conditions and one does not. The example used in the above sections illustrates a very noisy but very dense sampling. With the second example we demonstrate the algorithm performance on very rough sampling but which does not permit great amounts of noise. The last example demonstrates the result of the algorithm on an insufficient sampling.
4.8.1 Sampling Parameters.
The first step in the reconstruction procedure is to determine the sampling parameters. The parameters r andγ are given by the original shape. We assume for all three examples the values in proportion to r. So, we may sayr= 1 and γ= 1.42993ror simplyγ= 1.42993.
The reconstruction requires the following conditions to be fulfilled to result in correct shape.
1 :p < α≤r−q ∧ 2 : β=α+p+q ∧ 3 : γ≥β+q >2(p+q)
The next example illustrates a very sparse sampling with p = 0.16, q = 0.53. According to first condition 0.16< α≤1−0.53 = 0.47. According to the third conditionα≤0.20993. So, we choose for the first exampleα= 0.20493. The union of α-balls and the sampling is illustrated in Figure 4.11 (b).
At first we take our example from previous sections withp= 0.4512,q= 0.08939. According to first condition 0.4512< α≤1−0.08939 = 0.91061. According to the third conditionα≤0.79996. In this example we setα= 0.5. The sampling is demonstrated in Figure 4.11 (c).
Our last 2D-example demonstrates the consequences of an insufficient sampling. In this example the data acquisition device delivers only very noisy data withq= 0.54607 and is not able to ensure a dense boundary sampling. The only possible density is p = 0.3765. Here we badly choose α = 0.5684 (see Figure 4.11 (d) ). So, we break the first condition 0.3765 < α ≤ 1−0.54607 = 0.45393. The third condition requiresα≤ −0.0387 and, consequently, is not possible to be met.
4.8.2 (α, β )-Shape-Reconstruction.
The first step of the algorithm is to compute theα-shape on the sample points. Theα-shape is a subset of Delaunay triangulation and as we introduced in Section 4.4 theα-shape is dual to the union ofα-balls shown in Figure 4.11 as union of gray balls.
Anα-shape (thick black edges and dark gray triangles in Figure 4.12) is a subcomplex of the Delaunay triangulation (light gray edges). Once the Delaunay triangulation is computed the construction of the α-shape is done by selecting the Delaunay simplices which, first, are less thanαand, second, the circumball does not contain any other vertex. The second condition is easily checked since only vertices on adjacent simplices have to be tested and the neighborhood relation is already given by Delaunay construction.
Building the (α, β)-shape-reconstruction is the second step of the algorithm. The greatest simplex in the not-(α, β)-hole is less than β. So, the construction can be done by sorting the simplices in decreasing order. Starting with the greatest simplex greater than β the first greatest simplex of the greatest (α, β)-hole is found. In 2D the greatest simplex is a triangle and in 3D it is a tetrahedron.
A simplex of the same dimension as the greatest one is in the same (α, β)-hole if the incident simplex of the same dimension is in the (α, β)-hole and the intersection of the two simplices is not in theα-shape.
The intersections of simplices - the faces of simplices - are in the same (α, β)-hole if they are notα-shape.
In fact we described the traversal though the Delaunay triangulation starting in the greatest simplex greater thanβ and ending on the bounds of theα-shape. All simplices we pass through are in the same (α, β)-hole.
The (α, β)-shape-reconstruction is then the Delaunay trinagulation without the (α, β)-holes. The (α, β)-holes are illustrated as white spaces in Figure 4.12 and the (α, β)-shape is then the union of gray polygons (not-(α, β)-holes) and dark gray polygons and thick black edges (α-shape).
If we compare the results of the (α, β)-shape-reconstruction we notice that the not-(α, β)-holes are the consequence of great amounts of noise or, more precisely, the great ratio between the noise q and the chosenα.
In the first example the ratio is 0.53/0.20493 which allows moreα-holes to hide inside theq-dilation.
4.8. ALGORITHM 97
(a) (b)
(c) (d)
Fig. 4.11: (a)original space partition (b) Theα+q-dilation∂R⊕(light gray) of the space partition covers the union ofα-ballsU (the union of gray and dark gray balls).U covers∂R (black curve),∂Rhas the same number of components asU (one connected component only in
this case; but a circle is one connected component too).
Whereas in the second example the q-dilation with q = 0.08939 is too thin to include α-holes with α= 0.5.
The third example demonstrated consequences of two problems. First theα-value has been overes-timated. The consequence is, too great alpha balls intersect more than allowed which results in more connections between vertices in theα-shape. In the center of the example in Figure 4.12, the narrowing of the middle region develops a connection which in our 2D example divides the (α, β)-hole in two. So, the one-to-one mapping between the (α, β)-holes and the original regions is no longer valid.
In 3D there are two different ways in which a narrowing can develop in the boundary. First is the
“bottle neck”. The result of insufficient sampling or overestimatedα-value would close the narrowing with a surface membrane and divide the corresponding region in two.
The other kind of a narrowing in 3D develops if the circular surface of a disc is pressed together.
Invalid reconstruction of such a narrowing results in a connection, a chain of singular simplices in such a narrowing which connects the upper surface with the bottom one and does not divide the inner region in two but destroys the topology of the inner region and the surface. The reconstructed surface is then a nonmanifold and the inner region is a donut.
The second problem in the third example is the too great amount of noise. According to the parameter values the γ-value has to be γ =p+α+q+q = 0.3765 + 0.5684 + 0.54607 + 0.54607 = 2.03703 but the greatest inscribing ball in the bottom region has the radius 1.42993. Consequently, the computed β-value is too great and the bottom α-hole is computed to be a not-(α, β)-hole and is filled in the (α, β)-shape-reconstruction (greatest of gray polygons in Figure 4.12 (d) ).
(a) (b)
(c) (d)
Fig. 4.12: (a)original space partition. (b,c,d) Gray thin edges: Delaunay triangulation. Dark gray polygons and thick edges:α-shape. Gray polygons: not-(α, β)-holes. Union of thick edges, gray and dark gray polygons: (α, β)-shape-reconstruction. (b) Smallα, highly noise corrupted
sampling (c) very sparse sampling density, almost no noise (d) Overestimatedα, insufficient sampling density, highly noise corrupted sampling.
4.8.3 Topology Preserving Thinning
The original boundary is assumed to be infinitely thin. So, to achieve the topologically correct recon-struction the (α, β)-shape-reconrecon-struction has to be thinned. Homotopy preserving thinning is done by elementary collapse (see Definition 2.43).
Elementary collapse removes a simple simplex and its proper coface from the reconstructed simplicial complex. The (α, β)-shape-reconstruction infills the not-(α, β)-holes. So, it adds the interiors to the holes until it touches the boundary. Consequently, the filling elements are bound by theα-shape and cannot be simple. It follows that the first simple candidates to be collapsed on are simplices of theα-complex.
This fact can effectively be used in the implementation since only a subset of simplices needs to be checked for the property to be simple.
Thinning vs. Merging Consider a singular simplex which bounds a not-(α, β)-hole. After the filling this simplex becomes simple. The elementary collapse on the simplex deletes it and the first element of the interior of the hole, a Delaunay simplex greater thanα. So, the new simplices which become simple are in the interior of the not(α, β)-hole.
The elementary collapse does not change the homotopy type. So, proceeding on simple simplices inside the not-(α, β)-hole is homotopy equivalent to any other collapsing order. The thinning on simplices in the not-(α, β)-hole completely contracts its interior leaving the boundary, and as we concluded before the boundary is in theα-complex.
4.8. ALGORITHM 99
We notice that the thinning on simple simplices inside the not-(α, β)-hole can be seen as the merging of two reconstructed regions on their boundary simplex. In this case one of the reconstructed regions is the not-(α, β)-hole. We use this observation to compare the (α, β) reconstruction algorithm with the algorithm in the following chapter. (See in Section 5.13.1, Theorem 5.25)
Thinning Order
(a) (b)
Fig. 4.13: (a)original space partition. (b) (α, β)-shape-reconstruction thinned in increasing order - on a smaller simple simplex first.
If more than one simple simplex is to be deleted the order determines the geometrical properties of the outcome. The topological properties are preserved in any case.
The result of elementary collapse in increasing order is demonstrated in our first example in Fig-ure 4.13 (b). The thinning on the same (α, β)-shape-reconstruction in decreasing order is presented in Figure 4.14 (b).
The choice of the thinning order may depend on the difference between the sampling density in closer environment of the boundary compared with the sampling density at farther distance. In cases of laser range scanners the sampling points are very close to the boundary. The sampling points at greater distances from the boundaries are outliers and are great deal less probable. In this case the decreasing thinning is preferable. The algorithm prefers to connect vertices near the boundary and by doing so to approximate the boundary geometrically more precisely. Since the number of simplices connecting the more distant points is minimized the reconstructed boundary is smoother.
The volume based digitization methods based for example on edge detection may deliver samplings where the density in thep-environment of the boundary does not differ from the sampling density in the q-environment of the boundary. In such case the difference between the greatest simplex and the smallest simplex is not sufficient. The result of thinning in decreasing order is then hardly distinguishable from the thinning in increasing order. Such consequence may be observed on the boundary reconstruction of top left region in Figure 4.13 (b) and in Figure 4.14 (b).
4.8.4 Thinned-(α, β)-Shape-Reconstruction Method.
Summarizing the previously derived and discussed reconstruction steps into one method we obtain the following algorithm:
1. Reconstruction Parameters: Given the set of sample pointsS and the sampling parameters r,γ,pandq, chooseαsuch thatp < α≤r−q ∧ α≤γ−p−2q
2. (α, β)-Shape-Reconstruction
• Compute the Delaunay triangulation Dandα-complexDα onS
(a) (b)
(c) (d)
Fig. 4.14: (a)original space partition. (b,c,d) Thinned-(α, β)-shape-reconstruction. (b) Small α, highly noise corrupted sampling (c) very sparse sampling density, almost no noise (d)
Overestimatedα, insufficient sampling density, highly noise corrupted sampling.
• Compute the not-(α, β)-holes - the connected components ofD\Dαwith the greatest Delaunay simplex greater thanβ, withβ=p+α+q.
• The (α, β)-Shape-ReconstructionD⊕α,β is the union ofDαand the not-(α, β)-holes.
3. Homotopy Preserving Thinning. Put all simple simplices - simplices in D⊕α,β with only one adjacent proper coface inD⊕α,β - in priority queue Qin increasing order according to their circum-radius. UntilQis not empty do:
(a) σ= Top element ofQ.
(b) PopQ.
(c) Ifσis not simple, continue.
(d) Else removeσand its proper cofaceτ fromD⊕α,β (e) Add all simple simplices adjacent toτ toQ.
Extension: If it is assumed that the reconstructed boundary consists only of simplices which contribute to separate regions as it is in [Stelldinger, 2008c], then remove all singular edges fromDα.
Resulting Reconstruction. Let us observe now the result of the thinning. Notice that it is not the aim of the algorithm to connect all sample points. In very noise corrupted cases connecting all sample