In this section, we summarize the reconstruction steps and illustrate the results of the algorithm on three examples. Each example is a sampling of our non-manifold 2D example “fish” (see Figure 5.13 (a)). In (b) we demonstrate a very sparse noise-free homotopy stable sampling. The example in (c) is a very dense homotopy stable sampling which has been corrupted by high amounts if noise without losing the sampling stability conditions. The sparse sampling in the last example (d) lost its stability conditions as the consequence of noise.
5.12.1 Sampling Conditions
Obviously, the sampling sets in Figure 5.13 are artificial and the sampling parameters are known. In real applications, the sampling parameters are not always known in the reconstruction procedure and have to be determined or estimated.
Consider the illustration of noise-free sampling conditions in Figure 5.13 (b). The sampling points are on the boundary. The dark gray balls surrounding the sample points represent theψρlhfsvalue on this point. So, we want to assume that for each boundary point in this ball the nearest sample point is
5.12. ALGORITHM 133
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Fig. 5.13: (a) Space partition consisting of 4 regions and the background. Thick line is the boundary,⊕denotes the local continuous maxima,⊗denotes a saddle, the dashed line is the homotopical axis and the dotted line is the medial axis extension. (b) Noise-free homotopy stable
sampling withψ <0.5. (c) Homotopy stable sampling withψ <0.5 andρ <0.2. (d) Homotopy stable sampling (centers of dark gray circles) withψ <0.6,ρ <0.8 which has been corrupted by huge amount of noise (ρ <0.2). (b,c,d) the gray circles represent theψρlhfson the boundary point
where the center is the corresponding sample point. (c,d) the gray dilation is the (1−ρ)lhfs-dilation of the boundary.
the center of this ball and since the distance to it is less thanψρlhfs, then the sampling conditions are fulfilled. But the local homotopical feature size (lhfs) is not constant on the boundary. So, taking for our consideration the greatest dark gray ball we conclude that thelhfson boundary points to the right of the sampling point is decreasing and is increasing to the left of the sample point. The ball containing the boundary points for which the center is the nearest sample point fulfilling the sampling conditions should actually be an “egg” shape. However, for simplified illustration we chose the circular balls with variable radius. In our illustration the union of balls covers the boundary and is covered by theψρlhfs boundary dilation. In such a way we ensure the sampling conditions to be fulfilled. Notice, that in the examples (c) and (d) the irregular light gray envelope of the boundary is not the (1−ρ)lhfs-dilation, but the illustration of thelhfsdistribution.
So, the first example is noise-free. That implies that ρ= 1. Observing the radii of the gray balls, we state that the radii are less than 0.5lhfs. So, we estimateψ≈0.5. The second example (c) is highly noise corrupted. But there is a dense point set around the boundary such that the distances between the boundary points and the nearest sample points are less than 0.1lhfs. The sample point deviation from the boundary is in the 0.8lhfs-dilation of the boundary. So, we estimate the parameters with ρ <0.2, ψ <0.5.
The last example is a sparse sampling which has been corrupted by noise in the same way as in the previous example. The ψρlhfs balls have the radii of 0.5lhfs with estimated ψ-value of 0.5 and,
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Fig. 5.14: (a) Space partition consisting of 4 regions and the background. Thick line is the boundary,⊕denotes the local continuous maxima,⊗denotes a saddle, the dashed line is the
homotopical axis and the dotted line is the medial axis extension. (b,c,d) The results of elementary thinning (thick line).
consequentlyρ≈1 which does not allow any sample point deviation from the boundary. But the balls overlap and their centers are not exactly on the boundary. Furthermore the local homotopical feature size is equal to the distance to the nearest maximum in only a few points. The maximal deviation on such points is critical for reconstruction. The initially sparse sampling denoted by dark gray balls might have been fulfilling the sampling conditions but is now corrupted by huge amount of noise with maximal deviation of 0.8lhfs. So, we expect the reconstruction to lose topological properties. We observe furthermore that the regions which correspond to “fins” are represented by a soup of almost uniformly distributed sample points.
5.12.2 Elementary Thinning
The first step of our method is to build a refinement on the Delaunay triangulation by deleting all centered cells. The holes in the triangulation are the reconstructed regions of the first step. The boundary between the regions may be thick and consist of further Delaunay cells. The constructive retraction method applied on the simple simplices thins the boundary leaving the compatible reconstructed regions bounded by minimal simplices.
As stated previously the reconstructed regions contain at this step of the algorithm one discrete local maximum which makes the refinement elementary. It follows that since maxima are separated by minimal simplices the refinement is minimal.
The results of elementary thinning on our three examples is presented in Figure 5.14. The
recon-5.12. ALGORITHM 135
struction is an oversegmentation which correctly separates the local maxima. The reconstructed contour in (a) is a good approximation of the original boundary especially in curve segments with low curvature.
Unintentionally the sampling met the corners of “fins”, so, the contour is less erroneous in the sharp angles.
The results in (c) and (d) are built on point sets consisting of equal subsets. So, the reconstruction may consist of equal structures and, so, look similar. But in (c) the points near the boundary are much denser than in (d). Both reconstructions seem to consist of a soup of triangles especially in the areas of fins and the fishtail. But the ratio between the greatest triangles inside the fins and the edges around the original boundary is much higher in (c) than in (d). In (d) the size of triangles inside the fins and the fishtail is the same as the size of triangles around the boundary. That means that the distance values along the paths through the original region do not change significantly, which makes the separation between the maxima impossible.
5.12.3 Refinement Reduction
The last step of our algorithm reduces the oversegmentation by appropriate merging of reconstructed regions. Appropriate regions for a merge are reconstructed regions which belong to the same original region. Consider for example the two greatest empty areas in Figure 5.14 (c) or (d). The two areas belong to the same original region - the interior of the fish - and are separated by a slim triangle which for itself is a reconstructed region of the same original region. Breaking through the slim triangle and so to join the three areas into one is the appropriate merging.
The results of refinement reduction on our three examples are presented in Figure 5.15. In all exam-ples we see, why the reconstruction is only a refinement and cannot guarantee the topological correctness.
The narrowing between the eye and the fish boundary is a saddle on the distance transform. The re-finement reconstruction algorithm was built to separate the local maxima, but not all critical points.
The guarantee of homotopical stability in the sampling ensures that the simplices cutting the homo-topical axis (dashed line) may be removed from the reconstruction without destroying the topological correspondence. In our example we observe an edge cutting the dashed line. The circumcenter of that edge is a saddle point on the discrete distance transform. Deleting the edge results in a loop in the reconstructed region which corresponds to the loop of the original homotopical axis.
In (b) we observe that even though the sampling achieved data points on the sharp tips of the corners the reconstruction cut them off. Let us consider the reconstructed region containing the tip in its boundary. The sharp corner is a tip of a triangle with nearly equal sides and a very small angle between them. The edge opposite the acute angle is the smallest. Since the method pursues smaller boundary simplices, the smallest edge of the triangle is preferred to separate regions. So, merging of reconstructed regions containing this triangle in the boundary results in deleting the greater edge and, so, in cutting off the sharp tip of the region. The goal of our reconstruction method is the simplest way and conditions for preservation of topological properties. The result with cut off sharp tips and the result with sharp corners in the boundary are topologically equivalent, so the reconstruction problem is solved in both cases.
The result in (c) demonstrates the stability of the algorithm on a highly noisy data set. In (d), as already presumed, some of the reconstructed regions are lost in the reconstruction. Since the ratio between the local maximum of the reconstructed region and its boundary simplices is too small, the regions are assumed to belong to the same original region and, so, are merged in the reconstruction process. The interior of the fish does contain a discrete local maximum with distance value which is much greater than the sampling rate on the boundary. The result is, the interior region is surrounded by small enough simplices and, so, the reconstructed region is recognized as a discrete region corresponding to a different original region.
In (d) we also illustrated the result of the reconstruction (thin line) on a data set before it was corrupted by high amounts of noise as in (c). The boundary is sampled such that the distances between the boundary and the sample points are less than 0.5lhfs. The sampling is done with ψ ≈ 0.6 and 1> ρ >0.8. Here we also demonstrate robustness of the algorithm due to the choice of the reconstruction parameterψ. The sampling parameterψis expected to be passed to the reconstruction method. But it is
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Fig. 5.15: (a) Space partition consisting of 4 regions and the background. Thick line is the boundary,⊕denotes the local continuous maxima,⊗denotes a saddle, the dashed line is the
homotopical axis and the dotted line is the medial axis extension. (b,c,d) The results of refinement reduction withψ= 0.5 (thick line). (d) The refinement reconstruction on data set
(withψ <0.6, ρ <0.8) which was not corrupted by noise withρ <0.2
not always exactly measurable. Here the original parameter wasψ= 0.6. In the reconstruction method the parameter was set toψ= 0.5. In this case the underestimated parameter delivers a refinement with a reduced oversegmentation. The result is a correct separation of the local maxima. Why is that so?
The minimal density of the points depends on the maximal sample point deviation from the boundary.
Consider the sampling of a circle. If the maximal sample point deviation is allowed to be up to 0.8rwhere r is the radius of the original circle then the points might be placed so unfortunately that inside the circle develops the greatest triangle with circumradius approximately 0.2r. Our reconstruction method requires then the local maximum to be surrounded by simplices less thanψ times 0.2r.
Such an unfortunate constellation does not occur in our case. The maximum inscribing ball corre-sponding to the greatest continuous local maximum of the interior region touches the boundary in three points only. The distance to the homotopical axis at these points is less than the distance value of the maximum. So, even with the sample point deviation of 0.2lhfsthe points do not land too close to the local maximum. The greatest inscribing balls in the fins and the fishtail also touch the boundary in three points each. But the sample points do not reach the 0.2 closure of the maxima and so, the greatest sample point deviation is less than 0.2. Summing up, even if, the sampling simulation parameters are set toψ <0.6, ρ <0.8 the resulting data set can be modelled withψ≈0.5 andρ≈0.9.