axis. On the medial axis the gradient norm is smoothed and is equal to zero at critical points. In “small”
environments of the critical points the gradient values approximate 0.
The critical function maps a distance value to the infimum of the gradient norm on points at this distance. We can visualize the critical function as the boundary of a boundary dilation: the value of the dilation is the distance value, and the gradient values on the boundary of the dilation determine the value of the critical function. Again, according to Definition 2.17 the critical function has greater values if the boundary of the dilation does not intersect the medial axis, and lower values if it does. The critical function is zero if the boundary of the dilation intersects a critical point.
Definition 2.37(Critical Function [Chazal et al., 2009]). Given a compact set∂R, its critical function χ∂R : (0,+∞)→R+ is the real function defined by:
χ∂R(d) = inf
d−1∂R(d)k∇∂Rk
The function d−1∂R(d) is the inverse to the distance function and returns the set of points with distance valued. ∇∂R is the gradient function defined on the distance function of∂Randk∇∂Rkis its norm.
Theµ-reach is the infimum of all distances at which the critical function is less thanµ.
Definition 2.38 (µ-Reach [Chazal et al., 2009]). The µ-reach rµ(∂R) of a compact set ∂Ris defined by:
rµ(∂R) =) inf{d|χ∂R(d)< µ}
Setting µ= 0 theµ-reach becomes theweak feature size (see Definition 2.31.). Weak feature size, as we recall, is the minimal distance value of critical points. So, theµ-reach is some kind of weak feature size but it investigates the distance values on points around the critical points.
2.3 Digital Geometry
In our framework we assume that there is a process which converts a scene in the real world into a computer file. The scene in the real world is a space partition. The converting or as we call it digitization process performs a data acquisition device. The data acquisition device samples the boundary of the real space partition.
2.3.1 Boundary Sampling Points
The point set resulting from sampling the original boundary may be seen as the starting point for the reconstruction method. Our investigation starts earlier. As described previously, we can use shape descriptors like medial axis to determine the homotopy type of the shape. Using the shape descriptor we can define a function mapping each point on the boundary to a unique value called feature size.
According to the assumed feature size we determine the sampling density and the maximal sample point deviation from the boundary. Using these limits we can state that the guarantees on the reconstruction can be given if the sampling conditions do not exceed these limits.
Definition 2.39 (Boundary Sampling Points). Let ∂R be the boundary of a space partition R. Let f, g:∂R →Rbe well-defined functions. The set of pointsS is called boundary sampling points if
∀b∈∂R ∃s∈S: ||b−s|| ≤ f(b) and
∀s∈S∃b∈∂R: ||b−s|| ≤ g(b)
In our work we investigate the poorest sampling conditions under which the topological guarantees can be given. Consider our 2D fish example in Figure 2.15. The points illustrate the result of a very turbulent sampling. We observe that the sampling points deviate from the boundary, building not very dense clouds of points, some of which build shape-like groups, some of which quite uniformly distributed.
However, we also observe that the points densify the closer they are to the boundary.
The relative closure between the boundary and the points is measured and limited by thef-function in Definition 2.39. Phrasing the formula we say, the distance between each boundary point and its closest sampling is at mostf(b). So,f(b) measures thedensity of sampling points relative to boundary pointb.
The maximal deviation from the boundary is measured byg(b). Phrasing the formula, we say, the distance between each sampling point and its closest boundary point is at mostg(b). In our framework we do not differentiate between the case of sampling point deviation from the boundary and the sampling points which do not correspond to any point on the boundary. The latter points are known asoutliers.
We generalize the two concepts and call the input set as noisy or noise-corrupted if not all sampling points are on the boundary. So,gmeasures the maximalamountof noise which a reconstruction method can handle.
s
b
s
b
Fig. 2.14: Left: for all sampling points the maximal distance to the closest boundary point is zero. Right: the maximal distance between any boundary point and its closest sampling point
tends to zero.
In Figure 2.14 we illustrate the difference between the measurements f and g. In the left figure the maximal distance between any sampling point and its closest boundary point is zero. All sampling points are on the boundary. So,∀s∈S∃b∈∂R: ||b−s||=g(b) = 0. But there are no points around the boundary point b. The closest sampling point forb iss. This distance is ignored by the functiong which measures the sampling points deviation from the boundary.
The right illustration in Figure 2.14 demonstrates a very dense sampling. For each boundary point there is a sampling point a some close distance: ∀b∈∂R ∃s∈S: ||b−s||=f(b)→r. Here the function f ignores the fact that there is an outliersat great distance from the boundary.
2.3.2 Discrete Distance Transform
In Section 2.2.2 we learned the function which maps any point in space to the distance to its closest boundary point - the distance transform. The distance transform measures the Hausdorff distance between two sets, the set consisting of the current space point only and the continuous set of boundary points.
The concept of the distance transform is generalized to Hausdorff distance between a point in space and any set of points. According to the definition in Definition 2.16 we define the discrete distance transform as:
Definition 2.40 (Discrete Distance Transform). Let ∂Rbe a boundary of a space partition Rand let S be a finite set of boundary sampling points. The discrete distance transform dS of a set S ⊂R3 is defined asdS(x) = miny∈Skx−yk. Thereversed discrete distance transformis defined asrdS(x) ={y∈ S| kx−yk= dS(x)}.
Consider the very noisy set of sampling points in Figure 2.15 (a). The points are very dense on some parts of the boundary and strongly deviate from the boundary at others. Illustration of the discrete distance transform is given in Figure 2.15 (b). The greater the distance value the brighter is the gray value. Mapping each coordinate to its gray value we also illustrate in Figure 2.15 (d) the discrete distance transform as gray value mountains, which corresponds to the illustration of the continuous distance transform in Figure 2.8 (b).
2.3. DIGITAL GEOMETRY 33
(a) (b)
(c) (d)
Fig. 2.15: Boundary sampling points distributed around the original boundary.
Gradient
The gradient notation in Definition 2.17 introduced in [Lieutier, 2004] is based on the center of the smallest closed ball enclosing the nearest boundary points. In discrete case it is the center of the smallest ball enclosing the nearest points in the sampling which are given by rdS(x) for any pointxin space
Let Θ(x) be the center of the smallest closed ball enclosing rdS(x). Then gradient on x on the discrete distance transform is defined in the same way as in Definition 2.17, where the set of critical points is straightforwardly defined by replacing the continuous set of boundary pointsR by the finite set of sampling pointsS: set ofcritical pointsof∇is given byF(S) ={x| k∇(x)k= 0}for each pointx in space but not inS. The general definition is then: Fβ(S) ={x| k∇(x)k ≤β}.
The discrete distance transform is not everywhere smooth. So ∇ is not continuous, which can be seen by sharp “ridges” on the gray value “mountains” in Figure 2.15 (d). These sharp ridges are the points on the distance transform where the gradient value is less than 1. Everywhere else but on the sampling points the gradient value is 1.
Using the gradient we again know how to “climb the mountains” (compare Section 2.2.2). Gradient maximized the growth of the distance transform. The flow induced by the gradient is also defined here, as in [Lieutier, 2004] where the authors prove that using the vector field ∇ Euler schemes converge uniformly when the integration step decreases. In 3D the definition is then as follows:
C:R+×R3\S7→R3\S with C(t, x) =x+ Z t
0 ∇(C(τ, x))dτ
Starting in any point in space we reach for limt→∞a critical point on the distance transform. The
resultingflow line is thesteepest increasing path on the discrete distance transform. In Section 2.4.8 we will use the flow lines to specify elements of a combinatorial structure and in Section 2.4.9 to imitate the flow using a relation between these elements.
2.3.3 Discrete Medial Axis
Here we introduce for the first time the relation between the distance transform and the combinatorial structures which will be used for reconstruction in our framework. Consider the construction of the medial axis for a continuous shape in Section 2.2.3. As introduced in [Blum, 1967] we used the centers of maximal inscribed balls to define the medial axis. The points on the medial axis are exactly the points where the gradient value is not 1. In the 2D example in Figure 2.8 the medial axis is the sharp “ridges on the mountains”.
The radius of the maximum inscribed ball is the distance value of the distance transform on the center of this ball. So, we obviously can build the medial axis on our set of sample points. If the maximum inscribed ball touches at least two points we define its center as a point of thediscrete medial axis. So, for a finite set of sample pointsSthediscrete medial axis is defined as
MAS={x∈R3 |rdS(x)|>1}
The result for our 2D example is illustrated in Figure 2.15 (c). We observe that the lines of the discrete medial axis correspond to the sharp ridges of the discrete transform mountain in Figure 2.15 (d).
Each point in Figure 2.15 (c) is enclosed by the lines of the discrete medial axis. The seemingly open cells separate the corresponding points from the others by infinite lines. Suchcells are also called
“Voronoi cells” and the union lines of the discrete medial axis is called “Voronoi diagram”. We introduce this concept in Section 2.4.6.