Sagi Filin 
  
  
but also in respect to the internal hole. Thus the 
modified planar subdivision procedure creates a list of 
three object types, external faces (defining the inner 
part of a face) external faces (holes) and a separated 
list of counterpart free or linked edges. Next, the last 
two types are associated with the first. Since holes or 
edges cannot belong to more then one face it is 
sufficient to determine whether or not one of their 
Figure 5. Inner faces (holes) inside a face vertices is inside a face (by applying any point in 
polygon algorithm). Note that although holes can 
appear inside another hole, they should be associated only with the inner most one. Having the overall plane subdivided 
(equivalent to triangulating the plane in the point-base conflation) enables performing the local rubber-sheeting 
transformations. 
  
  
  
  
  
2.4 The Rubber-Sheeting Transformation 
The transformation of objects inside each face is based on the transformations between the vertices of the bounding 
polylines, just as it was in the matching of counterpart polylines. However, not all vertices participate in the transformation 
of each object. This happens when some of the reference vertices are occluded by other segments of the polygon or by other 
reference objects. Occlusions between or within reference elements can block the influence of regions on the transformation 
of a given point (Doytsher and Gelbman, 1995). To demonstrate this, consider for example a point in Figure 6, located 
under one of the two internal structures. As the upper part of the bounding polygon is hidden, it is reasonable to assume that 
this part will not affect the translation of that point. The influence of the upper part will be due to the internal structure. 
Self-occluded areas are also possible, for example with non-convex bounding polygons and certainly with holes, for which 
one of their sides is always hidden. Regions of influence define the reference vertices affecting the transformation of a given 
point, therefore their actual effect is in defining which vertices participate in the transformation and which do not. 
As the influence regions are a function of both self-occlusions and occlusions between reference objects, they should be 
computed separately and then superimposed to form a composite influence region. The suggested algorithm for their 
computation has a linear running time, and is based on scanning the polygon vertices in a clockwise order (the outcome of 
the face construction procedure) starting from a reference vertex that is visible from the object point. The visibility or 
invisibility of the consecutive points along the polygon is based on a relative topological analysis of their locations in regard 
to the object point. The algorithm also holds for the computation of the 
influence regions for holes, while in this case the order of the vertices 
pact - should be counterclockwise (i.e., the original order of vertices for holes). 
For computing influence regions for polylines the same algorithm holds as 
well, simply duplicating the polylines in an inverse order and treating them 
as closed holes. The superposition of influence regions checks the relations 
; between objects in terms of occluding parts. This is best captured by polar 
em X coordinate notation, i.e. distance from the given object point and angular 
i * 3 coordinates, so that for overlapping objects, the one nearest to the point will 
ir mm be the influential one. Figure 6 illustrates the output of the algorithm for a 
non-convex polygonal boundary and an internal hole. As can be seen, parts 
of the outer polygon do not influence the point due to self-occlusion, in 
addition, another occluded part is a result of the superposition with the 
internal hole. For this region, the influential displacements are due to the visible part of the hole. This combination would 
have most likely been a natural human interpretation of the influencing components. The computational complexity of the 
algorithm (assuming regular cases) is of linear order (O(n)) for the computation of the self-occluding parts (which is the best 
that can be achieved) and linear as well (O(m)), for the mutual occlusions, where ‘m’ is the number of elements (‘m’ is 
usually small). 
  
Figure 6. Influence regions 
The transformation of a given point inside a boundary is performed by computing the displacements of the object vertices 
according to the influencing reference vertices (computed previously). The transformation that was used for this purpose is 
based on the weighted-average of displacements of all related points. A measure for the weight that was found to be 
adequate is the reciprocal of the square distance between the point in question and a given reference vertex. 
  
286 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.