A new computational analysis is described that is capable of estimating the 3D shapes of continuously curved surfaces with anisotropic textures that are viewed with negligible perspective. This analysis assumes that the surface texture is homogeneous, and it makes specific predictions about how the apparent shape of a surface should be distorted in cases where that assumption is violated. Two psychophysical experiments are reported in an effort to test those predictions, and the results confirm that observers' ordinal shape judgments are consistent with what would be expected based on the model. The limitations of this analysis are also considered, and a complimentary model is discussed that is only appropriate for surfaces viewed with large amounts of perspective.

*The perception of the visual world*, Gibson (1950) introduced the concept of texture gradients as a potential source of optical information about the layout of surfaces in the environment. Since then, many different properties of optical texture have been identified that can be used to estimate various aspects of 3D surface structure (Gårding, 1992, 1993; Purdy, 1958), and many psychophysical experiments have been performed in an effort to determine which of these potential sources of information are most relevant to human perception. The literature on this topic contains many conflicting results, however, and there is no clear consensus as yet on how patterns of optical texture are perceptually analyzed by human observers.

*μ*(

*ϕ*,

*γ*) at a point on a planar surface in a visual direction (

*ϕ*,

*γ*) can be used to determine the optical slant

*σ*(

*ϕ*,

*γ*) at that point from the normalized gradient defined by the following equation:

*ϕ*,

*γ*) is the relative depth of a surface point in a visual direction (

*ϕ*,

*γ*),

*λ*(

*ϕ*,

*γ*) is the projected major axis of a texture element at that point, and

*λ*

_{Max}and

*λ*

_{Min}are the maximum and minimum major axis lengths. Note that this is similar to a normalized gradient, but it is designed to evaluate the variations in scaling over large regions of visual space, rather than small local neighborhoods.

*ϕ*on the cross-section is labeled

*ω*(

*ϕ*), and the local width maximum that occurs at the depth extremum is labeled

*ω*

_{Max}. Consider a vector

**N**(

*ϕ*) in the plane of the surface cross-section that is normal to its outer boundary at the position

*ϕ*. If the surface is depicted with negligible perspective and its texture is homogeneous, then the angle

*σ*(

*ϕ*) of this vector relative to the z-axis can be determined by the following equation:

**N**(

*ϕ*) will not in general be a surface normal. Rather, it is the component of the surface normal that is in the plane of the cross-section. Similarly, surface points at the local depth extrema will not generically have fronto-parallel orientations. They are the points along a cross-section that are closer to fronto-parallel than their neighbors. It should also be noted that edge density along a cross-section is the reciprocal of texture width, and could therefore provide an alternative measure for calculating slant in Equation 3. Indeed, this would be the preferred approach for surfaces with more fine scale textures whose elements are not easily individuated (see Thaler, Todd, & Dijkstra, 2007).

^{2}region of the display screen, which subtended 20° of visual angle when viewed at a distance of 83 cm. The displays were viewed monocularly with an eye patch, and a chin rest was used to constrain head movements.

*ω*(

*ϕ*)/

*ω*

_{Max}(

*ϕ*) will always be exactly one at the depth extrema, and between zero and one everywhere else.

*ω*

_{Max}(

*ϕ*) have been appropriately adjusted for each position

*ϕ*along a scan line, and the width maxima that correspond to depth extrema have been distinguished from those that occur at inflection points, it is then possible to estimate the surface depth gradients using the following equation:

*ω*(

*ϕ*)/

*ω*

_{Max}(

*ϕ*) is constrained to be between zero and one, and they are therefore insufficient on their own for estimating 3D shape. However, this problem can easily be resolved by combining Equation 5 with the simple rule described above that the sign of slant reverses at each depth extremum. It is then possible to estimate the surface depth profile by integrating the gradient function to determine the relative depths along each scan line.