Perceived azimuth direction is exaggerated: Converging evidence from explicit and implicit measures
Identifieur interne : 000072 ( Pmc/Checkpoint ); précédent : 000071; suivant : 000073Perceived azimuth direction is exaggerated: Converging evidence from explicit and implicit measures
Auteurs : Zhi Li [États-Unis] ; Frank H. Durgin [États-Unis]Source :
- Journal of Vision [ 1534-7362 ] ; 2016.
Abstract
Recent observations suggest that perceived visual direction in the sagittal plane (angular direction in elevation, both upward and downward from eye level) is exaggerated. Foley, Ribeiro-Filho, and Da Silva's (
Url:
DOI: 10.1167/16.1.4
PubMed: 26756174
PubMed Central: 4743713
Affiliations:
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<record><TEI><teiHeader><fileDesc><titleStmt><title xml:lang="en">Perceived azimuth direction is exaggerated: Converging evidence from explicit and implicit measures</title><author><name sortKey="Li, Zhi" sort="Li, Zhi" uniqKey="Li Z" first="Zhi" last="Li">Zhi Li</name><affiliation wicri:level="2"><nlm:aff id="aff1">Department of Psychology, Swarthmore College, Swarthmore, PA</nlm:aff><country xml:lang="fr">États-Unis</country><placeName><region type="state">Pennsylvanie</region></placeName><wicri:cityArea>Department of Psychology, Swarthmore College, Swarthmore</wicri:cityArea></affiliation></author><author><name sortKey="Durgin, Frank H" sort="Durgin, Frank H" uniqKey="Durgin F" first="Frank H." last="Durgin">Frank H. Durgin</name><affiliation wicri:level="2"><nlm:aff id="aff1">Department of Psychology, Swarthmore College, Swarthmore, PA</nlm:aff><country xml:lang="fr">États-Unis</country><placeName><region type="state">Pennsylvanie</region></placeName><wicri:cityArea>Department of Psychology, Swarthmore College, Swarthmore</wicri:cityArea></affiliation></author></titleStmt><publicationStmt><idno type="wicri:source">PMC</idno><idno type="pmid">26756174</idno><idno type="pmc">4743713</idno><idno type="url">http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4743713</idno><idno type="RBID">PMC:4743713</idno><idno type="doi">10.1167/16.1.4</idno><date when="2016">2016</date><idno type="wicri:Area/Pmc/Corpus">000523</idno><idno type="wicri:Area/Pmc/Curation">000523</idno><idno type="wicri:Area/Pmc/Checkpoint">000072</idno></publicationStmt><sourceDesc><biblStruct><analytic><title xml:lang="en" level="a" type="main">Perceived azimuth direction is exaggerated: Converging evidence from explicit and implicit measures</title><author><name sortKey="Li, Zhi" sort="Li, Zhi" uniqKey="Li Z" first="Zhi" last="Li">Zhi Li</name><affiliation wicri:level="2"><nlm:aff id="aff1">Department of Psychology, Swarthmore College, Swarthmore, PA</nlm:aff><country xml:lang="fr">États-Unis</country><placeName><region type="state">Pennsylvanie</region></placeName><wicri:cityArea>Department of Psychology, Swarthmore College, Swarthmore</wicri:cityArea></affiliation></author><author><name sortKey="Durgin, Frank H" sort="Durgin, Frank H" uniqKey="Durgin F" first="Frank H." last="Durgin">Frank H. Durgin</name><affiliation wicri:level="2"><nlm:aff id="aff1">Department of Psychology, Swarthmore College, Swarthmore, PA</nlm:aff><country xml:lang="fr">États-Unis</country><placeName><region type="state">Pennsylvanie</region></placeName><wicri:cityArea>Department of Psychology, Swarthmore College, Swarthmore</wicri:cityArea></affiliation></author></analytic><series><title level="j">Journal of Vision</title><idno type="eISSN">1534-7362</idno><imprint><date when="2016">2016</date></imprint></series></biblStruct></sourceDesc></fileDesc><profileDesc><textClass></textClass></profileDesc></teiHeader><front><div type="abstract" xml:lang="en"><p>Recent observations suggest that perceived visual direction in the sagittal plane (angular direction in elevation, both upward and downward from eye level) is exaggerated. Foley, Ribeiro-Filho, and Da Silva's (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>) study of perceived size of exocentric ground extent implies that perceived angular direction in azimuth may also be exaggerated. In the present study, we directly examined whether perceived azimuth direction is overestimated. In <xref ref-type="sec" rid="s2">Experiment 1</xref>, numeric estimates of azimuth direction (−48° to 48° relative to straight ahead) were obtained. The results showed a linear exaggeration in perceived azimuth direction with a gain of about 1.26. In <xref ref-type="sec" rid="s3">Experiment 2</xref>, a perceptual extent-matching task served as an implicit measure of perceived azimuth direction. Participants matched an egocentric distance in one direction to a frontal extent in nearly the opposite direction. The angular biases implied by the matching data well replicated Foley et al.'s finding and were also fairly consistent with the azimuth bias function found in <xref ref-type="sec" rid="s2">Experiment 1</xref>, although a slight overall shift was observed between the results of the two experiments. <xref ref-type="sec" rid="s4">Experiment 3</xref>, in which half the observers were tilted sideways while making frontal/depth extent comparisons, suggested that the discrepancy between the results of <xref ref-type="sec" rid="s2">Experiment 1</xref> and <xref ref-type="sec" rid="s3">2</xref> can partially be explained by a retinal horizontal vertical illusion affecting distance estimation tasks. Overall the present study provides converging evidence to suggest that the perception of azimuth direction is overestimated.</p></div></front><back><div1 type="bibliography"><listBibl><biblStruct><analytic><author><name sortKey="Avery, G C" uniqKey="Avery G">G. 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<front><journal-meta><journal-id journal-id-type="nlm-ta">J Vis</journal-id><journal-id journal-id-type="iso-abbrev">J Vis</journal-id><journal-id journal-id-type="hwp">jovi</journal-id><journal-id journal-id-type="pmc">jovi</journal-id><journal-id journal-id-type="publisher-id">jovi</journal-id><journal-title-group><journal-title>Journal of Vision</journal-title></journal-title-group><issn pub-type="epub">1534-7362</issn><publisher><publisher-name>The Association for Research in Vision and Ophthalmology</publisher-name></publisher></journal-meta><article-meta><article-id pub-id-type="pmid">26756174</article-id><article-id pub-id-type="pmc">4743713</article-id><article-id pub-id-type="doi">10.1167/16.1.4</article-id><article-id pub-id-type="sici">jovi-16-01-07</article-id><article-id pub-id-type="publisher-id">MS#: JOV-04769-2015</article-id><article-categories><subj-group subj-group-type="heading"><subject>Article</subject></subj-group></article-categories><title-group><article-title>Perceived azimuth direction is exaggerated: Converging evidence from explicit and implicit measures</article-title></title-group><contrib-group><contrib contrib-type="author"><name><surname>Li</surname><given-names>Zhi</given-names></name><xref ref-type="aff" rid="aff1"></xref><aff><email>zli1@zju.edu.cn</email></aff></contrib><contrib contrib-type="author"><name><surname>Durgin</surname><given-names>Frank H.</given-names></name><xref ref-type="aff" rid="aff1"></xref><aff><email>fdurgin1@swarthmore.edu</email><break></break><ext-link ext-link-type="uri" xlink:href="http://www.swarthmore.edu/SocSci/fdurgin1/publications.html">http://www.swarthmore.edu/SocSci/fdurgin1/publications.html</ext-link></aff></contrib><aff id="aff1">Department of Psychology, Swarthmore College, Swarthmore, PA</aff></contrib-group><author-notes><fn id="n101" fn-type="current-aff"><p>Present address: Department of Psychology and Behavioral Sciences, Zhejiang University, Hangzhou, China</p></fn></author-notes><pub-date pub-type="epub"><day>8</day><month>1</month><year>2016</year></pub-date><pub-date pub-type="collection"><year>2016</year></pub-date><volume>16</volume><issue>1</issue><elocation-id>4</elocation-id><history><date date-type="received"><day>29</day><month>3</month><year>2015</year></date></history><permissions><license license-type="cc-by-nc-nd" xlink:href="http://creativecommons.org/licenses/by-nc-nd/4.0/"><license-p>This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.</license-p></license></permissions><self-uri xlink:title="pdf" xlink:href="i1534-7362-16-1-4.pdf"></self-uri><abstract><p>Recent observations suggest that perceived visual direction in the sagittal plane (angular direction in elevation, both upward and downward from eye level) is exaggerated. Foley, Ribeiro-Filho, and Da Silva's (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>) study of perceived size of exocentric ground extent implies that perceived angular direction in azimuth may also be exaggerated. In the present study, we directly examined whether perceived azimuth direction is overestimated. In <xref ref-type="sec" rid="s2">Experiment 1</xref>, numeric estimates of azimuth direction (−48° to 48° relative to straight ahead) were obtained. The results showed a linear exaggeration in perceived azimuth direction with a gain of about 1.26. In <xref ref-type="sec" rid="s3">Experiment 2</xref>, a perceptual extent-matching task served as an implicit measure of perceived azimuth direction. Participants matched an egocentric distance in one direction to a frontal extent in nearly the opposite direction. The angular biases implied by the matching data well replicated Foley et al.'s finding and were also fairly consistent with the azimuth bias function found in <xref ref-type="sec" rid="s2">Experiment 1</xref>, although a slight overall shift was observed between the results of the two experiments. <xref ref-type="sec" rid="s4">Experiment 3</xref>, in which half the observers were tilted sideways while making frontal/depth extent comparisons, suggested that the discrepancy between the results of <xref ref-type="sec" rid="s2">Experiment 1</xref> and <xref ref-type="sec" rid="s3">2</xref> can partially be explained by a retinal horizontal vertical illusion affecting distance estimation tasks. Overall the present study provides converging evidence to suggest that the perception of azimuth direction is overestimated.</p></abstract><kwd-group><title>Keywords</title><kwd><italic>angular expansion hypothesis</italic></kwd><kwd><italic>perceived visual direction</italic></kwd><kwd><italic>size perception</italic></kwd><kwd><italic>distance perception</italic></kwd><kwd><italic>horizontal vertical illusion</italic></kwd></kwd-group></article-meta></front><body><sec id="s1"><title>Introduction</title><p>The geometrical structure of space is not always perceived veridically, even under full-cue viewing conditions. For example, it has been well documented that distance along a sagittal plane (in-depth distance) is perceptually foreshortened relative to distance along frontal parallel planes (frontal extent). This is known as distance anisotropy (Beusmans <xref rid="i1534-7362-16-1-4-Beusmans1" ref-type="bibr">1998</xref>; Foley, Ribeiro-Filho, & Da Silva, <xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>; Levin & Haber, <xref rid="i1534-7362-16-1-4-Levin1" ref-type="bibr">1993</xref>; Li & Durgin, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>; Li, Phillips, & Durgin, <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>; Li, Sun, Strawser, Spiegel, Klein, & Durgin, <xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>; Loomis, Da Silva, Fujita, & Fukusima, <xref rid="i1534-7362-16-1-4-Loomis1" ref-type="bibr">1992</xref>; Loomis & Philbeck, <xref rid="i1534-7362-16-1-4-Loomis2" ref-type="bibr">1999</xref>; Loomis, Philbeck, & Zahorik, <xref rid="i1534-7362-16-1-4-Loomis4" ref-type="bibr">2002</xref>; Toye, <xref rid="i1534-7362-16-1-4-Toye1" ref-type="bibr">1986</xref>; Wagner, <xref rid="i1534-7362-16-1-4-Wagner1" ref-type="bibr">1985</xref>). Whereas this spatial bias in distance perception has been mostly studied in the context of research on the geometry of visual space (e.g., Foley et al., <xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>; Gilinsky, <xref rid="i1534-7362-16-1-4-Gilinsky1" ref-type="bibr">1951</xref>; Levin & Haber, <xref rid="i1534-7362-16-1-4-Levin1" ref-type="bibr">1993</xref>; Toye, <xref rid="i1534-7362-16-1-4-Toye1" ref-type="bibr">1986</xref>; Wagner, <xref rid="i1534-7362-16-1-4-Wagner1" ref-type="bibr">1985</xref>; for an excellent review on the geometry of visual space, see Wagner, <xref rid="i1534-7362-16-1-4-Wagner2" ref-type="bibr">2006</xref>), it has also provided evidence of perceptual biases in angular variables.</p><p>For example, in studies of anisotropy in perceived exocentric extent (distance between two targets on the ground), a common finding is that the magnitude of the anisotropy increases with viewing distance (e.g., Kudoh, <xref rid="i1534-7362-16-1-4-Kudoh1" ref-type="bibr">2005</xref>; Li & Durgin, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>; Loomis et al., <xref rid="i1534-7362-16-1-4-Loomis1" ref-type="bibr">1992</xref>). Whereas the increased bias with distance is consistent with an affine model of visual space (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>; Wagner, <xref rid="i1534-7362-16-1-4-Wagner1" ref-type="bibr">1985</xref>), it is also in line with the assumption that the distance anisotropy is primarily a result of misperception of optical slant—surface orientation relative to the line of sight (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>, <xref rid="i1534-7362-16-1-4-Li3" ref-type="bibr">2012a</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>). That is, by the principles of shape and size constancy, when an exocentric in-depth ground extent is viewed from a distance, its size can be estimated by its visual angle, the viewing distance, and the optical slant (see <xref ref-type="fig" rid="i1534-7362-16-1-4-f01">Figure 1</xref>). Therefore, overestimation in perceived slant, which is itself distance-dependent (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>), could be partly responsible for the anisotropy in exocentric distance perception. Direct support for the slant account comes from the findings of Loomis and Philbeck (<xref rid="i1534-7362-16-1-4-Loomis2" ref-type="bibr">1999</xref>). By elevating their observers, Loomis and Philbeck showed that when optical slant was kept constant, perceived aspect ratio under monocular viewing became nearly invariant to viewing distance. This result is consistent with the idea that the perceived aspect ratio was based on apparent optical slant information. Moreover, the data of Loomis and Philbeck suggested that the effect was not completely invariant with distance.</p><fig id="i1534-7362-16-1-4-f01" orientation="portrait" position="float"><label>Figure 1</label><caption><p>Exocentric in-depth distance, <italic>L</italic>, can be estimated using the visual angle subtended, <italic>θ</italic>, the viewing distance, <italic>D</italic>, and the optical slant, <italic>β</italic>. That is, according to the law of Sines, (sin <italic>θ</italic>)/<italic>L</italic> = (sin <italic>β</italic>)/<italic>D</italic>. Thus, <italic>L</italic> = <italic>D</italic> sin <italic>θ</italic>/sin <italic>β</italic>. Overestimation in perceived <italic>β</italic> would result in underestimation in perceived L.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f01"></graphic></fig><p>A smaller distance anisotropy has been observed when egocentric distance (i.e., the distance between observer and a target on the ground) is compared to a frontal extent (Li et al., <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>). The magnitude of this second anisotropy is nearly constant with viewing distances: Li et al. found that, for egocentric distances from 5 to 30 m, a physically matched frontal extent was perceived to be about 1.2 times of the egocentric distance. This anisotropy is smaller than the one found for relatively small exocentric extents, such as studied by Loomis and Philbeck (<xref rid="i1534-7362-16-1-4-Loomis2" ref-type="bibr">1999</xref>), where the matching ratio often exceeds 2:1. The crucial difference between these cases could be interpreted to be due to the difference between egocentric and exocentric extents. However, Li et al. (<xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>) found that the perceived ratio between frontal and in-depth extents was also about 1.2 for in-depth ground extents that were relatively large in visual angle. Li et al. (<xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>; see also Li & Durgin, <xref rid="i1534-7362-16-1-4-Li3" ref-type="bibr">2012a</xref>) proposed that when an in-depth distance is angularly large (e.g., in the case of egocentric distance), its apparent length could be estimated by using visual direction in elevation to the two ends of the extent (rather than optical slant). It is already known that angular direction in elevation is a powerful source of information for the estimation of egocentric distance (e.g., Wallach & O'Leary, <xref rid="i1534-7362-16-1-4-Wallach1" ref-type="bibr">1982</xref>; Williams & Durgin, <xref rid="i1534-7362-16-1-4-Williams1" ref-type="bibr">2015</xref>). This variable has been variously called “slope of regard” (Wallach & O'Leary), “angular declination” (Ooi, Wu, & He, <xref rid="i1534-7362-16-1-4-Ooi1" ref-type="bibr">2001</xref>), and “gaze declination” (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>), but in the present paper we will refer to the entire axis as <italic>angular elevation</italic> to correspond with our focus on azimuth<xref ref-type="fn" rid="n1"><sup>1</sup></xref> and use <italic>angular declination</italic> only when discussing exclusively the downward component of the angular elevation axis of space. Even if the underestimation of egocentric distance is explained in terms of bias in perceived direction of angular declination (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>), that angular bias alone is insufficient to explain the observation that frontal extents are perceptually longer than egocentric distances of equal physical size. As shown in <xref ref-type="fig" rid="i1534-7362-16-1-4-f02">Figure 2</xref>, misperception of the ratio between egocentric and frontal extents suggests instead that perceived azimuth direction must be overestimated.</p><fig id="i1534-7362-16-1-4-f02" orientation="portrait" position="float"><label>Figure 2</label><caption><p>The perceptual anisotropy (Li et al., <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>) that makes frontal extent L appears larger (<italic>L</italic><sup>′</sup>) than the perceived egocentric distance, <italic>D</italic><sup>′</sup>, may be due to an exaggeration in perceived azimuth, <italic>φ</italic><sup>′</sup>. In fact, there is also evidence that perceived deviations in angular elevation, γ′, are exaggerated relative to straight ahead in a manner that predicts egocentric distance underestimation, but if <italic>φ</italic> were perceived correctly, underestimation of egocentric distance would not cause anisotropy on its own. Note, this diagram is plotted in a perspective view. The extents <italic>L</italic>, <italic>L</italic><sup>′</sup>, <italic>D</italic>, and <italic>D</italic><sup>′</sup> are all on a level ground plane.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f02"></graphic></fig><p>It may seem counterintuitive to assume that perceived visual direction is distorted. How can we be wrong about our visual direction? However, how could one know that perceived visual direction is not systematically biased? It is often suggested that accurate action implies accurate perception. But actions can be calibrated to stably distorted perception, as shown in prism adaptation studies (e.g., Harris <xref rid="i1534-7362-16-1-4-Harris1" ref-type="bibr">1963</xref>; Held & Freedman, <xref rid="i1534-7362-16-1-4-Held1" ref-type="bibr">1963</xref>). Just as people who wear glasses all day long do not normally notice the optical distortion introduced by the lens, we wouldn't be able to notice the stable distortions in perceived visual direction given that our perceived actions are coded in the same spatial framework as everything else.</p><p>A series of recent studies conducted in our lab suggest that perceived angular direction in elevation (i.e., visual direction in the sagittal plane) is distorted by a gain factor of about 1.5 (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>; Li, Phillips, & Durgin, <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>; Li et al., <xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>). This distortion in perceived visual direction can quantitatively account for the linear compression (with a gain factor of about 0.7) in the verbal estimation of egocentric distance (e.g., as observed by Foley et al., <xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>; see <xref ref-type="fig" rid="i1534-7362-16-1-4-f03">Figure 3</xref> left; see also Loomis & Philbeck, <xref rid="i1534-7362-16-1-4-Loomis3" ref-type="bibr">2008</xref>), and for the bias in the perceived distance-to-height ratio (e.g., Higashiyama & Ueyama, <xref rid="i1534-7362-16-1-4-Higashiyama3" ref-type="bibr">1988</xref>; Li et al., <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>) illustrated in the right panel of <xref ref-type="fig" rid="i1534-7362-16-1-4-f03">Figure 3</xref>. Given the evidence that perceived visual direction in the sagittal plane is indeed expanded, it seems reasonable to test whether perceived azimuth direction may also be distorted.</p><fig id="i1534-7362-16-1-4-f03" orientation="portrait" position="float"><label>Figure 3</label><caption><p>Diagrams suggesting that some perceptual biases in space perception can be well understood in terms of a distortion in perceived visual direction. Left: If the egocentric distance, <italic>D</italic>, is determined by angular elevation relative to straight ahead, γ, then perceptual exaggeration in perceived angular elevation, γ<sup>′</sup>, should result in nearly linear compression in perceived egocentric distance, D′. Right: Under a similar logic, the distance-to-height ratio (D/H) will be perceptually distorted if both perceived visual direction below the horizon, γ<sup>′</sup>, and perceived visual direction above the horizon, <italic>θ</italic><sup>′</sup>, are exaggerated.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f03"></graphic></fig><p>In a study of the geometry of visual space, Foley et al. (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>) systematically measured the perceived size of egocentric and exocentric ground extents in a large grass field in natural viewing condition. Perceived sizes were assessed using numerical estimation. Substantial compression (with a gain factor of 0.7) was observed in perceived egocentric distance. More importantly, for the present purpose, exocentric extents with varied ground orientations relative to the line of sight of the observer were also examined. To account for their empirical data, Foley et al. proposed two mathematical models (one for perceived egocentric distance and the other for perceived exocentric distance). They found that to best fit their data with models, they had to assume that, when computing perceived exocentric distance, the visual angle signal has to undergo a magnifying transform. In other words, their study suggested that perceived azimuth angle might be overestimated. However, perhaps because their study did not directly measure perceived azimuth angle, Foley and colleagues (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>) remained neutral about whether perceived azimuth direction was actually expanded, stating:</p><disp-quote><p>The common assumption is that perceived direction re straight ahead equals physical direction, but there is evidence that perceived direction may be slightly greater than this and that consequently perceived visual angles may be slightly greater than the corresponding physical angles… Either assumption can be incorporated into the model without affecting the nature of the space. (p. 153)</p></disp-quote><p>Several earlier papers by Foley (e.g., Foley, <xref rid="i1534-7362-16-1-4-Foley1" ref-type="bibr">1965</xref>, <xref rid="i1534-7362-16-1-4-Foley2" ref-type="bibr">1972</xref>; see also Bock, <xref rid="i1534-7362-16-1-4-Bock1" ref-type="bibr">1993</xref>) suggested exaggeration in perceived azimuth angle/direction. However, because those studies were conducted in dark environments, the angular distortions implied in those studies could have been due to impoverished visual information. Philbeck, Sargent, Arthur, and Dopkins (<xref rid="i1534-7362-16-1-4-Philbeck2" ref-type="bibr">2008</xref>) reported little bias in angular judgments of azimuth in near space in a lighted indoor environment, but several other authors have consistently found evidence of exaggeration in perceived visual direction (e.g., Fortenbaugh, Sanghvi, Silver, & Robertson, <xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>; Haun, Allen, & Wedell, <xref rid="i1534-7362-16-1-4-Haun1" ref-type="bibr">2005</xref>; Higashiyama, <xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>). Higashiyama (<xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>) directly examined perceived vertical and horizontal visual angles in an outdoor setting and found evidence of large angular expansion (by as much as 1.6 with a viewing distance of 30 m). One limitation of his study was that he used markers on the face of a building to indicate the visual angles to be estimated. He did find greater angular expansion in elevation than in azimuth, but since all the markers (for all the vertical and horizontal visual angles he tested) were visible to the participants throughout the experiment, the angular estimates may have been confounded with size judgments scaled by the features of the building such as the windows. A more complete quantitative review of prior evidence concerning azimuthal bias will be presented in the <xref ref-type="sec" rid="s5">General </xref><xref ref-type="sec" rid="s5">discussion</xref>.</p><p>The purpose of the present study was to establish a mathematical model to describe the perception of azimuth direction in an outdoor environment. To accomplish this, we systematically measured the perceived azimuth direction in an outdoor open field under full cue viewing conditions. Both an explicit measure (numerical estimation, <xref ref-type="sec" rid="s2">Experiment 1</xref>) and an implicit measure (ground extent matching, <xref ref-type="sec" rid="s3">Experiment 2</xref>) were used. As we will show, the two measures provided converging evidence that allowed us to model the perceived azimuth direction using a linear function. Because there was a small but systematic overall shift between the results of the two experiments, in <xref ref-type="sec" rid="s4">Experiment 3</xref>, we investigated whether this discrepancy could be partly due to a retinotopic, horizontal vertical illusion. The azimuth model established in the present study strengthens the explanation power of the <italic>Angular Expansion Hypothesis</italic> (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>) which has already offered quantitative accounts for several well-documented biases in space perception, including the uphill/downhill geographical slope exaggeration, the egocentric distance foreshortening, the exocentric distance anisotropy, and the bias in perceived distance-height ratio (Durgin, Li, & Hajnal, <xref rid="i1534-7362-16-1-4-Durgin5" ref-type="bibr">2010</xref>; Li & Durgin, <xref rid="i1534-7362-16-1-4-Li1" ref-type="bibr">2009</xref>, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>, <xref rid="i1534-7362-16-1-4-Li3" ref-type="bibr">2012a</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>; Li et al., <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>). Incorporating the azimuth model into the angular expansion hypothesis helps extend it beyond slants and distances in the sagittal plane (see also Li & Durgin, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>).</p></sec><sec id="s2"><title>Experiment 1: Explicit estimation of azimuth direction</title><p>Previous work has justified using magnitude estimation in the study of perceived slant, visual direction, and 2D orientation (e.g., Dick & Hochstein, <xref rid="i1534-7362-16-1-4-Dick1" ref-type="bibr">1989</xref>; Durgin et al., <xref rid="i1534-7362-16-1-4-Durgin5" ref-type="bibr">2010</xref>; Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>, <xref rid="i1534-7362-16-1-4-Durgin3" ref-type="bibr">2011b</xref>; Li & Durgin, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>). In the present study, to assess perceived azimuth direction, participants were asked to give numerical estimates of angular separation between a reference marker (always at 15 m in the straight ahead direction) and a target (at 5 or 15 m). We used angular separations in the range of 0°–48° (with both leftward and rightward azimuth direction). In order to be able to differentiate between exaggerated perception of head (or eye) rotation and a more general visual directional bias, we manipulated whether participants were allowed to move their head or eyes during the task. There is evidence suggesting that perceived egocentric distance to a target on the ground is not affected by whether the observer's gaze is directed to the target or not (Gajewski, Wallin, & Philbeck, <xref rid="i1534-7362-16-1-4-Gajewski2" ref-type="bibr">2014a</xref>, <xref rid="i1534-7362-16-1-4-Gajewski3" ref-type="bibr">2014b</xref>). We suspect that perceived azimuth direction may also be unaffected by gaze/head direction.</p><sec id="s2a"><title>Method</title><sec id="s2a1"><title>Participants</title><p>Thirty-five undergraduates from Swarthmore College (20 male, 15 female) participated in this experiment for payment. All the participants had normal or corrected-to-normal vision. All the experimental procedures used in the present study were approved by the local research ethics committee.</p></sec><sec id="s2a2"><title>Apparatus</title><p>A mobile eye-tracker (Positive Science, LLC, New York, NY) was used to monitor the participants' eye and head movements. The eye-tracker is composed of two digital camcorders, one for monitoring the visual scene, the other for monitoring the eye/pupil position. With a standard five-point calibration procedure, the eye fixation positions relative to the visual scene can be measured offline using the Yarbus eye-tracking software with subdegree precision.</p></sec><sec id="s2a3"><title>Stimuli and task</title><p>The experiment was conducted in a grass field at the campus of Swarthmore College. The participants stood facing the field from one side. There were trees and buildings at a distance of a few hundred meters. A red disk (15 cm in diameter) attached to the top of a thin stake (1.6 m tall) was set 15 m away in front of the participants, which served as a reference for the direction of straight ahead (<xref ref-type="fig" rid="i1534-7362-16-1-4-f04">Figure 4</xref>, solid circle). Twenty-four possible locations were used to place the target (<xref ref-type="fig" rid="i1534-7362-16-1-4-f04">Figure 4</xref>, open circles), which corresponded to six azimuth directions to the left and to the right (±8°, ±16°, ±24°, ±32°, ±40°, and ±48°) at two viewing distances (5 m, 15 m). The locations were marked so that the experimenter could quickly find them, but with markers invisible to the participants. In each trial, the experimenter stood at one of the target locations and served as the target. The participants' task was to estimate, in degrees, the angular deviation from the experimenter to straight ahead.</p><fig id="i1534-7362-16-1-4-f04" orientation="portrait" position="float"><label>Figure 4</label><caption><p>Diagram of the experimental setup used in <xref ref-type="sec" rid="s2">Experiment 1</xref> (top view). The participants gave verbal estimations of angular deviation from the target location (open circles) to straight ahead (direction to the solid circle). Only one target location was visible on each trial.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f04"></graphic></fig></sec><sec id="s2a4"><title>Design and procedure</title><p>Three viewing conditions were used between subjects. In the <italic>Head Free</italic> condition, participants were allowed to move their head or eyes as they wished. In the <italic>Head-Fixed</italic> condition, participants rested their head on a chinrest (the height of the chinrest was adjusted for each participant to ensure comfort.) The participants were asked to face their head toward the red disk at all times during the test, but they were allowed to move their eyes (i.e., gaze) freely. Finally, in the <italic>Gaze-Fixed</italic> condition, participants also rested their head on the chinrest and faced toward the red disk, but they were instructed to always keep their eyes fixated on the red disk during the test. Each participant was tested in one of these conditions: Eleven participants were assigned to the <italic>Head-Fixed</italic> condition; twelve participants were assigned to each of the other two conditions. Before the tests, participants were instructed using a diagram showing an overview of the sort of angles they were to be judging. Once it was certain that the participants understood the task, they were fitted with the mobile eye tracker. A standard five-point eye movement calibration was completed before the estimation began. At the beginning of each trial, the participants were asked to close their eyes while waiting for the experimenter to relocate to the new target location (the order of the locations was prerandomized for each participant). Then, the participants were signaled to open their eyes and to make their numeric angular estimate. Two-way radios were used for communication when necessary at the larger distances. After each estimate was recorded, the participants closed their eyes, and the experimenter moved to the next target location.</p></sec></sec><sec id="s2b"><title>Results</title><p>An initial examination of the overall results (with the mean verbal estimates collapsed across all viewing conditions and distances, but maintaining the sign of the angular deviations) suggested a fairly linear relationship between the perceived and actual azimuth directions. In order to conduct an analysis to test for possible effects of viewing condition and distance, for each participant, we first computed the angular gain (i.e., the slope of a linear regression) for targets presented at each viewing distance.</p><p>A mixed 3 between (Gaze Restriction: Gaze-Fixed, Head-Fixed, and Head Free) × 2 within (viewing distance: 5 and 15 m) factor ANOVA was conducted on the individual participant's gain data at each distance. The effect of Viewing Distance on angular gain differed as a function of Gaze Restriction, <italic>F</italic>(2, 32) = 7.90, <italic>p</italic> = 0.0016. This interaction is shown in <xref ref-type="fig" rid="i1534-7362-16-1-4-f05">Figure 5</xref> (left). Posthoc tests, with Bonferroni correction (<italic>α</italic> = 0.017), confirmed that the gain was reliably higher when the target was nearer (i.e., 5 m), when either eye movements were allowed (<italic>Head-Fixed</italic>), <italic>t</italic>(10) = 2.92, <italic>p</italic> = 0.015, or when both head and eye movements were allowed (<italic>Head Free</italic>), <italic>t</italic>(11) = 5.05, <italic>p</italic> < 0.001; but that there was no effect of viewing distance when gaze was required to be fixed on the reference disk throughout each trial, <italic>t</italic>(11) = 1.27, <italic>p</italic> = 0.23. Recall that the reference point, to which targets were compared, was at 15 m. Because the angular gain was unaffected by viewing distance when gaze refixation on the targets was not allowed, it seems possible that the higher gains for the 5 m condition might be artifacts resulting from changes in fixation between the 5 m target and the 15 m reference—or, more specifically, due to the greater physical distance between the ground contact points of the near targets and the far reference post—which would have been more visually accessible when gaze could shift. Although we cannot rule out the possibility that the angular gain would have been higher for nearer targets even if the reference point were at the nearer distance, there is no support for this idea in the literature (see <xref ref-type="sec" rid="s5">General </xref><xref ref-type="sec" rid="s5">discussion</xref>). Because Higashiyama's (<xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>) data suggest that there may be contaminating effects of linear distances on azimuthal estimation, and because there was no effect of viewing distance in the <italic>Gaze-Fixed</italic> condition, it seems safest to assume that something unusual is happening in the other conditions. In particular it seems likely that the data from the 5 m distance with a 15 m reference distance may contain artifacts based on increased linear ground extents that became particularly salient when fixation was not maintained at the far target and interfered with the evaluation of angular extent. We therefore focus our analysis on the 15 m condition alone, because viewing condition had no effect here. Mean verbal estimates for the 15 m targets (collapsed across viewing conditions) are plotted for the rightward and leftward viewing directions separately (<xref ref-type="fig" rid="i1534-7362-16-1-4-f05">Figure 5</xref>, right), which shows consistent overestimation in perceived azimuth. A linear fit to all 12 points indicates an angular gain of 1.26 in the perceived azimuth function.</p><fig id="i1534-7362-16-1-4-f05" orientation="portrait" position="float"><label>Figure 5</label><caption><p>Results of <xref ref-type="sec" rid="s2">Experiment 1</xref>. Left: mean gain of perceived azimuth direction as a function of viewing condition and viewing distance. Standard error bars are shown. Right: mean verbal estimates of perceived azimuth direction (collapsed across viewing conditions) as a function of actual azimuth direction. Standard errors are shown only for data dots of the rightward azimuth direction for clarity.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f05"></graphic></fig><sec id="s2b1"><title>A model of the perceived azimuth direction</title><p>The primary purpose of the present study is to model perceived azimuth direction. The verbal estimation data in <xref ref-type="sec" rid="s2">Experiment 1</xref> suggest that the perceived azimuth direction can be approximated by a linear function of the actual azimuth direction, with an angular gain of 1.26. If we leave the gain to be a free parameter, the azimuth model will take the form of <xref ref-type="disp-formula" rid="i1534-7362-16-1-4-e01">Equation 1</xref>.
<disp-formula id="i1534-7362-16-1-4-e01"><graphic xlink:href="i1534-7362-16-1-4-e01.jpg" position="float" orientation="portrait"></graphic></disp-formula></p><p>Where <italic>φ</italic>′ is the perceived azimuth direction, <italic>φ</italic> is the actual azimuth direction, and <italic>k</italic> is a constant gain factor. The results of <xref ref-type="sec" rid="s2">Experiment 1</xref> suggest that <italic>k</italic> is 1.26.</p></sec></sec><sec id="s2c"><title>Discussion</title><p>The results of <xref ref-type="sec" rid="s2">Experiment 1</xref> suggested that the numerically estimated azimuth direction can be approximated by a linear function of the actual azimuth direction in the range we tested (−48° to 48°), with the best estimated angular gain being 1.26. Whereas in previous studies, we have not differentiated between gaze direction and visual direction, the present results support the conclusion that visual direction is expanded even when the gaze directions are fixed. This is consistent with the observations by Gajewski, Wallin, and Philbeck (<xref rid="i1534-7362-16-1-4-Gajewski3" ref-type="bibr">2014b</xref>; see also Gajewski, Philbeck, Pothier, & Chichka, <xref rid="i1534-7362-16-1-4-Gajewski1" ref-type="bibr">2010</xref>) that showed that visual ground distance is perceived similarly even with visual presentation too brief for eye movements to occur.</p><p>Although numerical estimation tasks would require cognitive translations of perceptual variables into numbers, prior studies of slant and spatial orientation perception using numerical estimation have demonstrated rather consistent spatial patterns across multiple methods. For example, spatial biases are similar for surface slant and 2D line orientation whether orientation is numerically estimated relative to horizontal as zero or estimated relative to vertical as zero (Durgin et al., <xref rid="i1534-7362-16-1-4-Durgin5" ref-type="bibr">2010</xref>; Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin3" ref-type="bibr">2011b</xref>). Moreover, numeric angular estimates are also fairly consistent with nonverbal measures of space perception.<xref ref-type="fn" rid="n2"><sup>2</sup></xref> For example, angle (and direction) bisection tasks show that the midpoint is biased in the same way as the estimated 45° point (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>; Durgin et al., <xref rid="i1534-7362-16-1-4-Durgin5" ref-type="bibr">2010</xref>). The consistency of numeric estimates in orientation is probably because educated adults (such as college students) share a common conceptual scale of orientation with fixed numeric anchors (0°, 45°, 90°, etc.). For these reasons (and others), we think that numeric estimates of perceived direction and orientation provide an important source of information for estimating biases in angular variables.</p><p>Li and Durgin (<xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>) have shown that numerical estimation of perceived optical slant was substantially consistent with implicit slant estimation based on data from an aspect ratio task. Li et al. (<xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>) also showed that numeric estimates of angular elevation in the sagittal plane (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin3" ref-type="bibr">2011b</xref>; Li & Durgin, <xref rid="i1534-7362-16-1-4-Li1" ref-type="bibr">2009</xref>) predict findings from behavioral measures such as perceptual matching of height and egocentric distance These same errors in perceptual matching of height and egocentric distance were found even when participants had expertise that allowed them to verbally estimate egocentric distance more accurately (Durgin, Leonard-Solis, Masters, Schmelz, & Li, <xref rid="i1534-7362-16-1-4-Durgin1" ref-type="bibr">2012</xref>). Thus, spatial comparisons in locomotor space can act as converging measures of angular bias. We adopted a similar approach in <xref ref-type="sec" rid="s3">Experiment 2</xref>, using ground-extent-matching as an implicit measure of perceived azimuth direction. The extent-matching task can be an important source for providing converging evidence.</p></sec></sec><sec id="s3"><title>Experiment 2: Implicit measurement of perceived azimuth direction</title><p>As we have mentioned in the <xref ref-type="sec" rid="s6">I</xref><xref ref-type="sec" rid="s1">ntroduction</xref>, the data from egocentric-to-frontal, extent-matching task have suggested overestimation in perceived azimuth direction (e.g., Foley, <xref rid="i1534-7362-16-1-4-Foley2" ref-type="bibr">1972</xref>; Higashiyama, <xref rid="i1534-7362-16-1-4-Higashiyama2" ref-type="bibr">1996</xref>; Li et al., <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>). However, in the standard versions of this egocentric-frontal matching task, only a single azimuth direction, the perceived 45°, is measured because the task amounts to forming an equilateral right triangle, with the observer at one of the acute angles (<xref ref-type="fig" rid="i1534-7362-16-1-4-f06">Figure 6</xref>, left panel). Li et al. (<xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>; see also Philbeck, O'Leary, & Lew, <xref rid="i1534-7362-16-1-4-Philbeck1" ref-type="bibr">2004</xref>) used a modified version of the egocentric-frontal matching task, in which the frontal extent was matched to an egocentric distance along a line of sight that was oblique to the frontal extent (<xref ref-type="fig" rid="i1534-7362-16-1-4-f06">Figure 6</xref>, Right panel). The advantage of the modified version of the extent-matching task is that the size of the frontal extent becomes independent to its viewing distance, so that multiple azimuth directions can be assessed.</p><fig id="i1534-7362-16-1-4-f06" orientation="portrait" position="float"><label>Figure 6</label><caption><p>Diagrams showing the egocentric-frontal, extent-matching tasks. In the standard version (left), the observer set the egocentric distance, D, between himself/herself and the frontal extent, L, to match the length of the frontal extent. In the modified version (right), the observer set the length of an egocentric distance, D, along a diagonal path to the frontal extent, L, to match the length of the frontal extent. Note, in both cases, the egocentric and frontal extents are always on the ground plane.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f06"></graphic></fig><p>In <xref ref-type="sec" rid="s3">Experiment 2</xref>, we developed a variation of the egocentric-frontal matching task used by Li et al. (<xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>) to derive an implicit estimate of perceived visual direction. In this variant, the frontal extent was matched to an egocentric distance at a far oblique viewing direction (so as to avoid configural judgments such as those that might occur when both the egocentric and frontal extent<bold>s</bold> are visible in the same part of the visual field). Additionally, we used frontal extents that were presented at several different viewing distances. In conjunction with our quantitative model of perceived egocentric distance (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>), this extent-matching task can be interpreted as providing an implicit measure of perceived azimuth direction (<xref ref-type="sec" rid="sa1">Appendix</xref>).</p><sec id="s3a"><title>Method</title><sec id="s3a1"><title>Participants</title><p>Nineteen undergraduates from Swarthmore College (11 male, eight female) participated in this experiment for payment. None had participated in <xref ref-type="sec" rid="s2">Experiment 1</xref>. All participants had normal or corrected-to-normal vision. All the procedures used in this experiment were approved by the local research ethics committee.</p></sec><sec id="s3a2"><title>Stimuli and task</title><p>The experiment was conducted in a large, flat, outdoor grass field (200 m × 150 m). Participants stood at the central area of the field. Their task was to match a frontal extent <italic>(D</italic><sub>frn</sub>) to an egocentric distance (<italic>D</italic><sub>ego</sub>) at far oblique viewing direction (<xref ref-type="fig" rid="i1534-7362-16-1-4-f07">Figure 7</xref>). Combinations of a 9-m frontal extent with three viewing distances (21, 30, and 42 m) and combinations of a 15-m frontal extent with another three viewing distances (27, 42, and 54 m) were tested. These combinations correspond to six azimuth directions (i.e., 12.1°, 15.5°, 16.7°, 19.7°, 23.2°, and 29.1°). Each of the combinations was presented twice to each participant (one trial with a small initial <italic>D</italic><sub>ego</sub> and one trial with a large initial <italic>D</italic><sub>ego</sub>). Because only two physical frontal extents (i.e., 9 and 15 m) were presented in these combinations, we worried that participants might notice repetition, which could introduce more cognitive artifacts into their responses. To minimize this possibility, we added in six filler trials, in which the viewing distance to the frontal extent was one of 9, 15, 18, 27, 30, and 54 m, while the size of the frontal extent was randomly varied. No data were actually recorded on these filler trials to avoid slowing the experiment. The frontal extents in the filler trials were chosen with the constraint that their corresponding azimuth directions were always between 6° and 45°. Thus, each participant did 18 trials of extent-matching, including 12 test trials, for which matches were carefully measured, and six filler trials intended to disguise the experimental design. The order of the test trials was randomized for each participant, with fillers on trial 1, 4, 7, 10, 13, and 16. The first trial for each participant was a filler trial that also served as a practice trial. The standing location of the participant was fixed throughout the experiment. The standing locations of experimenters B and C in the test trials were marked in advance using labeled markers set in the grass that can be easily found by the experimenters, but were not visible from where the participants stood.</p><fig id="i1534-7362-16-1-4-f07" orientation="portrait" position="float"><label>Figure 7</label><caption><p>Diagram depicting the setup in <xref ref-type="sec" rid="s3">Experiment 2</xref>. Two experimenters (B and C) served as the targets for a frontal extent (D<sub>frn</sub>). Their distance (D<sub>view</sub>) to the participant (O) varied from trial to trial. The participant was asked to adjust the position of a third experimenter (A) along an opposite-viewing direction so that the egocentric distance (D<sub>ego</sub>) from the participant to experimenter A appeared the same as the frontal extent, D<sub>frn</sub>. Note, all the experimenters were on a level ground.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f07"></graphic></fig></sec><sec id="s3a3"><title>Procedure</title><p>The participants were instructed with the help of a diagram showing an overview of the matching task. Once it was certain that the participants understood the task, the experiment began. At the beginning of each trial, the participants were asked to hold a foam board to cover their face and chest while facing experimenter A, so that they could not see the experimenters. The foam board, in conjunction with a laser range finder, was also used to measure <italic>D</italic><sub>ego</sub>. Once all experimenters were in position, experimenter A signaled the participants, by voice, to start the trial. Two-way radios were used to facilitate communication, especially at farther distances. The participants, then, lowered the foam board and started to direct experimenter A to move closer or farther away. They were allowed to look back and forth between experimenter A and B and C at will. After the participants were satisfied with their distance adjustment, they turned to experimenter A and covered their face and chest with the foam board again. Experimenter A, then, measured <italic>D</italic><sub>ego</sub>, and all experimenters relocated to the predetermined locations for the next trial. Then, a new trial began.</p></sec></sec><sec id="s3b"><title>Results</title><p><xref ref-type="fig" rid="i1534-7362-16-1-4-f08">Figure 8</xref> shows the mean matched egocentric distance as a function of viewing distance and of size of the frontal extent (9 m and 15 m). We conducted a within-subjects ANOVA with viewing distance (Near, Medium, or Far) and Frontal extent (Short or Long) as factors. There was, of course, a highly reliable effect of Frontal extent that longer matches were made for longer frontal extent, <italic>F</italic>(1, 18) = 240.7, <italic>p</italic> < 0.0001. However, there was also a reliable effect of viewing distance, <italic>F</italic>(2, 36) = 6.14, <italic>p</italic> = 0.0051. Examination of <xref ref-type="fig" rid="i1534-7362-16-1-4-f08">Figure 8</xref> reveals that for the same-sized horizontal extent, the matched egocentric distances were greater at larger viewing distances. Such a result may tend to seem inconsistent with a constant angular gain, but it could also be a sign of overconstancy due to cognitive compensation (Carlson, <xref rid="i1534-7362-16-1-4-Carlson1" ref-type="bibr">1962</xref>) or other factors not accommodated by the present model (e.g., Higashiyama, <xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>).</p><fig id="i1534-7362-16-1-4-f08" orientation="portrait" position="float"><label>Figure 8</label><caption><p>Results of <xref ref-type="sec" rid="s3">Experiment 2</xref>. Mean matched egocentric distance as a function of viewing distance and of size of frontal extent. Standard errors are shown.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f08"></graphic></fig><p>The purpose of <xref ref-type="sec" rid="s3">Experiment 2</xref> was to use the extent-matching task as an implicit measure to assess the perceived azimuth direction. Details of deriving perceived azimuth direction from the extent-matching task are given in the <xref ref-type="sec" rid="sa1">Appendix</xref>. Deduced perceived azimuth direction is plotted as a function of actual azimuth direction in <xref ref-type="fig" rid="i1534-7362-16-1-4-f09">Figure 9</xref> (open circles). The best linear fit showed a slope of 1.14 and an intercept of 6.5°. For comparison, the result from <xref ref-type="sec" rid="s2">Experiment 1</xref> is also plotted as open triangles. It is clear that the trend of the results from the two experiments was fairly consistent, but the perceived azimuth function derived from the matching data is apparently shifted upwards by several degrees from the directly measured azimuth function. To further compare the present results to that of Foley et al. (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>), the imputed azimuth function developed by Foley et al. is also plotted in <xref ref-type="fig" rid="i1534-7362-16-1-4-f09">Figure 9</xref> (gray line). This imputed azimuth function was derived from Foley et al.'s distance estimation data (it was originally shown in figure 5 of Foley et al.'s paper). It is clear that the azimuth function derived from the extent-matching task of <xref ref-type="sec" rid="s3">Experiment 2</xref> is nearly identical to the findings of Foley et al. Therefore, both the size perception data (i.e., <xref ref-type="sec" rid="s3">Experiment 2</xref> and Foley et al.) and the explicit angular estimation data (<xref ref-type="sec" rid="s2">Experiment 1</xref>) seem to agree that perceived azimuth direction is exaggerated.</p><fig id="i1534-7362-16-1-4-f09" orientation="portrait" position="float"><label>Figure 9</label><caption><p>The results of <xref ref-type="sec" rid="s3">Experiment 2</xref> (the matching task) plotted in terms of the deduced perceived azimuth function (open circles). For comparison purpose, the deduced azimuth function from the explicit measure of <xref ref-type="sec" rid="s2">Experiment 1</xref> (open triangles) and from the study of Foley et al. (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>) (gray line) are also shown.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f09"></graphic></fig></sec><sec id="s3c"><title>Discussion</title><p><xref ref-type="sec" rid="s3">Experiment 2</xref>, using the modified egocentric-frontal, extent-matching task, replicated Foley et al.'s (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>) finding, based on numeric estimation, that perceived azimuth direction may be overestimated. The results were also fairly consistent with the explicit measure of perceived azimuth direction in <xref ref-type="sec" rid="s2">Experiment 1</xref>. But, there was an apparent overall shift between the results based on distance judgments (i.e., <xref ref-type="sec" rid="s3">Experiment 2</xref> and Foley et al.) and the findings based on explicit direction estimations (<xref ref-type="sec" rid="s2">Experiment 1</xref>). If we assume that the azimuth function must have a zero intercept (i.e., perceived straight ahead direction is accurate), then the explicit direction judgment data indicate a linear azimuth function, while the distance judgment data suggest a nonlinear azimuth function. (Foley et al.'s proposed azimuth function contained both a linear component and a power function component). It might be that either measure has some bias in it or that other factors contribute to either measure. For example, it might be that the numerical estimation task suffers from cognitive factors that produced an illusion of linearity. Conversely, it also could be that distance estimation of frontal ground extents show distance-dependent overcompensation.</p><p>In particular, it seems possible that the apparent upward shift in the deduced azimuth functions, based on the distance tasks, was partly due to the overcompensation of perceived size (the size overconstancy effect, Carlson, <xref rid="i1534-7362-16-1-4-Carlson1" ref-type="bibr">1962</xref>; Granrud, <xref rid="i1534-7362-16-1-4-Granrud2" ref-type="bibr">2012</xref>; Granrud et al., <xref rid="i1534-7362-16-1-4-Granrud1" ref-type="bibr">2009</xref>). That is, most adults know that the size of an object appears smaller at farther distance. Under objective instruction, people tend to overcompensate in the estimates of perceived size, especially when the target object is viewed at a large distance. Such overcompensation is suggested by <xref ref-type="fig" rid="i1534-7362-16-1-4-f08">Figure 8</xref>, in which the apparent size of the same-length frontal extent was judged longer when viewed at a farther distance. If the matching of frontal extents in <xref ref-type="sec" rid="s3">Experiment 2</xref> (and also the estimation of ground extents in Foley et al.'s, <xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref> study) included overcompensation, then the actual perceived azimuth direction should be smaller than the deduced values, and would be more similar to those directly measured in <xref ref-type="sec" rid="s2">Experiment 1</xref>.</p><p>However, another straightforward source of bias in the extent-matching task might be the horizontal vertical illusion (HVI). When comparing an egocentric distance to a frontal extent, the egocentric distance would correspond to a vertical line on the retina, whereas the frontal extent would correspond to a horizontal line. If vertical lines are overestimated by about 4%–6% relative to horizontal lines, as is typically found in the HVI, we would expect the egocentric settings to be slightly shorter than they should be. Similar bias could also affect Foley et al.'s model because that model was based on the estimation data of both egocentric and frontal extents. However, such a consideration shouldn't apply in the angular estimation task because only a single axis is considered. In a previous study of perceived aspect ratio of configurations presented on surfaces slanted in pitch or in yaw, we found evidence of a small (6%) but reliable HVI in extent matching (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>). Therefore, in <xref ref-type="sec" rid="s4">Experiment 3</xref>, we decided to investigate whether some of the differences between the angular gain estimates derived in <xref ref-type="sec" rid="s2">Experiments 1</xref> and <xref ref-type="sec" rid="s3">2</xref> might be partly due to the HVI affecting the matches in <xref ref-type="sec" rid="s3">Experiment 2</xref>.</p></sec></sec><sec id="s4"><title>Experiment 3: Egocentric-frontal extent matching with sideways observers</title><p>One way to study the effect of HVI on the extent-matching task is to tilt observers sideways so that their retinal image will be reoriented by 90°. This should reverse the HVI effect on the matching data (Avery & Day, <xref rid="i1534-7362-16-1-4-Avery1" ref-type="bibr">1969</xref>; Klein, Li, & Durgin, <xref rid="i1534-7362-16-1-4-Klein1" ref-type="bibr">in press</xref>; Künnapas, <xref rid="i1534-7362-16-1-4-Kunnapas1" ref-type="bibr">1958</xref>; Prinzmetal & Gettleman, <xref rid="i1534-7362-16-1-4-Prinzmetal1" ref-type="bibr">1993</xref>). If the HVI affected the results of the extent-matching in <xref ref-type="sec" rid="s3">Experiment 2</xref>, then when we tilt the observers to their side, their judgments on the extent-matching task should differ from those when they stand upright. Moreover, if we mathematically remove the HVI effect from the matching data, the difference in the extent-matching results between the upright and sideways conditions should disappear. In <xref ref-type="sec" rid="s4">Experiment 3</xref>, we tested this hypothesis by having half of the participants lie sideways at eye level while they did an egocentric-frontal, extent-matching task.</p><sec id="s4a"><title>Method</title><sec id="s4a1"><title>Participants</title><p>Thirty-five undergraduates from Swarthmore College (18 male, 17 female) participated in this experiment. Seventeen were assigned to the <italic>upright</italic> condition and the other 18 were assigned to the <italic>sideways</italic> condition. None of them had participated in <xref ref-type="sec" rid="s2">Experiment 1</xref> or <xref ref-type="sec" rid="s3">2</xref>. All had normal or corrected-to-normal vision. The procedures used in this experiment were approved by the local research ethics committee.</p></sec><sec id="s4a2"><title>Stimuli and task</title><p>The experiment was conducted in a different open grass field because the large grass field where we did <xref ref-type="sec" rid="s3">Experiment 2</xref> was now under construction. This field was relatively small (about 40 m × 50 m) and was surrounded by trees, buildings, and a parking lot. The standard egocentric-frontal, extent-matching task (i.e., <xref ref-type="fig" rid="i1534-7362-16-1-4-f06">Figure 6</xref>, left panel) was used, because when participants lie on their side, they cannot easily turn to look in a different direction. Two experimenters stood in front of the participants to form the frontal extent. The positions of the participant and the two experimenters formed a right triangle with the participant being always at an acute angle vertex. The participants were asked to direct the experimenter at the other acute angle vertex to move closer to (or farther away from) the experimenter at the right angle vertex, so that the distance between the two experimenters (the frontal extent) appeared the same as the distance between the participant and the experimenter at the right angle vertex (the egocentric distance). Two <italic>Body Orientations</italic> (upright vs. lying sideways) were examined (between subjects). Four egocentric distances (7, 10, 13, and 16 m), combined with two initial frontal extents (small vs. large), were tested with each participant. A wooden cart was built to support sideways participants approximately at eye height (see Klein et al., <xref rid="i1534-7362-16-1-4-Klein1" ref-type="bibr">in press</xref>). A cot was placed on top of the cart, and the participants lay on the cot on their side, with their head resting horizontally on a pillow. The eyes of the participants in the sideways condition were about 1.7 m above ground.</p></sec><sec id="s4a3"><title>Procedure</title><p>The participants were instructed with the help of a diagram showing an overview of the task. The instruction to the sideways participants was given after they were lying on the cot. Once it was certain that the participants understood the task, the trials began. A third experimenter stood near the participants throughout the experiment. At the beginning of each trial, the third experimenter blocked the view of the participants with a board, which was removed after the other two experimenters were in position. Then, the participants started to adjust the position of the experimenter at the acute angle vertex. Two-way radios were used to facilitate communication. Once the participant was satisfied with the adjustment, their view was blocked again, while the experimenters recorded the matched distance and relocated to new initial positions for the next trial. Then, a new trial began. The initial locations of the experimenters were marked in advance with labeled markers set in the grass that were not visible to the participants.</p></sec></sec><sec id="s4b"><title>Results</title><p><xref ref-type="fig" rid="i1534-7362-16-1-4-f10">Figure 10</xref> (left panel) shows the mean matched frontal extent as a function of viewing (i.e., egocentric) distance and body orientation. As we have reasoned above, if the HVI affected extent-matching, then we should expect differences in the matching results between the two body-orientation conditions. A mixed 2 between (<italic>Body orientation</italic>: upright or sideways) × 4 within (<italic>Egocentric distance</italic>: 7, 10, 13, or 16 m) × 2 within (<italic>Initial frontal extent</italic>: small or large) ANOVA confirmed that there was a reliable effect of <italic>Body orientation</italic> such that longer matches were made by upright participants, <italic>F</italic>(1, 33) = 31.4, <italic>p</italic> < 0.0001. There was also a reliable effect of <italic>Egocentric distance</italic>, <italic>F</italic>(3, 99) = 574.7, <italic>p</italic> < 0.0001, and a reliable effect of <italic>Initial frontal extent</italic>, <italic>F</italic>(1, 33) = 13.68, <italic>p</italic> < 0.001. An interaction between <italic>Body orientation</italic> and <italic>Egocentric distance</italic> was also found, <italic>F</italic>(3, 99) = 7.17, <italic>p</italic> < 0.001. These results are consistent with our assumption. That is, if the HVI caused a slight overestimation in perceived egocentric distance relative to perceived frontal extent, then the matched frontal extent would be set slightly larger by the upright participants relative to that set by the sideways participants. Moreover, if the difference in the extent-matching results of <xref ref-type="sec" rid="s4">Experiment 3</xref> (between the upright and sideways participant) was caused by HVI, then we should expect this difference to disappear once the effect of the HVI is removed from the matching data. Li and Durgin (<xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>) found that a HVI with a magnitude of about 6% affected their aspect ratio matching task. Therefore, we assume a HVI with similar magnitude may have affected the present extent-matching task. To remove the effect of this HVI, we modified the individual participant's matched frontal extent. For the upright participants, they may have set the frontal extent slightly too long, so we divided their matched frontal extent by 1.06 (i.e., the magnitude of the assumed HVI). For the sideways participants, they may have set the frontal extent too short, so we multiplied their matched frontal extent by 1.06. After this HVI correction, the extent-matching results of the two body orientation groups became very similar (<xref ref-type="fig" rid="i1534-7362-16-1-4-f10">Figure 10</xref>, right panel). The same ANOVA conducted on the modified matching data showed that the main effect of the <italic>Body orientation</italic> disappeared, <italic>F</italic>(1, 33) = 1.13, <italic>p</italic> = 0.293; and that the interaction between <italic>Body orientation</italic> and <italic>Egocentric distance</italic> also disappeared, <italic>F</italic>(3, 99) = 2.49, <italic>p</italic> = 0.065; but that the effect of <italic>Egocentric distance</italic>, <italic>F</italic>(3, 99) = 567.6, <italic>p</italic> < 0.0001, and the anchor effect of the <italic>Initial frontal extent</italic>, <italic>F</italic>(1, 33) = 13.28, <italic>p</italic> < 0.001, remained reliable.</p><fig id="i1534-7362-16-1-4-f10" orientation="portrait" position="float"><label>Figure 10</label><caption><p>The results of <xref ref-type="sec" rid="s4">Experiment 3</xref>. Left panel: matched frontal extent as a function of egocentric distance and of body orientation (upright vs. sideways). Right panel: the same matching data after the presumed effect of the (6%) HVI was removed from individual participants' matched frontal extents. Standard deviations are shown.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f10"></graphic></fig></sec><sec id="s4c"><title>Discussion</title><p>The perceived azimuth direction functions derived from the results of <xref ref-type="sec" rid="s2">Experiments 1</xref> and <xref ref-type="sec" rid="s3">2</xref> were slightly different (i.e., <xref ref-type="fig" rid="i1534-7362-16-1-4-f09">Figure 9</xref>, open circles vs. open triangles). We proposed that at least two factors may have contributed to this discrepancy: overconstancy and the HVI. In <xref ref-type="sec" rid="s4">Experiment </xref><xref ref-type="sec" rid="s4">3</xref>, we tested whether the HVI could cause a measurable change in the extent-matching task if we tilted half the observers sideways (which should reverse the HVI effect on the matching task). The results were consistent with our hypothesis.</p><p>Because the results from <xref ref-type="sec" rid="s4">Experiment 3</xref> confirmed that an HVI (with a magnitude of about 6%) seems to affect extent-matching tasks (see also Li & Durgin, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>), we can now mathematically remove this effect from the matching results of <xref ref-type="sec" rid="s3">Experiment 2</xref>. Because all participants were upright in <xref ref-type="sec" rid="s3">Experiment 2</xref>, we simply divided their matched egocentric distance by 1.06, on the assumption that the HVI has resulted in a slight distortion of perceived egocentric distance relative to frontal extent. After this HVI correction for each individual participant, we regenerated the implied azimuth function (<xref ref-type="fig" rid="i1534-7362-16-1-4-f11">Figure 11</xref>, gray circles) and plotted it with the azimuth function derived from the explicit direction estimation of <xref ref-type="sec" rid="s2">Experiment 1</xref> (<xref ref-type="fig" rid="i1534-7362-16-1-4-f11">Figure 11</xref>, open triangles). It is clear that after taking the HVI into account, the two azimuth functions have become much more similar. There are still residual differences between the two functions, but these additional differences might have resulted from size overconstancy (Carlson, <xref rid="i1534-7362-16-1-4-Carlson1" ref-type="bibr">1962</xref>) in the extent judgment process. It is also possible that additional unspecified perceptual factors contributed to this discrepancy.</p><fig id="i1534-7362-16-1-4-f11" orientation="portrait" position="float"><label>Figure 11</label><caption><p>Replot of the results of <xref ref-type="sec" rid="s2">Experiment 1</xref> (open triangles) and <xref ref-type="sec" rid="s3">Experiment 2</xref> (gray circles), after applying a correction for the HVI to the extent-matching data of <xref ref-type="sec" rid="s3">Experiment 2</xref>.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f11"></graphic></fig><p>The strategy employed here to eliminate the HVI assumes that there is a retinotopic HVI that increases the ratio between perceived egocentric and frontal distances for upright observers and decreases the ratio for sideways observers. We have recently employed the same strategy for the large-scale HVI to show that there is a large, ground-based component and a smaller, retinotopic component of the outdoor HVI (Klein et al., <xref rid="i1534-7362-16-1-4-Klein1" ref-type="bibr">in press</xref>).</p></sec></sec><sec id="s5"><title>General discussion</title><p>In the present study, we measured perceived azimuth direction using two very different tasks—an explicit numerical estimation task (<xref ref-type="sec" rid="s2">Experiment 1</xref>), and an extent-matching task that implicitly assesses perceived azimuth direction (<xref ref-type="sec" rid="s3">Experiment 2</xref>). The results from both experiments are consistent with the hypothesis that perceived azimuth direction is overestimated. The distance results are quite consistent with the exocentric distance estimation data of Foley et al. (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>), which also implied overestimation in perceived azimuth angles. This consistency seems important given that we used a nonverbal method to measure perceived extent, whereas Foley et al. used numeric estimates. Thus, the present work provides strong converging evidence concerning the estimated size of exocentric ground extents. Moreover, based on the results of <xref ref-type="sec" rid="s4">Experiment 3</xref> where some observers made size comparisons while lying sideways, we have suggested that one small additional source of bias in both of these types of distance judgments is a retinotopic HVI.</p><p>Whereas Foley et al. (<xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>) proposed that angular expansion might be a mathematical fiction in their model, our numerical estimation data suggest that at least part of the angular expansion assumed in Foley et al.'s model is evidently present in the direct estimations of visual azimuth direction (relative to straight ahead). In fact, the exaggeration in perceived visual direction in azimuth fits well with the framework of the angular expansion hypothesis (i.e., Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>; Li & Durgin, <xref rid="i1534-7362-16-1-4-Li3" ref-type="bibr">2012a</xref>) which proposes that several well-documented biases in space perception can be quantitatively understood by assuming perceptual biases in two other angular variables—visual direction along the sagittal plane, and optical slant. For example, it has been shown that exaggeration in angular elevation (both upward and downward relative to straight ahead) can quantitatively explain egocentric distance foreshortening (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>), bias in perceived egocentric distance to height ratio (Li et al., <xref rid="i1534-7362-16-1-4-Li6" ref-type="bibr">2011</xref>), and downhill slope exaggeration (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li1" ref-type="bibr">2009</xref>); and that overestimation in perceived optical slant can account for uphill slope exaggeration (Durgin et al., <xref rid="i1534-7362-16-1-4-Durgin5" ref-type="bibr">2010</xref>; Li & Durgin, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>), exocentric distance anisotropy (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li2" ref-type="bibr">2010</xref>, <xref rid="i1534-7362-16-1-4-Li5" ref-type="bibr">2013</xref>; Li et al., <xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>), and compressive foreshortening in the perceived exocentric in-depth extent (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li3" ref-type="bibr">2012a</xref>). Incorporating the new azimuth model (i.e., <xref ref-type="disp-formula" rid="i1534-7362-16-1-4-e01">Equation 1</xref>) into the angular expansion hypothesis confirms that even findings of anisotropy in size perception of ground extents (such as reported by Foley et al.) can be accounted for largely by angular bias. Moreover, Klein et al. (in press) have recently found evidence that the fact that the angular gain of perceived visual direction is apparently different for azimuth (1.26) than for elevation (1.50) accounts for the very large perceptual anisotropy between horizontal and vertical extents (about 20%) that has been reported for large outdoor objects (Chapanis & Mankin, <xref rid="i1534-7362-16-1-4-Chapanis1" ref-type="bibr">1967</xref>).</p><p>With regard to perceived slant, evidence of consistent and systematic orientation biases have been reported in the haptic perception of slant (Durgin & Li, <xref rid="i1534-7362-16-1-4-Durgin4" ref-type="bibr">2012</xref>; Hajnal, Abdul-Malak, & Durgin, <xref rid="i1534-7362-16-1-4-Hajnal1" ref-type="bibr">2011</xref>; Shaffer & McManama, <xref rid="i1534-7362-16-1-4-Shaffer1" ref-type="bibr">2015</xref>) as well as in the proprioception of hand orientation (Li & Durgin, <xref rid="i1534-7362-16-1-4-Li4" ref-type="bibr">2012b</xref>). With respect to perceived azimuth direction, angular scale expansion has additionally been discovered in the vestibular perception of heading direction (e.g., Cuturi & MacNeilage, <xref rid="i1534-7362-16-1-4-Cuturi1" ref-type="bibr">2013</xref>). Thus, this seems to be a general perceptual strategy that extends beyond vision to the multimodal perception of locomotor space.</p><p>Although there is a great deal of literature on the possibility of visual error in perceived angular direction in near space (e.g., Bock, <xref rid="i1534-7362-16-1-4-Bock1" ref-type="bibr">1993</xref>; Fortenbaugh, Sanghvi, Silver, & Robertson, <xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>; Temme, Maino, & Noell, <xref rid="i1534-7362-16-1-4-Temme1" ref-type="bibr">1985</xref>), much of this literature can be hard to interpret in terms of the <italic>perceived</italic> angular direction (i.e., in degrees from fixation). For example, Bock (<xref rid="i1534-7362-16-1-4-Bock1" ref-type="bibr">1993</xref>) used pointing to show angular expansion (overpointing) in perceived azimuthal direction. But this kind of measure compares two unknowns (perceived visual direction and perceived pointing direction). Even if pointing direction were accurately aligned with visual direction, it would be impossible to know the perceived direction of either the pointing direction or the visual direction from this method, though it would imply that the two percepts were consistent with each other. The difference between pointing direction and visual direction observed by Bock (which disappeared when gaze was toward the peripheral target) might indicate something about transformations between visual and pointed direction, but this is only speculative. In what follows, we seek to summarize prior relevant data concerning perceived azimuthal direction that can be interpreted as estimating bias.</p><p>Fortenbaugh et al. (<xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>, following the method of Temme et al., <xref rid="i1534-7362-16-1-4-Temme1" ref-type="bibr">1985</xref>) had people make numeric or spatial judgments of perceived eccentricity where they defined the range of their scale (0–100 scale units) in terms of the angular distance from the central fixation point to the edge of the visual field. Because they did not assess perceived direction to the edge of the visual field, their method would not be able to detect any overall linear bias, such as we (and Higashiyama, <xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>) have observed. This is because Fortenbaugh et al. assumed in their interpretation of their data that there is no error in the perceived direction to the extreme edge. This means their method can best pick up nonlinear components (nonuniform expansion or compression) between fixation (assumed to be zero) and the nominal edge (assumed to be perceived accurately). If the perceived direction to the edge of the visual field were exaggerated, however, their method could seem to show underestimation (relative to that edge) even when overestimation was present (overall). When no far boundary was visible, Fortenbaugh et al. did primarily observe expansion away from the fovea in azimuth and even greater expansion in angular elevation. When they added a visible outer boundary, the boundary also seemed to repel the judgments. Nonetheless, for monocular judgments of azimuth in the temporal direction, when no visible boundary was present, it is possible to interpret the azimuth data of Fortenbaugh et al. (<xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>) in terms of a presumed/perceived 90° limit of the field of view.<xref ref-type="fn" rid="n3"><sup>3</sup></xref> When so interpreted, their data (which are fairly linear out to 50° of azimuth) had a gain of 1.33 for eccentricities of 40° or less. These data are plotted in <xref ref-type="fig" rid="i1534-7362-16-1-4-f12">Figure 12</xref>.</p><fig id="i1534-7362-16-1-4-f12" orientation="portrait" position="float"><label>Figure 12</label><caption><p>Plots (limited to azimuthal eccentricities of 40° or less) of the results of several previous studies measuring perceived azimuthal direction. The data of Higashiyama (<xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>, <xref ref-type="sec" rid="s4">Experiment 3</xref>) represent verbal estimates in degrees of horizontal intervals. Clock-face estimates and verbal estimates diverged in the study of Philbeck et al. (<xref rid="i1534-7362-16-1-4-Philbeck2" ref-type="bibr">2008</xref>), and so are plotted separately (see text). The data of Fortenbaugh et al. (<xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>) are limited to temporal eccentricities where the (unbounded) monocular visual field is presumed to extend to 90°. The data of Haun et al. (<xref rid="i1534-7362-16-1-4-Haun1" ref-type="bibr">2005</xref>) were from memory, but appear consistent with judgments from perception. Gain values for the data from each study are shown for this range in the legend, and the gain for the data overall is plotted (the solid line). The dashed line represents unity.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f12"></graphic></fig><p>Haun et al. (<xref rid="i1534-7362-16-1-4-Haun1" ref-type="bibr">2005</xref>) have argued that verbal estimates of azimuthal direction from memory are biased away from categorical boundaries (including 90° eccentricity), and it is possible that the boundary effects observed by Fortenbaugh et al. (<xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>), when visible outer boundaries were present, were of a similar origin. When Fortenbaugh et al. imposed a visible outer boundary (or the body itself did—such as the nose does for nasal monocular eccentricity), they found relatively more evidence of (relative) expansion away from the outer boundary; but when the outer boundary was not a visible object, they primarily observed expansion away from fixation (which was visually marked in all conditions). This kind of pattern of results could be due to the kind of categorization effects proposed by Haun et al., but it could also occur as a result of participants changing reference frames when evaluating objects near to an outer (visible) boundary (see also Simmering, Spencer, & Schöner, <xref rid="i1534-7362-16-1-4-Simmering1" ref-type="bibr">2006</xref>). In the study of Haun et al., visible boundaries were available both near straight ahead and near 90° as the edges of a circular curtain used as a backdrop that approximated a quadrant from the front to the left of the observer. Thus, reference-frame switching remains a possible explanation even for the data of Haun et al., consistent with expansion away from both straight ahead and from 90°. If we consider only data for eccentricities of 40° or less, however, the estimates collected by Haun et al. look similar to those observed by Fortenbaugh et al., as shown in <xref ref-type="fig" rid="i1534-7362-16-1-4-f12">Figure 12</xref>. Thus, although Haun et al. attributed the biases in their estimation data to category effects in memory, those biases seem to be identical to biases observed in perception.</p><p>There is, moreover, additional evidence of reference-frame switching in the azimuth estimation data of Higashiyama (<xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>). When Higashiyama had participants estimate the angular extents of horizontal intervals on the face of a building from different viewing distances, he observed strong linear angular exaggeration at the farthest viewing distance (30 m). The range of angles formed by the intervals at this distance was from 4° to 25°. For the same frontal intervals viewed from a distance of 3 m, most of the visual angles were closer to 90°, and among those greater than 45° (ranging from 63° to 78°), Higashiyama's data suggest linear expansion away from 90° though the gain is somewhat less. In this sense, Higazhiyama's azimuth data resemble those of Fortenbaugh et al. (<xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>) in showing a (weaker) tendency for repulsion from outer boundaries (i.e., 90°). It is interesting that no such reversed repulsion is evident in Higashiyama's elevation angle estimation data, and this may be because the egocentric vertical (90° elevation) is not a visually salient boundary, whereas the egocentric 90° (in azimuth) is a salient perceptual boundary. We have plotted all of the estimation data from Higashiyama's study for azimuthal eccentricities of 40° or less in <xref ref-type="fig" rid="i1534-7362-16-1-4-f12">Figure 12</xref>. The overall gain for these data is 1.25.</p><p>Philbeck et al. (<xref rid="i1534-7362-16-1-4-Philbeck2" ref-type="bibr">2008</xref>) found that when using an exocentric pointer (e.g., a stick that rotated on a pivot in front of the observer) the perceived azimuthal direction of pointing was substantially misperceived—especially for directions that were behind the observer (see also Kelly, Loomis, & Beall, <xref rid="i1534-7362-16-1-4-Kelly1" ref-type="bibr">2004</xref>; Koenderink, van Doorn, & Lappin, <xref rid="i1534-7362-16-1-4-Koenderink1" ref-type="bibr">2003</xref>). Philbeck et al. contrasted this with egocentric judgments of azimuthal direction (in a 360° circle), allowing the observer to rotate fully. Verbal reports of azimuthal angle (sampled at 20° intervals in the frontal quadrants and 10° intervals behind) were fairly accurate when made egocentrically. Interestingly, however, when clock units rather than degrees were employed, the egocentric azimuthal estimation data of Philbeck et al. show angular expansion for eccentricities less than 45°. Both the clock face estimation data and the degrees estimation data of Philbeck et al. are also shown in <xref ref-type="fig" rid="i1534-7362-16-1-4-f12">Figure 12</xref>. The clock face data suggest a gain of about 1.18.</p><p>Philbeck et al. (<xref rid="i1534-7362-16-1-4-Philbeck2" ref-type="bibr">2008</xref>) are fairly unique in observing no evidence of azimuthal bias with verbal degree estimates. This might be partly because they asked participants to use 10° as the finest grain of judgment for the verbal degree judgments, whereas they encouraged a 10-minute (5°) level of precision in their clock face estimation task. However, this is unlikely to be a sufficient account, based on control experiments they conducted. It is also possible, however, that the corners of their testing room provided an additional 45° reference frame that counteracted other reference frame effects in their study when degree estimation was used (45° is “1:30” in clock-face terms, which seems less likely to serve as a reference frame for clock-face estimates). Overall, a number of different paradigms (including the clock-face data of Philbeck et al.) seem to show systematic biases in perceived azimuthal direction even in near space, but the exact shape and nature of these biases is not always easy to interpret. We have emphasized the possibility that fairly linear expansion exists relative to a given reference frame, and that the relevant reference frame can shift depending on whether the requested angle is nearer to straight ahead or to some other salient reference (Fortenbaugh et al., <xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>; Simmering et al., <xref rid="i1534-7362-16-1-4-Simmering1" ref-type="bibr">2006</xref>).</p><p>In summary, <xref ref-type="fig" rid="i1534-7362-16-1-4-f12">Figure 12</xref> shows the estimation data of all the studies discussed here for azimuthal eccentricities of 40° or less. There is some notable variability, but the general pattern suggests that the azimuthal angle tends to be exaggerated in the way our own data have suggested. Importantly, the magnitudes of azimuthal exaggeration observed by Fortenbaugh et al. (<xref rid="i1534-7362-16-1-4-Fortenbaugh1" ref-type="bibr">2012</xref>) in very near space and by Higashiyama (<xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>) in very far space are not terribly different. The overall gain for <xref ref-type="fig" rid="i1534-7362-16-1-4-f12">Figure 12</xref> is 1.22 (1.28 if the degree-estimation data of Philbeck et al., <xref rid="i1534-7362-16-1-4-Philbeck2" ref-type="bibr">2008</xref> are excluded), and a linear fit seems fairly reasonable out to 40°. Using the clock-face data of Philbeck et al., the gains implied by the other three studies for eccentricities up to 40°, and our own estimate of 1.26, suggests an overall azimuthal gain relative to straight-ahead of about 1.27, 95% CI [1.18, 1.34] across studies, though clearly there are factors that can affect the measured gain that are not understood at present.</p><p>As proposed by Durgin and Li (<xref rid="i1534-7362-16-1-4-Durgin2" ref-type="bibr">2011a</xref>; see also Higashiyama, <xref rid="i1534-7362-16-1-4-Higashiyama1" ref-type="bibr">1992</xref>), perceptual expansion in spatial angular variables may be a functional biological coding choice. Some spatial angular variables provide important information regarding the spatial relationship between the observer and potential target(s) in space. For example, angular declination (the downward component of elevation) to a target on level ground tells you how far it is from you; azimuth direction helps you to figure out your heading direction relative to the target direction; optical slant tells you the orientation of a surface relative to your view of the surface, etc. Perceptual scale expansion in those angular variables (in the range where they are used) provides higher perceptual resolution (i.e., sensitivity). This may enhance the precision of control in relevant actions that are guided by feedback provided in terms of those perceptual angular variables. For the purpose of action control, accuracy of perception is not necessary because actions can (and will) be calibrated to stable distortions in perception (such as induced by the lenses in glasses, for example). It is therefore the precision of perceptual feedback that limits action calibration rather than the accuracy of perception. Thus it is the sensitivity or precision of perceptual coding that provides the basis for accuracy (or efficiency) in visually guided action.</p></sec><sec id="s6"><title>Conclusion</title><p>We have previously proposed that many biases in the perception of spatial layout in locomotor space can be easily understood (and modeled) as errors in the perception of angular variables. In the present study, we further showed that visually perceived azimuth direction is also distorted. This finding has been supported by converging evidence concerning numeric angular estimation, verbal-distance estimation (i.e., Foley et al., <xref rid="i1534-7362-16-1-4-Foley3" ref-type="bibr">2004</xref>), and perceptual-extent matching. Our data suggest that systematic biases in the evaluation of both frontal and in-depth ground extents in locomotor space may be accounted for by the angular expansion hypothesis.</p></sec><sec sec-type="supplementary-material"><title>Supplementary Material</title><supplementary-material content-type="local-data" id="i1534-7362-16-1-4-icon"><media xlink:href="i1534-7362-16-1-4-icon.jpg" orientation="portrait" id="d36e80" position="anchor"></media></supplementary-material></sec></body><back><ack><title>Acknowledgments</title><p>We thank Cassandra Strawser, Brennan Klein, Ariana Spiegel, Elizabeth Cifuentes, and Christine Pham for their assistance in data collection. We also thank John Foley and John Philbeck for many thoughtful and constructive comments that have greatly improved this paper. This research was supported by Award Number R15 EY021026 to Frank H. Durgin from the National Eye Institute. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Eye Institute or the National Institutes of Health.</p><p>Commercial relationships: none.</p><p>Corresponding author: Frank H. Durgin.</p><p>Email: fdurgin1@swarthmore.edu.</p><p>Address: Department of Psychology, Swarthmore College, Swarthmore, PA, USA.</p></ack><fn-group><fn id="n1"><label>1</label><p>This follows the practice in aviation of using <italic>elevation</italic> and <italic>azimuth</italic> to indicate visual directions in the sagittal and the horizontal planes respectively. <italic>Angular elevation</italic> includes visual directions in the sagittal plane, both above and below eye level. “Angular declination” has become a standard term used to refer to the downward directions of angular elevation, but “declination” and “ascension” are also astronomical concepts that correspond roughly to latitude and longitude, whereas angular <italic>elevation</italic> and <italic>azimuth</italic> reference the terrestrial gravitational framework in which most human behavior takes place.</p></fn><fn id="n2"><label>2</label><p>Note that we do not regard matching measures such as manual actions (e.g., palm boards, pointing) as comparable measures of visual experience, because matching measures may often involve visual, haptic, or proprioceptive biases associated with the measure themselves. Inconsistencies between manual measures and perceptual reports may often be theoretically important (e.g., Philbeck et al., <xref rid="i1534-7362-16-1-4-Philbeck2" ref-type="bibr">2008</xref>), but it has also often turned out that manual measures are more susceptible to measurement artifacts (e.g., Li et al., <xref rid="i1534-7362-16-1-4-Li7" ref-type="bibr">2013</xref>; Shaffer, McManama, Swank, Williams, & Durgin, <xref rid="i1534-7362-16-1-4-Shaffer2" ref-type="bibr">2014</xref>), than are other nonverbal perceptual measures.</p></fn><fn id="n3"><label>3</label><p>The temporal limit of the visual field is certainly greater than 90° (e.g., Rønne, 1915, cited in Traquair, <xref rid="i1534-7362-16-1-4-Traquair1" ref-type="bibr">1938</xref>), but Fortenbaugh et al. 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</mixed-citation></ref></ref-list><app-group><app><title></title><sec id="sa1"><title>Appendix. Deriving perceived azimuth direction from the extent-matching task</title><p>In <xref ref-type="sec" rid="s3">Experiment 2</xref>, we used an egocentric extent-matching task as an implicit measure to assess perceived azimuth direction. Here, we show how the perceived azimuth direction was derived from the matching data.</p><p>The experiment setup used in <xref ref-type="sec" rid="s3">Experiment 2</xref> is shown in <xref ref-type="fig" rid="i1534-7362-16-1-4-f13">Figure 13</xref> (assume the ground is flat). The physical viewing distance from the observer, O, to the frontal extent, BC, is D<sub>1</sub>, and the corresponding perceived viewing distance is <inline-graphic xlink:href="i1534-7362-16-1-4-ilm01.jpg"></inline-graphic>. The egocentric distance from the observer to experimenter A is <italic>D</italic><sub>2</sub>, and the corresponding perceived egocentric distance is <inline-graphic xlink:href="i1534-7362-16-1-4-ilm02.jpg"></inline-graphic>. The physical size of the frontal extent is <italic>L</italic>, and the corresponding perceived size is <italic>L</italic>.<sup>′</sup> The physical azimuth angle subtended by the frontal extent is <italic>φ</italic>, and the corresponding perceived azimuth angle is <italic>φ</italic>′.
</p><fig id="i1534-7362-16-1-4-f13" orientation="portrait" position="anchor"><label>Figure 13</label><caption><p>Diagram illustrating the geometry of the spatial variables in the experimental setup used in <xref ref-type="sec" rid="s3">Experiment 2</xref>. Note that all the experimenters were on a level ground plane.</p></caption><graphic xlink:href="i1534-7362-16-1-4-f13"></graphic></fig><p>According to the second assumption of the angular expansion hypothesis (i.e., Laws of Euclidean geometry can still be applied to the perceived variables), the perceived size of the frontal extent, <italic>L</italic><sup>′</sup>, can be determined by the perceived azimuth angle, <italic>φ</italic>′, and the perceived viewing distance to the frontal extent, <inline-graphic xlink:href="i1534-7362-16-1-4-ilm03.jpg"></inline-graphic>.
<disp-formula id="i1534-7362-16-1-4-e02"><graphic xlink:href="i1534-7362-16-1-4-e02.jpg" position="float" orientation="portrait"></graphic></disp-formula></p><p>Because the participant's task was to match the perceived size of the frontal extent to the perceived egocentric distance to Experimenter A, i.e.
<disp-formula id="i1534-7362-16-1-4-e03"><graphic xlink:href="i1534-7362-16-1-4-e03.jpg" position="float" orientation="portrait"></graphic></disp-formula></p><p>Replace <italic>L</italic><sup>′</sup> with <inline-graphic xlink:href="i1534-7362-16-1-4-ilm04.jpg"></inline-graphic>in the first equation, and we have
<disp-formula id="i1534-7362-16-1-4-e04"><graphic xlink:href="i1534-7362-16-1-4-e04.jpg" position="float" orientation="portrait"></graphic></disp-formula></p><p>According to the angular expansion hypothesis, the perceived egocentric distances (<inline-graphic xlink:href="i1534-7362-16-1-4-ilm05.jpg"></inline-graphic>and <inline-graphic xlink:href="i1534-7362-16-1-4-ilm06.jpg"></inline-graphic>) are foreshortened due to the exaggeration in perceived angular declination (i.e., <italic>γ</italic><sup>′</sup> = 1.5 <italic>γ</italic>). The perceived egocentric distance is proportional to the physical egocentric distance when the distances are relatively large (because tan (<italic>γ</italic>) ≈ <italic>γ</italic> for small angular elevations). Thus, in <xref ref-type="disp-formula" rid="i1534-7362-16-1-4-e04">Equation A1</xref>, the ratio between the perceived distances, <italic>D</italic><sub>2</sub>′/<italic>D</italic><sub>1</sub>′, can be approximated by the ratio of the physical distances, <italic>D</italic><sub>2</sub>/<italic>D</italic><sub>1</sub>. Then, the perceived azimuth direction, <italic>φ</italic>′, can be determined by the ratio between <italic>D</italic><sub>2</sub> and <italic>D</italic><sub>1</sub>, i.e.,
<disp-formula id="i1534-7362-16-1-4-e05"><graphic xlink:href="i1534-7362-16-1-4-e05.jpg" position="float" orientation="portrait"></graphic></disp-formula></p></sec></app></app-group></back></pmc><affiliations><list><country><li>États-Unis</li></country><region><li>Pennsylvanie</li></region></list><tree><country name="États-Unis"><region name="Pennsylvanie"><name sortKey="Li, Zhi" sort="Li, Zhi" uniqKey="Li Z" first="Zhi" last="Li">Zhi Li</name></region><name sortKey="Durgin, Frank H" sort="Durgin, Frank H" uniqKey="Durgin F" first="Frank H." last="Durgin">Frank H. Durgin</name></country></tree></affiliations></record>Pour manipuler ce document sous Unix (Dilib)
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