Reconstruction of 3D objects from 2D cross-sections with the 4-point subdivision scheme adapted to sets

Reconstruction of 3D objects from 2D cross-sections is an intriguing problem with many potential applications. We approach this problem through a novel multi-resolution method based on iterative refinement of the sets representing the cross-sections. To that end, we introduce a new geometric weighted average of two sets, defined for positive weights (corresponding to interpolation) and when one weight is negative (corresponding to extrapolation). This new average can be used to interpolate between cross-sections of a 3D object in a piecewise way. To obtain a smoother reconstruction of the 3D object, we adapt to sets the 4-point interpolatory subdivision scheme using the new average with both positive and negative weights. The effectiveness of the new method is demonstrated by several examples.