# Construction of 3D Orthogonal Convex Hull of a Digital Object

## Abstract

Orthogonal convex hull of a digital object in 3D domain is defined as the minimum volume orthogonal polyhedron enclosing the object such that its intersection with an axis-parallel face plane is either empty or a collection of projection-disjoint convex polygons. A novel and efficient algorithm for construction of 3D orthogonal convex hull of a digital object is proposed. The algorithm is based on orthogonally slicing the object into slab polygons followed by connecting all possible slab polygons on a slicing plane and finding their 2D orthogonal convex hulls. The regions belonging to the 2D orthogonal convex hulls are replaced by the corresponding UGCs (unit grid cubes) and the exterior UGC-faces are merged to give the 3D orthogonal convex hull. The algorithm operates in integer domain and executes in time linear in the number of voxels on the object surface. The algorithm operates in exactly two passes irrespective of the object size or grid resolution. Experimentation with a wide range of objects has provided accurate results, some of which are presented here to demonstrate the effectiveness of the algorithm.

## Keywords

3D orthogonal convex hull Orthogonal slicing 3D orthogonal outer cover 3D concavity 2D orthogonal convex hull## Notes

### Acknowledgement

A part of this research is funded by CSIR, Govt. of India under SRF (File No. 08/03(0091)/2012-EMR-1) and Sponsored project (Scheme No. 22(0568)/12/EMR-II).

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