We examined the perception of 3D shape for surfaces folded, carved, or stretched out of textured materials. The textures were composed of sums of sinusoidal gratings or of circular dots, and were designed to differentiate between orientation and frequency information present in perspective images of the surfaces. Correct perception of concavities, convexities, saddles, and slants required the visibility of signature patterns of orientation modulations. These patterns were identical to those identified previously for developable surfaces (A. Li & Q. Zaidi, 2000; Q. Zaidi & L. Li, 2000), despite the fact that textures were statistically homogeneous on developable surfaces but not on carved or stretched surfaces. Frequency modulations in the image were interpreted as cues to distance from the observer, which led to weak but qualitatively correct percepts for some carved and stretched surfaces but to misperceptions for others, similar to the misperceptions for developable surfaces (A. Li & Q. Zaidi, 2003). Irrespective of whether texture on the surface is homogeneous or non-homogeneous, similar neural modules can be used to locate signature orientation modulations and thus extract shape from texture cues.

*correct*percepts, we explicitly mean that the perceived

*signs*of curvatures and

*directions*of slants are identical to those of the simulated 3D surface.)

*z*-axis (i.e., the axis of carved depth). The constant-x solid was formed by repeating identical planar patterns along the

*x*-axis, orthogonal to the axis of carved depth. The same planar patterns were also folded into developable corrugations, and stretched onto corrugated solids.

*z*-axis (Figure 2). In the images of the solids patterned with the horizontal-vertical plaid (Figure 10A) concavities, convexities, right and left slants can all be correctly identified. The orientation modulations of the horizontal component appear identical to those of the horizontal component on the developable surface. Because the carved solid’s axis of maximum curvature is horizontal, the horizontal component is not distorted on the surface and is identical to the undistorted horizontal component on the developable surface. Projection thus results in patterns of orientation modulations of the horizontal component that are identical for both kinds of surfaces.

*x*-axis (Figure 2), the texture on the surface is inhomogeneous, but the inhomogeneities and hence the perspective images are quite different from the carved constant-z solid. Figure 15 shows images of the corrugations carved from constant-x solids formed by grating patterns. Concavities, convexities, right and left slants are all identifiable for the images of the horizontal-vertical plaid in Figure 15A and the octotropic plaid in Figure 15B, and observers identify slants correctly (upper left and middle panels, Figure A3). In the images, the horizontal component gives rise to the same signature orientation modulations as the developable surface because the horizontal component is not distorted by the carving along the horizontal axis. When the horizontal component is subtracted from the planar pattern of the constant-x solid in Figure 15C, the image no longer contains sufficient information to distinguish signs of curvatures and slants. As a result, observers confuse left and right slants (upper right panel, Figure A3).

*x*-axis. As the angle of the cut increases, the frequency on the surface of the cut increases. Because increasing the slant also decreases the projected width of a unit surface length in the image, the projected frequency in the image increases much more with increasing slant than for the developable surface. The directions of the frequency gradients are similar for developable and constant-x solids, but the projected frequency for the constant-x solid will be zero when the slant of the cut is zero (i.e., where the surface is fronto-parallel). Concavities and convexities thus exhibit similar high-zero-high frequency gradients. In addition, portions of the surface that are at equal depths (e.g., the peaks of the convexities) cut through identical portions of the planar pattern along the

*x*-axis. Because the surface is periodic and presented with either a central concavity or convexity, the images are symmetric about the vertical mid-line (e.g., Figure 15B–C).

^{2}. Surfaces were presented in one of four different central phases as shown in Figure A0. For the two images on the right, the projection was centered, respectively, to the left and right of a concavity (phase = −pi/8 and pi/8), and for the two images on the right, the projection was centered, respectively, to the left and right of a convexity (phase = 7pi/8 and 9pi/8). Thus the images at phases −pi/8 and 9pi/8 were centered on rightward slanting portions of the surface, and at pi/8 and 7pi/8 they were centered on leftward slanting portions of the surface. Each image contained two thin, red, vertical lines, each of which subtended 0.5 deg, displaced 0.4 deg to the left and to the right of the central vertical mid-line (0.8 deg apart). (In Figure A0, the lines have been thickened and lengthened for visibility.) One of the lines was always located at the center of either the concavity or the convexity. Observers were told that these lines indicated two locations directly behind them on the surface. The task was to indicate which of the two locations on the surface appeared closer to them, or if they appeared at equal depths. If the surface presented in phases of −pi/8 or 9pi/8 (slanted to the right) was perceived correctly, observers should have responded that the left line appeared closer to them in depth. If the surface presented in phases of pi/8 or 7pi/8 (slanted to the left) was perceived correctly, observers should have indicated that the right line appeared closer to them. If any surface appeared fronto-parallel, observers indicated that the two red lines appeared at equal depths.

*xy*-plane at the angle of the texture component. The line is then slanted out of the

*xy*-plane about a vertical axis at an angle equal to the local slant of the surface and its perspective projection in the

*xy*-plane is computed. We also compute the perspective projection if the slanted line is additionally pitched about a horizontal axis at an angle equal to the pitch of the surface. The perspective coordinates of the slanted line in the

*xy*-plane then provide the projected orientation, and the projected frequency is equivalent to the inverse of the projected length of the line in the

*xy*-plane.

*d*(i.e., (0, 0, 0) is at eye-height). We start with the following parameters in radians:

*x, y, z*) (i.e.,

*y*units above eye-height), where

*z*is the difference in depth between the surface and the image plane. Figure B shows views of this line in both the

*xy*- (frontal) and

*xz*- (aerial) planes. If the line were lying in the

*xy*-plane at an angle of

*ω*radians from the horizontal, the coordinates of the rightmost end point would be .

*ω*from the horizontal. Changes in the length of this line as a function of

*θ*and

*α*provide changes in local spatial frequency of the texture component oriented at

*ω*+

*π*/2 radians.

*xy*-plane is repeated along the

*z*-axis, and we compute the perspective projection of the slanted carving through this solid. We also compute the perspective projection of the carving if it is pitched about a horizontal axis.

*d*(i.e., (0, 0, 0) is at eye-height). We start with the following parameters in radians:

*x, y, z*) (i.e.,

*y*units above eye-height), where

*z*is the difference in depth between the surface and the image plane. Figure C shows views of this line in both the

*xy*- (frontal) and

*xz*- (aerial) planes. If the line were lying in the constant-z plane at an angle of

*ω*radians from the horizontal, the coordinates of the rightmost end point would be .

*xy*-planes of the solid material to be carved are identical, the

*x*-projection of this line will be identical for all values of

*z*(indicated by the shaded area in the aerial view). The surface is carved at

*θ*away from the

*xy*-plane. The length of the cut in the

*xz*-plane is

*R*and its

*x*-projection will equal cos

*ω*. The projected orientation and frequency are computed from the endpoints of the cut

*R*= cos

*ω*/cos

*θ*. The coordinates of the endpoints of the cut will be (

*x, y, z*) and .

*ω*from the horizontal. Changes in the length of this cut as a function of

*θ*and

*α*provide changes in local spatial frequency of the texture component oriented at

*ω*+

*π*/2 radians.

*yz*-plane of the constant-x solid contains the same oriented texture component that is repeated along the

*x*-axis. This derivation computes the perspective projection of the slanted carving through this solid. We also compute the perspective projection if the carved solid is pitched about a horizontal axis.

*d*(i.e., (0, 0, 0) is at eye-height). We start with the following parameters in radians:

*x, y, z*) (i.e.,

*y*units above eye-height), where

*z*is the difference in depth between the surface and the image plane. Figure D shows views of this line in both the

*xy*- (frontal) and

*xz*- (aerial) planes. If the line were lying in the constant-z plane at an angle of

*ω*radians from the horizontal, the coordinates of the rightmost end point would be .

*yz*-planes of the solid material to be carved are identical, the

*x*-projection of this line will be identical for all values of

*x*(indicated by the shaded area in the aerial view). The surface is carved at

*θ*away from the

*xy*-plane. The length of the cut in the

*xz*-plane is

*C*and its

*z*-projection will equal cos

*ω*. The projected orientation and frequency are computed from the endpoints of the cut

*C*= cos

*ω*/tan

*θ*. The coordinates of the endpoints of the cut will be (

*x, y, z*) and .

*ω*from the horizontal. Changes in the length of this cut as a function of

*ϑ*and

*α*provide changes in local spatial frequency of the texture component oriented at

*ω*+

*π*/2 radians.