Surface Point Determination in Subvoxel Accuracy from Pre-Segmented Multi-Material Volume Data for Metrological Applications

When using X-ray computed tomography in coordinate metrology, methods for the determination of surface points at subvoxel accuracy (hereinafter subvoxeling) are essential. In recent years, the demand to measure objects consisting of several materials has increased. While subvoxeling edge detection operators proven for mono-material applications are rarely suitable for multi-material specimens, a number of algorithms enable segmentation at voxel resolution. To overcome the gap, a new method has been developed. In a first step, volumes of material probabilities are calculated using a (pre-)segmentation while obtaining the implicitly given subvoxel-accurate information. The surface points can then be calculated using common edge detection operators. Initial results of the validation based on selected examples are presented.


Problems applying subvoxel-accurate edge detection operators to multi-material volumes
In recent years, the principle of computed tomography has become established, since it offers a fast, comprehensive, and nondestructive metrological assessment of all internal and external geometrical characteristics.For many applications, the use of subvoxel accurate edge detection operators is decisive.These allow the interpolation of surface points in the spatially discrete grid of voxels containing the reconstructed attenuation coefficients (CT volume data).For measuring objects made of one material (mono-material), a multitude of methods exist which allow the subvoxel-accurate calculation of surface points [9,7].Furthermore, algorithms like Marching Cubes [1] do no only calculate surface points, but generate a closed description of the surface using triangulation.Such edge detection operators at least implicitly binarize the volume.For objects made of more than one material (multi-material), however, simply distinguishing between foreground and background is insufficient.For each material transition, the surface is rather described by the combination of adjacent materials and its orientation.The resulting, generally non-manifold topology cannot by definition be determined with such edge detection operators.For this reason, the application of most known edge detection operators to multi-material volume data is very limited, e.g. to local application.The term surface itself is questionable in this context.Subsequently, both the manifold interfaces between two materials and the non-manifoldness composed of them are referred to as surfaces.

Segmentation algorithms: implicit material surfaces at voxel resolution
In contrast to this, segmentation algorithms exist, which can be used to assign a material (label) to each voxel (e.g. the so-called watershed transformation [2]).By applying such a method, the material surfaces are implicitly resolved in the voxel grid and are topologically correct.However, these segmentation algorithms are only conditionally suitable for metrological applications, because the resolution of the implicitly calculated surfaces is determined by the voxel grid.The actual structural resolution of the system is therefore not exhausted.
3 Method for subvoxel accurate determination of surface points from multi-material volume data In the following, a method is presented that enables the subvoxel-accurate calculation of surface points from multi-material CT volume data.For this purpose, the subvoxeling edge detection operators proven in the mono-material case are combined with the voxel-accurate segmentation algorithms.Figure 1 illustrates the entire procedure using an example: In a first step, a material (label) is assigned to each voxel using common segmentation algorithms.From this pre-segmentation and the attenuation coefficient itself, a probability for each material is then assigned per voxel in a second step.Finally, the respective material surfaces can be determined with subvoxel accuracy applying edge detection operators to the corresponding volumes of the material probability.In the example in figure 1, closed surface descriptions in triangulated form (as STL) are generated for two materials.

Calculation of material probabilities
The calculation of the material probabilities combines the material information from the pre-segmentation with the implicitly subvoxel-accurate information from the volume of the attenuation coefficients.Figure 2 shows the procedure schematically.To calculate the material probabilities P i of a voxel, its local environment containing all voxels lying within a defined radius is analysed.This definition is more flexible than just taking voxels into account, that are direct adjacent in the voxel grid.In the example shown in figure 2, a radius of 2.24 times the size of a voxel was used, resulting in a local environment containing a total of 57 voxels, 21 of which can be seen and are framed in red in the shown slice.A distinction is made between three cases depending on whether the environment contains one, two or more different material labels.If all labels of this environment are identical (trivial case), the voxel is assigned the probability 1 for the corresponding material.However, if there are two labels, as is the case in the environment shown in the example in figure 2, the probabilities of the two corresponding materials in this voxel are determined taking into account the attenuation coefficients in the local environment.A simple approach is to calculate the two probabilities directly from the measured attenuation coefficient µ of the voxel in question.Assuming that it is proportionally composed from the attenuation coefficients of the materials µ i weighted with the probabilities P i to be determined, the following system of equations can be drawn up.
(1) By restricting the probabilities to values between 0 and 1 and considering random and systematic errors of the measured attenuation coefficient, more advanced models can be set up.If there are more than two materials in the environment, the problem can be reduced to one of the two cases mentioned above by reducing the size of the environment under consideration.In places, where three or more materials meet in the pre-segmentation, this recursive approach is not purposeful.The resulting gaps can be closed subsequently by a separate procedure.For instance, 10th Conference on Industrial Computed Tomography, Wels, Austria (iCT 2020), www.ict-conference.com/2020interpolating probability values of neighboured voxels where the recursive approach terminates regularly is a simple yet convenient approach.

Subvoxel-accurate determination of surface points from the volumes of material probabilities
For the subvoxel-accurate determination of the surface points from the probability volumes, any subvoxeling (mono-material) edge detection operator can be used.If the used edge detection operator requires a threshold value (e.g.Marching Cubes), this value is always 50% due to the definition of the material probabilities.Thus, such a procedure becomes practically parameterless when applied to probability volumes.

Results and outlook
For validation, the presented method was applied to mono-material volume data.The calculated surface points are identical to those resulting from the direct application of the edge detection operator to the CT volume data.When applied to multi-material volume data, subvoxel-accurate determination of surface points was possible close to locations where more than two materials meet While the meeting of four or more materials in the voxel grid of the pre-segmentation is rather a theoretical problem, places where three materials meet are present whenever the topology of the surface is non-manifold.They form closed curves in the volume, so called triple lines.The example of figure 2 contains such a triple line, resulting in a triple point in the two-dimensional cutting plane.At the triple lines themselves, the surface is not differentiable.The subvoxel-accurate determination of surface points by interpolation violates the sampling theorem at these points.

Performance
The calculation of the material probabilities could be implemented completely parallel and shows linear runtime complexity.The resulting calculation times for the example shown in figure 1  As the times for the calculation of the material probabilities and for the application of the edge detection operator are of the same order of magnitude, the method is suitable for industrial use.However, the method to calculate the pre-segmentation, which in our case was a single-threaded implementation of the watershed algorithm, has a major impact on the total computing time.Using one of the several approaches to parallel implementation [6,10,5,3,8] should therefore greatly reduce the overall computing time.

Further possible applications
The described procedure allows the combination of arbitrary segmentation algorithms (for pre-segmentation) with arbitrary subvoxeling mono-material edge detection operators.In addition, the adaptation of other methods or algorithms whose application has so far been limited to the mono-material case, is made possible for the multi-material case.For this purpose, first all material probabilities are calculated analogously.Then, the methods to be adapted are applied to the individual volumes of the material probability.Subsequently, the individual results must be superposed in a meaningful way.Instead of separately considering the volumes of the material probabilities, they can also be seen as the volume of a vectorial quantity.For instance, subvoxel-accurate triangulated surface points of all materials can be calculated in one step using algorithms such as Generalized Marching Cubes [4], thus both preserving non-manifold topologies and reducing the computing time even further.

Figure 1 :
Figure 1: summary of the procedure.Corresponding slices from a volume of attenuation coefficients and the resulting presegmentation assigning each voxel a material label are exemplarily shown.From this pre-segmentation and the attenuation coefficients itself, the probabilities of each material are calculated for each voxel: corresponding slices for two materials are shown here.From those, the rendered surfaces (point clouds) shown below where computed with a triangulating edge detection operator.

Figure 2 :
Figure 2: computation of material probabilities.Corresponding slices (i.e. a small area thereof, making the individual voxels visible) from the volume of attenuation coefficients, pre-segmentation, and resulting volumes of material probability are shown.The latter are framed in the respective colour used for the visualisation of the pre-segmentation.The analysed voxel and its local environment used for the calculation are marked in red.

Table 1 :
are shown in table1.theCPUtimes for the example shown in figure1are shown.Except for the pre-segmentation, for all steps parallel CPU implementations where used.