Bayesian Modeling of Perceived Surface Slant from Actively-Generated and Passively-Observed Optic Flow
Identifieur interne : 002403 ( Pmc/Curation ); précédent : 002402; suivant : 002404Bayesian Modeling of Perceived Surface Slant from Actively-Generated and Passively-Observed Optic Flow
Auteurs : Corrado Caudek [Italie] ; Carlo Fantoni [Italie] ; Fulvio Domini [Italie, États-Unis]Source :
- PLoS ONE [ 1932-6203 ] ; 2011.
Abstract
We measured perceived depth from the optic flow (a) when showing a stationary physical or virtual object to observers who moved their head at a normal or slower speed, and (b) when simulating the same optic flow on a computer and presenting it to stationary observers. Our results show that perceived surface slant is systematically distorted, for both the active and the passive viewing of physical or virtual surfaces. These distortions are modulated by head translation speed, with perceived slant increasing directly with the local velocity gradient of the optic flow. This empirical result allows us to determine the relative merits of two alternative approaches aimed at explaining perceived surface slant in active vision: an “inverse optics” model that takes head motion information into account, and a probabilistic model that ignores extra-retinal signals. We compare these two approaches within the framework of the Bayesian theory. The “inverse optics” Bayesian model produces veridical slant estimates if the optic flow and the head translation velocity are measured with no error; because of the influence of a “prior” for flatness, the slant estimates become systematically biased as the measurement errors increase. The Bayesian model, which ignores the observer's motion, always produces distorted estimates of surface slant. Interestingly, the predictions of this second model, not those of the first one, are consistent with our empirical findings. The present results suggest that (a) in active vision perceived surface slant may be the product of probabilistic processes which do not guarantee the correct solution, and (b) extra-retinal signals may be mainly used for a better measurement of retinal information.
Url:
DOI: 10.1371/journal.pone.0018731
PubMed: 21533197
PubMed Central: 3077406
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<front><div type="abstract" xml:lang="en"><p>We measured perceived depth from the optic flow (a) when showing a stationary
physical or virtual object to observers who moved their head at a normal or
slower speed, and (b) when simulating the same optic flow on a computer and
presenting it to stationary observers. Our results show that perceived surface
slant is systematically distorted, for both the active and the passive viewing
of physical or virtual surfaces. These distortions are modulated by head
translation speed, with perceived slant increasing directly with the local
velocity gradient of the optic flow. This empirical result allows us to
determine the relative merits of two alternative approaches aimed at explaining
perceived surface slant in active vision: an “inverse optics” model
that takes head motion information into account, and a probabilistic model that
ignores extra-retinal signals. We compare these two approaches within the
framework of the Bayesian theory. The “inverse optics” Bayesian
model produces veridical slant estimates if the optic flow and the head
translation velocity are measured with no error; because of the influence of a
“prior” for flatness, the slant estimates become systematically
biased as the measurement errors increase. The Bayesian model, which ignores the
observer's motion, always produces distorted estimates of surface slant.
Interestingly, the predictions of this second model, not those of the first one,
are consistent with our empirical findings. The present results suggest that (a)
in active vision perceived surface slant may be the product of probabilistic
processes which do not guarantee the correct solution, and (b) extra-retinal
signals may be mainly used for a better measurement of retinal information.</p>
</div>
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</subj-group>
</subj-group>
</subj-group>
<subj-group subj-group-type="Discipline-v2"><subject>Social and Behavioral Sciences</subject>
<subj-group><subject>Psychology</subject>
<subj-group><subject>Behavior</subject>
<subj-group><subject>Human Performance</subject>
</subj-group>
</subj-group>
<subj-group><subject>Experimental Psychology</subject>
<subject>Psychophysics</subject>
<subject>Sensory Perception</subject>
</subj-group>
</subj-group>
</subj-group>
</article-categories>
<title-group><article-title>Bayesian Modeling of Perceived Surface Slant from Actively-Generated
and Passively-Observed Optic Flow</article-title>
<alt-title alt-title-type="running-head">Bayesian Modeling of Perceived Surface
Slant</alt-title>
</title-group>
<contrib-group><contrib contrib-type="author"><name><surname>Caudek</surname>
<given-names>Corrado</given-names>
</name>
<xref ref-type="aff" rid="aff1"><sup>1</sup>
</xref>
<xref ref-type="aff" rid="aff2"><sup>2</sup>
</xref>
<xref ref-type="corresp" rid="cor1"><sup>*</sup>
</xref>
</contrib>
<contrib contrib-type="author"><name><surname>Fantoni</surname>
<given-names>Carlo</given-names>
</name>
<xref ref-type="aff" rid="aff2"><sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author"><name><surname>Domini</surname>
<given-names>Fulvio</given-names>
</name>
<xref ref-type="aff" rid="aff2"><sup>2</sup>
</xref>
<xref ref-type="aff" rid="aff3"><sup>3</sup>
</xref>
</contrib>
</contrib-group>
<aff id="aff1"><label>1</label>
<addr-line>Department of Psychology, Università degli Studi di Firenze, Firenze, Italy</addr-line>
</aff>
<aff id="aff2"><label>2</label>
<addr-line>Center for Neurosciences and Cognitive Systems, Istituto Italiano di Tecnologia, Rovereto, Italy</addr-line>
</aff>
<aff id="aff3"><label>3</label>
<addr-line>Department of Cognitive, Linguistic and Psychological Sciences, Brown University, Providence, Rhode Island, United States of America</addr-line>
</aff>
<contrib-group><contrib contrib-type="editor"><name><surname>de Beeck</surname>
<given-names>Hans P. Op.</given-names>
</name>
<role>Editor</role>
<xref ref-type="aff" rid="edit1"></xref>
</contrib>
</contrib-group>
<aff id="edit1">University of Leuven, Belgium</aff>
<author-notes><corresp id="cor1">* E-mail: <email>corrado.caudek@unifi.it</email>
</corresp>
<fn fn-type="con"><p>Conceived and designed the experiments: CC CF FD. Performed the experiments:
CF CC FD. Analyzed the data: CC FD CF. Wrote the paper: CC FD CF.</p>
</fn>
</author-notes>
<pub-date pub-type="collection"><year>2011</year>
</pub-date>
<pub-date pub-type="epub"><day>14</day>
<month>4</month>
<year>2011</year>
</pub-date>
<volume>6</volume>
<issue>4</issue>
<elocation-id>e18731</elocation-id>
<history><date date-type="received"><day>18</day>
<month>11</month>
<year>2010</year>
</date>
<date date-type="accepted"><day>11</day>
<month>3</month>
<year>2011</year>
</date>
</history>
<permissions><copyright-statement>Caudek et al. This is an open-access article distributed under the
terms of the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium, provided the original author and
source are credited.</copyright-statement>
</permissions>
<abstract><p>We measured perceived depth from the optic flow (a) when showing a stationary
physical or virtual object to observers who moved their head at a normal or
slower speed, and (b) when simulating the same optic flow on a computer and
presenting it to stationary observers. Our results show that perceived surface
slant is systematically distorted, for both the active and the passive viewing
of physical or virtual surfaces. These distortions are modulated by head
translation speed, with perceived slant increasing directly with the local
velocity gradient of the optic flow. This empirical result allows us to
determine the relative merits of two alternative approaches aimed at explaining
perceived surface slant in active vision: an “inverse optics” model
that takes head motion information into account, and a probabilistic model that
ignores extra-retinal signals. We compare these two approaches within the
framework of the Bayesian theory. The “inverse optics” Bayesian
model produces veridical slant estimates if the optic flow and the head
translation velocity are measured with no error; because of the influence of a
“prior” for flatness, the slant estimates become systematically
biased as the measurement errors increase. The Bayesian model, which ignores the
observer's motion, always produces distorted estimates of surface slant.
Interestingly, the predictions of this second model, not those of the first one,
are consistent with our empirical findings. The present results suggest that (a)
in active vision perceived surface slant may be the product of probabilistic
processes which do not guarantee the correct solution, and (b) extra-retinal
signals may be mainly used for a better measurement of retinal information.</p>
</abstract>
<counts><page-count count="12"></page-count>
</counts>
</article-meta>
</front>
<body><sec id="s1"><title>Introduction</title>
<p>The current models of active Structure-from-Motion (SfM) are based on the
Helmholtzian account of perception as inverse inference <xref ref-type="bibr" rid="pone.0018731-Rao1">[1]</xref>
–<xref ref-type="bibr" rid="pone.0018731-Zhong1">[5]</xref>
. According to this approach, the
goal of the perceptual system is to infer from the sensory evidence the
environmental three-dimensional (3D) shape most likely to be responsible for
producing the sensory experience. In order to obtain this goal, the current approach
inverts the generative model for the optic flow. Mathematically, this corresponds to
an application of Bayes' rule in which first-order optic flow information is
combined with information about the observer's motion provided by
proprioceptive and vestibular signals <xref ref-type="bibr" rid="pone.0018731-Colas1">[6]</xref>
. The solution of this
“inverse-optics” problem can produce the correct result if some
assumptions about the distal objects are satisfied and if the extra-retinal signals
are measured with high precision <xref ref-type="bibr" rid="pone.0018731-Wexler1">[3]</xref>
.</p>
<p>An alternative theory hypothesizes that the visual system estimates the metric
properties of local surface orientation by using retinal information alone. Retinal
information “directly” specifies the 3D affine properties of the distal
object (such as the parallelism of local surface patches or the relative-distance
intervals in parallel directions), but it does not allow a unique determination of
its Euclidean metric properties, such as slant <xref ref-type="bibr" rid="pone.0018731-Koenderink1">[7]</xref>
(see <xref ref-type="fig" rid="pone-0018731-g001">Figure 1</xref>
). <xref ref-type="bibr" rid="pone.0018731-Domini1">[8]</xref>
proposed that the perception of
local surface slant can be understood in terms of a probabilistic estimation
process. Consider a property <inline-formula><inline-graphic xlink:href="pone.0018731.e001.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of the optic flow
which is related to the distal property <inline-formula><inline-graphic xlink:href="pone.0018731.e002.jpg" mimetype="image"></inline-graphic>
</inline-formula>
through a one-to-many
mapping. Any estimate <inline-formula><inline-graphic xlink:href="pone.0018731.e003.jpg" mimetype="image"></inline-graphic>
</inline-formula>
based solely on
<inline-formula><inline-graphic xlink:href="pone.0018731.e004.jpg" mimetype="image"></inline-graphic>
</inline-formula>
will produce an error equal to
<inline-formula><inline-graphic xlink:href="pone.0018731.e005.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Through learning, however, the visual system may select the
function <inline-formula><inline-graphic xlink:href="pone.0018731.e006.jpg" mimetype="image"></inline-graphic>
</inline-formula>
that minimizes <inline-formula><inline-graphic xlink:href="pone.0018731.e007.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<inline-formula><inline-graphic xlink:href="pone.0018731.e008.jpg" mimetype="image"></inline-graphic>
</inline-formula>
indexes the instances <inline-formula><inline-graphic xlink:href="pone.0018731.e009.jpg" mimetype="image"></inline-graphic>
</inline-formula>
that could have
produced <inline-formula><inline-graphic xlink:href="pone.0018731.e010.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. This approach has proven adequate to explain human passive
SfM, but it could be applied to active SfM as well.</p>
<fig id="pone-0018731-g001" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g001</object-id>
<label>Figure 1</label>
<caption><title>The two degrees of freedom of surface orientation.</title>
<p>A set of circular patches is used to illustrate the slant
(<inline-formula><inline-graphic xlink:href="pone.0018731.e011.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) and tilt (<inline-formula><inline-graphic xlink:href="pone.0018731.e012.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) components of
surface orientation <xref ref-type="bibr" rid="pone.0018731-Stevens1">[40]</xref>
. The line at the center of each patch is aligned
in the direction of the surface normal. The slant
<inline-formula><inline-graphic xlink:href="pone.0018731.e013.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is defined by the tangent of the angle between the
normal to the surface and the line of sight <inline-formula><inline-graphic xlink:href="pone.0018731.e014.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The tilt
<inline-formula><inline-graphic xlink:href="pone.0018731.e015.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is defined as the angle between the
<inline-formula><inline-graphic xlink:href="pone.0018731.e016.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-axis of the image plane and the projection into the
image plane of the normal to the surface (<inline-formula><inline-graphic xlink:href="pone.0018731.e017.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). The dotted
lines identify patches with same slant but different tilt magnitudes.</p>
</caption>
<graphic xlink:href="pone.0018731.g001"></graphic>
</fig>
<p>A fundamental difference between these two approaches is that only the first one
makes use of information about ego-motion. This difference is very important. In
fact, empirical results show that perceived surface tilt depends critically on
ego-motion information. A particularly convincing demonstration in this respect has
been provided by <xref ref-type="bibr" rid="pone.0018731-Wexler2">[9]</xref>
. In the “active” condition, the observer
translated along the <inline-formula><inline-graphic xlink:href="pone.0018731.e018.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-axis while fixating a
planar surface with 90<inline-formula><inline-graphic xlink:href="pone.0018731.e019.jpg" mimetype="image"></inline-graphic>
</inline-formula>
tilt and undergoing
rotation about the horizontal axis. The rotation of the surface was paired with the
observer's motion, so as to generate a pure compression optic flow. In the
“passive” condition, the same optic flow was “replayed” to a
stationary observer. Tilt perception was veridical when ego-motion information was
available (perceived tilt was 90<inline-formula><inline-graphic xlink:href="pone.0018731.e020.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), but not for the
passive observer (perceived tilt was either 0<inline-formula><inline-graphic xlink:href="pone.0018731.e021.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
180<inline-formula><inline-graphic xlink:href="pone.0018731.e022.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) – see also <xref ref-type="bibr" rid="pone.0018731-Wexler3">[10]</xref>
–<xref ref-type="bibr" rid="pone.0018731-Zhong2">[13]</xref>
.</p>
<p>The purpose of the present investigation is to determine whether observers use
information about the speed of head motion to estimate surface slant. To this
purpose, we compared the judgments of local surface slant provided by active and
passive observers to the estimates provided by two Bayesian models. The two models
were constructed (a) by taking into account information about the observer head
motion, and (b) without taking into account information about the observer head
motion. The empirical data were obtained by asking observers to judge the local
slant of virtual and physical planar surfaces from the optic flows generated by
normal or slower head translation velocities.</p>
<sec id="s1a"><title>Surface slant and first-order optic flow</title>
<p>Consider a coordinate system centered at the observer's viewpoint, with the
<inline-formula><inline-graphic xlink:href="pone.0018731.e023.jpg" mimetype="image"></inline-graphic>
</inline-formula>
axis orthogonal to the observer's frontal-parallel
plane (see <xref ref-type="fig" rid="pone-0018731-g002">Figure 2</xref>
).
Suppose that the observer fixates the surface's point located at
<inline-formula><inline-graphic xlink:href="pone.0018731.e024.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where <inline-formula><inline-graphic xlink:href="pone.0018731.e025.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the viewing
distance. If the observer translates in a direction orthogonal to the line of
sight, with translational velocity <inline-formula><inline-graphic xlink:href="pone.0018731.e026.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, or the surface
rotates with angular velocity <inline-formula><inline-graphic xlink:href="pone.0018731.e027.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, then the texture
elements on the surface will project onto the image plane a velocity field which
can be locally described by the following equation:<disp-formula><graphic xlink:href="pone.0018731.e028"></graphic>
<label>(1)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0018731.e029.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the retinal
angular velocity, <inline-formula><inline-graphic xlink:href="pone.0018731.e030.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the angular
velocity resulting from the relative rotation between the observer and the
surface, and <inline-formula><inline-graphic xlink:href="pone.0018731.e031.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the relative depth of each surface point with respect
to the fixation point.</p>
<p>In the present investigation, we only consider planar surfaces slanted by an
angle <inline-formula><inline-graphic xlink:href="pone.0018731.e032.jpg" mimetype="image"></inline-graphic>
</inline-formula>
along the vertical (<inline-formula><inline-graphic xlink:href="pone.0018731.e033.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) dimension. Such
surfaces are defined by equation<disp-formula><graphic xlink:href="pone.0018731.e034"></graphic>
<label>(2)</label>
</disp-formula>
which, substituted in Eq. 1,
gives:<disp-formula><graphic xlink:href="pone.0018731.e035"></graphic>
<label>(3)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0018731.e036.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the vertical
elevation of a generic feature point. The deformation (<italic>def</italic>
)
component (<italic>i.e.</italic>
, the gradient) of the velocity field –
which is zero along the horizontal dimension for our stimuli – is given
by<disp-formula><graphic xlink:href="pone.0018731.e037"></graphic>
<label>(4)</label>
</disp-formula>
Eq. 4 is a good approximation of the
local velocity field produced by a surface patch subtending up to
8<inline-formula><inline-graphic xlink:href="pone.0018731.e038.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of visual angle. Importantly, Eq. 4 reveals that the
gradient of the velocity field is <italic>not</italic>
sufficient to specify the
slant of the surface. In order to specify <inline-formula><inline-graphic xlink:href="pone.0018731.e039.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, in fact, the
knowledge of the relative rotation <inline-formula><inline-graphic xlink:href="pone.0018731.e040.jpg" mimetype="image"></inline-graphic>
</inline-formula>
between the
observer and the planar surface is required. Note that, in general,
<inline-formula><inline-graphic xlink:href="pone.0018731.e041.jpg" mimetype="image"></inline-graphic>
</inline-formula>
depends both on the surface's rotation about the
vertical axis (<inline-formula><inline-graphic xlink:href="pone.0018731.e042.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) and on the
translation of the observer:<disp-formula><graphic xlink:href="pone.0018731.e043"></graphic>
<label>(5)</label>
</disp-formula>
where <inline-formula><inline-graphic xlink:href="pone.0018731.e044.jpg" mimetype="image"></inline-graphic>
</inline-formula>
denotes the
relative velocity of the surface resulting from the movement of the observer in
an egocentric reference frame.</p>
<fig id="pone-0018731-g002" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g002</object-id>
<label>Figure 2</label>
<caption><title>Schematic of stimulus geometry.</title>
<p>Top panel: bird-view of the viewing geometry. The
<inline-formula><inline-graphic xlink:href="pone.0018731.e045.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and the
<inline-formula><inline-graphic xlink:href="pone.0018731.e046.jpg" mimetype="image"></inline-graphic>
</inline-formula>
axes
represent the horizontal image dimension and the line of sight,
respectively; <inline-formula><inline-graphic xlink:href="pone.0018731.e047.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the
viewing distance, <inline-formula><inline-graphic xlink:href="pone.0018731.e048.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the
horizontal head translation velocity, and
<inline-formula><inline-graphic xlink:href="pone.0018731.e049.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the
relative angular velocity produced by the motion of the observer's
head. Bottom panel: side view of the viewing geometry.
<inline-formula><inline-graphic xlink:href="pone.0018731.e050.jpg" mimetype="image"></inline-graphic>
</inline-formula>
represents
the relative depth of the point <bold>P</bold>
with respect to the
fixation point; <inline-formula><inline-graphic xlink:href="pone.0018731.e051.jpg" mimetype="image"></inline-graphic>
</inline-formula>
represents
the elevation of the point <bold>P</bold>
with respect to the optical
axis; <inline-formula><inline-graphic xlink:href="pone.0018731.e052.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the tangent of the angle between the surface
(represented by the red segment) and the fronto-parallel plane.</p>
</caption>
<graphic xlink:href="pone.0018731.g002"></graphic>
</fig>
<p>In general, the ambiguity of <italic>def</italic>
could be solved if the visual
system were able to accurately measure the second-order optic flow
(<italic>i.e.</italic>
, the image accelerations), but several studies show
that this is not the case <xref ref-type="bibr" rid="pone.0018731-Calderone1">[14]</xref>
–<xref ref-type="bibr" rid="pone.0018731-Werkhoven1">[18]</xref>
. Alternatively,
<italic>def</italic>
can be disambiguated by combining the information
provided by the first-order optic flow and the extra-retinal signals, if some
assumptions are met (see next Section).</p>
</sec>
<sec id="s1b"><title>Bayesian slant estimation from retinal, vestibular, and proprioceptive
information</title>
<p>The ambiguity of <italic>def</italic>
can be overcome by the active observer
under the assumption that the object is stationary – a reasonable
assumption in many real-world situations <xref ref-type="bibr" rid="pone.0018731-Wexler4">[12]</xref>
. If the object is
stationary, <inline-formula><inline-graphic xlink:href="pone.0018731.e053.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and the relative rotation between the observer and the
surface is equal-and-opposite to the observer's motion:
<inline-formula><inline-graphic xlink:href="pone.0018731.e054.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. If information about <inline-formula><inline-graphic xlink:href="pone.0018731.e055.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is obtained from
proprioceptive and vestibular signals, it is thus possible to estimate
<inline-formula><inline-graphic xlink:href="pone.0018731.e056.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>The Bayesian model presented by Colas and collaborators formalizes this idea
<xref ref-type="bibr" rid="pone.0018731-Colas1">[6]</xref>
. The
uncertainty in the estimation of the relative motion
<inline-formula><inline-graphic xlink:href="pone.0018731.e057.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is described by a Gaussian distribution
<inline-formula><inline-graphic xlink:href="pone.0018731.e058.jpg" mimetype="image"></inline-graphic>
</inline-formula>
centered at <inline-formula><inline-graphic xlink:href="pone.0018731.e059.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and having an
arbitrary standard deviation <inline-formula><inline-graphic xlink:href="pone.0018731.e060.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(here and in the
following we use capital letters to indicate random variables). The spread of
this Gaussian distribution encodes the noise in the measurement of the
vestibular and proprioceptive signals and the possibility that the surface
undergoes a rotation independent from the observer's motion. By centering
this probability distribution at <inline-formula><inline-graphic xlink:href="pone.0018731.e061.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, Colas et al.
implement the <italic>stationarity assumption</italic>
, that is, they favor the
solutions in which the optic flow is produced by the observer's motion
<xref ref-type="bibr" rid="pone.0018731-Wexler4">[12]</xref>
.
Colas et al. also consider the possibility that the optic flow is not measured
accurately, or is produced by some degree of non-rigid motion. Under these
circumstances, the surface slant <inline-formula><inline-graphic xlink:href="pone.0018731.e062.jpg" mimetype="image"></inline-graphic>
</inline-formula>
combined with the
relative motion <inline-formula><inline-graphic xlink:href="pone.0018731.e063.jpg" mimetype="image"></inline-graphic>
</inline-formula>
does not produce a
unique <italic>def</italic>
value. This further source of uncertainty is
described by a Gaussian distribution <inline-formula><inline-graphic xlink:href="pone.0018731.e064.jpg" mimetype="image"></inline-graphic>
</inline-formula>
centered at
<inline-formula><inline-graphic xlink:href="pone.0018731.e065.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with an arbitrary standard deviation
<inline-formula><inline-graphic xlink:href="pone.0018731.e066.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. By centering this probability distribution at
<inline-formula><inline-graphic xlink:href="pone.0018731.e067.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, Colas et al. implement the <italic>rigidity
assumption</italic>
, that is, they favor the solutions in which
<italic>def</italic>
is produced by a rigid rotation. A further assumption
is that the slant of the surface does not depend on the relative motion between
the surface and the observer.</p>
<p>Under these assumptions, the problem of estimating local surface slant given the
knowledge of <italic>def</italic>
and <inline-formula><inline-graphic xlink:href="pone.0018731.e068.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(the
observer's motion) becomes the problem of identifying the density function
<inline-formula><inline-graphic xlink:href="pone.0018731.e069.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. This probability density function can be found through
Bayes' theorem by applying the rules of marginalization and probability
decomposition.</p>
<p>From the definition of the conditional probability
<inline-formula><inline-graphic xlink:href="pone.0018731.e070.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, by marginalizing over <inline-formula><inline-graphic xlink:href="pone.0018731.e071.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, we
obtain<disp-formula><graphic xlink:href="pone.0018731.e072"></graphic>
<label>(6)</label>
</disp-formula>
<disp-formula><graphic xlink:href="pone.0018731.e073"></graphic>
<label>(7)</label>
</disp-formula>
By
the chain rule, we can write<disp-formula><graphic xlink:href="pone.0018731.e074"></graphic>
<label>(8)</label>
</disp-formula>
Moreover,<disp-formula><graphic xlink:href="pone.0018731.e075"></graphic>
<label>(9)</label>
</disp-formula>
because, under the rigidity assumption,
<italic>def</italic>
depends only on the distal slant
<inline-formula><inline-graphic xlink:href="pone.0018731.e076.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and the relative rotation
<inline-formula><inline-graphic xlink:href="pone.0018731.e077.jpg" mimetype="image"></inline-graphic>
</inline-formula>
;<disp-formula><graphic xlink:href="pone.0018731.e078"></graphic>
<label>(10)</label>
</disp-formula>
because surface slant is independent from
the observer's relative motion and from the egocentric
motion;<disp-formula><graphic xlink:href="pone.0018731.e079"></graphic>
<label>(11)</label>
</disp-formula>
because of the chain rule.
Therefore, <inline-formula><inline-graphic xlink:href="pone.0018731.e080.jpg" mimetype="image"></inline-graphic>
</inline-formula>
can be re-written as<disp-formula><graphic xlink:href="pone.0018731.e081"></graphic>
<label>(12)</label>
</disp-formula>
By
virtue of Eq. 12, Eq. 7 takes the form<disp-formula><graphic xlink:href="pone.0018731.e082"></graphic>
<label>(13)</label>
</disp-formula>
<disp-formula><graphic xlink:href="pone.0018731.e083"></graphic>
<label>(14)</label>
</disp-formula>
In
conclusion, Eq. 14 provides a possible solution to the “inverse
optics” problem of estimating local surface slant from the deformation of
the optic flow (see <xref ref-type="fig" rid="pone-0018731-g003">Figure
3</xref>
). If <inline-formula><inline-graphic xlink:href="pone.0018731.e084.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and <italic>def</italic>
are measured with no error,
then <inline-formula><inline-graphic xlink:href="pone.0018731.e085.jpg" mimetype="image"></inline-graphic>
</inline-formula>
peaks at the true slant value
(<inline-formula><inline-graphic xlink:href="pone.0018731.e086.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) when the distal surface is stationary. In the presence
of measurement errors, instead, the estimated slant will be biased. The
magnitude of this bias depends on the precision with which
<inline-formula><inline-graphic xlink:href="pone.0018731.e087.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(the observer's motion) and <italic>def</italic>
are measured: the larger <inline-formula><inline-graphic xlink:href="pone.0018731.e088.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the larger the
under-estimation of slant.</p>
<fig id="pone-0018731-g003" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g003</object-id>
<label>Figure 3</label>
<caption><title>Recovery of surface slant according to <bold>Eq. 14</bold>
.</title>
<p>Intensity corresponds to probability. The values reported in the plot
refer to the case of a static plane slanted by
80<inline-formula><inline-graphic xlink:href="pone.0018731.e089.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(<inline-formula><inline-graphic xlink:href="pone.0018731.e090.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) around
the horizontal axis and viewed by an active observer performing a
lateral head translation at a speed that produces a relative
angular-rotation velocity of 0.32 rad/s (<inline-formula><inline-graphic xlink:href="pone.0018731.e091.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).
<bold>Panels a - e</bold>
: method for calculating the posterior
distribution. <bold>a</bold>
. Prior for frontal-parallel
<inline-formula><inline-graphic xlink:href="pone.0018731.e092.jpg" mimetype="image"></inline-graphic>
</inline-formula>
modeled as
a (half) Gaussian distribution centered at zero. <bold>b</bold>
.
Likelihood function <inline-formula><inline-graphic xlink:href="pone.0018731.e093.jpg" mimetype="image"></inline-graphic>
</inline-formula>
generated
by assuming that the <italic>def</italic>
measurements are corrupted by
Gaussian noise. <bold>c</bold>
. Uncertainty of the relative rotation
between the observer and the planar surface
<inline-formula><inline-graphic xlink:href="pone.0018731.e094.jpg" mimetype="image"></inline-graphic>
</inline-formula>
modeled as
a Gaussian distribution centered on the true value
<inline-formula><inline-graphic xlink:href="pone.0018731.e095.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.
<bold>d</bold>
. Product of the likelihood, the prior for
<inline-formula><inline-graphic xlink:href="pone.0018731.e096.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and the
prior for <inline-formula><inline-graphic xlink:href="pone.0018731.e097.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.
<bold>e</bold>
. Posterior distribution produced by the
marginalization over <inline-formula><inline-graphic xlink:href="pone.0018731.e098.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The
median of the posterior distribution (dotted line) gives the optimal
estimate of surface slant based on the knowledge of <italic>def</italic>
and <inline-formula><inline-graphic xlink:href="pone.0018731.e099.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The model's prediction (the value 5 in the
figure) gets more and more close to the “true” value of the
slant (<inline-formula><inline-graphic xlink:href="pone.0018731.e100.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) as
<inline-formula><inline-graphic xlink:href="pone.0018731.e101.jpg" mimetype="image"></inline-graphic>
</inline-formula>
decreases.</p>
</caption>
<graphic xlink:href="pone.0018731.g003"></graphic>
</fig>
</sec>
<sec id="s1c"><title>Bayesian slant estimation from retinal information alone</title>
<p>We propose that the visual system estimates surface slant without considering the
information about head translation velocity (see <xref ref-type="fig" rid="pone-0018731-g004">Figure 4</xref>
). With reference to the Bayesian
model discussed in the previous section, this means that
<inline-formula><inline-graphic xlink:href="pone.0018731.e102.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where <inline-formula><inline-graphic xlink:href="pone.0018731.e103.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the <italic>a
priori</italic>
distribution of a random variable representing the amount of
relative rotation between the observer and the surface. In this case, Eq. 14
reduces to<disp-formula><graphic xlink:href="pone.0018731.e104"></graphic>
<label>(15)</label>
</disp-formula>
</p>
<fig id="pone-0018731-g004" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g004</object-id>
<label>Figure 4</label>
<caption><title>Recovery of surface slant according to <bold>Eq. 15</bold>
.</title>
<p>The values reported in the figure refer to the stimulus conditions used
in <xref ref-type="fig" rid="pone-0018731-g003">Figure 3</xref>
.
Differently from the “inverse-optics” approach, in this case
the distribution <inline-formula><inline-graphic xlink:href="pone.0018731.e105.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is
non-informative. Note that, after computing the product of the
likelihood, the prior for <inline-formula><inline-graphic xlink:href="pone.0018731.e106.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and the
prior for <inline-formula><inline-graphic xlink:href="pone.0018731.e107.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the
marginalization over <inline-formula><inline-graphic xlink:href="pone.0018731.e108.jpg" mimetype="image"></inline-graphic>
</inline-formula>
produces a
posterior distribution that is very different from what is shown in
<xref ref-type="fig" rid="pone-0018731-g003">Figure 3</xref>
. The
estimate of surface slant is given by the median of the posterior
distribution (dotted line). The posterior median corresponds the point
on the hyperbola closest to the origin of the Cartesian axes. The
Bayesian posterior median estimator (which is equal to
<inline-formula><inline-graphic xlink:href="pone.0018731.e109.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) is
represented by the red dot in panel <inline-formula><inline-graphic xlink:href="pone.0018731.e110.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
</caption>
<graphic xlink:href="pone.0018731.g004"></graphic>
</fig>
<p>Domini and Caudek showed that this account is sufficient for predicting perceived
slant from the optic flow in the case of the passive observer <xref ref-type="bibr" rid="pone.0018731-Caudek1">[19]</xref>
–<xref ref-type="bibr" rid="pone.0018731-Domini7">[28]</xref>
. They
showed that the center of mass of the distribution described by Eq. 15 is equal
to <inline-formula><inline-graphic xlink:href="pone.0018731.e111.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, with <inline-formula><inline-graphic xlink:href="pone.0018731.e112.jpg" mimetype="image"></inline-graphic>
</inline-formula>
depending on the
spreads of the prior distributions of <inline-formula><inline-graphic xlink:href="pone.0018731.e113.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and of
<inline-formula><inline-graphic xlink:href="pone.0018731.e114.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0018731-Domini1">[8]</xref>
. The
center of mass as an estimate of <inline-formula><inline-graphic xlink:href="pone.0018731.e115.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is equivalent to
the posterior median, which is the Bayes estimator for the absolute error loss.
Indeed, it has been shown that Eq. 15 is a particular case of Eq. 14: The two
accounts are indistinguishable when information about the head's
translation is unavailable, like in the case of the passive observer <xref ref-type="bibr" rid="pone.0018731-Colas1">[6]</xref>
. Eqs. 14 and
15, instead, make different predictions for the active observer, when head
translation velocity is manipulated.</p>
<p>The importance of <inline-formula><inline-graphic xlink:href="pone.0018731.e116.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for the perceptual
recovery of local surface slant from the optic flow has been highlighted by
<xref ref-type="bibr" rid="pone.0018731-Domini1">[8]</xref>
, <xref ref-type="bibr" rid="pone.0018731-Domini3">[24]</xref>
.
<italic>def</italic>
is a one-parameter family of
<inline-formula><inline-graphic xlink:href="pone.0018731.e117.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(surface slant) and <inline-formula><inline-graphic xlink:href="pone.0018731.e118.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(relative angular
rotation) pairs, but not all possible <inline-formula><inline-graphic xlink:href="pone.0018731.e119.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula><inline-graphic xlink:href="pone.0018731.e120.jpg" mimetype="image"></inline-graphic>
</inline-formula>
pairs are equally likely. Under the assumption of
uniform prior distributions (bounded between 0 and
<inline-formula><inline-graphic xlink:href="pone.0018731.e121.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and between 0 and <inline-formula><inline-graphic xlink:href="pone.0018731.e122.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) for
<inline-formula><inline-graphic xlink:href="pone.0018731.e123.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and <inline-formula><inline-graphic xlink:href="pone.0018731.e124.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the conditional
probability of a <inline-formula><inline-graphic xlink:href="pone.0018731.e125.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula><inline-graphic xlink:href="pone.0018731.e126.jpg" mimetype="image"></inline-graphic>
</inline-formula>
pair given <italic>def</italic>
is not uniform, but it
has a maximum equal to <inline-formula><inline-graphic xlink:href="pone.0018731.e127.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, with
<inline-formula><inline-graphic xlink:href="pone.0018731.e128.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(see <xref ref-type="bibr" rid="pone.0018731-Domini3">[24]</xref>
).</p>
</sec>
<sec id="s1d"><title>Rationale of the Experiments</title>
<p>Eqs. 14 and 15 provide two alternative models for the perceptual derivation of
surface slant from the optic flow in active vision. The purpose of the present
investigation is to contrast them by comparing their predictions to the
behavioral data obtained when head translation velocity is manipulated.</p>
<p>In the present experiments, observers were required to produce two different head
translation velocities. The first was comparable to the peak horizontal head
velocity during normal locomotion <xref ref-type="bibr" rid="pone.0018731-Moore1">[29]</xref>
, the second was 80%
slower. This experimental manipulation (a) does not affect the estimate of local
surface slant according to Eq. 14 (by assuming that the measurement noise of
<inline-formula><inline-graphic xlink:href="pone.0018731.e129.jpg" mimetype="image"></inline-graphic>
</inline-formula>
remains unaltered), and (b) can strongly affect the
estimate of local surface slant according to Eq. 15 (because head translation
velocity is proportional to <inline-formula><inline-graphic xlink:href="pone.0018731.e130.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
</sec>
</sec>
<sec id="s2"><title>Results</title>
<sec id="s2a"><title>Perceived surface slant</title>
<p>Active and passive observers judged the perceived slants of virtual or physical
planar surfaces. The results indicate that the judgements made by the observers
are systematically biased by the head translation velocity (see <xref ref-type="fig" rid="pone-0018731-g005">Figure 5</xref>
). The same
qualitative trends are found for the active and passive viewing of a virtual
surface, and for the active viewing of a physical surface.</p>
<fig id="pone-0018731-g005" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g005</object-id>
<label>Figure 5</label>
<caption><title>Perceived slant as a function of simulated slant.</title>
<p>Fast and slow head translation velocities are coded by red and black,
respectively. The values are expressed in terms of the tangent. The
dashed lines indicate veridical performance. Vertical bars represent
<inline-formula><inline-graphic xlink:href="pone.0018731.e131.jpg" mimetype="image"></inline-graphic>
</inline-formula>
1 standard
error of the mean. (a) <italic>Passive-viewing with a virtual
surface</italic>
. The interaction Slant
<inline-formula><inline-graphic xlink:href="pone.0018731.e132.jpg" mimetype="image"></inline-graphic>
</inline-formula>
Velocity
is significant, <inline-formula><inline-graphic xlink:href="pone.0018731.e133.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 51.04,
<inline-formula><inline-graphic xlink:href="pone.0018731.e134.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001. The marginal effect of
the head's translation velocity is significant,
<inline-formula><inline-graphic xlink:href="pone.0018731.e135.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 7.48,
<inline-formula><inline-graphic xlink:href="pone.0018731.e136.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .006: On average, the amount
of reported slant is 51% lower for the slow than for the fast
head's translation velocity. (b) <italic>Active-viewing with a
virtual surface</italic>
. The interaction Slant
<inline-formula><inline-graphic xlink:href="pone.0018731.e137.jpg" mimetype="image"></inline-graphic>
</inline-formula>
Velocity
is significant, <inline-formula><inline-graphic xlink:href="pone.0018731.e138.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 47.31,
<inline-formula><inline-graphic xlink:href="pone.0018731.e139.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001: The effect of simulated
slant on the response is larger for the faster head's translation
velocity. The marginal effect of head's translation velocity is
significant, <inline-formula><inline-graphic xlink:href="pone.0018731.e140.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 5.62,
<inline-formula><inline-graphic xlink:href="pone.0018731.e141.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .018: On average, the amount
of reported slant is 60% lower for the slow than for the fast
head's translation velocity. (c) <italic>Active-viewing with a
physical surface</italic>
. The interaction Slant
<inline-formula><inline-graphic xlink:href="pone.0018731.e142.jpg" mimetype="image"></inline-graphic>
</inline-formula>
Velocity
is significant, <inline-formula><inline-graphic xlink:href="pone.0018731.e143.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 40.81,
<inline-formula><inline-graphic xlink:href="pone.0018731.e144.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001. The marginal effect of
head's translation velocity is significant,
<inline-formula><inline-graphic xlink:href="pone.0018731.e145.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 12.96,
<inline-formula><inline-graphic xlink:href="pone.0018731.e146.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001: On average, the amount
of reported slant is 75% lower for the slow than for the fast
head's translation velocity.</p>
</caption>
<graphic xlink:href="pone.0018731.g005"></graphic>
</fig>
<p>According to the Bayesian model described in Eq. 15, perceived surface slant
depends only on the square root of <italic>def</italic>
. For the active and
passive viewing of virtual planar surfaces, the observers' judgments of
slant complied with this prediction (see <xref ref-type="fig" rid="pone-0018731-g006">Figure 6</xref>
). For the active viewing of physical
surfaces, <inline-formula><inline-graphic xlink:href="pone.0018731.e147.jpg" mimetype="image"></inline-graphic>
</inline-formula>
was not the only determinant of the perceptual response,
but the additional contribution of the head translation velocity was negligible.
In the present investigation, therefore, there is no evidence that simulated
slant contributes to the perceptual response beyond what
<inline-formula><inline-graphic xlink:href="pone.0018731.e148.jpg" mimetype="image"></inline-graphic>
</inline-formula>
can explain.</p>
<fig id="pone-0018731-g006" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g006</object-id>
<label>Figure 6</label>
<caption><title>Perceived slant as a function of <inline-formula><inline-graphic xlink:href="pone.0018731.e149.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</title>
<p>Fast and slow head translation velocities are coded by red and black,
respectively. The values are expressed in terms of the tangent. Vertical
bars represent <inline-formula><inline-graphic xlink:href="pone.0018731.e150.jpg" mimetype="image"></inline-graphic>
</inline-formula>
1 standard
error of the mean. (a) <italic>Passive-viewing with a virtual
surface</italic>
. There is no significant interaction between
<inline-formula><inline-graphic xlink:href="pone.0018731.e151.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and head
translation Velocity, <inline-formula><inline-graphic xlink:href="pone.0018731.e152.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 0.43,
<inline-formula><inline-graphic xlink:href="pone.0018731.e153.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .513. There is no significant
effect of head translation Velocity, <inline-formula><inline-graphic xlink:href="pone.0018731.e154.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 0.57,
<inline-formula><inline-graphic xlink:href="pone.0018731.e155.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .449. In a no-intercept model
with <inline-formula><inline-graphic xlink:href="pone.0018731.e156.jpg" mimetype="image"></inline-graphic>
</inline-formula>
as the only predictor, the slope is 1.04,
<inline-formula><inline-graphic xlink:href="pone.0018731.e157.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 20.95,
<inline-formula><inline-graphic xlink:href="pone.0018731.e158.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001, and
<inline-formula><inline-graphic xlink:href="pone.0018731.e159.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .74. No improvement of fit is
found if simulated slant is added to such model,
<inline-formula><inline-graphic xlink:href="pone.0018731.e160.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 0.455. For a baseline model
with only the individual differences, <inline-formula><inline-graphic xlink:href="pone.0018731.e161.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is equal
to .45. (b) <italic>Active-viewing with a virtual surface</italic>
.
There is no significant interaction between
<inline-formula><inline-graphic xlink:href="pone.0018731.e162.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and head
translation Velocity, <inline-formula><inline-graphic xlink:href="pone.0018731.e163.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 2.50,
<inline-formula><inline-graphic xlink:href="pone.0018731.e164.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .114; there is not
significant effect of head translation Velocity,
<inline-formula><inline-graphic xlink:href="pone.0018731.e165.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 0.25,
<inline-formula><inline-graphic xlink:href="pone.0018731.e166.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .620. In a no-intercept model
with <inline-formula><inline-graphic xlink:href="pone.0018731.e167.jpg" mimetype="image"></inline-graphic>
</inline-formula>
as the only predictor, the slope is 1.88,
<inline-formula><inline-graphic xlink:href="pone.0018731.e168.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 16.46,
<inline-formula><inline-graphic xlink:href="pone.0018731.e169.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001, and
<inline-formula><inline-graphic xlink:href="pone.0018731.e170.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .69. No improvement of fit is
found if simulated Slant is added to such model,
<inline-formula><inline-graphic xlink:href="pone.0018731.e171.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= −0.22. For a baseline
model with only the individual differences,
<inline-formula><inline-graphic xlink:href="pone.0018731.e172.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is equal
to .33. When analyzing together the data of the passive and active
viewing of the virtual surfaces, significant effects are found for
<inline-formula><inline-graphic xlink:href="pone.0018731.e173.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula><inline-graphic xlink:href="pone.0018731.e174.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula><inline-graphic xlink:href="pone.0018731.e175.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001, and condition
(indicating a smaller response for the passive observer),
<inline-formula><inline-graphic xlink:href="pone.0018731.e176.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula><inline-graphic xlink:href="pone.0018731.e177.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001. For this model,
<inline-formula><inline-graphic xlink:href="pone.0018731.e178.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is equal
to .90. When controlling for <inline-formula><inline-graphic xlink:href="pone.0018731.e179.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the
effect of simulated Slant is not significant,
<inline-formula><inline-graphic xlink:href="pone.0018731.e180.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. For a
baseline model with only the individual differences,
<inline-formula><inline-graphic xlink:href="pone.0018731.e181.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is equal
to .43. The estimated standard deviation of the residuals is equal to
0.218 on the scale of the tangent of the slant angle. Given that mean
perceived slant is 0.493, the coefficient of variation is equal to
<inline-formula><inline-graphic xlink:href="pone.0018731.e182.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 0.44. (c)
<italic>Active-viewing with a physical surface</italic>
. For a
no-intercept model with the square root of <italic>def</italic>
as the
only predictor, <inline-formula><inline-graphic xlink:href="pone.0018731.e183.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .72. For a baseline model
with only the individual differences, <inline-formula><inline-graphic xlink:href="pone.0018731.e184.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .44. If head translation
velocity is added to the model including <italic>def</italic>
as
predictor, <inline-formula><inline-graphic xlink:href="pone.0018731.e185.jpg" mimetype="image"></inline-graphic>
</inline-formula>
increases
to .74; <inline-formula><inline-graphic xlink:href="pone.0018731.e186.jpg" mimetype="image"></inline-graphic>
</inline-formula>
increases
to .75 if the interaction between the two predictors is allowed. Even
though this increase in the model's fit is statistically
significant, <inline-formula><inline-graphic xlink:href="pone.0018731.e187.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 30.15,
<inline-formula><inline-graphic xlink:href="pone.0018731.e188.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001, the effect size (as
measured by <inline-formula><inline-graphic xlink:href="pone.0018731.e189.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) is very
small. No improvement of fit is found when adding the simulated Slant
predictor, <inline-formula><inline-graphic xlink:href="pone.0018731.e190.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 0.42,
<inline-formula><inline-graphic xlink:href="pone.0018731.e191.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .518. In the simpler
(no-intercept) model with the <inline-formula><inline-graphic xlink:href="pone.0018731.e192.jpg" mimetype="image"></inline-graphic>
</inline-formula>
predictor,
the slope is 2.41, <inline-formula><inline-graphic xlink:href="pone.0018731.e193.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= 12.29,
<inline-formula><inline-graphic xlink:href="pone.0018731.e194.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .001.</p>
</caption>
<graphic xlink:href="pone.0018731.g006"></graphic>
</fig>
</sec>
<sec id="s2b"><title>Implementation of the Bayesian models</title>
<sec id="s2b1"><title>Estimation of surface slant by taking into account head translation
velocity</title>
<p><xref ref-type="fig" rid="pone-0018731-g007">Figure 7</xref>
illustrates the
process of slant estimation according to Eq. 14: The Bayesian estimator is
plotted as a function of simulated slant (bottom left panel) and as a
function of <inline-formula><inline-graphic xlink:href="pone.0018731.e195.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(bottom right
panel). When the head translation velocity <inline-formula><inline-graphic xlink:href="pone.0018731.e196.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is measured
accurately, the Bayesian estimator is veridical; as the uncertainty about
<inline-formula><inline-graphic xlink:href="pone.0018731.e197.jpg" mimetype="image"></inline-graphic>
</inline-formula>
increases, the Bayesian estimator becomes more and
more biased. The slant estimates expressed as a function of
<inline-formula><inline-graphic xlink:href="pone.0018731.e198.jpg" mimetype="image"></inline-graphic>
</inline-formula>
lie on two different curves, regardless of the size
of <inline-formula><inline-graphic xlink:href="pone.0018731.e199.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(bottom right panel). The offset between these two
curves depends on the amount of measurement error and on the mean of the
distribution <inline-formula><inline-graphic xlink:href="pone.0018731.e200.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, which depends
on the head translation velocity.</p>
<fig id="pone-0018731-g007" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g007</object-id>
<label>Figure 7</label>
<caption><title>Monte Carlo simulation for the model of <bold>Eq.
14</bold>
.</title>
<p><bold>Top panels</bold>
: posterior distributions
<inline-formula><inline-graphic xlink:href="pone.0018731.e201.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for
five simulated slant magnitudes. Intensity corresponds to
probability. The posterior distributions are computed by setting the
mean of the Gaussian distribution <inline-formula><inline-graphic xlink:href="pone.0018731.e202.jpg" mimetype="image"></inline-graphic>
</inline-formula>
to the
values of 0.32 or 0.07 rad/s. These values correspond to the
empirical average of, respectively, the normal (right) or slow
(left) head translation velocity in our experiment. <bold>Middle
panels</bold>
: marginalization of the posterior distributions
over <inline-formula><inline-graphic xlink:href="pone.0018731.e203.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The
simulated slant magnitudes are coded by color, ranging from black
(20<inline-formula><inline-graphic xlink:href="pone.0018731.e204.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) to
saturated blue/red (80<inline-formula><inline-graphic xlink:href="pone.0018731.e205.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). The
dotted lines identify the medians of the marginalized posterior
distributions. <bold>Bottom panels</bold>
. Optimal estimation for
surface slant plotted as a function of the simulated slant
magnitudes (left) and <inline-formula><inline-graphic xlink:href="pone.0018731.e206.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(right). Full lines: results obtained by setting
<inline-formula><inline-graphic xlink:href="pone.0018731.e207.jpg" mimetype="image"></inline-graphic>
</inline-formula>
;
dashed lines: results obtained by setting
<inline-formula><inline-graphic xlink:href="pone.0018731.e208.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or
<inline-formula><inline-graphic xlink:href="pone.0018731.e209.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Blue
and red colors code the slow and fast head translation velocities,
respectively.</p>
</caption>
<graphic xlink:href="pone.0018731.g007"></graphic>
</fig>
</sec>
<sec id="s2b2"><title>Estimation of surface slant from retinal information alone</title>
<p><xref ref-type="fig" rid="pone-0018731-g008">Figure 8</xref>
illustrates the
process of slant estimation according to Eq. 15: The Bayesian estimator is
plotted as a function of simulated slant (bottom left panel) and as a
function of <inline-formula><inline-graphic xlink:href="pone.0018731.e210.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(bottom right
panel). When plotted as a function of <inline-formula><inline-graphic xlink:href="pone.0018731.e211.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the slant
estimates for the two translation velocities lie on the same straight line.
The parameter <inline-formula><inline-graphic xlink:href="pone.0018731.e212.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the slope
of the linear relation between perceived slant and
<inline-formula><inline-graphic xlink:href="pone.0018731.e213.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<fig id="pone-0018731-g008" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g008</object-id>
<label>Figure 8</label>
<caption><title>Monte Carlo simulation for the model of <bold>Eq.
15</bold>
.</title>
<p>The Monte Carlo simulation was carried out by setting the stimulus
parameters to the same values as in <xref ref-type="fig" rid="pone-0018731-g007">Figure 7</xref>
. <bold>Top
panels</bold>
: posterior distributions
<inline-formula><inline-graphic xlink:href="pone.0018731.e214.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The
outputs of the simulations for the slow and fast head translation
velocities are shown on the left and on the right, respectively.
<bold>Middle panels</bold>
: marginalization of the posterior
distributions over <inline-formula><inline-graphic xlink:href="pone.0018731.e215.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The
dotted lines identify the medians of the marginalized posterior
distributions. <bold>Bottom panel</bold>
: estimates of surface slant
plotted as a function of simulated slant (left) and as a function of
<inline-formula><inline-graphic xlink:href="pone.0018731.e216.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(right). Blue and red colors code the slow and fast head translation
velocities, respectively. The parameter
<inline-formula><inline-graphic xlink:href="pone.0018731.e217.jpg" mimetype="image"></inline-graphic>
</inline-formula>
represents the square root of the ratio between the standard
deviations of the prior distributions for
<inline-formula><inline-graphic xlink:href="pone.0018731.e218.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula><inline-graphic xlink:href="pone.0018731.e219.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
</caption>
<graphic xlink:href="pone.0018731.g008"></graphic>
</fig>
</sec>
<sec id="s2b3"><title>Perceived surface slant and Bayesian modeling</title>
<p>Eq. 15 offers a clear advantage over Eq. 14 in predicting the observers'
responses (see <xref ref-type="fig" rid="pone-0018731-g009">Figure
9</xref>
). If the uncertainty about <inline-formula><inline-graphic xlink:href="pone.0018731.e220.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is not
negligible, the Bayesian estimates of Eq. 14 expressed as a function of
<inline-formula><inline-graphic xlink:href="pone.0018731.e221.jpg" mimetype="image"></inline-graphic>
</inline-formula>
lie on two separate curves and are unable to
reproduce the qualitative trends in the experimental data (see <xref ref-type="fig" rid="pone-0018731-g005">Figures 5</xref>
, <xref ref-type="fig" rid="pone-0018731-g006">6</xref>
, and <xref ref-type="fig" rid="pone-0018731-g007">7</xref>
, <xref ref-type="fig" rid="pone-0018731-g008">8</xref>
). This lack of fit can be contrasted
with the excellent correspondence between the slant estimates of Eq. 15 and
the observers' judgments.</p>
<fig id="pone-0018731-g009" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g009</object-id>
<label>Figure 9</label>
<caption><title>Perceived slant as a function of the predictions of <bold>Eq.
15</bold>
.</title>
<p>Points of the same color represent different simulated Slant
magnitudes. The predicted values are obtained by setting
<inline-formula><inline-graphic xlink:href="pone.0018731.e222.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(the
square root of the ratio between the standard deviations of the
prior distributions for <inline-formula><inline-graphic xlink:href="pone.0018731.e223.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula><inline-graphic xlink:href="pone.0018731.e224.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) equal
to the slope of the linear relation between perceived slant and
<inline-formula><inline-graphic xlink:href="pone.0018731.e225.jpg" mimetype="image"></inline-graphic>
</inline-formula>
in
each viewing condition. AVV: Active Viewing of a Virtual surface;
PVV: Passive Vision of a Virtual surface; AVP: Active Viewing of a
Physical surface. For the data in the figure, the least-squares
regression line has an intercept of −0.03, 95%
<italic>C.I.</italic>
[−0.09, 0.02], and a slope
of 1.07, 95% <italic>C.I.</italic>
[0.97, 1.16];
<inline-formula><inline-graphic xlink:href="pone.0018731.e226.jpg" mimetype="image"></inline-graphic>
</inline-formula>
= .95.</p>
</caption>
<graphic xlink:href="pone.0018731.g009"></graphic>
</fig>
</sec>
</sec>
</sec>
<sec id="s3"><title>Discussion</title>
<p>Under some assumptions, the optic flow can be used, together with other signals, to
infer both the ordinal properties (<italic>e.g.</italic>
, tilt) and the Euclidean
metric properties (<italic>e.g.</italic>
, slant) of the visual scene. By using
sophisticated head-tracking techniques with high spatiotemporal resolution, we
manipulated the information content of the stimuli to generate optic flows
corresponding to (a) the active viewing of a virtual surface, (b) the passive
viewing of a virtual surface, and (c) the active viewing of a physical surface. We
also varied the head translation velocity (normal or slower). The observers'
judgments of perceived surface slant were then compared to the Bayesian estimates
computed with and without taking into account the translational velocity of the head
(Eqs. 14 and 15, <xref ref-type="fig" rid="pone-0018731-g007">Figures 7</xref>
and
<xref ref-type="fig" rid="pone-0018731-g008">8</xref>
).</p>
<p>The observers' responses are markedly different from the Bayesian estimates
derived by combining optic flow and head velocity information (Eq. 14, <xref ref-type="fig" rid="pone-0018731-g005">Figures 5</xref>
and <xref ref-type="fig" rid="pone-0018731-g006">6</xref>
). The empirical data from the active and
passive viewing of virtual planar surfaces, conversely, are consistent with the
Bayesian estimates computed without considering head velocity information (Eq. 15,
<xref ref-type="fig" rid="pone-0018731-g009">Figures 9</xref>
).</p>
<p>For the slant judgments of physical planar surfaces, the Bayesian model of Eq. 15
explains a large amount of the variance, but a very small portion of additional
variance is accounted for by the head translation velocity (<xref ref-type="fig" rid="pone-0018731-g006">Figure 6</xref>
, bottom panel). The Bayesian model of
Eq. 14, which takes head velocity into account, fits the data much worse. In the
present research, this effect is small but warrants further research. In a follow-up
experiment (not described here), we found that the monocular cues provided by our
physical stimuli were not sufficient for an immobile observer to successfully
discriminate between two surfaces slanted +45 <inline-formula><inline-graphic xlink:href="pone.0018731.e227.jpg" mimetype="image"></inline-graphic>
</inline-formula>
or −45
<inline-formula><inline-graphic xlink:href="pone.0018731.e228.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(surface tilt was constant). Together with the findings of
our main experiment, these results suggest that, although uninformative by
themselves, monocular cues can produce some form of “enhancement” of the
perceptual response, when they are presented together with the optic flow and with
vestibular and proprioceptive information <xref ref-type="bibr" rid="pone.0018731-Domini8">[30]</xref>
.</p>
<p>The slope of the linear relation between perceived surface slant and
<inline-formula><inline-graphic xlink:href="pone.0018731.e229.jpg" mimetype="image"></inline-graphic>
</inline-formula>
varies across the three viewing conditions: it is shallower
for the passive viewing of a virtual planar surface, it increases for the active
viewing of a virtual surface, and it is the largest for the active viewing of a
physical surface. We may expect a different visual performance for passive and
active SfM, and for virtual and physical stimuli. The present results suggest,
however, that more complete stimulus information does not necessarily result in
better (more veridical) performance: <italic>A stronger effect of def does not
guarantee</italic>
a more accurate response. Perceived slant is strongly
affected by <italic>def</italic>
despite the fact that there is no a
“one-to-one” correspondence between <italic>def</italic>
and distal
surface slant.</p>
<p>Animal studies <xref ref-type="bibr" rid="pone.0018731-Bradley1">[31]</xref>
and human experiments <xref ref-type="bibr" rid="pone.0018731-Orban1">[32]</xref>
, <xref ref-type="bibr" rid="pone.0018731-Vanduffel1">[33]</xref>
identify MT (MT+ in
humans) as the brain area involved in SfM processing. It has also been shown that
MST integrates MT inputs with vestibular signals originating from a different
(currently unidentified) neural pathway <xref ref-type="bibr" rid="pone.0018731-Assad1">[34]</xref>
, <xref ref-type="bibr" rid="pone.0018731-Gu1">[35]</xref>
. The integration of visual and
vestibular information in MSTd is consistent with both the Bayesian models discussed
here (Eqs. 14 and 15). Such integration could mean that (a) the visual system uses
extra-retinal signals to discount head motion from the optic flow in order to encode
a world-centered representation of the 3D objects <xref ref-type="bibr" rid="pone.0018731-vanBoxtel1">[11]</xref>
, or (b) non-visual
information about self-motion is used as a retinal stabilization factor for a better
measurement of the optic flow <xref ref-type="bibr" rid="pone.0018731-CornilleauPeres1">[36]</xref>
, <xref ref-type="bibr" rid="pone.0018731-Fantoni1">[37]</xref>
. The present behavioral
results, however, favor this second hypothesis.</p>
<p>In conclusion, the present data and simulations do not indicate so much that, by
disregarding vestibular and proprioceptive information, the visual system uses a
suboptimal strategy for estimating surface slant from the self-generated optic flow.
Instead, they suggest that, even though it does not always guarantee a veridical
solution to the SfM problem, the mapping between the deformation component of the
optic flow and the perceived surface slant may be the most efficient choice for a
biological system <xref ref-type="bibr" rid="pone.0018731-DiLuca1">[38]</xref>
, <xref ref-type="bibr" rid="pone.0018731-Thaler1">[39]</xref>
. An issue that remains to be investigated is whether and
how learning provides effective visual and haptic feedback for scaling
<italic>def</italic>
information.</p>
</sec>
<sec sec-type="methods" id="s4"><title>Methods</title>
<sec id="s4a"><title>Ethics Statement</title>
<p>Experiments were undertaken with the understanding and written consent of each
subject, with the approval of the Comitato Etico per la Sperimentazione con
l'Essere Umano of the University of Trento, and in compliance with national
legislation and the Code of Ethical Principles for Medical Research Involving
Human Subjects of the World Medical Association (Declaration of Helsinki).</p>
</sec>
<sec id="s4b"><title>Participants</title>
<p>Thirty-four undergraduate students at the University of Parma, Italy,
participated in this experiment. All participants were naïve to the
purposes of the experiment and had normal or corrected-to-normal vision.</p>
</sec>
<sec id="s4c"><title>Apparatus</title>
<p>The orientation of the participant's head and the translational head
displacements were recorded by an Optotrak 3020 Certus system. Two sensors
recovered the 3D position data of two infrared emitting diodes (markers on an
eyeglass frame) aligned with the observer inter-ocular axis. The signals emitted
by the markers were used to calculate the <inline-formula><inline-graphic xlink:href="pone.0018731.e230.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula><inline-graphic xlink:href="pone.0018731.e231.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, <inline-formula><inline-graphic xlink:href="pone.0018731.e232.jpg" mimetype="image"></inline-graphic>
</inline-formula>
coordinates of the
observers' viewpoints in order to update the geometrical projection of a
random-dot planar surface in real time. Displays were monocularly viewed through
a high-quality front-silvered mirror (150 <inline-formula><inline-graphic xlink:href="pone.0018731.e233.jpg" mimetype="image"></inline-graphic>
</inline-formula>
150 mm) placed at
eye-height in front of the observer's central viewing position and slanted
45 <inline-formula><inline-graphic xlink:href="pone.0018731.e234.jpg" mimetype="image"></inline-graphic>
</inline-formula>
away from the monitor and the observer's
inter-ocular axis. The effective distance from the pupil to the center of the
screen was 860 mm. Only the central portion of the surface was left visible to
the observer through a black mask with an irregularly-shaped central aperture
(about 70 <inline-formula><inline-graphic xlink:href="pone.0018731.e235.jpg" mimetype="image"></inline-graphic>
</inline-formula>
70 mm) superimposed on the screen. A chin-rest was used
to prevent head movements in the passive-vision condition.</p>
<p>A custom Visual C++ program supported by OpenGL libraries and Optotrak
API routines was used for stimulus presentation and response recording. The same
program also controlled the orientation of a physical planar surface that, in a
separate block of trials, was placed at a distance of 760 mm in front of the
observer. The boundary of the physical surface was occluded by the same mask
used for the virtual displays. This aperture was closed when the surface's
orientation was changed.</p>
</sec>
<sec id="s4d"><title>Stimuli</title>
<p>The simulated displays were random arrangements of (1
<inline-formula><inline-graphic xlink:href="pone.0018731.e236.jpg" mimetype="image"></inline-graphic>
</inline-formula>
1 mm) antialiased red dots simulating the projection of
a static planar surface centered on the image screen and with a variable slant
about the horizontal axis (virtual planar surfaces:
20<inline-formula><inline-graphic xlink:href="pone.0018731.e237.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, 35<inline-formula><inline-graphic xlink:href="pone.0018731.e238.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
50<inline-formula><inline-graphic xlink:href="pone.0018731.e239.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, 65<inline-formula><inline-graphic xlink:href="pone.0018731.e240.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
80<inline-formula><inline-graphic xlink:href="pone.0018731.e241.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; physical planar surfaces:
10<inline-formula><inline-graphic xlink:href="pone.0018731.e242.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, 20<inline-formula><inline-graphic xlink:href="pone.0018731.e243.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
40<inline-formula><inline-graphic xlink:href="pone.0018731.e244.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and 50<inline-formula><inline-graphic xlink:href="pone.0018731.e245.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). The surface tilt
was constant (90<inline-formula><inline-graphic xlink:href="pone.0018731.e246.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). About 100 dots
were visible through the irregular aperture occluding the outer part of the
screen. To remove texture (non-motion) cues, the dots were randomly distributed
into the projected image (not on the simulated surface). On each frame of the
stimulus sequence, the 2D arrangement of the dots was varied depending on the
observer's head position and orientation with respect to the simulated
surface. The dots on the simulated planar surface were projected onto the image
plane (CRT screen) by using a generalized perspective pinhole model with the
observer's right eye position as the center of projections. The position of
the observer's right eye was sampled at the same rate as the monitor
refresh and stimulus update rate.</p>
<p>The translation of the observer's head produced a relative rotation of the
simulated planar surface of about 3.32<inline-formula><inline-graphic xlink:href="pone.0018731.e247.jpg" mimetype="image"></inline-graphic>
</inline-formula>
about the vertical
axis, regardless of surface slant. The maximum lateral head shift was equal to
50 mm. In the passive-vision condition, the optic flows were generated by
replaying the 2D transformations generated by the corresponding active-vision
trials. The horizontal translation component of the optic flow was removed by
assuming that the cyclopean line of sight of the active observers was always
aligned with the centre on the planar surface, regardless of actual head
position and surface slant <xref ref-type="bibr" rid="pone.0018731-Fantoni1">[37]</xref>
.</p>
<p>The physical planar surface was painted black and randomly covered with
phosphorescent dots. With respect to the virtual surface, the physical surface
was covered by larger dots (about 5 mm) having an irregular shape, a lower
density (about 13 dots were visible through the irregular aperture), and
providing texture cues (<italic>i.e.</italic>
, dot foreshortening) consistent
with a slanted 3D planar surface. Given the smaller viewing distance (760 mm),
the constant amount of lateral head shift produced a relative rotation of the
surface about the vertical axis of 3.76<inline-formula><inline-graphic xlink:href="pone.0018731.e248.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>During the experiment, the room was completely dark. Peak head translation
velocity was either 285.6 mm/s or 57.7 mm/s. Depending on the head translation
velocity, on each trial the stimulus was visible for about 3.0 s or 11.1 s.</p>
</sec>
<sec id="s4e"><title>Design</title>
<p>Each observer participated in three experimental blocks in the following order:
Active-Vision with a Virtual surface (AVV), Passive-Vision with a Virtual
surface (PVV), and Active-Vision with a Physical surface (AVP). Participants
were randomly assigned either to the “normal” or to the
“slow” head translation velocity conditions. Each AVV and PVV block
comprised 25 trials (5 repetitions of 5 simulated slants magnitudes). The AVP
block comprised 16 trials (4 repetitions of 4 slant magnitudes). In the PVV
block, the stimuli generated in the AVV block were shown again in random order.
The completion of each block of trials required about 30 minutes.</p>
</sec>
<sec id="s4f"><title>Procedure</title>
<p>Participants were tested individually in total darkness, so that only the
stimulus displays shown on the CRT screen, or the luminous dots on the physical
surface, were visible. In the AVV and AVP blocks, observers viewed the stimuli
while making back-forth lateral head translations. The observer's head was
supported by an horizontally extended chin-rest allowing lateral movements of
<inline-formula><inline-graphic xlink:href="pone.0018731.e249.jpg" mimetype="image"></inline-graphic>
</inline-formula>
60 mm. An acoustic feedback signaled whether the average
head shift velocity exceeded the range of 83 mm/s
<inline-formula><inline-graphic xlink:href="pone.0018731.e250.jpg" mimetype="image"></inline-graphic>
</inline-formula>
40 mm/s (“normal” speed) or 20 mm/s
<inline-formula><inline-graphic xlink:href="pone.0018731.e251.jpg" mimetype="image"></inline-graphic>
</inline-formula>
10 mm/s (“slow” speed). The stimulus display
appeared on the screen when participants completed 2 consecutive back-and-forth
translations at the required velocity and disappeared after 5.5 back-and-forth
translations. After the stimulus disappeared, participants stopped moving their
head and provided a verbal judgment of the amount of perceived surface slant
(0° indicating a frontal-parallel surface, 90° indicating a surface
parallel to the <inline-formula><inline-graphic xlink:href="pone.0018731.e252.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,<inline-formula><inline-graphic xlink:href="pone.0018731.e253.jpg" mimetype="image"></inline-graphic>
</inline-formula>
plane) – see
<xref ref-type="fig" rid="pone-0018731-g010">Figure 10</xref>
. In the PVV
condition, participants were required to remain still for the entire duration of
each trial.</p>
<fig id="pone-0018731-g010" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g010</object-id>
<label>Figure 10</label>
<caption><title>Temporal structure of the sequence in each trial.</title>
<p>Schematic representation of one trial of our experiment. Here we consider
the case of the acting viewing of a virtual surface.</p>
</caption>
<graphic xlink:href="pone.0018731.g010"></graphic>
</fig>
<p>Each experimental session was preceded by a preparatory session in which the
participant's inter-pupillary distance was measured, the instructions were
provided, and training about the appropriate head translation velocity and the
magnitude estimation task was provided. Participants were trained in the
magnitude estimation task by completing two blocks of 20 trials each. In one
block, they were required to generate an angle between two segments on a
computer screen after being prompted by a random number in the range
0–360. In the other block, they were required to estimate a random angle
depicted to the screen. The relationship between the response and the test
values was analyzed with a linear regression. Only participants who met
performance criteria of a slope in the interval [0.9, 1.1] and an
intercept in the interval [−0.3, 0.3] entered the experimental
session.</p>
<p>The maximum value of <italic>def</italic>
was extracted in each trial from the
instantaneous profile of the deformation component of the optic flow by
following the procedure illustrated in <xref ref-type="fig" rid="pone-0018731-g011">Figure 11</xref>
. These <italic>def</italic>
values
were then used to test the prediction of Eq. 15.</p>
<fig id="pone-0018731-g011" position="float"><object-id pub-id-type="doi">10.1371/journal.pone.0018731.g011</object-id>
<label>Figure 11</label>
<caption><title>Example of instantaneous <italic>def</italic>
as a function of
time.</title>
<p>The maximum of this temporal profile (averaged across the
<italic>def</italic>
cycles) was computed for each trial of the
experiment and was used to test the Bayesian model of Eq. 15.</p>
</caption>
<graphic xlink:href="pone.0018731.g011"></graphic>
</fig>
</sec>
<sec id="s4g"><title>Statistical Analyses</title>
<p>Statistical analyses were performed by means of Linear Mixed-Effects models with
participants as random effects and <inline-formula><inline-graphic xlink:href="pone.0018731.e254.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, simulated slant,
and head translation velocity (“normal”, “slow”) as
fixed effects. We evaluate significance by computing the deviance statistic
(minus 2 times the log-likelihood; change in deviance is distributed as
chi-square, with degrees of freedom equal to the number of parameters deleted
from the model) and with the help of 10,000 samples from the posterior
distributions of the coefficients using Markov chain Monte Carlo sampling. From
these samples, we obtained the 95% Highest Posterior Density confidence
intervals, and the corresponding two-tailed <inline-formula><inline-graphic xlink:href="pone.0018731.e255.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-values. Several
indexes have been proposed to measure the prediction power and the
goodness-of-fit for linear mixed models (<italic>e.g.</italic>
, Sun, Zhu,
Kramer, Yang, Song, Piepho, & Yu, 2010). Here, we measure the goodness of
fit as <inline-formula><inline-graphic xlink:href="pone.0018731.e256.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where <inline-formula><inline-graphic xlink:href="pone.0018731.e257.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is an
<inline-formula><inline-graphic xlink:href="pone.0018731.e258.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula><inline-graphic xlink:href="pone.0018731.e259.jpg" mimetype="image"></inline-graphic>
</inline-formula>
1 vector, <inline-formula><inline-graphic xlink:href="pone.0018731.e260.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are the fitted
values, <inline-formula><inline-graphic xlink:href="pone.0018731.e261.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the mean of <inline-formula><inline-graphic xlink:href="pone.0018731.e262.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula><inline-graphic xlink:href="pone.0018731.e263.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the mean of <inline-formula><inline-graphic xlink:href="pone.0018731.e264.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(Vonesh,
Chinchilli, & Pu, 1996). The <inline-formula><inline-graphic xlink:href="pone.0018731.e265.jpg" mimetype="image"></inline-graphic>
</inline-formula>
statistic can be
interpreted as a measure of the degree of agreement between the observed values
and the predicted values. The possible values of <inline-formula><inline-graphic xlink:href="pone.0018731.e266.jpg" mimetype="image"></inline-graphic>
</inline-formula>
lie in the range
<inline-formula><inline-graphic xlink:href="pone.0018731.e267.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
</sec>
</sec>
</body>
<back><ack><p>We thank John A. Assad for helpful comments on a previous version of the
manuscript.</p>
</ack>
<fn-group><fn fn-type="conflict"><p><bold>Competing Interests: </bold>
The authors have declared that no competing interests exist.</p>
</fn>
<fn fn-type="financial-disclosure"><p><bold>Funding: </bold>
The authors have no support or funding to report.</p>
</fn>
</fn-group>
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