Where is the light? Bayesian perceptual priors for lighting direction
Identifieur interne : 001F91 ( Pmc/Checkpoint ); précédent : 001F90; suivant : 001F92Where is the light? Bayesian perceptual priors for lighting direction
Auteurs : J. V. Stone [Royaume-Uni] ; I. S. Kerrigan [Royaume-Uni] ; J. Porrill [Royaume-Uni]Source :
- Proceedings of the Royal Society B: Biological Sciences [ 0962-8452 ] ; 2009.
Abstract
Perception of shaded three-dimensional figures is inherently ambiguous, but this ambiguity can be resolved if the brain assumes that figures are lit from a specific direction. Under the Bayesian framework, the visual system assigns a weighting to each possible direction, and these weightings define a prior probability distribution for light-source direction. Here, we describe a non-parametric maximum-likelihood estimation method for finding the prior distribution for lighting direction. Our results suggest that each observer has a distinct prior distribution, with non-zero values in all directions, but with a peak which indicates observers are biased to expect light to come from above left. The implications of these results for estimating general perceptual priors are discussed.
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DOI: 10.1098/rspb.2008.1635
PubMed: 19324801
PubMed Central: 2674484
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Where is the light? Bayesian perceptual priors for lighting direction</title><author><name sortKey="Stone, J V" sort="Stone, J V" uniqKey="Stone J" first="J. V." last="Stone">J. V. Stone</name><affiliation wicri:level="1"><nlm:aff id="aff1"><institution>Department of Psychology, University of Sheffield</institution><addr-line>Sheffield S10 2TP, UK</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Sheffield S10 2TP</wicri:regionArea></affiliation></author><author><name sortKey="Kerrigan, I S" sort="Kerrigan, I S" uniqKey="Kerrigan I" first="I. S." last="Kerrigan">I. S. Kerrigan</name><affiliation wicri:level="1"><nlm:aff id="aff2"><institution>School of Psychology, University of Southampton</institution><addr-line>Southampton SO17 1BJ, UK</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Southampton SO17 1BJ</wicri:regionArea></affiliation></author><author><name sortKey="Porrill, J" sort="Porrill, J" uniqKey="Porrill J" first="J." last="Porrill">J. Porrill</name><affiliation wicri:level="1"><nlm:aff id="aff1"><institution>Department of Psychology, University of Sheffield</institution><addr-line>Sheffield S10 2TP, UK</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Sheffield S10 2TP</wicri:regionArea></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">19324801</idno><idno type="pmc">2674484</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2674484</idno><idno type="RBID">PMC:2674484</idno><idno type="doi">10.1098/rspb.2008.1635</idno><date when="2009">2009</date><idno type="wicri:Area/Pmc/Corpus">000E32</idno><idno type="wicri:Area/Pmc/Curation">000E32</idno><idno type="wicri:Area/Pmc/Checkpoint">001F91</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Where is the light? Bayesian perceptual priors for lighting direction</title><author><name sortKey="Stone, J V" sort="Stone, J V" uniqKey="Stone J" first="J. V." last="Stone">J. V. Stone</name><affiliation wicri:level="1"><nlm:aff id="aff1"><institution>Department of Psychology, University of Sheffield</institution><addr-line>Sheffield S10 2TP, UK</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Sheffield S10 2TP</wicri:regionArea></affiliation></author><author><name sortKey="Kerrigan, I S" sort="Kerrigan, I S" uniqKey="Kerrigan I" first="I. S." last="Kerrigan">I. S. Kerrigan</name><affiliation wicri:level="1"><nlm:aff id="aff2"><institution>School of Psychology, University of Southampton</institution><addr-line>Southampton SO17 1BJ, UK</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Southampton SO17 1BJ</wicri:regionArea></affiliation></author><author><name sortKey="Porrill, J" sort="Porrill, J" uniqKey="Porrill J" first="J." last="Porrill">J. Porrill</name><affiliation wicri:level="1"><nlm:aff id="aff1"><institution>Department of Psychology, University of Sheffield</institution><addr-line>Sheffield S10 2TP, UK</addr-line></nlm:aff><country xml:lang="fr">Royaume-Uni</country><wicri:regionArea>Sheffield S10 2TP</wicri:regionArea></affiliation></author></analytic><series><title level="j">Proceedings of the Royal Society B: Biological Sciences</title><idno type="ISSN">0962-8452</idno><idno type="eISSN">1471-2954</idno><imprint><date when="2009">2009</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Perception of shaded three-dimensional figures is inherently ambiguous, but this ambiguity can be resolved if the brain assumes that figures are lit from a specific direction. Under the Bayesian framework, the visual system assigns a weighting to each possible direction, and these weightings define a prior probability distribution for light-source direction. Here, we describe a non-parametric maximum-likelihood estimation method for finding the prior distribution for lighting direction. Our results suggest that each observer has a distinct prior distribution, with non-zero values in all directions, but with a peak which indicates observers are biased to expect light to come from above left. The implications of these results for estimating general perceptual priors are discussed.</p></div></front></TEI><pmc article-type="research-article" xml:lang="EN"><pmc-comment>The publisher of this article does not allow downloading of the full text in XML form.</pmc-comment>
<front><journal-meta><journal-id journal-id-type="nlm-ta">Proc Biol Sci</journal-id><journal-id journal-id-type="publisher-id">RSPB</journal-id><journal-title>Proceedings of the Royal Society B: Biological Sciences</journal-title><issn pub-type="ppub">0962-8452</issn><issn pub-type="epub">1471-2954</issn><publisher><publisher-name>The Royal Society</publisher-name><publisher-loc>London</publisher-loc></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">19324801</article-id><article-id pub-id-type="pmc">2674484</article-id><article-id pub-id-type="publisher-id">rspb20081635</article-id><article-id pub-id-type="doi">10.1098/rspb.2008.1635</article-id><article-categories><subj-group subj-group-type="heading"><subject>Research Article</subject></subj-group></article-categories><title-group><article-title>Where is the light? Bayesian perceptual priors for lighting direction</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Stone</surname><given-names>J.V.</given-names></name><xref ref-type="aff" rid="aff1">1</xref><xref ref-type="corresp" rid="cor1">*</xref></contrib><contrib contrib-type="author"><name><surname>Kerrigan</surname><given-names>I.S.</given-names></name><xref ref-type="aff" rid="aff2">2</xref></contrib><contrib contrib-type="author"><name><surname>Porrill</surname><given-names>J.</given-names></name><xref ref-type="aff" rid="aff1">1</xref></contrib></contrib-group><aff id="aff1"><label>1</label><institution>Department of Psychology, University of Sheffield</institution><addr-line>Sheffield S10 2TP, UK</addr-line></aff><aff id="aff2"><label>2</label><institution>School of Psychology, University of Southampton</institution><addr-line>Southampton SO17 1BJ, UK</addr-line></aff><author-notes><corresp id="cor1"><label>*</label>Author for correspondence (<email>j.v.stone@shef.ac.uk</email>)</corresp></author-notes><pub-date pub-type="epub"><day>25</day><month>2</month><year>2009</year></pub-date><pub-date pub-type="ppub"><day>22</day><month>5</month><year>2009</year></pub-date><volume>276</volume><issue>1663</issue><fpage>1797</fpage><lpage>1804</lpage><history><date date-type="received"><day>10</day><month>11</month><year>2008</year></date><date date-type="rev-recd"><day>13</day><month>1</month><year>2009</year></date><date date-type="accepted"><day>14</day><month>1</month><year>2009</year></date></history><permissions><copyright-statement>© 2009 The Royal Society</copyright-statement><copyright-year>2009</copyright-year></permissions><abstract><p>Perception of shaded three-dimensional figures is inherently ambiguous, but this ambiguity can be resolved if the brain assumes that figures are lit from a specific direction. Under the Bayesian framework, the visual system assigns a weighting to each possible direction, and these weightings define a prior probability distribution for light-source direction. Here, we describe a non-parametric maximum-likelihood estimation method for finding the prior distribution for lighting direction. Our results suggest that each observer has a distinct prior distribution, with non-zero values in all directions, but with a peak which indicates observers are biased to expect light to come from above left. The implications of these results for estimating general perceptual priors are discussed.</p></abstract><kwd-group><kwd>perception</kwd><kwd>Bayes</kwd><kwd>lighting direction</kwd></kwd-group></article-meta></front><floats-wrap><fig id="fig1" position="float"><label>Figure 1</label><caption><p>Typical stimulus presented to an observer on a single trial. The observer's task is to indicate whether or not the quadrant marked with a cross (×) appears convex or concave. This response implicitly defines the perceived direction of the light source. For example, if the marked quadrant is perceived as convex, then this implies that the light originates from the lower right (i.e. approx. 300°), but if it is perceived as concave, then this implies that the light originates from the upper left (i.e. approx. 120°).</p></caption><graphic xlink:href="rspb20081635f01"></graphic></fig><fig id="fig2" position="float"><label>Figure 2</label><caption><p>Evaluating cross-validation method for estimating the value of the smoothing parameter <italic>λ</italic>. Results for the simulated observer's data shown in (<italic>c</italic>) for <italic>σ</italic>=1. See <xref ref-type="sec" rid="sec16">appendix B</xref>. (<italic>a</italic>) At each sampled value of <italic>λ</italic>, three quarters of the data were used to estimate the prior. Using this prior, the likelihood of the remaining quarter was evaluated using equation <xref ref-type="disp-formula" rid="fd2.23">(2.23)</xref>, where <italic>σ</italic>=1. This was repeated once for each of four disjoint quarters, and the mean of the four resultant likelihood functions is plotted here. The minimum with respect to the negative log likelihood corresponds to <italic>λ</italic>≈700. (<italic>b</italic>) To evaluate the success of cross-validation for each sampled value <italic>λ</italic><sub><italic>j</italic></sub> of <italic>λ</italic>, the Kullback–Leibler (KL) divergence between the known prior <inline-formula><inline-graphic xlink:href="rspb20081635e01.jpg" alternate-form-of="M1"></inline-graphic><mml:math id="M1"><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mi>θ</mml:mi><mml:mo>*</mml:mo></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> of this simulated observer and the prior <inline-formula><inline-graphic xlink:href="rspb20081635e02.jpg" alternate-form-of="M2"></inline-graphic><mml:math id="M2"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mi>θ</mml:mi></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> obtained with <italic>λ</italic><sub><italic>j</italic></sub> was calculated as <inline-formula><inline-graphic xlink:href="rspb20081635e03.jpg" alternate-form-of="M3"></inline-graphic><mml:math id="M3"><mml:mrow><mml:msub><mml:mi>E</mml:mi><mml:mrow><mml:mtext>KL</mml:mtext></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>1000</mml:mn><mml:mo>×</mml:mo><mml:mi mathvariant="normal">Δ</mml:mi><mml:mi>θ</mml:mi><mml:mo>∑</mml:mo><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mn>θ</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mtext>log</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:msub><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mn>θ</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>/</mml:mo><mml:msubsup><mml:mi>p</mml:mi><mml:mi>θ</mml:mi><mml:mo>*</mml:mo></mml:msubsup><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>θ</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mo>,</mml:mo></mml:mrow></mml:math></inline-formula> where <italic>i</italic>=(0, 10, …, 350), <italic>θ</italic><sub><italic>i</italic></sub>=10×<italic>i</italic> and Δ<italic>θ</italic>=10°. The minimum with respect to the KL distance also corresponds to <italic>λ</italic>≈700, confirming that cross-validation chooses a value of <italic>λ</italic>, which provides a good estimate of the true prior. (<italic>c</italic>) Estimating the light-from-above prior <italic>p</italic><sub><italic>θ</italic></sub>(<italic>θ</italic>) for a simulated observer. The lighting direction varies around the circle, and the probability that the stimulus was judged to be convex varies with distance from the origin. The graph shows (i) a sample from the posterior <italic>p</italic>(<italic>c</italic><sub>0</sub>|<bold><italic>x</italic></bold>) as the proportion of convex responses (dashed), (ii) the known prior (dotted), (iii) the estimated prior (solid), based on the proportion of convex responses, as it would be for a human observer, and (iv) the mean vector (solid line), which is the mean of the prior (see <xref ref-type="sec" rid="sec17">appendix C</xref>). The direction of this vector indicates the bias in the prior, and its length shows the amount of bias (see <xref ref-type="sec" rid="sec13">appendix A</xref>). The simulated observer was exposed to the same 36 lighting directions and the same number of trials per lighting direction (32) as the human observers used in the experiment described in the text, a discrimination parameter that was set at <italic>σ</italic><sup>*</sup>=1 dB and a concavity preference set at <italic>p</italic>(<italic>c</italic><sub>1</sub>)=0.5. The value of the smoothing parameter estimated from cross-validation is <italic>λ</italic>=700 (see (<italic>a</italic>) and (<italic>b</italic>)). Using <italic>λ</italic>=700 and <italic>σ</italic>=1, the concavity preference was estimated as <inline-formula><inline-graphic xlink:href="rspb20081635e04.jpg" alternate-form-of="M4"></inline-graphic><mml:math id="M4"><mml:mrow><mml:mover accent="true"><mml:mi>p</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mo stretchy="false">(</mml:mo><mml:msub><mml:mi>c</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mn>0.507</mml:mn></mml:mrow></mml:math></inline-formula>.</p></caption><graphic xlink:href="rspb20081635f02"></graphic></fig><fig id="fig3" position="float"><label>Figure 3</label><caption><p>Cross-validation. Result for estimating the value of the smoothing parameter <italic>λ</italic> for observer <italic>a</italic> in <xref ref-type="fig" rid="fig4">figure 4</xref>, with <italic>σ</italic>=2. The minimum with respect to the negative log likelihood corresponds to <italic>λ</italic>≈400. This curve is typical of that obtained for other observers, and a value of <inline-formula><inline-graphic xlink:href="rspb20081635e05.jpg" alternate-form-of="M5"></inline-graphic><mml:math id="M5"><mml:mrow><mml:mover accent="true"><mml:mi>λ</mml:mi><mml:mo>ˆ</mml:mo></mml:mover><mml:mo>=</mml:mo><mml:mn>400</mml:mn></mml:mrow></mml:math></inline-formula> was therefore used for all human observers.</p></caption><graphic xlink:href="rspb20081635f03"></graphic></fig><fig id="fig4" position="float"><label>Figure 4</label><caption><p>Polar plots of estimated priors for eight observers. Each graph shows the frequency of convex responses (dashed) as a function of light-source direction. This is essentially a sample from the observer's posterior, and is used to estimate the prior (solid). For display purposes, the lengths of all mean vectors (solid line) have been scaled by the same factor across all graphs, and all graphs are drawn to the same scale (see <xref ref-type="sec" rid="sec17">appendix C</xref>). Note that all biases are to the left, with values of 20°, 7°, 9°, 18°, 34°, 14°, 28° and 16°, respectively (mean 18°). These results were obtained using all the data for each observer with <italic>λ</italic>=400 and <italic>σ</italic>=2.</p></caption><graphic xlink:href="rspb20081635f04"></graphic></fig><fig id="fig5" position="float"><label>Figure 5</label><caption><p>The discrimination parameter <italic>σ</italic> is undetermined. (<italic>a</italic>) Graph of an example log prior, log <italic>p</italic><sub><italic>θ</italic></sub>(<italic>θ</italic>) (solid horizontal sinusoid curve), as a function of lighting direction <italic>θ</italic>. Given two hypothetical neurons with preferred lighting directions <italic>θ</italic> and <inline-formula><inline-graphic xlink:href="rspb20081635e06.jpg" alternate-form-of="M6"></inline-graphic><mml:math id="M6"><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></inline-formula>, their responses are determined by their log probability density functions, log <italic>p</italic>(<italic>θ</italic>) and <inline-formula><inline-graphic xlink:href="rspb20081635e07.jpg" alternate-form-of="M7"></inline-graphic><mml:math id="M7"><mml:mrow><mml:mtext>log</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, indicated by the vertical dashed and solid curves, respectively. For a given stimulus, the larger of the two observed values from the probability density functions log <italic>p</italic>(<italic>θ</italic>) and <inline-formula><inline-graphic xlink:href="rspb20081635e08.jpg" alternate-form-of="M8"></inline-graphic><mml:math id="M8"><mml:mrow><mml:mtext>log</mml:mtext><mml:mspace width="0.25em"></mml:mspace><mml:mi>p</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mover accent="true"><mml:mi>θ</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> determines the lighting direction assumed by the observer, and this, in turn, determines the concave/convex observer response. These two observed values are noisy estimates of the probability density function means, so the choice probability <italic>q</italic> (see equation <xref ref-type="disp-formula" rid="fd2.19">(2.19)</xref>) is determined by the relative overlap of the probability density functions (vertical dashed and solid curves) for these two quantities. (<italic>b</italic>) A log prior with amplitude variations <italic>k</italic> times smaller than in (<italic>a</italic>) leads to the same choice probabilities as in (<italic>a</italic>), provided the noise level <italic>σ</italic> is also reduced by a factor <italic>k</italic>. (For simplicity, this analysis assumes a concavity preference of 0.5.)</p></caption><graphic xlink:href="rspb20081635f05"></graphic></fig></floats-wrap></pmc><affiliations><list><country><li>Royaume-Uni</li></country></list><tree><country name="Royaume-Uni"><noRegion><name sortKey="Stone, J V" sort="Stone, J V" uniqKey="Stone J" first="J. V." last="Stone">J. V. Stone</name></noRegion><name sortKey="Kerrigan, I S" sort="Kerrigan, I S" uniqKey="Kerrigan I" first="I. S." last="Kerrigan">I. S. Kerrigan</name><name sortKey="Porrill, J" sort="Porrill, J" uniqKey="Porrill J" first="J." last="Porrill">J. Porrill</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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