Although the orientation of line segments and simple shapes is a well-studied area of vision, little is known about geometric factors that influence perceived orientation of complex multipart shapes. The study of these factors is of interest because it allows for an insight into the basic problem of how local geometric attributes are integrated perceptually into a global shape representation. We examined the perceived orientation of two-part shapes using an adjustment method and a 2AFC task. In particular, we investigated the influence of the perceptual salience, or distinctiveness, of a part—as defined by the turning angles at its boundaries—and its area relative to the main “base” part. In contrast to previous results on simple shapes, our results exhibited large and systematic deviations of perceived orientation from the principal axis of the shape. For shapes with sharp part boundaries, perceived global orientation deviated maximally from the principal axis and was approximated by the axis of the main base part of the shape. With weakening part boundaries, the perceived orientation gradually approached the principal axis of the entire shape, reflecting that both parts were taken into account in estimating orientation. The results are consistent with a differentially weighted principal-axis computation in which the attached part is given systematically lower weighting with increasing turning angles at the part boundaries. They thus allow a quantitative characterization of part salience in terms of *part independence*: Turning angles at a part's boundaries determine the extent to which its influence is perceptually separable from the rest of the shape. We suggest that Robust Statistics may provide a useful framework for quantifying the influence of part segmentation on visual estimation.

^{1}have the benefit of allowing for greater robustness under changes in viewing conditions—for instance, involving changes in articulated pose—because the representation of an object's shape can be dissociated from any particular spatial configuration that its parts may take.

*minima rule*was motivated by the

*principle of transversality,*according to which the intersection of two smooth surfaces almost surely (i.e., with probability 1) generates a concave tangent discontinuity at their locus of intersection

^{2}(Guillemin & Pollack, 1974).

*salient*perceptually than the one on the right. This example illustrates two geometric factors proposed by Hoffman and Singh (1997) in their theory of part salience:

*part-boundary strength*and

*part protrusion*. The part on the left is delineated by negative minima that have substantially greater turning angles than the one on the right,

^{3}and its protrusion—defined by the ratio of its perimeter to the length of its part cut—is also substantially higher.

*representational strength*of a part in a shape's description: The higher the part's salience is, the stronger is its representation (e.g., Hoffman & Singh, 1997; Rom & Medioni, 1993; Siddiqi & Kimia, 1995). Such a view is consistent with claims in the literature that perceptually salient parts enjoy special status, in terms of being remembered, attended, or named more readily (e.g., Bower & Glass, 1976; Hoffman & Singh, 1997; Palmer, 1977; Reed, 1974).

*part independence*: Highly salient parts have a higher likelihood of being segmented and are, thus, more readily perceived as independent units. Under the part-independence interpretation, part salience may be construed as the extent to which a part's representation is perceptually separable from the rest of shape. Parts with sharper turning angles at their boundaries, for instance, are more likely to be perceptually segmented and represented as distinct units.

*global*property of a complex shape (i.e., when integration over the entire shape is required). A highly salient part is likely to be perceived as more separate from the rest of the shape. Hence, it is likely to be weighted

*less strongly*in the estimation of a global property because its contribution is less compatible with the bulk of the shape.

*weakly*the part's contribution should be incorporated into the global estimate. In the experiments reported here, we tested this hypothesis in the context of visual estimation of shape orientation.

^{4})—the orientation is determined by its second-order statistics—that is, the “spread” of its points around the center of mass. Specifically, it is computed based on the direction of the first principal axis (i.e., the eigenvector of the covariance matrix corresponding to the largest eigenvalue). This is equivalent to the direction of the line that minimizes the sum of the squared (perpendicular) distances to all points that constitute the interior of the shape.

^{5}

*implicit orientation*

^{6}of simple symmetric shapes (such as ellipses or the letter “X”) and that estimates of implicit orientation exhibit many of the same characteristics as the explicit orientation of a line segment. For instance, implicit orientation was shown to produce an oblique effect (see also Liu, Dijkstra, & Oomes, 2002). Orientation was perceived with the greatest speed and accuracy when the shape was vertical or horizontal, despite the presence of oblique edges within the shape. (Note that the implicit orientation of an ellipse, or the letter “X,” corresponds precisely to the direction of its principal axis.) Based on their results, Li and Westheimer suggested that the automatic computation of global shape orientation may be performed by mechanisms closely related to those computing the explicit orientation of a line segment. Consistent with this suggestion, Boutsen and Marendaz (2001) have shown, using a visual-search paradigm, that the principal axis of a complex shape (with a clear dominant axis but no salient part structure) is computed sufficiently early to produce pop-out effects based on orientation differences.

*core model*(see Burbeck & Pizer, 1995)—must play a role in the perception of object orientation. (Their manipulation of relative phase and spatial frequency had a systematic influence on the medial axis). Although Burbeck and Zauberman did not consider such a model, the computation of principal axis also predicts an influence of relative phase and frequency modulation in their shapes. (1 shows the results of the principal-axis computation applied to their shapes.)

*silhouette*of the 3D objects.

*homogenous computation hypothesis*—postulates that all points within a shape are treated uniformly in computing the principal axis, irrespective of part structure. In other words, the computation of principal axis proceeds just as it does for simpler shapes. According to the second—

*part-based computation hypothesis*—the computation of shape orientation explicitly takes into account the decomposition of the shape into parts. Principal axes may be computed separately for individual parts and then integrated into a global orientation estimate or the principal-axis computation may proceed by assigning different weights to different points within the shape, depending on which part they belong to.

*r*/15,

*r*/15], where

*r*is the radial distance of the point from the center of the ellipse.

^{2}) and presented against a black background (0.03 cd/m

^{2}). Sixteen instances were generated for each of the 16 shape types, yielding a total of 256 test shapes. Figure 4 shows an instance of a shape for each of the 4 × 4 combinations of turning angle and part size.

*part size*and

*turning angle*at the part boundaries, each with four levels (see Stimuli section). Thus, 16 shape types were possible.

^{7}The orientation settings exhibited a significant dependence on turning angle at the part boundaries,

*F*(3, 45) = 42.29,

*p*< .0001. As the turning angles at the part boundaries increase, observers' orientation settings deviate increasingly from the shape's principal axis, in the direction of the base-part axis. The settings also exhibited a systematic dependence on part size,

*F*(3, 45) = 21.35,

*p*< .0001. The larger the attached part is, the greater is the deviation of perceived orientation from the shape's principal axis.

*F*(9, 135) = 8.73,

*p*< .0001. As is evident in Figure 5a, the influence of turning angle on observers' settings is considerably greater for larger parts. (See 2 for polar histograms of observers' orientation settings, for each of the 16 conditions.)

*geometric*influence that these variables have on the principal-axis orientation—in particular, on the angular separation between the principal axis and the base-part axis. As part size increases, for instance, the principal-axis orientation deviates increasingly from the base-part orientation. Thus, if observers' orientation settings corresponded to the orientation of the base part, for instance, their settings expressed as deviations from the principal axis would exhibit a systematic increase with part size. This would clearly not constitute a genuine perceptual influence of part size on shape orientation, however, beyond simply its geometric influence on the shape's principal axis.

*perceptual*influence, we normalized the orientation settings in Figure 5a by the respective angular separations between principal-axis and base-part orientation (i.e., by the

*geometric*influence that the presence of the part has on the shape's principal axis, or “maximal impact”). Figure 5b plots the data from Experiment 1 in terms of this normalized orientation.

*y*-axis, 0 corresponds to the orientation of the principal axis and 1 to the orientation of the base-part axis.

^{8}When the analyses were performed on normalized orientation, the influence of part size became marginal,

*F*(3, 45) = 2.84,

*p*= .0485. Thus, part size does not exert a reliable influence on perceived orientation, beyond its geometric influence on the principal axis. The influence of turning angle, however, continued to be highly significant,

*F*(3, 45) = 10.469,

*p*< .0001. This indicates that turning angle at the part boundaries has a genuine perceptual influence on orientation estimates, beyond any geometric influence.

^{9}Finally, the interaction between turning angle and part size was significant,

*F*(9, 135) = 2.83,

*p*< .005. The influence of turning angle on normalized orientation tends to be greater for larger parts.

^{10}

*psignifit*software, version 2.5.6 for Matlab (see Wichmann & Hill, 2001). The PSE (50% threshold) was used to measure perceived orientation, and the slope of the Weibull fit was used to measure the precision of the orientation estimate.

^{11}

*w*(where 0 ≤

*w*≤ 1). When

*w*= 1, points within the attached part are on an equal footing with the base part and, as a result, the computed orientation coincides simply with that of the standard principal axis of the entire shape (see Figure 10a).

*w*= 0, the attached part is entirely deweighted and not used at all (or ignored) in the principal-axis computation; the computed orientation thus coincides with the principal axis of the base part alone. As the weight

*w*decreases from 1 to 0, the computed orientation shifts gradually from the principal axis of the entire shape to that of the base part alone (see Figure 10a).

*w*for the attached part, which, when used in a differentially weighted principal-axis computation for that shape, would generate the observed orientation. We then collapsed these computed weights across all shape instances within a given condition (i.e., a combination of turning angle and part size).

*y*-axis in the two cases have different meanings. The normalized orientation data in Figure 5b indicate where the observed orientation falls on the unit scale between the principal-axis orientation and base-part orientation, whereas the values on the

*y*-axis of Figure 10b indicate the differential weight to points within the attached part, necessary in a principal-axis computation, to generate that observed orientation.

^{12}Nevertheless, the basic conclusion remains unchanged: Turning angle at the part boundaries has a genuine perceptual influence on perceived orientation that is not reducible to a geometric influence on the principal axis.

*Robust Statistics*approach to statistical estimation. This branch of statistics aims to develop estimators that are robust against violations of one's assumptions concerning the underlying model (e.g., the data points being sampled independently from a single Gaussian distribution). The methods developed allow for principled ways to deal with the presence of outliers—data points that either involve gross errors (e.g., in entry or coding) or were otherwise generated from a different process than the one under study.

*p*reflecting the perceptual independence of a part may be quantified simply as

*p*= 1

*−*

*w,*where

*w*is the relative weight assigned to it in the differentially weighted principal-axis scheme.) Robust Statistics is thus likely to provide a useful quantitative framework within which to capture the influence of perceptual segmentation on visual estimation.

*Structural descriptions*have sometimes been taken to be synonymous with volumetric-component approaches to object recognition (e.g., Biederman, 1987; Marr & Nishihara, 1978). The notion of a structural description is considerably more general, however, and does not logically entail either that the parts be three-dimensional or that they belong to a predefined class of shape primitives (such as generalized cones).

*shape*denoted by the distribution (see Melcher & Kowler, 1999; Vishwanath & Kowler, 2003).

*mutual regression,*in that the squared errors to be minimized are measured perpendicular to the candidate line, rather than in a fixed “vertical” direction as in standard linear regression (i.e., the squared residuals (ŷ − y)

^{2}of the predicted variable).

*r*

^{2}= .996 for means,

*r*

^{2}= .998 for standard deviations), which is to be expected given the small angular range within which the measurements fall. We thus report only the standard statistics throughout.