1, the contour segment is convex if the region on the left (darker gray) is inside, but it is concave if the region on the right (lighter gray) is inside. Although there are always two regions, one convex and one concave, the ambiguity is resolved on the basis of which is the inside. The short segment within the aperture (circle) separates two regions.

The example is that of a large letter C, but a segment of the contour is also shown in isolation. Furthermore, a point of zero curvature along the contour will also mark any smooth transition from convexity to concavity (inflection).įigure 1 provides a visual example of contour curvature along a shape. We will come back to this when talking about figure–ground organization.Ī flat line is a special case of contour it has zero curvature, but as long as we can measure some nonzero curvature, we can also assign both a magnitude and a sign to it. Without a rule to select the side, there is ambiguity about convexity and concavity. For example, we can say that a circle has positive (convex) curvature only because we choose the sign with respect to the inside area of the circle. This can be done by choosing to label contours with respect to the “inside,” but this convention requires that we know which is the inside and which is the outside of the region of interest. To uniquely label a region as convex or concave, we need to choose only one of the two sides as we move along the contour. That is, for every convexity, we have a corresponding concavity on the opposite side of the contour, and vice versa. If we were to label both regions, we would have a redundant and complementary sequence of convexities and concavities. This sign allows us to distinguish the convex and concave regions, and by convention, a positive sign refers to convexity and a negative one to concavity.

For this segment of a contour, the region within the aperture that includes all chords connecting any pair of points on the contour is labeled as convex, and the complementary region is labeled as concave. We select a short segment so that it curves smoothly and does not include any inflections. To understand the meaning of convexity and the need for a sign, we start by thinking of a segment of a contour as a line that divides two adjacent portions of the plane, within a local aperture. What is important to note is that in addition to a magnitude, curvature needs a sign. For example, curvature along a circle is constant and is equal to the reciprocal of its radius and, therefore, decreases as the radius increases with scale. We conclude that there is good evidence for the role of convexity information in figure–ground organization and in parsing, but other, more specific claims are not (yet) well supported.įor any smooth contour in the image, it is possible to measure curvature at any location along the contour: the change in tangent direction as we move along the curve. The focus is on convexity and concavity along a 2-D contour, not convexity and concavity in 3-D, but the important link between the two is discussed. Despite some broad agreement on the importance of convexity in these areas, there is a lack of consensus on the interpretation of specific claims-for example, on the contribution of convexity to metric depth and on the automatic directing of attention to convexities or to concavities. A review is necessary to analyze the evidence on how convexity affects (1) separation between figure and ground, (2) part structure, and (3) attention allocation. This includes evidence from behavioral, neurophysiological, imaging, and developmental studies.

We review a large body of evidence on the role of this information in perception of shape and in attention. For smooth contours in an image, it is possible to code regions of positive (convex) and negative (concave) curvature, and this provides useful information about solid shape. Interest in convexity has a long history in vision science.

Point of contour definition