A new convexity measurement for 3D meshes

Abstract

This paper presents a novel convexity measurement for 3D meshes. The new convexity measure is calculated by minimizing the ratio of the summed area of valid regions in a mesh's six views, which are projected on faces of the bounding box whose edges are parallel to the coordinate axes, to the sum of three orthogonal projected areas of the mesh. The complete definition, theoretical analysis, and a computing algorithm of our convexity measure are explicitly described. This paper also proposes a new 3D shape descriptor CD (i.e., Convexity Distribution) based on the distribution of above-mentioned ratios, which are computed by randomly rotating the mesh around its center, to better describe the object's convexity-related properties compared to existing convexity measurements. Our experiments not only show that the proposed convexity measure corresponds well with human intuition, but also demonstrate the effectiveness of the new convexity measure and the new shape descriptor by significantly improving the performance of other methods in the application of 3D shape retrieval.

Investigation on Geometrical Complexity Techniques for Assessing AM Feasibility

“…Even though there is a lack of standard objective complexity metrics, several researchers proposed objective metrics for assessing the shape/geometrical complexity, assigning quantitative values. One of the most used in literature, which is considered the reference technique in this study and here-in named convex envelope complexity (CEC), is proposed by Lian et al [10] and used by Fera et al [3] in their study. It is based on the ratio between part's volume and the volume of its convex envelope, which from a mathematical point of view is complex to evaluate.…”

Convex Decomposition And Efficient Shape Representation Using Deformable Convex Polytopes

“…Therefore, the proposed global shape concavity depends on the number of convex components the shape has and the relative size of the unique region each polytope represents compared to the largest convex component of the shape. Notice that, the global shape concavity measure is not necessary for the proposed approximate convex decomposition (which only requires local convexity measure); however, convexity of a shape has many applications of its own [12,11,13].…”

“…Convexity Measure: Convexity is one of the most basic shape descriptor with many applications [11]. The two most common definitions of convexity of a shape are the regionbased (RB) and the perimeter-based (PB) approaches.…”

“…The two most common definitions of convexity of a shape are the regionbased (RB) and the perimeter-based (PB) approaches. RB defines the convexity of a shape as the ratio of the shape's area to the area of its convex hull; whereas, PB convexity is the ratio of the perimeter of the convex hall of the shape to the perimeter of the shape [11,12]. However, the major problem of the RB convexity methods are their insensitivity to deep and thin protrusions, while the PB methods are intolerant to small boundary deformations [13].…”

“…However, the major problem of the RB convexity methods are their insensitivity to deep and thin protrusions, while the PB methods are intolerant to small boundary deformations [13]. Recent papers in the literature that addresses the above problems of the convexity measures of shapes can be found in [12,11,13,14,15].…”