Visualization of proximity objects and refinement of characteristic line from biased samples

Applications in many applied areas of modelling are based on recurrent sampling of geometric patterns. These specimens are usually biased, thus a preliminary goal in order to obtain a clear and comprehensible image is refinement of a characteristic object from a collection of distorted samples. This is done by establishing a visualization method which enables the simultaneously representation of multiple occurrences of elements. Generally, objects that are linearly defined and embedded in 3-dimensional space can be characterized by their edges ("slices" of lines). Consider, a collection of lines generated by a family of proximity edges. The question discussed here is this: how can we visualize and then approximate (in terms of line's coefficients) this collection by a single line ? Namely, refine (i.e. recover) the source line. A solution to this issue is the basis for refinement of objects. The visualization of linearly defined objects in Parallel Coordinates is constructed from the fundamental point ↔ line duality, thus their representations are determined by the coefficients of their defining equations. Composing this representational advantage with additional geometric methods yields simple planar visualization, and an efficient solution to the refinement problem, where both can be applied to any space dimension. Algorithm based on this concept of representation which uses dual images of lines and determines an approximated line is also obtained. Additionally, it turns out that our approach can be nicely generalized to more complex multi-dimensional regions and approximated curves.