method ψρ 1−ρ comments
topologically equivalent smooth manifold reconstruction
“crust” 0.06 0 parameter free
[Amenta et al., 1998]
“power crust” 0.1 0 parameter free
[Amenta et al., 2000b, Amenta et al., 2001]
“co-cone” 0.06 0 parameter free
[Amenta et al., 2000a]
“Modified Power Crust” 0.1 0.1 parameter: smallestlfs [Mederos et al., 2005]
“Robust Co-Cone” 0.1 0.1 not parameter free [Dey and Goswami, 2004]
topologically equivalent
“Noisy, Non-Uniform Approximation” 0.1 0.1 wfs, not parameter free [Chazal and Lieutier, 2008] no reconstruction method
proof for union of balls only
limited topological guarantees
“Refinement Reconstruction” <1 <1 refinement reconstruction,
[Stelldinger and Tcherniavski, 2009b] topologically correct on (p, q)-sampling sets [Tcherniavski et al., 2012] the greater noise, the denser sampling
Table 5.1: Comparison of different surface reconstruction algorithms based on locally adaptive sampling conditions
Our method also requires theψ-parameter to be set. However for higher sampling densities or lower amounts of noise the parameter may arbitrarily be set to a guaranteed overestimated value and so ensure the guaranteed outcome. The usual value in our practical experiments isψ= 0.5. The setting limits the sampling density but is still a great advantage over the results in related work.
5.14. EXPERIMENTS 143
The results in (c) and (d) demonstrate the advantage of refinement reconstruction over related methods. The data sets are here corrupted by outliers: the salt-and-pepper noise. More than 10%
of points in the armadillo data set are outliers. In the dragon data set more than 20% are salt-and-pepper noise. The number of points and times are collected in Table 5.2. Time in sec. is the elapsed time including loading of points, reconstruction and saving the mesh into a text-file. Notice, noα-shape-based method is able to obtain a comparative result as is demonstrated in (c) and (d).
data set points noise time (s) comments
Armadillo 172974 0 30.43 smooth manifold reconstruction Armadillo 192974 >10% 31.91 more than 10% outliers
Armadillo 34006 0.361r 6.81 globally set density and sample point deviation Dragon 437645 0 118.58 smooth manifold reconstruction
Dragon 528575 >20% 119.77 more than 20% outliers
Dragon 28395 0.12r 3.86 globally set density and sample point deviation
Table 5.2: Comparison of reconstruction parameters on “Armadillo” and “Dragon” data sets.
(See Figure 5.16 for illustration) First is a very dense noise-free sampling. Second is salt-and-pepper noise-corrupted sampling. Third is a sampling with globally set minimal density
and maximal sampling point deviation.
The last experiment in (e) and (f) serve as comparison to the result of thinned-(α, β)-shape-recon-struction. The data sets are sparse and the sampling points deviate from the boundary. The density and the maximal sample point deviation are uniformly set for the whole shape according to the r-stability value.
5.14.2 Volume-Based Sampling Sets
Volume-based methods enable insight into the object. The data acquisition device samples the scene at any point. The result is a sampled 3D interval, for example, with a regular grid. The typical examples are X-ray computed tomography (CT) and magnetic resonance imaging (MRI). The interior of the object can be subdivided into two or more regions which can have common boundaries.
We present our results on two similar data sets. Two different walnuts are scanned by computed topography resulting in a sequence of 2D gray color images. Using the 3D Canny edge detection algo-rithm2(compare [B¨ahnisch et al., 2009]), point sets are extracted which contain volumetric information on outer surface as well as boundaries between interior regions.
data set points ψ ρ time (s) comments
Sparse Walnut 156198 0.5 0.5 39.26 arbitrary set (ψ, ρ)-parameters Dense Walnut 2362275 0.5 0.5 824.86 arbitrary set (ψ, ρ)-parameters
Table 5.3: Comparison of reconstructions of two similar shapes. (top) Sparsely and nearly noise-free sampling of a walnut. (bottom) Dense and noise-corrupted sampling.
In Figure 5.17 (a) and (e) we illustrate the data sets. The first line demonstrates results of refinement reconstruction on a sparse nearly noise-free data set illustrated in (a). The example in (e) is a very dense strongly noise-corrupted data set. The second image in the set demonstrates the outer shell of the walnut.
The third image is the combination of the underlying point set and the extracted kernel. The extracted kernel, which is one of the interior regions of the walnut, is presented in the image on the extreme right of the sequence. In Table 5.3 we give the running time parameters. Notice that we assumed the (ψ, ρ) parameters to be unknown for both reconstructions. The default setting isψ=ρ= 0.5.
The reconstruction on walnut data sets results in 8 regions on sparsely sampled walnut and 21 regions on dense data set. The reconstructed regions are the thin outer shell ( the boundary between the walnut and the infinite background ), the kernel, the thick outer shell ( the walnut shell has certain spatial
2Implementation in the scope of Deutsche Forschungsgemeinschaft (DFG) project STI 147/2-1
(a) (b)
(c) (d)
(e) (f)
Fig. 5.16: Noise-free, “salt-and-pepper-corrupted” and noise-corrupted reconstruction. Left:
Armadillo. Right: Dragon. Compare the reconstruction parameters in Table 5.2
5.14. EXPERIMENTS 145
(a) (b) (c) (d)
(e) (f) (g) (h)
Fig. 5.17: Refinement Reconstruction of two different walnut shapes. (top) sparse nearly noise-free data set. (bottom) dense noise-corrupted data set. (a) (e) sampling points. (b) (f) reconstruction of the outer shell. (c) (g) combination of points and kernel. (d) (h) kernel
reconstruction
expansion ), the seed coat ( interior layer which envelopes the kernel preventing rancidity ), and the region corresponding to the hollow of the walnut. The specified regions are subdivided. The kernel consists of two regions in the sparse data set and of three regions in dense data set. The seed coat consists of the most regions and is not reconstructed properly since the sampling is insufficient. In Figure 5.17 we extracted the most relevant to us and most illustrative boundary reconstructions.
The dense sampling of a walnut in the bottom illustration in Figure 5.17 is very noisy. The data set consists of a great amount of outliers as well as of sampling point displacements. The reconstruction performs excellently on outliers ( see the kernel extraction and the outer shell ) but the systematic sample point displacements result in inaccurate constructions.
Observe the kernel boundary illustration in Figure 5.18. The left image shows salient spurious features. The displaced points correspond to the sampling of the seed coat. The density of displaced points is higher than the density of sample points near the boundary. The corresponding but even more illustrative spurious feature is demonstrated in the right image. The kernel is merged with a part of the seed coat. The narrowing between the kernel and the seed coat itself is insufficiently sampled. The sample points of the seed coat are dense enough to build a boundary. However the size of boundary triangles is related to the size of the kernel which results in satisfactory ratio. The narrowing is tighter than the sampling density of the boundary. Since the reconstruction is minimal we obtain an equivalent result for different (ψ, ρ) settings.
In Figure 5.19 we demonstrate the refinement reconstruction result on volume-based sampling of an orange. The reconstruction consists of 17 regions: 10 slices ( two of them are merged couples as can be seen in the right image ), two seeds and the rind. The rind is not sufficiently sampled and consequently is subdivided into several regions. The transparent outer boundary of the orange and the reconstructed slices - one of the slices is selected - are illustrated in the right image.
data set points ψ ρ time (s) comments
Orange 260215 0.5 0.5 84.35 arbitrary set (ψ, ρ)-parameters
Table 5.4: Reconstruction settings on orange data set.
The presented experiments enable us to compare the results of the refinement reconstruction to the thinned-(α, β)-shape-reconstruction as well as to present advantages over related methods. The great
(a) (b)
Fig. 5.18: Effects of insufficient sampling on dense walnut data set. Parts of the seed coat region is merged to the kernel.
(a) (b) (c)
Fig. 5.19: Reconstruction on “orange” data set. (a) point set (b) surface of the orange slices.
(c) transparent boundary between the rind and slices.
amount of the salt-and-pepper noise is not processable by thinned-(α, β)-shape-reconstruction, and no surface-based method results in the reconstruction of the interior regions of the object.
However the samplings used for experiments are uniform, i.e. the density does not vary on the boundary. The locally adaptive samplings, known to us from related works, are very dense-samplings of smooth surfaces with <0.1. To demonstrate the advantage of refinement reconstruction over related results we need to ensure the data set to be sufficiently sparse. In Chapter 6 we introduce new criteria for data set decimation, according to which the resulting data set preserves topological properties and is local homotopy stable.