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As an important geometric property of many structures or structural components, convexity plays an important role in computer vision and image understanding. In this paper, we describe a general approach that can force various edge-grouping algorithms to detect only convex structures from a set of boundary fragments. The basic idea is to remove some fragments and fragment connections so that, on the remaining ones, a prototype edge-grouping algorithm that detects closed boundaries without the convexity constraint can only produce convex closed boundaries. We show that this approach takes polynomial time and preserves the grouping optimality by not excluding any valid convex boundary from the search space. Choosing the recently developed ratio-contour algorithm as the prototype grouping algorithm, we develop a new convex-grouping algorithm, which can detect convex salient boundaries with good continuity and proximity in a globally optimal fashion. To facilitate the application of this convex-grouping algorithm, we develop a new fragment-connection method based on four-point Bezier curves. We demonstrate the performance of this convex-grouping algorithm by conducting experiments on both synthetic and real images. In addition, we provide a comparison with some prior edge-grouping algorithms. Finally, we show that the proposed convex-grouping algorithm can be further extended to detect convex open boundaries, derive region-based image hierarchies, and incorporate some simple human-computer interactions. [ABSTRACT FROM AUTHOR]