Shape Encoding and Subjective Contours

Ullman proposes that subjective contc,.us consist of two circles which meet smoothly and which arc tangential to the contrast boundaries from which they originate. The foim of the solution derives from a number of premises, one of which Ullman calls “the locality hypothesis”. ‘Ihis is “based in part on cxpcrimcntal obscrvntions, and partly on a theoretical consideration” ([I 51 ~2). ‘I’hc “cxpcrimcntal observation” rcfcrrcd to is the following: suppose that A’ is a point near A on the filled-in contour AB as shown in Figure 1. If the process by which AB was constructed is applied to A’B, it is claimed that it gcneratcs the portion of AJ3, bctwccn A’ and B. Let us call this property “cxtcnsibility”. Ullman argues that cxtcnsibility, togcthcr with the propcrtics of isotropy, smoothness, and having minimal integral curvature, logically entails a solution consisting of two circles which meet smoothly. In the fit11 version of this paper, we analyze the two-cti solution and formulate the condition for minimal integral curvature. This can bc solved by any descent method such as Newton-Raphson. A program has been written which computes the minimum integral curvature two-circle solution given the boundary angles 4, 8, and AB, and which returns as a result the point at which they meet and at which the curvature is discontinuous (the ‘knot point’). A tortuous version of this simple proof and program recently appcarcd in [13]. We then show by example that the two circle solution is not in fact extensible.