⊕ ⊕
Fig. 5.1: The thick line is the boundary of the original space partitionR: two 2D balls and the background. The triangles separate the centers of the 2D balls from each other and the
background.
Associated Discrete Maximum
In our framework, we require the sample points to be placed in such a way that the circumcenter of the corresponding tetrahedron is reachable by a simple increasing path on the distance transform of the sample points. But following the increasing path we do not necessarily stop in the circumcenter of the tetrahedron, because it is not necessarily a local maximum of the distance function. A tetrahedron which does not contain its own circumcenter is such a case. Here, the increasing path passes though the circumcenter in the direction of an infinite maximum ( compare the introduction in Section 2.4.8 ).
To avoid this problem, we also require the corresponding tetrahedron to contain its own circumcenter which is then a local maximum. This requirement establishes a correspondence between the original local maxima and the local maxima of the distance transform defined on the sample points. We call the discrete local maximum reachable by steepest path starting on an original maximum theassociated maximum:
Definition 5.1 (Associated Discrete Maximum). Given the space partition R, the continuous distance transform dR on R, the set of points S, and the discrete distance transform dS on S, let x be the local maximum of dR and H(S, x) be its set of reachable local maxima on dS. Then, we call y = arg maxy0∈H(S,x)dS(y0) the associated discrete maximum ofx.
The associated maximum is the greatest reachable maximum, where the set of maxima reachable by steepest paths is given by Definition 2.33.
Now, let us consider our original scene containing two balls in space. The reconstruction is said to separate the local maxima correctly, if for any two originally separated local maxima their associated maxima are also separated. We call such reconstruction arefinement.
Definition 5.2 (Refinement). Given two space partitions RandR0 with two distance transform func-tions dR and dR0 defined respectively on R and R0. R0 is called a refinement of R, if for any two local maxima xi, xj of dR lying inside different regions Ri, Rj of R, the discrete maxima x0i, x0j being associated to xi, xj lie in different regionsRi0, Rj0 of R0.
According to the new defined notation, the original space partition consists of three regions: two open balls and the infinite background space. The boundary of the two tetrahedrons which envelopes the original maxima partitions the space into three regions, too. The associates of the original maxima are correctly separated, so, the new space partition is arefinement of the original.
5.3. (ψ, ρ)-SAMPLING 117
⊕
⊕
⊕
Fig. 5.2: The thick line is the boundary of the original space partitionR. The edges of the triangles divide the space into a refinement ofR. The two most left continuous maxima are associated with the same discrete maximum. The maxima corresponding to the dotted circles are
not associated with any continuous maximum.
The definition of refinement does not make any statements about the possibility of regions with two or more local maxima. Let us extend our scene for example by adding a new local maximum and connecting the balls to a simplified shape of a “barbell”.
In the 2D illustration presented in Figure 5.2 the thick line is the boundary of the original scene.
The boundary divides the space into two regions: the interior of the barbell and the infinite background.
The distance transform defined on the boundary has three local maxima denoted by ⊕and, obviously, the maximum of the infinite background. The edges of the triangles divide the space into a refinement of the original space partition. The distance transform defined on the points has four local maxima denoted by plus the infinite maximum. The two left continuous maxima ⊕are associated with the same discrete maximum. The discrete maxima corresponding to the dotted circles are not associated with any continuous maximum.
We notice, that by Definition 5.2 for each continuous maximum there exists exactly one discrete maximum, but this mapping is neither injective nor surjective.
5.3 ( ψ, ρ )-Sampling
In the previous chapter on thinned-(α, β)-shape-reconstruction we introduced sampling conditions based on globally set parameterspandq(compare Definition 4.2). The parameterspandqcontrol sampling density and maximal sample point deviation from the boundary. p is the maximal distance between any boundary point and its nearest sampling point, and,qis the maximal distance between any sample point and its nearest boundary point.
Here we generalize the concept for locally adaptive sampling conditions. Analogously to (p, q)-sampling we want parameters to control density and noise amount, but now they should be locally adaptive. The local adaptation is done by a feature size (f(.)) defined for every point of the boundary.
The feature size may vary depending on curvature, size of the adjacent regions or a metric of a shape descriptor like distance to the medial or homotopical axis. Notice, the distance to the homotopical axis is another measure of curvature for smooth shapes. This property is partly lost for non-manifold shapes, see Section 2.2.5.
The feature size is scaled by parameters which correspond to pand q. The maximal sample point deviation depends on the sampling density. The greater the sample point deviation the denser has to be the sampling. This dependence is modeled by scaling parameter ρ. The greater ρ the sparser is the sampling and the less noise is allowed. The ψ parameter is an auxiliary parameter for the
r ψρr ρr (1−ρ)r
ψρr ψρr ψρr
Fig. 5.3: Dependency between theψρ-sampling parameters withras the local feature size. The sample points are in the (1−ρ) dilation of ther-circle. The maximal distance values on the
boundary of ther-circle areψρr.
sampling density only and lies in the interval (0,1). ψis used in our reconstruction method for definition of simplices which are too great to belong to the boundary, which we call undersampled (compare Definition 5.10).
Definition 5.3(Non-Uniform (ψ, ρ)-Sampling). Let∂Rbe the boundary of a space partitionRand let f : ∂R →R be a function which maps any point on the boundary to its feature size. Let S⊂R3 be a finite set of points. ThenS is said to be a (ψ, ρ)-sampling of∂R, if
∀b∈∂R: dS(b) ≤ ψρf(b) and
∀s∈S∃b∈∂R: ||b−s|| = dR(s) ≤ (1−ρ)f(b)
Notice, in the definition Definition 5.3 dS(b) is the distance between a boundary pointb and the set of sample pointsS, whereas dR(s) is the distance between a sample pointsand the continuous boundary of the space partition∂R.
In Figure 5.3 we find an equivalent illustration to Figure 4.3 introduced for (p, q)-sampling. Here we see again anr-ball (thick circle). We simplify the shape to a ball, so the feature size isrfor each point of the boundary. Ther-ball is dilated (gray in the illustration) by the maximal sampling point deviation (1−ρ)rdefined byf(b) =rfor each boundary pointb.
For two points on the boundary we visualize the maximal distance to the nearest sampling point.
The distance isψρrand is also visualized in the center of ther-ball. The maximal distance to the nearest sampling point ψρr necessarily is less than ρr. The circumcenter of the r-ball is the local continuous maximum of the region. Theψρrball is inscribed into the (1−ρ)r-dilation of ther-ball.
The consequence of the (ψ, ρ)-sampling conditions is, the discrete distance value on the local maxi-mum is enveloped by lesser distance values in the dilation. This is the quintessence of our work and the basis for the guarantees of the reconstruction method.