3.3.1 Sampling Conditions for Nonsmooth Manifolds by Critical Points The-ory
[Chazal et al., 2009]
Prerequisites:
• Shape: nonsmooth manifold surface
• Parameter: κand µ-reach of the boundary: the distance of µ-critical points to the boundary.
The norm of the gradient onµ-critical points isµ.
• Samplingglobally uniform, noisy,κ, µ-sampling where the Hausdorff distance between the sample points and the boundary is less thanκtimes theµ-reach (see Section 2.2.6) of the boundary. For the definition ifµ-reach consult
Definition 3.7 (Noisy, uniformκ, µ-sampling). Given κ≥0, µ≥0,S⊂R3is the κ, µ-samplingof ∂R iff the Hausdorff distance between∂RandS does not exceedκtimes theµ-reach of∂R.
Guaranties: Let the union of balls be the union of α-balls with 1µ4dH ≤ α < rµ −3dH, where dH is the Hausdorff distance between the point set and the boundary and rµ is the µ-reach (compare Definition 2.38). Then, given κandµ such that the critical point stability holds, the complement and respectively the of the union of balls is homotopy equivalent to the sufficiently small dilation of the boundary and respectively its complement.
Comments: The paper states the homotopy equivalence between the dilation of the sample points by the certain parameter and the dilation of the boundary by certain values computed by κ and µ.
Our interpretation of the guarantee follows from the homotopy equivalence between the dilation and the boundary [Chazal and Lieutier, 2005b] and the homotopy equivalence between theα-shape and its dual shape [Edelsbrunner, 1993].
Idea: The main idea here is based on thecritical point stability theorem which states that the critical points of the one compact set is not more than a certain value distant from the critical point of the other compact set if the Hausdorff distance between the two surfaces of the compact sets are not greater than another certain value. The two surfaces may be for instance the original and the approximated one. So, having the critical points of the one we can know the interval where the critical points of the other are to find.
Using the result of the theorem, perturbations on the surface may be assumed which do not change the intervals where the new critical points are. The deviation of sample points from the boundary can
3.3. GLOBALLY UNIFORM SAMPLING CONDITIONS 71
be seen as perturbation of the approximated surface from the original. The trick here is to bound the maximal perturbation in such a way that the critical point stability holds.
This is done first by bounding the maximum perturbation uniformly for each point on the surface.
Second, to ensure the Hausdorff distance between the surfaces not to exceed a certain fraction of the weak feature size of the original compact set.
Algorithm: α-shape.
Extensions: The authors propose a method to estimate the input parameter by analyzing the critical function. The critical function maps a certain distance from the boundary to the infimum of the gradient norm on points at this distance.
3.3.2 Reconstructing r-Regular Manifold Contours with α-Shapes
[Bernardini and Bajaj, 1997]
• • • • •
• • • •••••••••
••
••
••
(a)
•
(b) (c)
Fig. 3.6: Sampling construction and conditions for 1-manifold (a) The sampling density does not allow the disc to “fall” through the contour. (b) The disc radius is chosen in such a way that
the intersection between the disc and the contour is either empty or one point or one connected component the counterexample in (c)
Prerequisites:
• Shape: 2D only. smooth,r-regular manifold
• Parameter: r
• Sampling: globally uniform, noise-free
Guaranties: The contour approximation homeomorphic to the original boundary.
Comments: The construction of the proof assumes a disc such that the intersection of the disc and the boundary is exactly one connected component. The condition on both sides of the contour is equivalent to the definition of ther-regularity. Notice, disc radius is less thanr.
The authors state a conjecture with equivalent conditions for 3D reconstruction of a 2-manifold. But as it is shown in [Stelldinger, 2008c] this conjecture is not true. The counter example can be constructed by placing four nearly coplanar points on the surface in a concavity such that they build almost a square. By construction the all points lie on the surface of theα-ball and the reconstruction method leaves the tetrahedron in the surface. Consequently, the surface is not everywhere thin and cannot be homeomorphic to the original.
Idea: Here the proof is based on the idea to construct a set of points such that the α-shape is
homeomorphic to the original. To do that one can think of a disc (2D reconstruction) of certain radius.
First point can be placed arbitrary on the contour. The second one is place onto the contour in such a way that the disc touches the both but does not “fall” through the contour. So, the center of the disc must always be on the same of the contour, while the distance between the points is less than the diameter of the disc. The successive placement of sample points onto the boundary is illustrated in Figure 3.6 (a).
Second, the intersection between the disc and the contour must be either empty or one connected component (see Figure 3.6 (b) and the counterexample in Figure 3.6 (c)). Third the placing the next point and the disc between the points the disc is not allowed to cover any other point.
The proof the authors show that for each intersection of the disc with the boundary there is a hoeomorphic component in theα-shape.
Algorithm: α-shape
Extensions: Finding an optimalα-value in [Bernardini et al., 1999a]
3.3.3 α-Shapes, Normalized Mesh and Ball Pivoting
[Stelldinger, 2008c]
(a) (b)
(c) (d) (e)
Fig. 3.7: (a) Four nearly coplanar points on the surface with circumradius less thanα. (b) noise-freeα-sampling. (c) “Normalized mesh” in [Attali, 1997]. (d) and (e) outer and inner
boundary of theα-shape.
Prerequisites:
• Shape: smoothr-regular manifold
• Sampling: global, no noise,α-sampling: the greatest distance between a boundary point and the nearest sample points is lessα
• Parameter: α
3.3. GLOBALLY UNIFORM SAMPLING CONDITIONS 73
Guaranties: If 2α < r then the the reconstructed polytope is homotopy equivalent to the original boundary, the outer and the inner boundary of the polytope are homeomorphic to the original surface.
Comments: The authors present a counterexample to prove that the conjecture for 3D reconstruction in [Bernardini and Bajaj, 1997] is not true.
With permission of the author, the illustrations in Figure 3.7 are taken from [Stelldinger, 2008a].
Idea: The author describes a correspondence and guaranteed topological expectations on algorithm result between the “normalized mesh” in [Attali, 1997], the “ball pivoting” in [Bernardini et al., 1999b]
and the “α-shape” in [Edelsbrunner and M¨ucke, 1994] on noise freeα-samplings.
The normalized mesh consists of Delaunay simplices whose dual Voronoi simplices intersect the original boundary. Since any simplex of the normalized mesh has an empty circumball less than α, the normalized mesh is a subset of theα-shape on α-samplings. In Figure 3.7 the insufficiency of the normalized mesh in 3D is demonstrated. The normalized mesh in (c) onα-sampling results in a hole in the boundary.
The correspondence between the α-shape and the “ball pivoting” method is given by two proven properties. First, for ar-regular set (see Definition 2.30) and anα-sampling on it, theα-shape triangle normals build with the surface normals on surface points in the projection of the triangle onto the surface an angle not greater than π/6. Second the α-shape edges have at least two adjacent triangles. These two properties are use by ball pivoting to build the mesh.
Since the α-balls cannot break through the boundary without touching a sample point of an α-sampling the pivoting ball always stays on the same side of theα-shaoe and, consequently, reconstructs the outer boundary of theα-shape.
The next trick to prove the correctness o the algorithm results is to show that the outer and the inner boundary of theα-shape on anα-sampling of anr-regular shape with 2α <2 is homeomorphic to the original boundary.
Algorithm: α-shape and ball pivoting.
Extensions: The drawbacks of normalized mesh are holes in the reconstruction which develop in exactly the cases if four almost planar points lie on the surface of an α-ball. The α-shape closes such holes which can be used for detection of such cases.
3.3.4 Finding Homology
[Niyogi et al., 2008]
Prerequisites:
• Shape: smooth manifold,r-regular
• Sampling: noise-free (p, q)-sampling mitp <0.48randq= 0, noisy (p, q)-sampling mitp <0.172r andq <0.172r
• Parameter: p, q, r, α
Guaranties: For p < 0.48r, q = 0 the union of balls deformation retracts to the original smooth manifold. Therefore homology of the union of balls equals to the homology of the manifold.
Forp <(√ 9−√
8)r <0.172randq <(√ 9−√
8)r <0.172r the union of balls deformation retracts to the original smooth manifold. Therefore homology of the union of balls equals to the homology of the manifold.
Comments: Recommended reading on concept of homology can be found in [Hatcher, 2002] and [Munkres, 1984]. In [Dey et al., 1998], [Dumas et al., 2003] and in [Kaczy´nski et al., 2004] is the sec-ondary literature on omputing homology from simplicial complexes.
The compact manifold in this work is a smooth manifold surface (limited maximal curvature). The conditional number τ defined for each manifold relates to the maximal curvature of the surface and corresponds tor-value in case ofr-stability.
The authors give the lower bound for the number of sampling points. The guarantees are given first, for assumed sampling conditions and second, for the probability of fulfilled sampling conditions for rising number of sample points. Above, we mention only the guaranteed result for assumed sampling conditions.
Notice, we use the notations from our framework for simpler results comparison. The correspondence between our work notations and the notations in [Niyogi et al., 2008] areα∼, r∼τ,p∼s, q∼r Idea: Here is the intention to prove the homotopy equivalence between the union of balls placed on sample points of a certain radius and the manifold. The deformation retract of the union of balls is proven byfibers. Fiber connects any point of the surface with the boundary of the union of balls by a subset of the normal space on this point. The fibers correspond to the inverse distance transform for the boundary points on the union of balls and are defined as the intersection of union of balls with the normal space and a closedr-Ball on any boundary point. Under the assumption of the fulfilled sampling conditions the fibers do not intersect and contract uniquely to their boundary point.
In case of noise-free sampling, the union of balls is a simple expression of union ofα-balls with 2p <
α <0.96r. In case of a noise sampling the union of balls is a simple expression of union ofα-balls withα∈ 1
2
(q+r)−p
q2+r2−6r ,12
(q+r) +p
q2+r2−6r
. In limit: α ∈ (2−√
2)r,(2 +√ 2)r
. Here the centers of the balls are not necessarily on the surface.
Algorithm: According to homotopy equivalence between the union of balls and its dual α-shape in [Edelsbrunner, 1993], the appropriate algorithm isα-shape
Extensions: The authors give probability statements in cases if the sampling conditions are not guaranteed. So, the authors compute the probability for the sampling to fulfill the conditions for a rising number of sample points.
In noise-free case the points are sampled from theuniform probability distributionon the surface. In case of noisy sampling Here the probability measure has to satisfy two regularity conditions. First, the probability is not zero only in ther-dilation of the manifold. Second, the probability for a sample point to occur in aq0-ball (0< q0 < r) around a boundary point depends on valueq0 only, not the boundary point. The second condition ensures the probability measure to be independent on curvature or manifold conditions like surface narrowing. The distance to the boundary determines the probability measure.
Given the sampling criteria for the guaranteed deformation retract of the union of balls and the confidence that the sampling criteria are met, the authors prove the homology equivalence between the union of balls and the manifold under the confidence that the sampling conditions are fulfilled. So, the proof is valid without the assumption on the sampling conditions but its probability.
3.3.5 Thinned-(α, β)-Shape-Reconstruction
[Stelldinger and Tcherniavski, 2009c], see also Chapter 4 Prerequisites:
• Shape: “r-stable” space partition with nonmanifold boundary (The thin line in Figure 3.8(a) ). r is the greatest value by which the boundary can be dilated without to change the homotopy type.
Each region contains aγ-ball.
3.3. GLOBALLY UNIFORM SAMPLING CONDITIONS 75
r
(a) (b)
Fig. 3.8: (a) r-stable space partition. (b) The union of thick edges, black and dark gray triangles is (α, β)-shape reconstruction. The thin line segments are Delaunay edges.
• Sampling: globally uniform density and noise. “(p, q)-sampling” with sampling density p less than 0.5 of the radiusrand maximal deviation of sample pointsqless than r−2p.
• Parameter: The parametersαandβ are computed depending on givenr, γ, p, q.
Guaranties: If the p <0.5r, q < r−pand the parameter αis chosen such thatp < α≤r−q and each region contains an open γ-ball with γ ≥ p+α+ 2q than the resulting reconstruction preserves connectivity and neighbourhood relations of reconstructed regions and defines a one-to-one mapping of reconstructed and original regions.
Comments: The maximal sampling deviation is the difference between the radius of the tightest narrowing in the shape and the sampling density. Consequently, the denser the sampling the greater can the points deviate from the boundary and the closer the points are to the boundary the sparser can be the sampling.
The reconstructed boundary may consist of chains of edges without any adjacent triangles.
Idea: The valuercan equivalently be seen as the radius of the maximal inscribing ball into the tightest narrowing in the shape (the radius of the circle in Figure 3.8 (a) ) or theHausdorf-distancebetween the boundary and the set of critical points on the distance transform, or the smallest distance value on the critical points of the distance transform. It can also be seen as the value of the greatest dilation ( gray thick line in Figure 3.8 (a) ) of the boundary which does not change the original homotopy type.
The trick here is to know the tightest narrowing (ther-parameter) and the size of the smallest region (theγ-parameter) in the original shape as well as the sampling parameters pandq. Then the minimal sampling density pcan be limited in such a way that connecting the points at distance greater thanp somehow encloses the regions. Which is possible because the sampling conditions do not change on the boundary.
Of course if we connect the points at too great distance the regions will be completely covered. So, the other trick is to limit the maximal distance of connected sample points such that the regions are enclosed by edges or triangles but still recall the shape of the origonal region. This is done by limiting the maximal sample point deviationqto be less thanr−pand setting the connecting parameterαless than or equal tor−q.
The union of connections between points consisting of triangles and edges leaves spaces which are not covered by any edge or triangle. These free spaces are calledholes. The next trick here is to set up a parameter to distinguish between too great holes (union of white triangles in Figure 3.8 (b)) which correspond to original regions and the too small holes (union of gray triangles in Figure 3.8 (b)) which
are result of noise. The limitation which guarantees this differentiation is beside r the of the size of minimal orignal region (theγ). If the minimal region is big enough the too small holes are less than the β-parameter.
The last step is intuitive. It is to remove the too small reconstructed regions. This is done by elementary collapse after filling the too small regions with Delaunay simplices.
Extensions: Under assumption that the edges of the resulting reconstructed boundary have at least two adjacent triangles the set of singular edges may be removed from the reconstruction.
Minimal extension of reconstructed boundary fills the chains of singular edges with Delaunay triangles less thanβ. The subsequent elementary collapsing on simple edges removes the topological distortion.
Extended β-Filling fills the boundary with Delaunay tetrahedrons such that non-orientable surface patches contain simple boundary triangles.