
Illusions and their underlying percepts. In particular, modeling the illusion
effect as it is perceived is a challenging task in Computer Vision models.
Some of the explanations for the Café Wall illusion are based on “high-level”
models Gregory79 ; Kitaoka07 , but many rely on “low-level bioplausible”
models of simple filtering techniques Morgan86 ; Earle93 .

Figure 1: Left: The Spiral Café Wall illusion Kitaoka07 , Center: The Café
Wall pattern, Right: The Zollner illusion ZollnerWeb16 .

One of the first explanations for the Café Wall illusion was the _Irradiation
Hypothesis_ Moulden79 . This hypothesis was first introduced by Helmholtz
Helmholtz11 , proposing that a compressive transform causes a shift of high
contrast boundary towards the dark side of its inflection Morgan86 . The
limitation of the Irradiation Hypothesis as a sole explanation of the Café
Wall illusion is that it does not explain why the illusion is enhanced by the
mortar lines and why the pattern does not need to be of a very high contrast
as explicitly required by the Irradiation Theory. So this explanation is
incomplete on its own.

_Border Locking_ is another hypothesis used in high-level explanations for
Tilt Illusions Gregory79 . The appearance of small wedges with local asymmetry
is attributed to the luminance contrast of the light/dark half tiles and their
integration along the rows, so that they form long wedges. Gregory and Heard
Gregory79  state that the effect depends critically on luminance and that it
disappears if the mortar line is either brighter than the light tiles, or
dimmer than the dark ones.

_Brightness Induction_ (BI) describes a change in the direction of perceived
brightness, and is another explanation for the Café Wall illusion. The change
of direction can be towards the surrounds (Brightness Assimilation) or away
from it (Brightness Contrast) Penacchio13 . McCourt McCourt83  has shown that
a periodic field of a sine or square wave varying brightness would result in
inducing brightness on a homogeneous narrow grey field positioned on top of
it. He generated a new version of the Café Wall based on replacing mortar
lines with patches of light, dark and grey and explained the tilt effect as
Brightness Induction, noting that alternately-reversing diagonal line elements
can produce the characteristic of illusory convergence in the whole image.

Fraser Fraser08  proposed another hypothesis, connecting the _Twisted Cord_ to
the Café Wall illusion without using filtering but his hypothesis is not a
complete explanation of the illusion. His idea was based on the fact that the
tilt is perceived when two identically coloured tiles joined by the mortar
lines at their opposite corners create a Twisted Cord element. There are some
other proposed explanations such as _Fourier-based Models_ , but it seems that
the effect arises from small local wedge shapes rather than global features.
Therefore, what is needed is local frequency analysis rather than the global
transformation Morgan86 .

_Bandpass Spatial Filtering_ is said to be performed by the retinal ganglion
cells of ON-center and OFF-center receptive fields. Morgan and Moulden
Morgan86  modelled this by a simple discrete Laplacian filter mask viewed as
an approximation to Difference of Gaussians (DoG, noting that DoG gives the
same results), and Earle and Maskell Earle93  by a Difference of Gaussians
viewed as “a very close approximation” to a Laplacian of Gaussian (LoG).

There are some more recent explanations for the Café Wall illusion. For
example the _Irradiation Effect_ involving enlargement of bright regions to
encroach on their dark sides was identified for the shifted chessboard pattern
by Westheimer Westheimer07 , who proposed a model including a combination of
_‘light spread’, ‘compressive nonlinearity’, ‘center surround organization’
and ‘border locations’_ , to explain the Café Wall illusion. For retinal
spatial distribution of excitation, he first calculated the convolution of
image with the retinal point spread function, which then passed through a
certain nonlinearity equation (Naka-Rushton), and then to the spatial center-
surround transformation with the excitatory and inhibitory zones to get the
final result.

A similar psychological approach to that of Gregory and Heard Gregory79  was
proposed by Kitaoka as the _Phenomenal Model_ Kitaoka07 ; Kitaoka04 , which is
an elemental composition approach to explain a wide variety of Tilt Illusions.
Their psychological approach is based on the “contrast polarities of a solid
square and its adjacent line segment” to explain the Café Wall illusion. They
have suggested that when dark/light squares are adjacent to dark/light line
segments, the direction of tilt is to counteract the square angle, otherwise
the square angle is going to expand.

Fermüller and Malm Fermuller04  proposed a _Point and Line Estimation_ Model
for finding the displacement of points and lines as the elementary units in
images and they used their techniques to explain certain Geometrical
Illusions. As applied to the Café Wall, this theory is based on categorizing
the edges by saying that if mortar lines border both dark tile and bright
tile, then the two edges move towards each other. On the other hand, mortar
lines between two bright regions or two dark regions cause the edges to move
away from each other. Their explanation is similar to Gregory and Heard
Gregory79  and Kitaoka et al. Kitaoka04 .

Arai Arai05  proposed a computational nonlinear model for _Contrast
Enhancement_ in early visual processing based on discrete maximal overlap bi-
orthogonal wavelets. Arai Arai05  investigated the output of their system on
Brightness/Lightness Illusions such as the Hermann Grid, the Chevreul, Mach
Bands, the Todorovic, and the Café Wall.

As noted before, an important class of low-level models are motivated by the
_retinal ganglion cell (RGC) responses_ , including the _Bandpass Spatial
Filtering_ proposed by Morgan and Moulden Morgan86 . These retinal cells can
be modeled by Differences of Gaussian (DoG) filters as an effective filtering
technique for identifying the edges in the early stages of visual processing
Enroth-Cugell66 ; Rodieck65 . The model we have used presents the simple cell
responses using an _edge map representation at multiple scales_ derived from
the Differences of Gaussians Nematzadeh15 ; Nematzadeh16a ; Nematzadeh16b ;
Nematzadeh16c ; Nematzadeh17a ; Nematzadeh17b ; Nematzadeh18 . We show here
that the model can not only detect the tilts in variations of the Café Wall
tested, but also allows us to simply and correctly predict the strength of the
illusion and quantify the degree of tilt in these variations. The ‘predicted
tilt results’ are explained in the experimental result section and are
consistent with previous psychophysical findings across a full range of Café
Walls tested.

Elimination of the tilt effect in the Café Wall illusion is possible through
enlargement of the black tiles (Westheimer07, , Fig.11). This results in
compensating for the border shifts and corner distortions. Similar elimination
techniques can be used with additional superimposed dots positioned in the
corners (black/white dots on white/black tiles), eliminating the corner effect
(Fermuller04, , Fig.5) and (Kitaoka04, , Figs 5C,5D). The elimination of the
illusion is also possible by replacing the black and white tiles with
equiluminant but highly contrasting colour tiles McCourt83 ; Westheimer08 .

There are approaches that connect ‘Brightness Induction’ illusions and
‘Geometrical’ illusions. The Café Wall pattern and its variations are
accounted ‘second-order’ tilt patterns Nematzadeh15 ; Nematzadeh17b  involving
‘Brightness Assimilation and Contrast’ Jameson89  as well as ‘Border shifts’
Morgan86 ; Westheimer07 . In a thorough review of lightness, brightness, and
transparency (LBT) Kingdom Kingdom11  has noted that one of the most promising
approaches for modelling brightness coding is multiscale filtering Blakeslee99
in conjunction with contrast normalization. Lateral inhibition is a common
feature of most of the above models, including in particular _Irradiation_
Moulden79 , _Brightness Induction_ McCourt83 , and _Bandpass filtering_
Morgan86 .

###  1.2 Multiscale Representation in Vision

Psychophysical and physiological findings have suggested a multiscale model of
processing in the mammalian visual cortex as well as early stage processing
within the retina Field07 ; Gauthier09 ; Gollisch10 . Kuffler Kuffler52  was
the pioneer who recorded the activity of the retinal ganglion cells (RGCs)
that exit as the optic nerve, and found two types of center surround cells.
Hubel and Wiesel Hubel79 ; Hubel62  performed many pioneering experiments that
increased our understanding of cortical visual processing. Daugman Daugman80
showed an approximation of these cortical cells by using Gaussian windows
modulated by a sinusoidal wave for this impulse response. A specific spatial
orientation tuning of these cells arises from dilation of modulated Gaussian-
Gabor functions Lindeberg13 .

Pyramidal image representations and scale invariant transforms are used in
Computer Vision (CV) applications Marr80 ; Burt83 ; Marr82 ; Watt91 ;
Rosenfeld71 ; Mallat96 ; Lowe99  and are well matched to human visual
encoding. A scale-space analysis is an emergent result of image decomposition
by finding the differences between pairs of scaled filters with different
parameterizations, notably the Laplacian or Difference of Gaussian filters
(LoG/DoG) Lindeberg13 ; Jacques11 ; Lindeberg11 , giving rise to Ricker and
Marr wavelets.

Note further that self-organization models of repeated patterns of edge
detectors at particular angles are well-established Malsburg73 . Higher-level
spatial aggregation of regularly spaced spots or edges in turn automatically
gives rise to analogues of DCT and DWT type bases, the latter with
localization determined by the higher-level lateral interaction functions or
the constraints of an underlying probabilistic connectivity model Powers83 .

In our visual system, light is focused into receptors which transduce it to
neural impulses that are further processed in the retina Smith03 . Then the
middle layer of the retina, which is the focus of our study, enhances the
neural signals through the process of lateral inhibition Ratliff69 , causing
an activated nerve cell in the middle layer to decrease the ability of its
nearby neighbors to become active. This biological convolution with its
specific ‘Point Spread Function’ (PSF) enables feature/texture encoding of the
visual scene but also, we show, leads to optical illusions.

The first layer of the retina has a nonlinear mechanism for retinal gain
control, flattening the illumination component, and making it possible for the
eye to see under poor light conditions Smith03 . The lateral inhibition in the
middle layer of the retina thus evidences both a bandpass filtering property
and an edge enhancement capability. In the final layer, we find ganglion cells
whose axons exit the eye and carry the visual signals to the cortex (_inter
alia_).

The contrast sensitivity of the retinal ganglion cells can be modeled based on
Classical Receptive Fields (CRFs), involving circular center and surround
antagonism. To reveal the edge information it uses the differences and second
differences of Gaussians Enroth-Cugell66 ; Rodieck65  or Laplacian of Gaussian
(LoG) Ghosh07 . Marr and Hildreth Marr80  proposed an approximation of LoG
with DoG based on a specified ratio of the standard deviations ($\sigma$) of
the Gaussians Enroth-Cugell66 ; Rodieck65 .

Symmetrical DoGs at multiple scales have been used for spatial filtering in
our study to generate edge maps, modelling ON-center OFF-surround receptive
fields (RFs). We show the model can predict the illusory tilt in the Café Wall
patterns by providing both qualitative and quantitative results.

In Section 2 we describe the characteristics of a simple classical model based
on _Difference and Laplacian of Gaussians_ (DoG, LoG), and utilize this
bioplausible model to explain the Café Wall illusion qualitatively and
quantitatively (Sections 2.1 and 2.2) followed by the details of patterns
investigated (Section 2.3). The experimental results on variations of the Café
Wall illusion are presented in Section 3 along with a summary diagram in Fig.
8 based on the strength of illusion. This is the first time a quantitative
tilt predictions of a model is reported for variations of the Café Wall
illusions. We conclude by highlighting the advantages and disadvantages of the
model and proceed to outline a roadmap of our ongoing and future work (Section
4).

##  2 Material and Method

We hypothesize that visual perception of a scene starts by extracting the
multiscale edge map, and propose that a bioplausible implementation of a
contrast sensitivity of retinal RFs using DoG filtering produces a stack of
multiscale outputs Romeny08 . One of the first models for foveal retinal
vision was proposed by Lindeberg and Florack Lindeberg94  and our model is in
most respects inspired by it. Their model is based on simultaneous sampling of
the image at all scales, and the edge map in our model is generated in a
similar way. There are also possibilities of involvement of higher order
derivatives of Gaussians, which can be seen in retinal to cortical visual
processing models such as Marr80 ; Ghosh07 ; Lourens95 ; Young87  but there is
no biological evidence for them.

Our model has some similarities with ‘Brightness Assimilation and Contrast’
theory of early models developed by Jameson and Hurvich Jameson89  based on
DoG filters with multiple spatial scales. They noted the existence of parallel
processes that occur in our visual system as the result of the simultaneous
appearance of sharp edges and mixed colours that define delimited regions.
They proposed that a contrast effect happens when the stimulus components are
relatively large in size compared to the center of the filter, and an
assimilation effect happens when components of the stimulus are small compared
to the filter center.

Recent physiological findings on retinal ganglion cells (RGCs) have
dramatically extended our understanding of retinal processing. Previously, it
was believed that retinal lateral inhibition could not be a major contributor
to the Café Wall illusion because the effect is highly directional and arises
from both orientation as well as spatial frequency. Neuro-computational eye
models Lindeberg13 ; Romeny08 ; Lindeberg94 ; Young87  have been proposed
based on biological findings by considering the size variation of RGCs due to
variations of the eccentricity and dendritic field size Shapley86 . It was
also shown that, some RGCs have been found to have orientation selectivity
similar to the cortical cells Barlow63 ; Weng05 . Also there is evidence for
the existence of other types of retinal cells such as horizontal and amacrine
cells which have elongated surround beyond the CRF size. This has lead to
orientation selectivity for modelling of the eye, as retinal non-CRFs (nCRFs)
implementation Carandini04 ; Cavanaugh02 ; Wei11 .

The diversity of intra-retinal circuits and different types of RGCs Field07 ;
Gauthier09  are the biological evidence for the underlying mechanisms of
retinal multiscale processing of visual data at fine to coarse scales. Also
there are variations of the size of each individual RGC due to the retinal
eccentricity which is the distance from the fovea Lourens95 . This indicates a
high likelihood of the involvement of retinal/cortical simple cells and their
early stages of processing in revealing the tilt cues inside Tile Illusion
patterns and in the Café Wall in particular.

A complete vision theory, modelling the first stages of visual processing, has
been presented by Lindeberg Lindeberg13 . This reflects the properties of the
receptive field (RF) profiles, in particular, ‘scale covariance and affine
covariance’, ‘temporal causality’, ‘time-recursive updating mechanism’ and
‘mutual consistency of an internal representation at different spatial and
temporal scales’. He demonstrated that these types of receptive field models
that cover spatial, spatio-chromatic and spatio-temporal receptive field
properties have a very close similarity to the cell recording in biological
vision. These cover a range of ON-center OFF-surround and vice versa RFs in
the fovea and the LGN , then simple cells with spatial directional preference
in V1, and even space-time separable spatio-temporal receptive fields in the
LGN and V1 within the same unified theory. Lindeberg Lindeberg13  demonstrates
how these mathematical models of the linear receptive fields (derived from the
Gaussian kernels and its derivatives and closely related operators) can model
the nonlinearity behaviour in their responses.

###  2.1 The Proposed Model

The features of our bioplausible approach are intended to model the
characteristics of a human’s early visual processing. Based on numerous
physiological studies e.g. Field07 ; Gauthier09 ; Gollisch10  there is a
diversity of receptive field types and sizes inside the retina, resulting in
multiscale encoding of the visual scene. This retinal representation is
believed to be “scale invariant” in general, and there is an adaptation
mechanism for the Receptive Field sizes to some textural elements (pattern
features) inside our field of view Romeny08 ; Craft07 .

Marr and Hildreth Marr80  are well known for proposing the idea of multiscale
edge analysis for producing edge maps of the image at different scales. In
their proposal they argued that the optimal smoothing filter for images should
be localized in both spatial and frequency domains. Gaussian filter is the
only operator that satisfies this uncertainty principle Basu02  and has the
best trade-off between the localization in both spatial and frequency domains
simultaneously. The Gaussian operator in two dimensions is given by the
following formula:

$G(x,y)={\frac{1}{2\pi{\sigma}^{2}}}e^{-{\left(x^{2}+y^{2}\right)}/{\left(2\sigma^{2}\right)}}$
(1)

where $\sigma$ is the standard deviation of the Gaussian function (referred to
as scale) and $(x,y)$ are the Cartesian coordinates of the image. They have
suggested that by applying Gaussian filters at different scales to an image, a
set of images with different levels of smoothness can be obtained Marr80 ;
Basu02 , these are referred to as edge maps. In order to detect the edges in
the edge map the zero-crossings of their second derivatives need to be found.
Marr and Hildreth achieved this by applying the _Laplacian of a Gaussian_
function described below:

$\displaystyle\nabla^{2}G(x,y)={\frac{d^{2}}{dx^{2}}}G(x,y)+{\frac{d^{2}}{dy^{2}}}G(x,y)=$
$\displaystyle{\frac{\left(x^{2}+y^{2}-2\sigma^{2}\right)}{2\pi{\sigma}^{6}}}e^{-{\left(x^{2}+y^{2}\right)}/{\left(2\sigma^{2}\right)}}$
(2)

where $G(x,y)$ is the Gaussian function with standard deviation of $\sigma$,
and $(x,y)$ are the Cartesian coordinates of the image. The LoG can be
estimated as the Differences of two DoGs, estimating the second derivative of
Gaussian as given in Eq. (2). For the DoG filtering, Cope et al., Cope13  used
the following equation to represent the difference of two circular Gaussians:

$f(x,y)={\frac{1}{2\pi{\sigma_{c}}^{2}}}e^{-{\left(r^{2}\right)}/{\left(2\sigma_{c}^{2}\right)}}-{\frac{\beta_{cs}}{2\pi{\sigma_{s}}^{2}}}e^{-{\left(r^{2}\right)}/{\left(2\sigma_{s}^{2}\right)}}$
(3)

Here $r$ is the radial distance from the center of the receptive field and is
equal to $\sqrt{x^{2}+y^{2}}$, $\sigma_{c}$ is the center sigma, $\sigma_{s}$
is the surround sigma, and the parameter $\beta_{cs}$ is noted to be the
dimensionless ‘balance parameter’ in their model that has a range between
$0<\beta_{cs}\leq 1$. They have explained that the total volume of $f$ is
$1-\beta_{cs}$, and $f$ is said to be balanced if $\beta_{cs}=1$ (results in a
zero total volume of the filter).

In our model, we have used simple circular DoG filters with zero total volume
in the sense of Cope13 . These are thus perfectly ‘balanced’, and we refer to
these filters as ‘normalized DoGs’. The condition $0<\sigma_{c}<\sigma_{s}$
ensures that the regions of excitation is positioned in the center and
therefore surrounded by the inhibition region of the surround Gaussian.

For an input image $I(x,y)$ with non-negative values, the DoG filters actually
map the input pattern $I$ to output pattern $O$, using a ‘convolution
operator’:

$O(\sigma_{c},\sigma_{s},…)=\int_{R\texttimes
R}{f_{DoG}(\sigma_{c},\sigma_{s},…)I(x,y)dxdy}$ (4)

For a 2D signal such as image $I$, the DoG output modelling the retinal
ganglion cell responses with center surround organization in our
implementation is given by:

$\displaystyle\Gamma_{\mathrm{\sigma,s\sigma}}(x,y)=I\ast{\frac{1}{2\pi{\sigma}^{2}}}e^{-{\left(x^{2}+y^{2}\right)}/{\left(2\sigma^{2}\right)}}-$
$\displaystyle
I\ast{\frac{1}{2\pi{s\sigma}^{2}}}e^{-{\left(x^{2}+y^{2}\right)}/{\left(2s^{2}\sigma^{2}\right)}}$
(5)

where $x$ and $y$ are the distance from the origin in the horizontal and
vertical axes respectively and $\sigma$ is the standard deviation/scale of the
center Gaussian ($\sigma_{c}$). This is exactly the same as Eq. (3)
considering $\beta_{cs}=1$, for a normalized/zero total volume filter, and the
size ratio of the center and surround Gaussians. As shown in Eq. (6),
$\sigma_{s}$ indicates the standard deviation (or scale) of the surround
Gaussian ($\sigma_{s}=s\times\sigma_{c}$). $s$ is referred to as _Surround
ratio_ in our model.

$s=\sigma_{\mathrm{surround}}/\sigma_{\mathrm{center}}=\sigma_{\mathrm{s}}/\sigma_{\mathrm{c}}$
(6)

As indicated in Eq. (5) by increasing the value of s, we reach a wider area of
surround suppression, although the height of the surround Gaussian declines.
We also tested a broader range of _Surround ratio_ from 1.4 to 8 but this made
little difference to our results. For modelling the receptive field of retinal
ganglion cells, DoG filtering Enroth-Cugell66 ; Rodieck65  is a good
approximation of LoG, if the ratio of dispersion of center to surround is
close to 1.6 Earle93 ; Marr80  ($s=1.6\approx\varphi$, the ubiquitous ‘Golden
Ratio’).

In addition to the $s$ factor, the filter size is another parameter to be
considered in the model. The DoG is only applied within a window in which the
value of both Gaussians are insignificant outside the window (less than 5% for
the surround Gaussian). A parameter is defined called _Window ratio_ ($h$) to
control window size. The size is determined based on this parameter ($h$) and
the scale of the center Gaussian ($\sigma_{c}$) as given in Eq. (7):

$windowSize=h\times\sigma_{\mathrm{c}}+1$ (7)

Parameter $h$ determines how much of each Gaussian (center and surround) is
included inside the DoG filter (+1 as given in Eq. (7) guarantees a symmetric
filter). Variations of DoGs have been tested with different values of $h$ (2
$\leq h\leq$ 12) while keeping $\sigma_{c}$ and $s$ constant. For the
experimental results, in order to capture both excitation and inhibition
effects, the _Window ratio_ ($h$) is set to 8 in this paper.

A sample DoG filter is shown in Fig. 2(a) for scale $\sigma_{c}=8$ and for the
values of $s=2$, $h=8$. For more clarity, the center and surround Gaussians
have been displayed in separate 2D-graphs in part (b) of the figure. Fig. 2(c)
shows the relationship between the scale of the center Gaussian in our model
and the filter sizes at each scale that are defined based on Eq. (7) and the
characteristics of the pattern.

Figure 2: (a) 3D surface representation of Difference of Gaussian filters with
standard deviations as $\sigma_{c}=8$ and $\sigma_{s}=16$ (_Surround ratio_ is
2.0), displayed in jetwhite colormap Powers16 . (b) 2D representation of the
center and surround Gaussians for scale $\sigma_{c}=8$ and non-critical
parameters of the model at $s=2$ and $h=8$ (_Surround_ and _Window ratios_
respectively). (c) The relationship between the scale of a DoG filter and the
filter size, based on Eq. (7).

A sample DoG edge map of a Tile Illusion is shown in Fig. 3. The DoG is highly
sensitive to spots of matching size, but is also responsive to lines of
appropriate thickness and to contrast edges. A cropped section of the
Trampoline pattern Kitaoka00  was selected as an input image (84$\times$84px).
The edge map is shown at five different scales, $\sigma_{c}=0.5$, 1.0, 1.5,
2.0 and 2.5 to capture important features from the image (this is related to
the texture/object sizes in the pattern and by applying Eq. (7); we can
determine a proper range for $\sigma_{c}$ in the model for any arbitrary
pattern). The DoG filters in the figure are shown in the _jetwhite colormap_
Powers16 .

Scale invariant processing in general is not sensitive to the exact parameter
setting; ideally the model’s parameters should be set in a way that, at fine
scales, they capture high frequency texture details and at coarse scales, the
kernel has appropriate size relative to the objects within the scene. The
appearance of the Twisted Cord elements has previously been shown as the
filtered output after either DoG or LoG on a Café Wall image at specific
filter sizes (scales) Morgan86 ; Earle93 . We used a DoG derived model
Nematzadeh15 ; Nematzadeh16a ; Nematzadeh17a ; Nematzadeh17b  that highlights
the perception of divergence and convergence of mortar lines in the “Café
Wall” illusion. We show that the model is capable of revealing the illusory
cues in the pattern qualitatively, as well as allowing us to measure the
strength and orientation of the tilt effect quantitatively in a wide range of
Café Wall patterns.

Figure 3: DoG edge map of a cropped section of the Trampoline pattern
Kitaoka00  as a sample Tile Illusion with the tile sizes of 84$\times$84px.
The scales of DoG filters are $\sigma_{c}=0.5$, 1.0, 1.5, 2.0, 2.5 to detect
the important features in the pattern. Other parameters of the model are:
$s=2$, and $h=8$ (_Surround_ and _Window ratios_ respectively).

###  2.2 Edge map at multiple scales

The DoG transformation, modelling retinal ganglion cell (RGC) responses,
creates an edge map representation at multiple scales for the pattern,
consisting of a set of tilted line segments for the Café Wall stimulus
Nematzadeh15 ; Nematzadeh16a ; Nematzadeh17a ; Nematzadeh17b  at its fine
scales, noted as Twisted Cord elements in the literature Morgan86 ; Bressan85
; Fraser08 . Then for quantitative measurement of tilt angles in the edge map
to compare them with the tilt perceived by a human observer, we embed the DoG
model in a processing pipeline using Hough space Nematzadeh16a ; Nematzadeh16b
(details in Appendix A).

The most fundamental parameter in the model is the scale of the center
Gaussian ($\sigma_{c}$). Defining the scales in our model is highly correlated
with characteristics of the pattern and its features, in particular the mortar
lines and tile sizes in the Café Wall pattern. Based on the fixed parameters
of the _Surround and Window ratios_ , relative to $\sigma_{c}$ ($s=2$ and
$h=8$) and the pattern characteristics, an illustrative range for $\sigma_{c}$
to detect both mortar information and tiles is a range of $0.5\emph{M}=4$px to
$3.5\emph{M}=28$px (refer to Eq. (7)). At scale 28 the DoG filter has a size
of $8\times\sigma_{c}=8\times 28=224$, close to Tile size$=200$px. Increasing
the scale from this point results in a very distorted edge map, due to the DoG
averaging and the filter size. We have selected incremental steps of 0.5 _M_
between scales. This allows for the extraction of both mortar lines (at fine
scales) and the Café Wall tiles (at coarse scales) in the edge map, as well as
revealing the tilt cues and different perceptual groupings Nematzadeh17b  at
multiple scales through the gradual increase of the DoG scales. So the output
of the model is a DoG edge map at multiple scales for any arbitrary pattern.

Since we have used normalized Gaussians in our model (Section 2.1), the curve
between the center and surround Gaussians in the DoG filter shows the
activation response and that the surround Gaussian intersects the center
Gaussian at its inflection points (for the 2.0 ratio). Therefore,
$\sigma_{c}=4$ in our model corresponds to a filter in which the mortar size
of 8px lies within one standard deviation ($\sigma_{c}$) away from the mean of
the filter, so we are able to detect the mortar lines with high accuracy in
the filtered response. Accordingly, the tilts detected near the horizontal at
$\sigma_{c}=4$ in the quantitative tilt results (Appendix C) show the
_prediction of tilt angles in the foveal vision_ with high acuity in the
center of our current gaze direction. We explain how to measure the slope of
the detected tilted lines in the DoG edge maps in Appendix A.

For a thorough investigation of tilt, we need to consider not just the
horizontal mean tilts detected by our model from the edge maps at scale
$\sigma_{c}=4$ that reflect the _‘foveal tilt effect’_ (shown in red boxes in
Figs C1, C2), but also more features from the edge maps and the predicted
tilts by our model.

One of the main scientific contributions of this study is to show a very
important correlation that exists between the strength of the tilt effect in
the Café Wall illusion and the persistence of mortar cues in the edge maps
across multiple scales of the DoGs. We argue that the _‘persistence of mortar
cues’_ (PMC) plays a major role in determining how strongly we perceive the
induced tilt in the Café Wall illusion. Fig. 5 shows the edge map for one of
the stimulus tested ($ML=0.50$ in Mortar-Luminance variations). The PMC
feature reflects the appearance of tilt and the generation of the Twisted Cord
elements in the edge map of the Café Wall pattern that begins to emerge at
scale 4 ($\sigma_{c}=4$) and persists till scale 16 ($\sigma_{c}=16$). In the
next section, we will explain more about this feature, which we called PMC,
and the information we can extract from the edge maps (Section 3.1.2).

###  2.3 Patterns Investigated

To evaluate the tilts predicted by the model and to find out how close these
predictions are to reported psychophysical experiments in the literature, we
generated different variations of the Café Wall pattern similar to the
previously tested ones. All the generated patterns have the same configuration
(number of rows and columns of tiles) as Café Walls of 3$\times$8 tiles with
200$\times$200px tiles as given in Fig. 4. The patterns investigated include
_Mortar-Luminance_ (_ML_) variations in which the mortar lines have a relative
luminance in the range of Black ($\emph{ML}=0.00$) to White
($\emph{ML}=1.00$), and three shades of Grey in between Black and White (0.25,
0.50 and 0.75).

We also investigate _Mortar-Width_ (_MW_) variations, ranging from no mortar
lines ($\emph{MW}=0$px) through $\emph{MW}=4$, 8, 16, 32 and 64px.

The bottom of Fig. 4 show three patterns involving tiles with two shades of
Grey (0.25 and 0.75) separated by mortar with one of three levels of luminance
(0, 0.5, 1), and below that two variations showing different degrees of
displacement (_phase shifts_ of 1/3 and 1/5), and finally a _mirrored image_
(_Direction Change_) used to demonstrate that predicted tilts of the pattern
reverses as expected and a _hollow square_ version that is used as an extreme
case.

##  3 Experimental Results

The tilt perception in the Café Wall pattern is not only affected by the
foveal and peripheral view of the pattern Nematzadeh16a ; Nematzadeh16b ;
Nematzadeh16c , but also by the pattern characteristics such as mortar
luminance, size, phase of tile displacement, tiles contrast and so forth. We
analyse the effect of these parameters on the magnitude and orientation of the
tilt effect in eighteen different variations of the Café Wall pattern (Fig.
4).

###  3.1 Mortar Luminance variations

####  3.1.1 Pattern Characteristics

Figure 4: Patterns investigated. All the variations are Café Walls of
3$\times$8 tiles with 200$\times$200px tiles. Top: ‘Mortar Luminance’ (_ML_)
variations from Black ($\emph{ML}=0.00$) to White ($\emph{ML}=1.00$) mortars,
and three shades of Grey in between the Black and White ($\emph{ML}=0.25$,
0.50, and 0.75). Middle: ‘Mortar Width’ (_MW_) variations, from no mortar
lines ($\emph{MW}=0$) and $\emph{MW}=4$, 8, 16, 32, 64px patterns (*: The
original Café Wall with Black and White tiles, the mortar of intermediate
luminance between the luminance of the tiles, and the $mortarwidth=8$px in our
samples). Bottom: ‘Significant variations’ investigated from Grey-Tiles with
three luminance levels for mortar lines, below, between, or above the
luminance of tiles, then phase of tile displacement for shifts of 1/3 and 1/5
of a tile between consecutive rows of the Café Wall, and finally mirrored
image (Direction Change) inducing opposite direction of tilt as well as the
Hollow Square pattern. Reproduced by permission from Nematzadeh18 . Figure 5:
The binary DoG edge map at seven scales ($\sigma_{c}=4$, 8, 12, 16, 20, 24,
28) for the variation of $\emph{ML}=0.50$ (200$\times$200px tiles and 8px
mortar). The edge map has been overlayed by Hough lines displayed in green
(See Appendix A for the procedure behind the extraction of the Hough lines).

The patterns under investigation are five variations given in Fig. 4-top.
These patterns are Café Walls with 200$\times$200px tiles (_T_) and 8px mortar
(_M_). To generate the samples, we have used a value that can be interpreted
as ‘reflectance’ (viewed printed) or ‘relative luminance (viewed on screen) to
represent grey levels in the range [0,1]. Together with ambient direct and
indirect illumination, these combine to give a subjective effect of relative
brightness. Kingdom Kingdom11  defines ‘brightness’ as the perceptual
correlate of the perceived luminance. Blakeslee and McCourt Blakeslee15
explain that ‘luminance’ is the physical intensity of a stimulus and that in
achromatic vision, we see patterns of varying luminance, in which the
judgement of ‘brightness’, ‘lightness’ and ‘contrast’ would be identical in
this case. In this paper we refer to the grey level of mortar lines as simply
‘luminance’, and denote Mortar Luminance with _ML_ for an easier referral to
this pattern characteristic in the rest of this paper.

By looking at the Mortar-Luminance variations, we see very minor tilt effects
for the Black ($\emph{ML}=0.00$; the Munsterberg pattern) and White
($\emph{ML}=1.00$) mortar patterns, compared to the Grey mortar variations
(with three luminance levels: $\emph{ML}=0.25$, 0.50, 0.75) and we investigate
the predicted tilts by our model for these patterns.

####  3.1.2 DoG Edge Maps and Quantitative Tilt Results

The binary DoG edge map at seven scales for $\emph{ML}=0.50$ in this category
is given in Fig. 5. The range of DoG scales starts from $0.5\emph{M}=4$px and
continues to $3.5\emph{M}=28$px with incremental steps of $0.5\emph{M}$
($\sigma_{c}=4$, 8, 12, 16, 20, 24, 28; Section 2.2). The edge map has been
overlayed by Hough lines displayed in green (check Appendix B-Fig. B1 for a
complete set of edge maps for these patterns with the overlayed Hough lines.
Blue lines indicate the longest lines detected). Note that for the detection
of near horizontal tilted line segments, as the perceived mortar lines in the
Café Wall illusion, the DoG scale should be close to the mortar size,
$\sigma_{c}\sim M=8$px Nematzadeh16a ; Nematzadeh16b . We suggest that
$\sigma_{c}=4$ is appropriate for predicting the _foveal tilt effect_ and
$\sigma_{c}=8$ for predicting the tilt at the edge of the image in the
periphery of the retina. Comparing the edge maps at coarse scales
($\sigma_{c}=20$, 24, 28) shows very similar DoG outputs in most the
variations tested. This is because, at the coarse scales, the scale of the DoG
is large enough to capture the tile information.

The difference between the DoG edge maps of the investigated patterns is
mainly at fine to medium scales ($\sigma_{c}=4$, 8, 12, 16). As shown in Fig.
5 at scale 16, we see a transient state between detecting near horizontal
tilted line segments connecting tiles with the Twisted Cord elements along the
mortar lines, to zigzag vertical grouping of tiles at the coarse scales, in a
completely opposite direction. At this scale, we still see some mortar cues
left in the edge map of this stimulus, while in others, those mortar cues
completely disappear.

A key factor in making a predictive model for the illusory tilt in the Café
Wall pattern is to consider essential visual cues that contribute to our
perception of tilt in this illusion. We show that one of these informative
cues for a correct prediction of tilt here is the _persistence of mortar cues_
(PMC) in the edge maps across multiple scales. For the prediction of the
illusory tilt along the mortar lines, the main focus is on near horizontal
tilts and generally at fine to medium scales of the edge maps.

To _predict the tilt effect_ in the Café Wall illusion, we need to investigate
both the ‘edge maps’ and the ‘detected tilt angles’ across multiple scales of
the edge maps. As given in Appendix C, we use the predicted mean tilts in the
edge maps to extract the _foveal tilt effects_ (we called FTE) which is the
detected tilt angles at scale 4 ($\sigma_{c}=4$, highlighted in red in Figs
C1, C2), and also the _range of detected tilt angles_ at fine to medium scales
(highlighted in blue boxes). Also from the DoG edge maps (Appendix B), we
specify the _persistence of mortar cues_ (PMC) across multiple scales. If no
mortar cues are detected in the edge maps at scale 4, and similarly no Hough
lines are detected in the Hough processing stage (Appendix C), then we set the
PMC feature to _None_ and the foveal tilt angle to zero ($FTE=0.0^{\circ}$).
If the PMC lasts till scale 8, the PMC is set to _ML_ (Medium-Low). If the PMC
continues till the next scale ($\sigma_{c}=12$), we set the PMC value to _M_
(Medium), and for strong tilt effects with long-lasting PMCs reaching scale
16, we set the PMC to _MH_ (Medium-High). Above this scale ($\sigma_{c}>16$),
if the PMC continues from medium to coarse scales, we set this feature to _H_
(High).

To extract the PMC feature, an easy option is to use the mean tilts provided
in Figs C1, C2 (instead of checking the edge maps given in Appendix B). For
this, we should consider the range of scales from $\sigma_{c}=4$ till the
scale with maximum mean tilt. For a scale to be considered with a maximum tilt
angle, the minimum increment of tilt from the previous scale should be at
least 1∘. We highlight the range of mean tilts in blue boxes in these figures
that also indicate the range of corresponding scales for evaluating the PMCs
in the Café Walls tested (for samples with no detected Hough lines at scale 4,
the PMC is set to None and the FTE to $0.0^{\circ}$).

Figure 6: The summarized output of the model showing the ‘predicted tilt
results’ for the illusory tilt effect in the Café Wall patterns tested (Fig.
4). The results are provided in the Grey shaded areas next to each stimulus
and on their right hand side. We have shown the _‘foveal tilt effect’_ (FTE)
in degrees (_White_), which is the mean tilt angles predicted by our model at
scale 4 of the edge maps. We have also presented a ‘qualitative feature’ we
called ‘ _persistence of mortar cues_ ’ (PMC) that indicates the strength of
the tilt effect in these patterns (_Blue_), and their modified/corrected
versions for the _Mortar-Width_ variations (_Yellow_). The PMC feature can
have values of _None_ , _L_ : Low, _ML_ : Medium-Low, _M_ : Medium, _MH_ :
Medium-High and _H_ : High, based on how strongly the mortar cues are
presented in the DoG edge maps and across multiple scales.

Fig. 6 shows the _predicted tilt results_ for these variations in Grey shaded
areas, on the right hand side of each stimulus. This includes the foveal tilt
effect (FTE) at scale 4 of the edge maps along the horizontal, presented in
_White_ (from red boxes in Figs C1, C2), as well as the PMC values, presented
in _Blue_ (from the corresponding range of scales shown by blue boxes in Figs
C1, C2). We will explain later how the PMC values for the _Mortar-Width_
variations will be modified in relation to the mortar width, as shown in
_Yellow_ in the figure. We have not reflected the _range of tilt angles_ at
fine to medium scales in the figure and whenever it is required for our
explanations, we refer to the Appendix (C). We have rounded up the mean tilts
to one decimal point and eliminated the standard errors of calculations by
considering only the horizontal deviations for this figure.

As the mean tilts in Fig. C1 show, there are no near horizontal lines detected
at any of the DoG scales in the Munsterberg pattern ($\emph{ML}=0.00$). Based
on the predicted tilt result shown in Fig. 6 (top-section) with a zero foveal
tilt effect ($FTE=0.0^{\circ}$) and a _None_ PMC value, we note that there is
no tilt illusion in the Munsterberg pattern ($\emph{ML}=0.00$). For the White
mortar variation, there are no detected lines at the finest scale
($\sigma_{c}=4$) similar to the Munsterberg pattern. But a few lines are
detected around 1∘ of horizontal deviation at the next scale ($\sigma_{c}=8$)
as per the details are given in Fig. C1. If there is an illusion in the White
mortar pattern, the tilts predicted by the model are quite negligible. Based
on the predicted tilt results in Fig. 6 with $FTE=0.0^{\circ}$ and $PMC=None$,
we conclude that as with the Black mortar, there is no illusion in the White
mortar variation as well.

For the Dark-Grey ($\emph{ML}=0.25$) and Mid-Grey ($\emph{ML}=0.50$) mortars,
the foveal tilt effects (FTE) are $\sim$3.5∘ compared to $\sim$4.5∘ for the
$\emph{ML}=0.75$ (Light-Grey), as shown in Fig. 6. The PMC values show that we
are still able to detect horizontal lines at scale $\sigma_{c}=16$ for two
patterns of $\emph{ML}=0.50$ and $\emph{ML}=0.75$ ($PMC=MH$), but not for
$\emph{ML}=0.25$ ($PMC=M$). Here we need to consider the range of detected
tilt angles at fine to medium scales (Fig. C1) for a reliable tilt prediction
in our investigations. We note that the range of detected mean tilts is up to
1∘ more for the $\emph{ML}=0.50$ compared to the $\emph{ML}=0.75$ pattern
(between $\sim$3.5∘ to 12∘ compared to $\sim$4.5∘ to 12∘ at fine to medium
scales). This could be an indication for a stronger tilt effect in the Mid-
Grey pattern compared to the Light-Grey variation. This supports previous
psychophysical findings that the highest strength for the tilt effect in the
Café Wall illusion is when the luminance of the mortar is in the intermediate
luminance of the tiles McCourt83 ; Morgan86 . We will explain later about the
vertical and diagonal tilt angles detected by the model (as given in Figs C1,
C2) for the phase displacement patterns, and here we keep the focus of our
explanations on the tilt effects in the horizontal direction.

###  3.2 Mortar Width variations

####  3.2.1 Pattern Characteristics

The variations of Mortar-Width are shown in Fig. 4 and are denoted with _MW_
for easier referral to this pattern characteristic. We investigate following
samples from the _Mortar-Width_ category: $\emph{MW}=0$, 4, 8, 16, 32, 64px.
It has been reported that by increasing the mortar width, an inverse of the
Café Wall illusion happens Earle93 , which is discussed here.

####  3.2.2 DoG Edge Maps and Quantitative Tilt Results

Since the mortar widths are different in these variations while the tile sizes
are the same, we have selected a pattern with mid-size mortar width,
$\emph{MW}=8$px as a base, to define the scales of the DoG edge maps for these
variations. So considering the mortar size of 8px (the DoG scales vary from
0.5 _M_ to 3.5 _M_ with incremental steps of 0.5 _M_), the appropriate scales
are $\sigma_{c}=4$, 8, 12, 16, 20, 24, 28 which we use for all the Mortar-
Width variations. Fig. 7 shows the DoG edge maps at seven scales in the
jetwhite colormap for the $\emph{MW}=16$, 32 and 64px patterns (thick mortar
lines). For a complete set of the DoG edge maps for these variations at fine
to medium scales please refer to Appendix B (Figs B2, B3).

As explained before, and due to the same sized tiles for all of the
investigated patterns, the coarsest scales ($\sigma_{c}=24$, 28) show very
similar DoG outputs. However, we see some substantial changes in the last two
variations of very thick mortar lines ($\emph{MW}=32$, 64px) at these scales.
The cues of mortar lines are still available for the $\emph{MW}=64$px pattern
even at scale $\sigma_{c}=28$, but since no lines are detected at
$\sigma_{c}=4$, the PMC value in Fig. 6 is set to _None_ for this pattern. The
mortar in two patterns of $\emph{MW}=32$, 64px is quite thick, wide enough to
separate the tiles completely, and our only perception when viewing these
patterns, are the changes of brightness along the mortar lines. The Brightness
Induction can be seen more clearly on the jetwhite colormap representations of
the edge maps in Fig. 7 (also this can be seen in Fig. B4 in the binary edge
maps). These brightness artifacts can be seen at scales 8 and 12 for the
$\emph{MW}=32$px pattern and at scales 12, 16 and 20 for the $\emph{MW}=64$px.
We want to emphasize that this is how _‘Brightness Induction’_ connects to the
_‘Tilt Effect’_ in the Café Wall illusion and our explanation of illusory
tilt. It becomes more apparent from Figs B3, 7 that before the stages of
Brightness Induction, we see breakage of the Twisted Cord elements into two
parallel line segments at finer scales of the edge maps (at $\sigma_{c}=4$ for
$\emph{MW}=32$px and at $\sigma_{c}=4$, 8 for $\emph{MW}=64$px). At these
scales, the directions of groupings of identically coloured tiles are not as
clear as they were in other patterns with thinner mortar lines (comparing Figs
B2, B3). We refer to this effect as ‘Enlarged Twisted Cord’. The directions of
these brightness changes are similar to what is detected at their fine DoG
scales. The thick mortar variations have a strong brightness induction, but
not a strong tilt effect.

Figure 7:  DoG edge maps at seven scales ($\sigma_{c}$) for the thick mortar
variations of $\emph{MW}=16$, 32, and 64px, displayed in the jetwhite colormap
Powers16 . This figure highlights the connection between the ‘tilt effect’ and
the ’brightness induction’ in the Café Wall illusions. This also shows how the
Twisted Cord elements splits into two line segments and the underlying tilt
effect and its direction in the thick mortar variations. The other DoG
parameters of the model are $s=2$ and $h=8$ for the _Surround_ and _Window
ratios_ respectively. Reproduced by permission from Nematzadeh18 .

In the literature it has been shown that when the diameter of the DoG operator
is larger than the mortar width, an opposite phase brightness induction
appears Foley85 . This has been reported as a Reverse of Café Wall illusion by
Earle and Maskell Earle93 . Lulich and Stevens Lulich89  also reported: “a
reversal of the traditional Café Wall effect that is dependent upon mortar
width” (p. 428). The ‘Reversed Twister Cord’ in thick mortar variations of the
pattern is also called ‘Twisted Rope’ Earle93  with an opposite direction to
the Twisted Cord elements along mortar lines to distinguish these two. The
term ‘Enlarged Twisted Cord’ emphasizes the continuity with the Twisted Cord
unlike the usage of Twisted Rope by Earle and Maskell Earle93 . They note that
the effect of Reversed Twisted Cord occurs for a limited range of spatial
frequency that is acting as bandpass filters. Outside this limit, the Twisted
Cord elements break into two parallel segments aligned with the mortar lines
Earle93 , and thus no Twisted Cord elements are presented. The breakage of
Twisted Cords have been shown in the edge map representation explained above
for $\emph{MW}=32$, 64px. The brightness induction observed and explained here
is also consistent with Morgan and Moulden’s conclusion Morgan86  that the
effect is a consequence of Bandpass filtering.

Lulich and Stevens Lulich89  note that by increasing mortar width the induced
tilt along the mortar is further diminished and disappears when the mortar
width is about twice the size of the DoG operator. In our results, the
diminishing of the tilt cues can be viewed for the $\emph{MW}=8$px pattern at
scale 16 of the edge map, but varies in other samples, with a higher range for
the $\emph{MW}=4$px, and a much lower range for the thicker mortar lines (from
the $\emph{MW}=16$px upwards). The scale of the DoG edge maps in the Mortar-
Width variations does not directly relate to the mortar width of each
individual stimulus in this class. This certainly affects our evaluations for
the PMC feature for these patterns.

We should note that for the very thick mortar patterns ($\emph{MW}=32$, 64px),
if the PMC feature is extracted from the DoG edge maps (Figs B3, B4), we get a
different value compared to using the mean tilt tables (Fig. C1). Therefore
for this we use the detected mean tilts and the specified rule we defined in
Section 3.1.2. As explained before, we should consider a scale with maximum
tilt angle considering that the minimum increment of tilt from the previous
scale is at least 1.0∘. These PMC values are shown in Blue in Fig. 6 and we
will explain later the necessity for modifying the PMC values in the Mortar-
Width variations.

As shown in Fig. C1, there are no detected Hough lines around the horizontal
orientation at any scales for $\emph{MW}=0$px. Here, the only grouping of
pattern features that can be seen in the edge map across multiple scales are
the zigzag vertical groupings of tiles (Figs B2, C1). Thus based on the
predicted tilt results shown in Fig. 6 (middle-section) with a 0.0∘ FET and a
_None_ PMC, we conclude that there is no tilt illusion in this pattern.

The foveal tilt effects (FTE) for the Mortar-Width variations as shown in Fig.
6 indicate that by increasing mortar width from 4px to 32px, there will be an
increase of the detected tilt angles from approximately 3.3∘ in the
$\emph{MW}=4$px pattern to roughly 5.8∘ in the $\emph{MW}=32$px variation.
After this point we reach a 0.0∘ FTE and a _None_ PMC value for the
$\emph{MW}=64$px. The reason for detecting larger tilt angles in the edge maps
of thick mortar variations is that due to positioning thick mortars in between
the shifted rows of tiles, the space between the same coloured tiles with the
inducing tilt effect from the Twisted Cord elements increases as the the
mortar width increases. This results in larger tilt angles that are correctly
predicted by the model. This can be seen clearly in Figs B3, B4 and 7. This
supports previous findings that in Café Wall patterns with thick mortar, the
tilt inducing bands of Twisted Cords appear at a steeper angle to the
horizontal compared to thinner mortar Lulich89 .

For the thickest mortar variation ($\emph{MW}=64$px) with the PMC value of
_None_ and a 0.0∘ foveal tilt, we can conclude that there is no tilt effect in
the pattern. The range of mean tilts of the thick mortars are larger than the
range of horizontal mean tilts in the thinner mortar variations at fine to
medium scales of the DoGs (which is $\sim$3.5∘\- 9∘ in the $\emph{MW}=4$px,
$\sim$3.5∘\- 12∘ in the $\emph{MW}=8$px and $\sim$5∘\- 14∘ in the
$\emph{MW}=16$ and 32px patterns). When we compare the tilt effect in
$\emph{MW}=64$px pattern with the thinner mortar variations such as the
$\emph{MW}=8$px, we see that the tilt effect is very weak here and we see
dominant brightness induction on the mortars. Checking the edge maps of the
patterns with the overlayed Hough lines in Fig. B3 reveals that at scales 12
and 16, we have both negative and positive tilt angles detected along each
mortar line. This also can be seen in the edge map of $\emph{MW}=32$px at
$\sigma_{c}=8$. In the real vision, these contradictory tilts tend to cancel
each other and result in a lower tilt range than the predicted tilt results.

As shown in Fig. 6, the PMC values for the thick mortar variations for
$\emph{MW}=16$, 32px are _M_ and _MH_ respectively, and the foveal tilt
effects are the largest tilt angles detected in our samples. Despite these
results, we do not see a strong tilt effect in them compared to the
$\emph{MW}=4$, 8px. We will investigate further the very thick mortar
variations in the next section to address this issue and to show how we are
going to modify the PMC values for these variations to obtain a reliable tilt
prediction.

####  3.2.3 Very Thick mortar variations

In the experiments reported so far, we have assumed the common hypothesis that
for detection of near horizontal tilted line segments along mortar lines or
the appearance of Twisted Cord elements in the literature, the DoG scale
($\sigma_{c}$) should be close to the mortar size ($\sigma_{c}$ $\sim$M )
Morgan86 ; Nematzadeh16a ; Nematzadeh16b ; Nematzadeh16c ; Nematzadeh17a ;
Lulich89 . We show now that this is not precisely true when mortar size
exceeds 16px in our samples (the Café Wall stimuli have tiles of
200$\times$200px). For two patterns of $\emph{MW}=32$ and 64px this hypothesis
is not valid. We show here that the mortar cues have completely disappeared in
the DoG edge maps of these patterns at scales much smaller than the mortar
size (see Figs 7, B4).

For very thick mortar variations ($\emph{MW}=32$, 64px), we found that in the
defined range of scales for the DoGs, the mortar cues still exist at scale 28.
In addition, we have detected brightness induction in the DoG edge maps of
these variations, much stronger than patterns with thinner mortar lines
although the perception of tilt in these variations is very weak. To predict
the tilts in these variations based on the summarized features in Fig. 6, we
see that despite our perception of weak tilt effect, the prediction solely
upon the FTE shows strong tilts. The FTE actually shows the angles related to
the appearance of the Twisted Cord elements in the edge maps and its steeper
angles for the thick mortars is being supported by previous reports Lulich89
(Section 3.2.2).

As indicated in Figs 7 and B4 for the $\emph{MW}=64$px pattern, we have mortar
cues at scale 28 and a large deviation from the horizontal at this scale along
the mortar lines. The zigzag vertical grouping of tiles which appeared clearly
at coarse scales of the DoG edge maps for the $\emph{MW}=16$ and 32px patterns
are not shown for the $\emph{MW}=64$px pattern in the predefined range of DoG
scales. So we have examined a few scales above 28 for these patterns, and
gathered some of the important results in Fig. B4 in an extended range of
scales for the edge maps. We see a change of groupings of tiles from the near
horizontal to the zigzag vertical at scale 24 for the $\emph{MW}=16$px
pattern, and at scale 32 for the $\emph{MW}=32$px pattern, this happens around
scale 48 for the $\emph{MW}=64$px (The emergence of the zigzag vertical
groupings of tiles are highly correlated to the PMC feature).

The other thing worth mentioning from Fig. B4 is that the mortar lines are
detectable at scale 16 ($\sigma_{c}=M$) for the $\emph{MW}=16$px pattern, but
not at scale 32 ($\sigma_{c}=M$) for the $\emph{MW}=32$px pattern (they are
detected at scale 24). There are no mortar cues available at scale
$\sigma_{c}=64=M$ with the $\emph{MW}=64$px. In the edge map, the mortar cues
exist till scale 32 ($\sigma_{c}=32$) and then disappear at coarse scales,
with a filter size of half the mortar lines! This might be an indication for a
very weak tilt effect, if there is any, for the $\emph{MW}=32$ and 64px
patterns compared to the $\emph{MW}=16$px variation, despite the PMC values
presented for them in Fig. 6 as _M_ and _MH_ respectively.

If we look back to the edge map of $\emph{MW}=8$px pattern (Fig. B2), we see
the mortar cues are not only detectable at scale 8 ($\sigma_{c}=M$), but also
at the following two scales ($\sigma_{c}=12$, 16). The PMC value of _MH_ for
this pattern (Fig. 6) shows this. Comparing two variations of $\emph{MW}=8$px
with $\emph{MW}=16$px (Figs B2, B3), we see that for the $\emph{MW}=16$px
pattern, the mortar cues disappear in just one scale after the mortar size (at
$\sigma_{c}=20$) except for very small dots in their place. Therefore, similar
to $\emph{MW}=8$px, the PMC value for the $\emph{MW}=16$px is also _MH_ , but
why the perceived tilt effect is weaker? For the very thin mortar
($\emph{MW}=4$px-Fig. B2), the edge map shows persistent mortar cues from the
finest scale ($\sigma_{c}=4=M$) till scale 12 and the PMC was set to _M_ in
Fig. 6 (in Blue) although we see a very strong tilt effect in this pattern.
Therefore, to keep this feature as a reliable indicator of the strength of the
tilt effect, we need to modify the PMC values for the Mortar-Width variations.

Using the same set of DoG scales for all the Mortar-Width variations to
utilize the same set of Hough parameters for detecting tilt angles, resulting
in unreliable PMC values for these stimuli. If we consider a proper range for
the PMC feature in relation to the width of the mortar, then this problem will
be fixed. For $\emph{MW}=4$px, the mortar cues last till scale 12 and this is
actually three times of the mortar width in this pattern. Therefore, the PMC
feature should be considered _H_ (Yellow - modified) instead of _M_ (Blue).
The reason is that the PMC for the $\emph{MW}=4$px is even stronger than the
PMC for the $\emph{MW}=8$px, that only last till scale $\sigma_{c}=16$ which
is twice of the mortar size in the $\emph{MW}=8$px pattern. We have modified
the PMC values in Fig. 6 with _Yellow_ colour considering the relation of the
PMC to the mortar size for the Mortar-Width variations (and by comparing of
the PMC feature of these variations with the original Café Wall pattern). By
checking the modified PMC values (Yellow) for the Mortar-Width variations, we
can see that the problem of _overestimation_ of the PMC values for the thick
mortar lines ($\emph{MW}=16$, 32px), as well as the _underestimation_ of the
PMC value for a very thin mortar lines ($\emph{MW}=4$px), are now resolved
based on this strategy.

Our DoG edge map representation supports the previous findings that the
strongest tilt effect in the Café Wall illusion occurs with the thin mortar
lines ($\emph{MW}=4$px with $PMC=H$ and then 8px with $PMC=MH$ in our
samples). Also, the edge map at multiple scales reveals the underlying cues
involved in thick mortar variations of the Café Wall illusion and indicates
how the tilt effect degrades here while the brightness induction increases in
these patterns. Our model seems to be a good fit for our biological
understanding of the early stages of vision.

###  3.3 Significant Variations of the Pattern

One of the main factors in illusions is the brightness or lightness of the
elements, and it is important to see whether the precise shade of squares is a
significant factor. We have also checked removing the cell shades completely
to verify that the illusion disappears. Similarly the illusions shown have all
been based on 1/2 shift of tiles. This raises the question of how differently
these shifts effect the illusion. These patterns are shown at the bottom of
Fig. 4 which contains _Grey Tiles_ variations, _Phase of tile displacement_ ,
_mirrored image_ and _Hollow Square_.

####  3.3.1 Grey Tiles variations

Grey-Tiles are variations of Café Wall patterns with lower contrasted tiles,
in which instead of the Black and White tiles with maximum contrast, the tiles
here are two shades of Grey. The relative luminance of tiles in these
variations are equal to 0.25 for Dark-Grey and 0.75 for Light-Grey). So in
these variations the luminance contrast between the tiles is half of the
luminance contrast of the original Café Wall pattern with the Black and White
tiles. The mortar lines have one of the three levels of luminance here, either
below, between, or above the luminance of both of the Grey tiles selected as
$\emph{ML}=0.00$ (Black), 0.50 (Mid-Grey), and 1.00 (White). The foveal tilt
effects (FTEs) and the PMC values for these variations are extracted from Fig.
C2, similar to the previously investigated patterns shown in Fig. 6.

For the Black mortar pattern ($\emph{ML}=0.00$), the foveal tilt effect is
much lower than the original Café Wall, and close to the $\emph{ML}=1.00$.
This is less than 1∘ for $\emph{ML}=0.00$ and $1.00$ (Black and White mortar)
and $\sim$2.5∘ for Mid-Grey mortar. Although for the Grey-Tiles, the FTEs are
smaller than the majority of the other variations tested with high contrasted
tiles of Black and White tiles, their PMC features are _M_ to _H_ (_M_ for
$\emph{ML}=0.00$, 0.50 and _H_ for $\emph{ML}=1.00$). It has been reported
that the strength of illusion in low contrasted tile variations of the pattern
is less than the original Café Wall with high contrasted tiles Gregory79 ;
Morgan86 . Here we need to check the range of detected tilt angles at fine to
medium scales to address the overall tilt effect in these patterns. Fig. C2
shows that the range of detected tilt angles for $\emph{ML}=0.50$ is between
2.5∘ to 7.5∘ (compared to 3.5∘ to 12∘ in the original Café Wall), that is a
good indication for a weak tilt effect. Therefore checking the range of mean
tilt angles (at fine to medium scales) seems to be essential in some cases to
lead us towards a reliable prediction for the tilt effect in the Café Wall
illusion. Again for the $\emph{ML}=1.00$, we find a lower range for the
predicted mean tilts at fine to medium scales, indicating a weak tilt effect
compared to the original Café Wall pattern.

Gregory and Heard Gregory79  noted that the effect depends critically on the
luminance and that it disappears if the mortar line is either brighter than
the light tiles, or dimmer than the dark ones, and our results support these
psychophysical findings. Also it has been reported that Café Walls with lower
contrasted tiles need thinner mortar lines to have the same degree of apparent
convergence/divergence in the illusion percept Gregory79 . This has not yet
been tested by our model (But for Mortar-Width variations with contrast
luminance of one unit for the tiles we have illustrated that patterns with
thinner mortar lines produce a wider range of tilt angles indicating stronger
tilt effect-Section 3.2).

####  3.3.2 Phase displacement effect on detected tilt

Two more patterns are generated based on two different phase of tile
displacements with more details as follows. The phase of tile displacement is
the amount of tile shift between consecutive rows in the Café Wall pattern.
One of these patterns has a phase shift of 1/3 (shift of 1/3rd of a tile) and
the other one a phase shift of 1/5. Since there is no near horizontal line
detected along the mortar at the finest scale for the shift of 1/5 of a tile,
we set the PMC feature to _None_ and the FTE to 0.0∘ (refer to Sections
3.1.2). For the phase shift of 1/3, the foveal tilt effect is 4.0∘ and the PMC
feature is _M_. Checking the mean tilts at fine to medium scales
($\sigma_{c}=4$, 8, 12-Fig. C2) for the phase shift of 1/3 with the original
Café Wall pattern shows that we can detect a lower range of mean tilts which
is $\sim$4∘-9.5∘ compared to $\sim$3.5∘-12∘, that highlights again a weaker
tilt effect in this pattern compared to the original Café Wall.

Now, we explain the effect of _‘vertical and diagonal mean tilts’_ in these
variations as an example in between our samples in this section and leave
similar investigations for the rest of the patterns to the reader. As shown in
Fig. C2, the vertical mean tilts is much smaller in the pattern with the phase
shift of 1/5 around 2.5∘-3.5∘ at different scales. It is slightly higher in
the phase shift of 1/3 ($\sim$4∘-6∘); recall that for the near vertical and
diagonal tilts, we should consider coarse scale DoGs in our model;
$\sigma_{c}>16$. Checking the edge maps in Fig. B6 shows that the vertical
grouping of tiles appears at scale 8 in the two variations of tile shift
displacement compared to scale 12 and even 16 in the original Café Wall
pattern. This is a good indication of a weaker tilt effect along the mortar
lines in these variations compared to the original Café Wall because the tilt
cues are more persistent along mortar lines at fine to medium scales in the
original Café Wall pattern ($PMC=\emph{MH}$ in Fig. 6). In addition, we see
sharper vertical lines with less deviation from the vertical in the phase
shift of 1/5, highlighting the vertical grouping of tiles that emerges at
smaller scales. The near diagonal mean tilts for these two patterns (phase
shifts of 1/3 and 1/5) are more than 3∘ lower than the predicted tilt ranges
along the diagonal axes compared to the original Café Wall ($\sim$9.5∘
compared to $\sim$12.5∘-13.5∘-Fig. C2). Our results are consistent with
previous reports that the tilt effect is maximal with a phase shift of half a
cycle between consecutive rows and minimal or no distortion when it is in a
checkerboard phase McCourt83 .

Figure 8: The predicted tilt effects for the Café Walls tested (Fig. 4) that
are classified in an ascending order from ‘No illusion’ to the ‘Strong tilt
effect’ based on the model predictions. The features on the left hand side of
the figure that were used for classification are the _foveal tilt effect_
(FTE) in degrees and the _persistence of mortar cues_ (PMC) in the range of
_None, ML, M, MH, H_ as explained in Fig. 6.

####  3.3.3 Tilt effect direction and Hollow Square pattern

The last two patterns we investigate in this work are the _mirrored image_ and
the variation of the _Hollow Square_ that are shown at the bottom of Fig. 4.
The _mirrored image_ of the original Café Wall pattern (which is the mirrored
image of either $\emph{ML}=0.50$ or $\emph{MW}=8$px, asterisked in figure)
results in an opposite direction of induced tilt. This version is referred to
as the _Direction Change_ variation of the Café Wall in the figures. The final
pattern we consider is the _Hollow Square_ pattern Woodhouse87 ; Bressan85 ,
which consists of hollow tiles with the same size of the Café Wall tiles. The
outlines of tiles are thinner than the mortar size but the border thickness of
these hollow tiles produces a roughly similar size to the mortar size of 8px
in the Café Wall pattern, since two hollow tiles are adjacent to each other.
If the outlines of the hollow squares are thickened in this version, we
ultimately reach a similar pattern to the Café Wall without any mortar lines
Woodhouse87 . As shown in Fig. 6 (bottom-section) for the Direction Change
pattern (Mirrored Image), the foveal tilt effect (FTE) is quite similar to the
original Café Wall pattern. Even the near horizontal tilt range is quite
similar to the original Café Wall pattern as shown in Fig. C2, ranging from
$\sim$3.5∘ to 12∘ at fine to medium scales in the edge maps (shown in Fig. 6
with the PMC value of _MH_). The mean tilts along the vertical and diagonal
orientations are again very similar to the original Café Wall pattern. Slight
changes less than a degree ($˂1^{\circ}$) are in the acceptable mean tilt
range due to the standard errors around 0.5∘-0.6∘ in the original Café Wall,
and this error indicates that the results are statistically very close to each
other in these two variations. This is what we expected, and we have shown
that the tilt effect has an opposite direction of divergence/convergence tilts
in the detected Hough lines (Fig. B7).

The last pattern we consider is the _Hollow Square_. Comparing the foveal tilt
effect for the Hollow Square (Fig. 6) with the original Café Wall pattern
shows that the FTE is negligible ($<1^{\circ}$) here compared to 3.5∘ in the
original Café Wall pattern. Despite the weak tilt effect in this pattern, its
PMC value is _MH_ that is overestimated. By further investigation of Figs B7,
C2, we found that at scale 4, the mean tilt is quite negligible
(0.5${}^{\circ}\pm$0.5) and it is $\sim$6∘ at scale 8. But what is important
to consider is that although the detected lines have a mean tilt deviation
around 6∘, the lines have both positive and negative orientations along each
mortar position at $\sigma_{c}=8$ (connecting of hollow tiles in rows)
compared to a single tilt orientation (either positive or negative tilt) for
the detected lines along each mortar in the original Café Wall pattern. In the
real vision, at this transition resolution, these contradictory tilts tend to
cancel each other and result in a lower tilt range than the predicted tilt
results ($\sim$6∘).

We suggest that for the investigation of tilt in a more general class of Tile
Illusions, we need to modify the measurement of tilt angles in the model to be
able to consider the sign of detected angles in the calculations. Currently,
for calculating the mean tilts in the model (Hough processing stage), we only
consider the absolute values of the detected mean tilts. Among all the
variations of the Café Walls tested, the Hollow Square and the very thick
mortars of $\emph{MW}=32$ and 64px were the only patterns with both positive
and negative tilt angles detected along the position of each mortar line (at
scales 8 and 12). For the Hollow Square version, it has been also reported
that the decrease of contrast reduces the apparent tilt of the _Hollow Square_
illusion Woodhouse87 . This has not yet been tested by our model.

###  3.4 Summary for the predicted tilt effects

As shown in Fig. 8, we have classified the tilts predicted by our model for
all the variations tested based on the strength of the Café Wall illusion. We
emphasize that all the features used for the investigation of tilt in the
illusion (Fig. 6), including the ‘foveal tilt effect’ (FTE), the ‘persistence
of mortar cues’ (PMC) and the ‘range of detected mean tilts’ at fine to medium
scales (Appendix C), are detected by the low- to mid-level model employed in
our study. A systematic approach was designed as a classifier for the tilt
effect that is described in the following:

  * o

If $FTE=0.0^{\circ}$ and $PMC=None$ $\Rightarrow$ ‘No illusion’ (e.g.
$ML=0.00$, 1.00 in Mortar-Luminance variations and $shift=1/5$tile)

  * o

If $FTE$ is negligible ($<1^{\circ}$), regardless of the $PMC$ value
$\Rightarrow$ ‘Very Weak’ tilt effect (e.g. $ML=0.00$ and 1.00 in the Grey-
Tiles patterns)

  * o

If $PMC=ML$ (for $FTE>1^{\circ}$) $\Rightarrow$ ‘Medium-Low’ tilt effect (e.g.
the Thick-mortar variations: $MW=16$px, 32px)

  * o

If $PMC=M$ (for $FTE>1^{\circ}$) $\Rightarrow$ ‘Medium’ strength tilt effect
(e.g. $shift=1/3$tile and $ML=0.25$)

  * o

If $PMC=MH$ (for $FTE>1^{\circ}$) $\Rightarrow$ ‘Medium-High’ strength tilt
effect (e.g. the original Café Wall: $ML=0.50$/$MW=8$px and the
$DirectionChange$/mirrored image)

  * o

If $PMC=H$ (for $FTE>1^{\circ}$) $\Rightarrow$ ‘Strong’ tilt effect (here we
predicted that the $MW=4$px shows the strongest tilt effect in between our
samples)

Therefore, the most substantial factor to detect ‘None’ to ‘Very Weak’ tilt
effect is when $FTE=0.0^{\circ}$ or quite negligible ($<1^{\circ}$, such as
$ML=1.00$ in Grey-Tile variations). Otherwise, for the Café Walls with
apparent tilt effects (from weak to strong), we rely on the $PMC$ feature, as
a key indicator to classify them based on the strength of the illusion. As
previously explained for the Café Walls investigated, $FTE$ and ‘range of
detected mean tilts’ at fine to medium scales provide additional information
to let us compare the tilt effects in different stimuli. For instance, in the
category of ‘Medium-High’ strength tilt effects (Fig. 8), the
$ML=0.50$/$MW=8$px both indicate the original Café Wall in our samples and
they have similar tilt effects to the ‘Direction Change’ pattern based on the
detected features (provided in Fig. 6). Also, $ML=0.75$ has been classified in
this category but the ‘range of detected mean tilts’ for this pattern shows a
weaker tilt effect compared to the original Café Wall pattern.

We have made quantitative predictions of tilt for a wide range of conditions,
and specifically, the predictions of the strength of illusion as conditions
vary. For the first time, we have early stage vision that makes quantifiable
predictions about a family of illusions.

##  4 Conclusion

It is increasingly clear that information in the visual systems is processed
at multiple levels of resolution, perhaps simultaneously, perhaps sequentially
in some sense. We examined the capability of a bioplausible vision model,
simulating retinal/cortical simple cells to address the illusory percept in
variations of the Café Wall pattern. Exploring the tilt effect in the Café
Wall illusion, we have shown that a simple DoG model of lateral inhibition in
retinal/cortical simple cells leads to the emergence of tilt in the pattern.
Our model generates an intermediate representation at multiple scales that we
refer to as an edge map. For the recognition of a line at a particular angle
of tilt, further processing by orientation selective cells in the retina
and/or cortex is assumed Grossberg85 ; Moulden79  but we have exploited an
image processing pipeline for quantitative measurement of tilt angle using
Hough transform. In our study, the possible location of edges in the edge maps
is determined by using the Hough technique. This procedure is closely related
to template matching, generates data concerning curve detection, i.e. line
detection in the given solution Ballard82 . The biological evidence for
template matching has been proposed by Hubel and Wiesel Hubel79  that the
mammalian visual system responds to edges based on a special low-level
template matching edge detectors. A further way where the model should be
developed is to replace the use of the Hough line detection with the proposed
successive layers of biologically plausible neurons, similar to the approach
of Von der Malsburg Malsburg73 .

We have shown in this paper that the DoG edge map not only shows the emergence
of tilt in the Café Wall illusion but can also explain different degrees of
tilt effect in variations of the Café Wall illusion based on their
characteristics. The qualitative and quantitative tilt results of the Café
Wall variations investigated support previous psychophysical and experimental
findings Gregory79 ; McCourt83 ; Morgan86 ; Earle93 ; Woodhouse87 ;
Grossberg85 ; Moulden79 ; Foley85 ; Lulich89  on these stimuli.

We have shown that explanatory models and hypotheses for the Café Wall
illusion such as _Irradiation_ , _Brightness Induction_ , and _Bandpass
filtering_ appear to share the central mechanism of lateral inhibition that
ultimately underlies the tilt effect in this illusory pattern. We further
expect that these retinal filter models will prove to play an important role
in higher–level models of simulating depth and motion processing. This
supports the use of Gaussian Filters and their differences or derivatives in
Computer Vision. We have also shown empirically that this model has a high
potential to reveal the underlying mechanism connecting low-level filtering
approaches to mid- and high-level explanations such as ‘Anchoring theory’ and
‘Perceptual grouping’ Nematzadeh17b .

Although we have covered many of the aspects involved in the illusory tilt
perceived in variations of the Café Wall pattern by our model in this work
(through relying on the available psychophysical reports in the literature),
many things are left to be explored. These include psychophysical experiments
as a priority in our future study to confirm the predictions implicit in our
results, and they are expected to lead us to a more precise multiple scale
filtering which is adaptable to the pattern characteristics.

## Appendix

##
A. Quantitative Measurement of Tilt Angles-Analysis with Hough

Figure A1: Flowchart of the model and analytical tilt processing. Reproduced
with permission from [25].

The DoG edge map at seven scales for a cropped section of a Café Wall pattern
with 200$\times$200px Tiles (_T_) and 8px Mortar (_M_) is shown in Fig. A1, as
the output of the model, revealing the Twisted Cord elements along the mortar
lines. The quantitative tilt measurement includes three stages (Fig. A1),
EDGES, HOUGH and ANALYSIS implemented in MATLAB.
EDGES: At each scale, first the edge map is binarized and then Hough Transform
(HT) [73] is applied which allows us to measure the tilt angles in detected
line segments in the binary edge map. HT uses a two-dimensional array called
the accumulator ($H_{A}$) to store lines information with quantized values of
$\rho$ and $\theta$ in its cells. $\rho$ represents the distance between the
line passing through the edge point, and $\theta$ is the counter-clockwise
angle between the normal vector ($\rho$) and the x-axis, with the range of [0,
$\pi$). So based on new parameters of $(\rho,\theta)$ every edge pixel $(x,y)$
in the image space corresponds to a sinusoidal curve in the Hough space as
given by $\rho=x.cos\theta+y.sin\theta$.
HOUGH: All possible lines that could pass through every edge point in the edge
map, are accumulated inside the $H_{A}$ matrix. We are more interested in the
detection of tilt inducing line segments inside the Café Wall pattern. Two
MATLAB functions called houghpeaks and houghlines were employed for this
reason. The houghpeaks function finds the peaks in the $H_{A}$ matrix, which
are the dominant line segments. It has parameters of NumPeaks (maximum number
of lines to be detected), Threshold (threshold value for searching the $H_{A}$
for the peaks), and NHoodSize (neighborhood suppression size which is set to
zero after the peak is identified). The houghlines function, however, extracts
line segments associated with a particular bin in the $H_{A}$. It has
parameters of FillGap (maximum gap allowed between two line segments
associated with the same Hough bin), and MinLength (minimum length for merged
lines to be kept). Sample outputs of the HOUGH analysis stage are presented in
Figs B1 to B3, and B5 to B7 for different variations of the Café Wall pattern
investigated (Detected Hough lines are shown in green, displayed on the binary
edge maps).
ANALYSIS: To categorize detected line segments, we have considered four
reference orientations of horizontal (H), vertical (V), positive diagonal
(+45∘, D1), and negative diagonal (-45∘, D2). An interval of [-22.5∘, 22.5∘)
around each reference orientation was chosen to cover the whole space. The
information from HOUGH is saved inside four orientation matrices based on how
close they are to one of these reference orientations for further tilt
analysis. The statistical tilt measurements of the detected Hough lines in the
neighborhood of each reference orientation is the output of this final stage.

To quantify the mean tilts in the edge maps for these patterns using the Hough
analysis pipeline in our model, the same set of parameters of Hough are
applied for all of these variations, and for every scale of the edge maps to
attain reliable tilt results, which are comparable between these variations.
The fixed Hough parameters are: NumPeaks= 1000, FillGap= 40px, and MinLength=
450px.

##
B. Detected Hough lines on the Edge Maps

Figure B1: DoG edge maps at seven scales ($\sigma_{c}$) for five Mortar-
Luminance variations, from Black (_ML_ 0.00) to White (_ML_ 1.00) mortar
lines, in which the edge maps are overlayed by the detected Hough lines
displayed in green (Blue lines indicate the longest lines detected). Hough
parameters are kept the same in all experiments to detect near Horizontal,
Vertical and Diagonal tilted lines in the edge maps (Appendix A). Other
parameters of the DoG model are kept constant at $s=2$ (_Surround ratio_),
$h=8$ (_Window ratio_). The last row shows each variation of the pattern
investigated. Reproduced by permission from [25]. Figure B2: DoG edge maps at
fine to medium scales ($\sigma_{c}$) for three variations of Mortar-Width,
from no mortar lines (_MW_ 0px) to 8px mortar, with the overlayed Hough lines
displayed in green. Other parameters of the model and Hough processing stage
are the same as Fig. B1. Figure B3: DoG edge maps at fine to medium scales
($\sigma_{c}$) for three variations of Mortar-Width, from mortar size of 16px
to 64px, with the overlayed Hough lines displayed in green. Other parameters
of the model and Hough processing stage are the same as Fig. B1. Figure B4:
Binary DoG edge maps at multiple scales ($\sigma_{c}$) for three thick-mortar
variations with mortar width of $MW=16$, 32 and 64px for the Café Walls of
3$\times$8 tiles with 200$\times$200px tiles. The other parameters of the DoG
model are $s=2$ (_Surround ratio_) and $h=8$ (_Window ratio_). Reproduced by
permission from [25]. Figure B5: DoG edge maps at fine to medium scales
($\sigma_{c}$) for three Grey-Tiles variations with relative luminances of
0.25-Dark Grey, and 0.75-Light Grey, with the overlayed Hough lines displayed
in green. Other parameters of the model and Hough processing stage are the
same as Fig. B1. Figure B6: DoG edge maps at fine to medium scales
($\sigma_{c}$) for three patterns of the original Café Wall and two variations
of phase of tile displacements with the shifts of 1/3 and 1/5 of a tile, with
the overlayed Hough lines displayed in green. Other parameters of the model
and Hough processing stage are the same as Fig. B1. Figure B7: DoG edge maps
at fine to medium scales ($\sigma_{c}$) for three patterns of the original
Café Wall, the _mirrored image_ (_Direction Change_) and the _Hollow Square_ ,
with the overlayed Hough lines displayed in green. Other parameters of the
model and Hough processing stage are the same as Fig. B1.

The DoG edge maps at multiple scales are presented here for all the variations
tested in this study, in which the edge maps have been overlayed by the
detected Hough lines displayed in green. Blue lines indicate the longest lines
detected (see Appendix A for the Hough analysis). The edge maps are shown at 7
scales for the ‘Mortar-Luminance’ variations in Fig. B1. The range of DoG
scales starts from 0.5 _M_ = 4px and continues to 3.5 _M_ = 28px with
incremental steps of 0.5 _M_ ($\sigma_{c}=$4, 8, 12, 16, 20, 24, 28, Section
2.2). Note that for the detection of near horizontal tilted line segments, as
the perceived mortar lines in the Café Wall illusion, the DoG scale should be
close to the mortar size, $\sigma_{c}\sim M=8$px [20,21].

The difference between the DoG edge maps in between all the investigated
patterns are mainly at fine to medium scales of the DoGs ($\sigma_{c}=$4, 8,
12, 16). As shown in Fig. B1 at medium scales ($\sigma_{c}=$12, 16), we see a
transient state between detecting near horizontal tilted line segments
connecting tiles with the Twisted Cord elements along the mortar lines, to
zigzag vertical grouping of tiles at the coarse scales, in a completely
opposite direction. The edge maps at coarse scales ($\sigma_{c}=$20, 24, 28)
show very similar DoG outputs. This is because at the coarse scales, the scale
of the DoG is large enough to capture the tile information. Therefore for the
rest of the investigated patterns, we just show the edge maps _at fine to
medium scales_ with the overlayed Hough lines. The edge maps for the ‘Mortar-
Width’ variations are presented in Figs B2 and  B3 as well as in the extended
version of the DoGs for the very thick mortar lines in Fig. B4 to cover up not
just the illusory tilts along the mortar lines, but also the groupings of
tiles in zigzag vertical orientation at coarse scales in the thick mortar
variations. For the rest of the stimuli tested, noted as ‘Significant
Variations’ in Fig. 4, the edge maps at fine to medium scales with the
overlayed Hough lines are shown in Figs B5 to B7. We show the mean tilt angles
measured in the DoG edge maps across multiple scales in Appendix C.

##
C. Quantitative mean tilts

The absolute mean tilts and the standard errors of detected tilt angles for
the Café Wall variations tested are provided here in Figs Figs C1 and C2. For
the ‘foveal tilt effect’ (_FTE_ , explained in Sections 3.1.1 and Section
3.1.2), we used the near horizontal mean tilts at scale 4, and reflected these
values to Fig. 6 (in White) for all the variations tested. The FTEs are
highlighted within ‘red boxes’ in Figs C1, C2. We used another feature for our
evaluations of tilt in the Café Wall illusion, called “persistence of mortar
cues” (_PMC_), that indicates the strength of the tilt effect in variations of
the Café Wall illusion tested. Please check Section 3.1.2 to find out how to
evaluate the PMC values corresponding to each stimulus tested. The values for
this feature include _None, ML:Medium Low, L:Low, M:Medium, MH:Medium High,
and H:High_ as reflected in Fig. 6 (in Blue). This feature provides a
qualitative metric for us showing the strength of the tilt effect in this
illusion. The PMC value can be extracted from the corresponding scales next to
the highlighted blue boxes in Figs C1, C2. For a complete analysis of the tilt
effect in the Café Wall, we need these two features (_FTE, PMC_) as well as
the ‘range of detected mean tilts’ as highlighted by the blue boxes. Please
check Section 3.4 and Fig. 8 for the final outcome of our tilt investigations
for all the Café Wall stimuli tested in this study.

Figure C1: Mean tilts and the standard errors of detected tilt angles of Hough
lines for (Top): Mortar-Luminance variations from Black (_ML_ 0.00) and White
(_ML_ 1.00) mortars on Top, to Grey mortar lines ($ML=0.25$, 0.50, 0.75) for
DoG edge maps at seven scales ($\sigma_{c}$). (Bottom): Mortar-Width
variations from $MW=0$ to 64px. NaN: not a number means no lines are detected
(Hough parameters are kept the same in all experiments; Appendix A). We have
highlighted the ‘foveal tilt effects’ (FTE) in ‘red’ and the ‘range of
detected mean tilts’ at fine to medium scales in ‘blue’ boxes for the
variations tested. Blue boxes also indicate the corresponding values for the
‘persistence of mortar cues’ (PMC) in these variations (Please check Section
3.1.2 and Fig. 6 for more information). If there are no lines detected at
scale 4, the PMC feature is set to _None_ and the FTE to $0.0^{\circ}$. If the
PMC last till scale 8, we set the PMC to _ML_. For Café Wall patterns with the
PMC reaching scale 12, the PMC is set to _M_. When the PMC reaches scale 16,
we set the PMC to _MH_ , and above this scale ($\sigma_{c}>16$), the PMC value
is set to _H_. These features are used to predict the tilt effect in
variations of the Café Walls tested. Figure C2: Mean tilts and the standard
errors of detected tilt angles of Hough lines for (Top): Three variations of
Grey-Tiles, (Middle): Three patterns of the original Café Wall pattern and two
variations of phase of tile displacements with the shifts of 1/3 and 1/5 tile,
(Bottom): The mirrored image and the Hollow Square version. The calculations
are done for the edge maps at seven scales ($\sigma_{c}$) with fixed Hough
parameters (Appendix A) to detect near horizontal, vertical and diagonal
tilted lines (H, V, D1, D2) in the edge maps (NaN means no lines are
detected). For more information on the two features of FTE and PMC that are
derived from these mean tilt tables please refer to Fig. C1 and the
information provided in Section 3.1.2.

###### Acknowledgements.

Nasim Nematzadeh was supported by an Australian Research Training Program
(RTP) award. The authors are grateful to Mrs. Sue Powers and Prof.
Shanbehzadeh for their critical reading and careful comments on a previous
draft of this paper. The authors declare that there is no conflict of interest
regarding the publication of this paper.

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