Minimum points and views for the recovery of three-dimensional structure.

Abstract

Mathematical analyses of motion perception have established minimum combinations of points and distinct views that are sufficient to recover three-dimensional (3D) structure from two-dimensional (2D) images, using such regularities as rigid motion, fixed axis of rotation, and constant angular velocity. To determine whether human subjects could recover 3D information at these theoretical levels, vie presented subjects with pairs of displays and asked them to determine whether they represented the same or different 3D structures. Number of points was varied between two and five; number of views was varied between two and six; and the motion was fixed axis with constant angular velocity, fixed axis with variable velocity, or variable axis with variable velocity. Accuracy increased with views, decreased with points, and was greater with fixed-axis motion. Subjects performed above chance levels even when motion was eliminated, indicating that they exploited regularities in addition to those in the theoretical analyses.Theoretical investigations of visual motion have provided a number of specific analyses of the minimum number of points and views required to recover three-dimensional (3D) structure from two-dimensional (2D) images. Recovery of 3D structure, in this context, is denned as determining the x, y, and z coordinates of each point, up to a scale factor. These analyses differ in the constraints that are imposed. UUman (1979) showed that under a rigidity constraint, three views of four noncoplanar points are sufficient to recover structure in an orthographic projection, up to a reflection about the frontal plane. The required numbers of points and views are reduced by adding further constraints, such as planarity (Hoffman & Flinchbaugh, 1982), fixed axis of rotation (Hoffman & Bennett, 1986;Webb & Aggarwal, 1981), and constant angular velocity (Hoffman & Bennett, 1985). These proofs are summarized in Table 1.A number of empirical studies have addressed issues related to theoretical analyses of the recovery of structure from motion. Several studies (e.g., Braunstein & Andersen, 1986;Schwartz & Sperling, 1983;Todd, 1985) have questioned the generality of the rigidity constraint. Other studies have considered the recovery of structure with small numbers of views or with small numbers of points. Lappin, Doner, and Kotlas (1980) found that subjects could make accurate judgments based on 3D structure This research was supported by a contract to D. Hoffman from the Office of Naval Research, Cognitive and Neural Sciences Division, Perceptual Sciences Group. We thank Joseph Lappin and James Todd for helpful comments on an earlier version of this article and Johnna Eastbum and James Tittle for assistance in various aspects of this research.

Kinetic depth effect and optic flow—I. 3D shape from Fourier motion

“…Examples of algorithms that derive a 3D configuration from a set of n points (or similar features) displayed in each of m frames are Hoffman and Bennett (1985) and Ullman (1979Ullman ( , 1985, or see Braunstein, Hoffman, Shapiro, Andersen and Bennett (1987) for a more empirical treatment. A list of visual features is identified and located in 2D space on each frame.…”

Monocular discrimination of rigidly and nonrigidly moving objects

“…These experiments show that various types of nonrigidity are detectable with the theoretical minimum number ofelements and views. In some experiments, performance improved with an increase in the number of views or angle of rotation (Braunstein et al, 1987;Todd, 1982). In other experiments, performance did not improve with an increase in the number of views (Todd & Bressan, 1990).…”

“…This is a more restricted form of the processing rule proposed by Jansson and Johansson (1973) concerning the principle of minimum object change. In many experiments (e.g., those of Braunstein, Hoffman, Shapiro, Andersen, & Bennett, 1987;Johansson, 1974;Wallach & O'Connell, 1953), subjects indeed have reported perceiving a single rigid interpretation while viewing displays that have rigid interpretations. The rule generally appears to accord with reality, although exceptions are known, such as the rubber pencil illusion (Pomerantz, 1983), the Ames window (Ames, 1951), and This research was supported by the Netherlands Organization for Scientific Research.…”

“…Several authors have attempted to deduce the minimum number of views and number of elements that are required by the visual system to detect nonrigidity from parallel projections (Braunstein, Hoffman, & Pollick, 1990;Braunstein et al, 1987;Norman & Todd, 1993;Perotti, Todd, & Norman, 1996;Todd, 1982;Todd & Bressan, 1990). These experiments show that various types of nonrigidity are detectable with the theoretical minimum number ofelements and views.…”

The perceptual buildup of three-dimensional structure from motion

“…Early perceptual work was typically focused on the recovery of qualitative aspects of an object's structure, such as its apparent rigidity, volume, or coherence. More recently, studies have addressed, quantitatively, the accuracy of perceived 3-D structure (e.g., Braunstein, Hoffman, Shapiro, Andersen, & Bennett, 1987; Dosher, Landy, & Sperling, 1989;Lappin & Fuqua, 1983; Loomis & Eby, 1988 Sperling, Landy, Dosher, & Perkins, 1989;Todd, 1982Todd, , 1984Todd, , 1985.In this paper, we present a set of psychophysical experiments in which we have examined both the accuracy We thank Tomaso Poggio, Shimon Ullman, and Whitman Richards for valuable comments on this paper, and Mike Landy for useful discussions. This report describes research done within the Artificial In- The experiments were motivated in part by a computational model proposed by Ullman (1984), called the incremental rigidity scheme, in which an accurate 3-D structure is built up incrementally, as images of moving objects are considered over an extended time period.…”