Donald D. Hoffman

Salience of visual parts

Many objects have component parts, and these parts often differ in their visual salience. In this paper we present a theory of part salience. The theory builds on the minima rule for defining part boundaries. According to this rule, human vision defines part boundaries at negative minima of curvature on silhouettes, and along negative minima of the principal curvatures on surfaces. We propose that the salience of a part depends on (at least) three factors: its size relative to the whole object, the degree to which it protrudes, and the strength of its boundaries. We present evidence that these factors influence visual processes which determine the choice of figure and ground. We give quantitative definitions for the factors, visual demonstrations of their effects, and results of psychophysical experiments.

Parsing silhouettes: The short-cut rule

Many researchers have proposed that, for the purpose of recognition, human vision parses shapes into component parts. Precisely how is not yet known. The minima rule for silhouettes (Hoffman & Richards, 1984) defines boundary points at which to parse but does not tell how to use these points to cut silhouettes and, therefore, does not tell what the parts are. In this paper, we propose the short-cut rule, which states that, other things being equal, human vision prefers to use the shortest possible cuts to parse silhouettes. We motivate this rule, and the well-known Petter’s rule for modal completion, by the principle of transversality. We present five psychophysical experiments that test the short-cut rule, show that it successfully predicts part cuts that connect boundary points given by the minima rule, and show that it can also create new boundary points.

The interpretation of biological motion

The term biological motion has been coined by Johansson (1973) to refer to the ambulatory patterns of terrestrial bipeds and quadripeds. In this paper a computational theory of the visual perception of biological motion is proposed. The specific problem addressed is how the three dimensional structure and motions of animal limbs may be computed from the two dimensional motions of their projected images. It is noted that the limbs of animals typically do not move arbitrarily during ambulation. Rather, for anatomical reasons, they typically move in single planes for extended periods of time. This simple anatomical constraint is exploited as the basis for utilizing a “planarity assumption” in the interpretation of biological motion. The analysis proposed is: (1) divide the image into groups of two or three elements each; (2) test each group for pairwise-rigid planar motion; (3) combine the results from (2). Fundamental to the analysis are two “structure from planar motion” propositions. The first states that the structure and motion of two points rigidly linked and rotating in a plane is recoverable from three orthographic projections. The second states that the structure and motion of three points forming two highed rods constrained to move in a plane is recoverable from two orthographic projections. The psychological relevance of the analysis and possible interactions with top down recognition processes are discussed.

The computation of structure from fixed-axis motion: Rigid structures

We show that three distinct orthographic views of three points in a rigid configuration are compatibel with at most 64 interpretations of the three-dimensional structure and motion of the points. If, in addition, one assumes that the three points spin about a fixed axis over the three views, then with probability one there is a unique three-dimensional interpretation (plus a reflection). Moreover the probability of false targets is zero. In the special case that the axis of rotation is parallel to the image plane three views of the three points are sufficient to obtain at most two interpretations (plus reflections)-unless one assumes the angular velocity about the axis is constant, in which case three views of two points are sufficient to determine a unique interpretation. Closed form solutions are obtained for each of these cases. The systems of equations studied here are in each case overconstraining (i.e. there are more independent equations than unknowns) and are amenable to solution by nonlinear programming. These two properties make possible the construction of noise insensitive algorithms for computer vision systems. Our uniqueness proofs employ the Principle of upper semicontinuity, a principle which underlies a general mathematical framework for the analysis of solutions to overconstraining systems of equations.

Parts of Visual Objects: An Experimental Test of the Minima Rule

Three experiments were conducted to test Hoffman and Richardss (1984) hypothesis that, for purposes of visual recognition, the human visual system divides three-dimensional shapes into parts at negative minima of curvature. In the first two experiments, subjects observed a simulated object (surface of revolution) rotating about a vertical axis, followed by a display of four alternative parts. They were asked to select a part that was from the object. Two of the four parts were divided at negative minima of curvature and two at positive maxima. When both a minima part and a maxima part from the object were presented on each trial (experiment 1), most of the correct responses were minima parts (101 versus 55). When only one part from the object—either a minima part or a maxima part—was shown on each trial (experiment 2), accuracy on trials with correct minima parts and correct maxima parts did not differ significantly. However, some subjects indicated that they reversed figure and ground, thereby changing maxima parts into minima parts. In experiment 3, subjects marked apparent part boundaries. 81% of these marks indicated minima parts, 10% of the marks indicated maxima parts, and 9% of the marks were at other positions. These results provide converging evidence, from two different methods, which supports Hoffman and Richardss minima rule.

Inferring three-dimensional shapes from two-dimensional silhouettes

Although an infinity of three-dimensional (3-D) objects could generate any given silhouette, we usually infer only one 3-D object from its two-dimensional (2-D) projection. What are the constraints that restrict this infinity of choices? We identify three mathematical properties of smooth surfaces plus one simple viewing constraint that seem to drive our preferred interpretation of 3-D shape from 2-D contour. The constraint is an extension of the notion of general position. Taken together, our interpretation rules predict that “dents” in a 3-D surface should never be inferred from a smooth 2-D silhouette.

Part-Based Representations of Visual Shape and Implications for Visual Cognition

ABSTRACT Human vision organizes object shapes in terms of parts and their spatial relationships. Converging experimental evidence suggests that parts are computed rapidly and early in visual processing. We review theories of how human vision parses shapes. In particular, we discuss the minima rule for finding part boundaries on shapes, geometric factors for creating part cuts, and a theory of part salience. We review empirical evidence that human vision parses shapes into parts, and show that parts-based representations explain various aspects of our visual cognition, including figure-ground assignment, judgments of shape similarity, memory for shapes, visual search for shapes, the perception of transparency, and the allocation of visual attention to objects.

Discriminating rigid from nonrigid motion: Minimum points and views

Theoretical investigations of structure from motion have demonstrated that an ideal observer can discriminate rigid from nonrigid motion from two views of as few as four points. We report three experiments that demonstrate similar abilities in human observers: In one experiment, 4 of 6 subjects made this discrimination from two views of four points; the remaining subjects required five points. Accuracy in discriminating rigid from nonrigid motion depended on the amount of nonrigidity (variance ofthe interpoint distances overviews) in the nonrigid structure. The ability to detect a rigid group dropped sharply as noise points (points not part of the rigid group) were added to the display. We conclude that human observers do extremely well in discriminating between nonrigid and fully rigid motion, but that they do quite poorly at segregating points in a display on the basis of rigidity.

Inferring local surface orientation from motion fields

The problem of inferring local surface orientation from changing images is studied computationally by deriving conditions under which the motion information is sufficient for an information processing system, biological or otherwise, to infer unique descriptions of the local surface orientation. The analysis is based on shape-from-motion proposition, which states, that given the first spatial derivatives of the orthographically projected velocity and acceleration fields of a rigidly rotating regular surface, then the angular velocity and the surface normal at each visible point on that surface are uniquely determined up to a reflection. The proof proceeds in two steps. First it is shown that surface tilt and one component of the angular velocity are uniquely determined by the first spatial derivatives of the velocity field. Then it is shown that surface slant and the remaining two components of the angular velocity are uniquely determined if the first spatial derivatives of the acceleration field are also available.

Completing visual contours: The relationship between relatability and minimizing inflections

Visual completion is a ubiquitous phenomenon: Human vision often constructs contours and surfaces in regions that have no sharp gradients in any image property. When does human vision interpolate a contour between a given pair of luminance-defined edges? Two different answers have been proposed: relatability and minimizing inflections. We state and prove a proposition that links these two proposals by showing that, under appropriate conditions, relatability is mathematically equivalent to the existence of a smooth curve with no inflection points that interpolates between the two edges. The proposition thus provides a set of necessary and sufficient conditions for two edges to be relatable. On the basis of these conditions, we suggest a way to extend the definition of relatability (1) to include the role of genericity, and (2) to extend the current all-or-none character of relatability to a graded measure that can track the gradedness in psychophysical data.