The determination of implicit surfaces, which are contained in three-dimensional
(3D) image data and numerical 3D simulation data that are
based on regularly shaped, cubic lattices, has become an important tool within
many fields of science, industry and medicine (cf., e.g.,
Ref.s RUSS98 ; LOHM98 ).
Such (usually triangular) surfaces can be used for visualization
purposes FOLE93 , e.g., when 3D shapes that are contained in the data
should be rendered, and/or they may represent the basis for further numerical
To this date, many articles have been written on the generation of triangular
surfaces from 3D digital data sets (cf., e.g., Ref.s LOHM98 ; LORE87 ; NVID07 ; DOI91 ; GUEZ95 ; THIR96 ; BLOO94 ; HO05 ; HILL95 ; CHER95 ; LEWI03 ; NEWM06 ).
A very popular algorithm for surface construction from 3D image data has been provided through the Marching Cubes algorithm (MCA), which has been developed by Lorensen and Cline in the mid-1980s LORE87 . Note, that this tool is nowadays still advertised (e.g., by the multinational NVIDIA Corporation NVID07 ) as one of the state-of-the-art technologies for digital surface construction. Alternate approaches for 3D surface construction include – but are not limited to – the Marching Tetrahedrons DOI91 ; GUEZ95 (MTA), Marching Lines THIR96 (MLA; cf., also Ref. BLOO94 ), the Cubical Marching Squares HO05 (CMSA) algorithms, and VESTA BRS03 ; VEST04 , which is described in this paper.
The MCA is a template-based approach, and as a consequence of the non-trivial topology in 3D HILL95 , several surface templates have been initially overlooked, resulting in the accidental generation of holes in some data sets LOHM98 . Meanwhile, this problem has been fixed (cf., e.g., Ref.s CHER95 ; LEWI03 ). For a recent and very detailed discussion on the MCA’s history, cf., Ref. NEWM06 . The MTA, unfortunately, has directional ambiguities, because it subdivides a cube with tetrahedrons, which cannot be done symmetrically DOI91 ; GUEZ95 . The MLA – and the similar techniques – are not template based THIR96 ; BLOO94 , but they require further processing, so that apparent cell ambiguities are properly resolved (cf., e.g., Ref. HILL95 ). The CMSA is very similar to the previous, cubically based approaches, however templates are used for the faces of the voxel neighborhoods under consideration, and further work to resolve cell ambiguities is done as well HO05 .
The ”Volume-Enclosing Surface exTraction Algorithm” (VESTA), which is presented in this paper, allows one to numerically compute – very fast and totally robust (i.e., without the accidental generation of any holes) – non-degenerate, mathematically well oriented triangular surfaces in 3D BRS03
. VESTA constructs surfaces – so to speak – from the grounds up, because it continuously transforms the initial surface that consists of the boundary faces (squares) of selected voxels into the final isosurface. Instead of using many different surface templates, VESTA uses a single building block that is based on the vector decomposition of a single voxel face (cf., Fig. 2.b). VESTA collects all of the participating vectors and groups them into closed vector cycles, while resolving 3D cell ambiguities properly. In fact, this is done in analogy to the DICONEX algorithm BRS09 , which allows one to construct gap-free contours for 2D digital data.
The original VESTA algorithm is not limited to the processing of voxel cell neighborhoods, and it therefore minimizes the generation of redundant information. Since VESTA surfaces can be generated in a global “disconnect”, a global “connect”, and a “mixed” connectivity mode, and since all of these three modes can be executed either in a low resolution (“L”) or in a high resolution (“H”) mode, VESTA can produce in total six different types of surfaces on demand. Since all VESTA surface cycles are confined to voxel cell neighborhoods at all times (cf., below), a marching -cell variant is presented in this paper as well. The marching VESTA makes use of a very simple, but explicit table (cf., Table 1, subsection II.F.) of directed vector paths for the construction of closed vector cycles, and it is therefore very easy to implement into computer code.
This paper is organized as follows. First, we shortly review the DICONEX algorithm BRS09 , which uses a 2D digital data set as input, because VESTA can be viewed as an extension of DICONEX from 2D into 3D. Then we shall explain VESTA in great detail. In particular, we shall compare VESTA with both: (i) the original MCA LORE87 , which can be found in many text books (cf., Ref.s LOHM98 ; NVID07 ), but which may generate holes; and (ii) a state-of-the-art MCA implementation BOUR94 by Bourke et al., which represents an extended version of the original MCA that does not generate any holes and that produces exactly the same surface as VESTA, when it is executed in its low resolution (“L”), global “disconnect” mode. The in-depth comparison with the original MCA is done within the theoretical section, whereas the comparison with the extended MCA is done within three VESTA application related subsections. Finally, this paper will conclude with a short summary.
Ii The Surface Extraction Framework
Before we explain the surface extraction with VESTA, we first consider contour extraction in 2D with DICONEX.
ii.1 Diconex - DIlated CONtour EXtraction
In Fig. 1.a, we show a binary image with pixels (i.e.,
Let us assume, that the gray pixels have been segmented, i.e., they have
been selected for an enclosure.
In the following, we shall denote segmented pixels as “active” and the
other pixels as “inactive” pixels.
The objective is now to enclose the active (gray) pixels with contours
while making use of the DICONEX algorithm BRS09 .
First all initial contour vectors (ICV) ought to be collected.
The ICVs are here oriented pixel edges that separate an active pixel from
an inactive one.
E.g., for a single active pixel that has no active next neighbor at all (cf., Fig. 1.b), we have a maximum of four single ICVs (in the figure, they are numbered from to ). Note, that an active pixel lies always to the left of its corresponding ICV. Because of this construction, it is possible to forward the information of the interior and/or exterior of a shape (i.e., a collection of segmented pixels) that needs to be enclosed by a finite number of contours. Note that a shape is enclosed counterclockwise, while its potential holes are enclosed clockwise.
In a second step, the ICVs will be connected to oriented shape-enclosing contours. Note, that each ICV both starts and ends in a single juncture (cf., Fig. 1.c). Most of the times it is trivial to find the successor of an ICV, if we attempt to determine the particular IVC connectivities. But sometimes it is possible that a single active pixel is in contact with another active pixel through only one single point (cf., white dot in Fig. 1.c). The latter results in a situation where we have two incoming and two outgoing ICVs for this particular point of contact (cf., Figs. 1.c and 1.d). In the following, we shall call such a juncture “point of ambiguity” (POA). In order to avoid gaps in the final set of contours, one has to treat these POAs in a special way (cf., Ref. BRS09 ).
In Fig. 1.d, a connectivity diagram is depicted. If one connects to an incoming IVC an outgoing IVC that belongs to the same active pixel as for the incoming IVC, then one performs a left turn, i.e., one follows one of the paths of the white bent vectors. Conversely, if one connects to an incoming IVC an outgoing IVC that belongs to another active pixel as for the incoming IVC, then one performs a right turn, i.e., one follows one of the paths of the black bent vectors instead. The particular turns will either lead to a separation (“disconnect” mode), or to a joining (“connect” mode) of shapes next to a POA, respectively.
A user can interactively make this consistent choice of either left or right turns within one given image, e.g., if only binary information is available, in order to ensure the construction of gap-free contours. Note that this rather global decision making process may be replaced and automated by a local decision on the turning of the IVCs, while using, e.g., the average gray level of the four pixels, which surround a particular POA, provided that gray level information is available. In Fig.s 1.e and 1.f, the final DICONEX contours are depicted for the “disconnect” mode and for the “connect” mode, respectively. These contours have been obtained from a displacement of the origins of the IVCs to the middle points of their corresponding pixel edges.
In essence, the DICONEX contours result from a displacement of the initial pixel edges that separate active pixels from inactive ones. Note that the set of these initial pixel edges already provides a perfect enclosure of the segmented pixels. In 3D, we shall proceed now analogously.
ii.2 Initial Considerations for Voxels
VESTA will enclose voxels that have an inherent field value, e.g., a shade
of gray, above an initially given threshold.
A voxel (i.e., VOlume piXEL) is a 3D object.
To be more specific, it is represented by a cube and, hence, it has six
squares as faces.
For an active voxel, i.e., a voxel that should be enclosed by a surface
section, VESTA will check all of its six nearest neighbors.
If there is a transition from an active to an inactive voxel, the
corresponding voxel face will be recorded.
In Fig. 2.a, two neighboring voxels are shown. One voxel, i.e., the active voxel, is marked with a sphere at its center. The second voxel has no sphere at its center, because it is considered inactive, i.e., this voxel has a field value below the initially given threshold. Between an active and an inactive voxel we shall have a contribution to the enclosing surface. Therefore, the face that separates these two voxels has to be considered.
In Fig. 2.b, we show such a boundary face. The center of this face is marked with a black dot. Such voxel face centers will finally be support points of the VESTA surface. In case an isosurface should be constructed, the voxel face centers may be moved within the bounds of its corresponding range vector, , which has its origin at the center of the active voxel and which ends in the center of the neighboring inactive voxel. Fig. 2.c helps to illustrate, how VESTA surfaces can be transformed into isosurfaces while considering the 2D analog of two neighboring pixels.
Two pixels - one with a gray-level value of , the other one with a zero valued gray-level - are initially separated by a contour section that is located exactly in the middle between them (dotted line). A range vector connects the centers of the two pixels. The centers of each pixel are assumed to correspond exactly with their gray-level values. Since the isocontour is supposed here to represent a gray-level of value , it should not be positioned at the middle of the range vector. This medium position actually represents a gray-level of value
while assuming a linear interpolation between the gray-level bounds. In fact, the “true” location of the support point for the isocontour is located closer - and therefore has to be shifted - towards the center of the pixel with the gray-level of value. Hence, the isocontour (solid line) is supported by a point, which is located within the pixel with the gray-level of value .
In Fig. 2.d, we show the dislocation of the boundary face center in 3D due to the previous outline for 2D222More complicated types of interpolations than just simple linear interpolations are possible. E.g., one could use (higher-dimensional) B-SplinesBOEH07 ; SALO05 instead, etc.. Note that the dashed gray lines each continue to provide a connection from the boundary face center to one of the four edge middle points of the boundary face. In the following, these four edge middle points, which are numbered counterclockwise from to in Fig. 2.b (i.e., if we apply a right-hand-rule to the perpendicular range vector), will assume the role of junctures (see below). The following ansatz will help to provide a connectivity among the voxel face vectors (VFV) within a given boundary face. Let us unite the eight black VFVs of Fig. 2.b into four vector pairs as follows: connect the juncture () via the face center to the juncture (). As a consequence, one obtains for each boundary voxel face the four internal paths , , , and , respectively.
Any single isolated active voxel or any active voxel cluster will initially be fully enclosed by a certain number of boundary faces. Hence, each boundary face will be in contact with at least another boundary face through one of its four edges or – to be more specific – through one of its four junctures. Without loss of generality, we shall discuss in the following how pairs of boundary faces may connect in 3D. In Fig. 3, the various possible configurations are shown. Through each juncture (white dots) two VFV pairs may be connected (cf., Fig.s 3.a, 3.f, and 3.j). While doing so, it is only permitted to attach the origin of a given VFV to another ones tip. Hence, these vector pairs will yield oriented paths from one boundary face center to another one.
If we ignore the junctures, the newly formed vector pairs can be replaced by single vectors, which each will connect one VESTA surface support point with another one (cf., Fig.s 3.b, 3.g, and 3.k, respectively). In the following, the latter vectors will be called “VESTA cycle vectors”. Throughout this paper, vector pairs of antiparallel vectors will sometimes simply be drawn as single lines (cf., Fig.s 3.c, 3.d, 3.e, 3.h, 3.i, 3.l, and 3.m). Furthermore, it may not be specified in one of the following drawings, on which side of a given boundary face the active voxel resides. Then again, single vectors may be drawn as well as single line segments where no particular orientation will be indicated, since it should be obvious from the particular context.
ii.3 VESTA - a Volume-Enclosing Surface exTraction Algorithm
In the previous subsection, we have discussed the basic
processing steps that will allow for a proper surface extraction.
We shall begin with the processing of a single active, isolated voxel
(cf., Fig. 4.a).
In Fig. 4.b, this voxel is represented by its six boundary faces.
As an example, only for three of the boundary faces, and only for a single
quadrant of each of these three faces, those VFVs are shown in Fig. 4.c, which
form a closed vector cycle.
On the one hand, Fig. 2.b has made clear that VFVs are pairwise connected by
the boundary face centers;
on the other hand, Fig. 3.f suggests, that VFVs can be pairwise connected
through the junctures.
After the replacement of VFV pairs that are connected through junctures by the proper VESTA cycle vectors (cf., Fig. 3.g), one obtains in Fig. 4.d a closed VESTA -cycle, which represents a single oriented triangle. The processing of all VFVs yields seven further triangles. As a result, one obtains a fully closed and oriented VESTA surface (cf., the octahedrons in Fig.s 4.e and 4.f). In Fig. 4.f, the octahedron is superimposed with the four normal vectors of the four visible surface triangles. These normal vectors point to the exterior of the enclosed shape.
All initially given VFVs are unique. In total, one obtains after their processing VESTA cycle vectors for the edges of the octahedron, i.e., each edge represents two antiparallel vectors. Note, that VESTA will always reproduce 2D DICONEX contours whenever a final 3D VESTA surface is intersected with a principal plane at the corresponding centers of the active voxels (cf., Fig.s 31.a – 31.c).
In general, one has to process more complex shapes than just single voxels. Let us assume for the remainder of this subsection that the 3D data set is simply binarized (as in “active” and “inactive” voxels). The more general situation will be discussed in one of the next subsections. As in 2D, we shall encounter 3D junctures in the process of VESTA surface cycle formation that will play the role of POAs. In Fig. 5, we show all possible nine configurations of boundary faces that can be in direct contact. Gray solid lines represent single VFVs in the figure. Most of the configurations show paths, which do not self-intersect. However, in Fig.s 5.e and 5.g, we observe self-intersections, which are due to the fact that two voxels are in contact with one another through only one single edge (cf., Fig. 6.a).
Fig. 6.a also shows a dotted square, which has the common juncture of the
four boundary faces that are in direct contact at its center.
As an example, one incoming and two outgoing VFVs are shown for this juncture.
The upper left connection diagram of Fig. 6.b shows this configuration once
However, due to the 3D nature of the problem we actually have four incoming
and four outgoing VFVs at this particular juncture.
Clearly, this juncture is a POA, and Fig. 6.b shows for each of the four
incoming VFVs a connection diagram with two valid outgoing VFVs.
In order to avoid holes in the final set of surface tiles, one has now
uniformly to choose among the two following possibilities for all four
Either, one selects as successor to a given VFV the one that is connected with its origin and that belongs to the same voxel; then one generates the “disconnect” mode while following the paths of the white bent vectors (cf., Fig. 6.c). Or, one selects as a successor the one that is connected with its origin and that belongs to the other voxel; then one generates the “connect” mode while following the paths of the black bent vectors (cf., Fig. 6.d). This selection process is the 3D analog of the 2D selection process (cf., Fig. 1.d). A consistent selection of these successors is the key step within VESTA, which prevents a tearing of holes into the final surface. The user has to perform a global selection of one of the two modes, if only binary data are available. In Fig.s 6.e and 6.f, the final VESTA surface wire frames are depicted for the “disconnect” mode and for the “connect” mode, respectively. Note, that each edge of the wire frames consists of two antiparallel VESTA surface cycle vectors.
In Fig. 7, all possible nine VESTA surface cycles are shown (cf., Fig. 5), which can be generated while uniformly either choosing the “disconnect” or the “connect” mode for the whole 3D data set under consideration. Note, that these planar and nonplanar cycles can be traversed both ways, depending on the orientation of the range vectors (not shown here) of the contributing active voxels. In particular, the cycles in Fig.s 7.e and 7.g are the VESTA surface cycles of maximum length; their separated counterparts can be seen in Fig.s 13.b and 13.a, respectively. Furthermore, and as an important result, all of the cycles are confined to a voxel neighborhood at all times. They are supported alone by the initial boundary face centers, which may vary – if necessary – within the bounds defined by their corresponding range vectors (cf., Fig. 2). Finally, it should be noted that more than one VESTA surface cycle can appear within a voxel neighborhood (cf., Fig. 13).
ii.4 VESTA Surface Cycle Decomposition
The VESTA surface cycles, which have been introduced in the previous
subsection, have to be processed further, if one wants to obtain surfaces,
which are decomposed into triangles.
In fact, the only cycle that requires no further processing is the
one that is shown in Fig. 7.a.
If we demand to use no further points while inserting edges, we may end
up with a result that is shown in Fig. 8.
Note that instead of inserting single edges, one rather has to insert
antiparallel vector pairs for the proper breakup of the -cycles ()
This allows one to pass on the initial orientation of the -cycles to the
newly formed -cycles.
In Fig. 8, the triangular surface tiles have been chosen similar to the template tiling, which has been proposed by various authors of the MCA (cf., Ref.s LORE87 and BOUR94 ). However, this tiling is not unique, since other subdivisions are possible. If no further points are used for the decomposition of VESTA -surface with , biases may be introduced with respect to the convexity and/or concavity of the local surface sections (cf., e.g., Fig.s 18.e and 18.f). In cases, when more numerical accuracy is desired, it may be advisable to introduce for each -cycle with an additional point that lies within the cycle (cf., e.g., Fig.s 18.c and 18.d).
In Fig. 9, the -cycles () of Fig. 7 have been broken down into -cycles (i.e., oriented triangles) while using as additional point the average of all involved cycle support points. Note that in the case where the initial boundary face centers are moved within the bounds of the range vectors, the average point should be determined after this movement in order to save computing time. Admittedly, this choice of the average is a simple one, but it will be used for the remainder of this paper. Ultimately, it is left to the designer of a particular VESTA implementation, whether a more elaborated approach should be used for the breakup of the surface cycles.
ii.5 More on Voxel Connectivity
In the case, that two neighboring active voxels only share a single point (cf., Fig. 10.a), VESTA will not connect the two volumes, since none of the involved VFVs of one voxel will ever meet one of the VFVs of the other voxel. In Fig. 10.b, we show the wire frames (i.e., two octahedrons) for this configuration. Apparently, this connection is in the discretized 3D space too weak that it should matter. If one – nonetheless – desires to establish a link between voxels that only share a single common point one may, e.g., insert a tubular template (while removing the two opposite triangles, i.e., -cycles, at the same time) as indicated through the six dashed lines in the figure. However, we shall not use such an approach here, since VESTA by itself does not establish this kind of connectivity.
So far, we have discussed only the uniform and global usage of either the
“disconnect” or the “connect” mode for the processing of the 3D data
set under consideration.
However, if more than just plain binary information (such as “do enclose”
and “do not enclose a voxel with a surface”) is contained in the data, e.g.,
gray-level information, we can locally define through a threshold,
which mode should be (consistently) applied.
Ultimately, the interaction of the user is no longer required, since the
thresholding could now be applied through automation.
In Fig. 10.c, we show two pairs of active voxels, which each just share a single edge, and that are in direct contact to each other. Here, we consider the case that for each of the two neighboring pairs a different connectivity mode has been selected as it is indicated in the figure. As a consequence of the tracing of VFVs (cf., Fig.s 11.a), a new VESTA surface cycle comes into existence. This -cycle is shown in Fig 12.a. The corresponding wire frame of the two voxel pairs is depicted in Fig 10.d where each edge of the wire frame consists of two antiparallel VESTA surface cycle vectors. Note, that if both connection modes would have been the same, e.g., “connect mode”, we would have obtained the two VESTA -cycles as shown in Fig. 13.c. In Fig. 14.a, the newly formed -cycle is decomposed into -cycles while using an additional average point of the involved cycle support points.
In fact, many more VESTA surface cycles can appear, dependent on the internal features of the considered 3D data. All of these are shown in Fig. 12; and in Fig. 14, we show their decomposition into triangles while using average points. In contrast to VESTA, the original MCA LORE87 does not propose templates for the here described further configurations. However, various extensions (cf., Ref.s CHER95 ; LEWI03 ; NEWM06 ) of the MCA use additional templates, which also use an additional average point for their triangular decomposition. In general, there is now a total of different types of VESTA -cycles (). In particular, -cycles with a length of or do not exist, since (complementary) -cycles with a length of or , respectively, do not exist either.
We would like to stress that VESTA can produce in total six different types of surfaces. There are the surfaces, which will be processed either in the global “disconnect” or “connect” modes. Then only VESTA -cycles of length up to seven will be generated. A third kind is generated while using the “mixed” mode, which has been described in this subsection. Furthermore, each of the three modes can be either executed in a so-called low resolution mode (“L”) where no additional points are used for the decomposition of the surface cycles; or in a so-called high resolution mode (“H”) where additional average points are used for the decomposition of surface cycles. Up to this point, we have described what we will later refer to as the “original VESTA”. Note, that this type of VESTA has been implemented successfully into efficient software VEST04 . In the following subsection, we shall discuss a marching variant of VESTA.
ii.6 The Marching 3-Cell Version of VESTA
Using the original VESTA, one processes the 3D data set under consideration
This has the advantage that redundant information can be mostly avoided
during the generation of surface tiles.
However, a clear disadvantage is that a large amount of computer
memory may be required, in particular, if large amounts of voxels have to
As a consequence, a computer with insufficient RAM may be forced to go into
a swapping mode, and the execution time may increase considerably.
Therefore, we shall describe in the following an alternate implementation
of VESTA, which we will name the “marching VESTA”.
In this subsection, we are going to describe the details of a scan of a given 3D data set with a marching voxel neighborhood (-cell, or cube). All of the surface tiles, which finally will represent a single or multiple VESTA surface(s), will be generated and collected during such a scan. Note, that this scanning of the data is the same as in case of the MCA LORE87 ; BOUR94 . However, we would like to stress, that we are confident to use such an approach only, because the ansatz of using the active voxels’ boundary faces as shown in Fig. 2.b has resulted in VESTA surface -cycles, which are all confined to -cells (cf., Fig.s 7 and 12). In the following, we shall outline the necessary steps for a more memory efficient VESTA implementation while using a few examples.
In Fig. 15.a, a marching -cell at position is shown.
The indices , , and , refer to the position of voxel no.
(cf., Fig. 15.b) in the -, -, and -directions,
Note that the coordinate system, , coincides with the range
vectors (cf., Fig. 2) of voxel no. for positive -,
-, and -directions.
The VESTA surface support points (cf., Fig. 15.c) can move within
the bounds of the range vectors, which coincide with the edges of the marching
Since an active voxel may show up in a marching -cell only with an
eighth of its volume (i.e., an octant), its initial corresponding boundary
faces may show up only as quadrants (cf., Fig. 2).
In Table 1, the paths of VFV pairs are listed, which each correspond to a quarter (i.e., a quadrant) of a potential boundary face of an active voxel within the -cell . These paths connect junctures (cf., Fig. 15.d) via the boundary face centers (cf., Fig. 15.c) for the given boundary faces. Since each boundary face is shared by two side by side voxels, two orientations, and , exist for the VFV based paths with respect to the positive and negative spatial directions. In Fig.s 15.e and 15.f together, all possible VFVs are shown for the marching -cell.
As an example, in Fig. 16.a, a -cell is shown with the single active site, no. , together with an octant of the active voxel, representing its partial volume. The three black dots in the figure represent the corresponding boundary face centers, and the six dashed lines connect these with the corresponding junctures. I.e., for a single active site, only three boundary face quadrants have to be considered at maximum. Note, that each of the boundary face quadrants are oriented. In Fig.s 16.b, 16.c, and 16.d, the corresponding (gray) VFV pairs are drawn, together with their corresponding boundary face centers, i.e., the paths , , and (cf., Table 1), respectively. The three paths connect to the closed point sequence, . After the removal of points with identities (i.e., junctures), the sequence will transform into the VESTA -cycle, . As a result, we obtain a single oriented triangle within the -cell.
In another example, we shall process the three active sites as shown in
First, all of the contributing boundary face quadrants are determined
with their proper orientation.
In Fig. 17.a, the according seven boundary face quadrants are shown,
together with the corresponding boundary face centers.
In a next step, one considers all relevant VFV paths (cf.,
If we focus on the six faces of the -cell, we notice that a face of a
-cell can always contain either zero, two, or four VFVs.
In this given example, five of the six -cell faces contain only
two VFVs each.
With the help of Table 1, the connection of the vectors is straightforward
But, one -cell face contains four VFVs, and at its center we observe a POA. Note that for the POA the application of the connection diagram as shown in Fig. 1.d will suffice for the proper execution of a previously specified “disconnect” and/or “connect” mode. In other words, for the marching VESTA, the much more complicated usage of the four connection diagrams as shown in Fig. 6.b is not necessary, hence, lesser computational decisions have to be made. Finally, considering the chosen connectivities among the VFVs, and after omission of the junctures, we end up with two solutions for the VESTA surface cycles. We obtain either a single -cycle and a single -cycle that are resulting from the “disconnect” mode (cf., Fig. 17.c), or we obtain a single -cycle that results from the “connect” mode (cf., Fig. 17.d).
For the sake of completeness, we demonstrate in Fig. 18 for two active sites the recommended processing steps for boundary face center displacement (which is in general necessary for isosurface generation, cf., Fig.s 18.a and 18.b), and the breakup of the VESTA -cycles () into -cycles (cf., Fig.s 18.c – 18.f). In this given example, we have one single VESTA -cycle. Note, that the contributing range vectors coincide with the respective edges of the -cell (cf., Fig.s 18.a and 18.b). Furthermore, it is recommended that the boundary face centers will be displaced before the breakup into -cycles, because then one is not required to reevaluate the 3D position of the possibly inserted average point, which once again may save computing time. Fig.s 18.e and 18.f illustrate the introduction of biases for the shape of the generated surface section, if no additional support points, e.g., average points, will be used.
This concludes the description of the marching VESTA.
ii.7 VESTA vs. the Marching Cubes Algorithm
In this subsection, we provide an in-depth comparison between VESTA
and the original MCALORE87 , because the original MCA can produce
holes in the final surfaces, whereas VESTA does not.
Since the marching VESTA scans the given 3D data set exactly like the
MCA, we can compare the results of VESTA -cell by -cell with
the various templates, which the original MCA uses.
Since VESTA can distinguish between a “disconnect” and a “connect”
mode whenever POAs are encountered, we shall present both solutions
for all given -cells.
In Fig. 19, the complete surface section template set of the original MCA is shown for its configurations of different -cell site occupancies (cf., also, e.g., Ref.s LOHM98 ; NVID07 ). The original MCA uses the same set of templates for its correspondingly inverted -cell sites (cf., Fig. 20). In comparison to Fig. 19, we show the computed surface -cycles for VESTA in global “disconnect” (“connect”) mode in Fig. 21 (Fig. 23); and in comparison to Fig. 20, we show the computed surface -cycles for VESTA in global “disconnect” (“connect”) mode in Fig. 22 (Fig. 24). Several differences can be observed.
First, we provide a comparison between VESTA in its global “disconnect” mode and the original MCA. VESTA reproduces in Fig. 21 perfectly the perimeters of the MCA template set (cf., Fig. 19). However, if we compare Fig. 22 with Fig. 20, there are differences between the cases (d), (g), (h), (l), (m), and (n). In particular, the original MCA does not provide templates like the processed VESTA -cycles shown in Fig.s 8.e and 8.g.
Secondly, we provide a comparison between VESTA in its global “connect” mode and the original MCA. VESTA reproduces in Fig. 24 perfectly the perimeters of the MCA template set for inverted sites (cf., Fig. 20). However, if we compare Fig. 23 with Fig. 19, there are differences – once again – between the cases (d), (g), (h), (l), (m), and (n). As already pointed out, the original MCA misses templates like the processed VESTA -cycles shown in Fig.s 8.e and 8.g.
Finally, we show in Fig. 25.a an example where the combination of the original MCA’s templates leads to the creation of a hole in the surface. On the contrary, VESTA consistently applies the “disconnect” and/or “connect” modes for the two side-by-side -cells under consideration, and it does not create any holes (cf., Fig.s 25.b and 25.c). The reason for VESTA’s success lies in the ansatz based on the VFVs of Fig. 2.b, and in the careful resolution of ambiguities at the junctures (cf., Fig. 25.d). In Fig.s 25.e and 25.f, VESTA’s high resolution surface tilings are shown in addition for this particular example.
Apparently, it is wrong to just take the same template set also for inverted -cells. Nowadays, various extensions of the original MCA account for the missing surface templates (cf., e.g., Ref.s CHER95 ; NEWM06 ; BOUR94 ). In fact, some authors seem to provide too many additional surface templates (cf., Ref. LEWI03 ), i.e., some of these are actually not necessary for an adequate surface extraction.
Instead of using many surface templates, VESTA only uses a single building block (cf., Fig. 2.b) as the basis for surface construction. This concludes the theoretical section of this paper.
|Technique||original VESTA||marching VESTA||extended MCA|
|L: Time [s]|
|H: Time [s]|
In this application section, VESTA is used to create isosurfaces for
3D image data that have been generated from CT-scans HERM09 and
x-ray microtomography STOC08 , respectively.
Furthermore, it is used to create a freezeout hypersurface
from a set of 3D numerical simulation data ORNI92 ; BRS93 that
result from the field of theoretical relativistic heavy-ion physics.
We also provide benchmarks between the original VESTA, the marching
VESTA, and an extended computer code implementation BOUR94 of
the original MCA (cf., “extended MCA”, in the benchmark
The software by Bourke et al. BOUR94 fixes the problems of the original MCA, and it uses a surface template set that exactly agrees to VESTA, when it is executed in a low resolution (“L”), global “disconnect” mode. In this particular mode, VESTA will generate surface -cycles up to length seven only (cf., Fig.s 7 and 8). In order to be able to better compare the execution times of the codes under consideration, all software implementations have been prepared in the following ways.
All codes first load a full 3D data set into the computers’ memory, and an initial isovalue has been provided. Then the start time is taken and the stop watch begins to run. The codes perform their various tasks and create lists of surface support points and surface triangles, respectively. Before all of the generated data are stored into a file that eventually may be used for, e.g., rendering purposes, the stop watch is halted and the stop time is taken. The time, which is the difference time between the stop and start times, is listed as an average in Tables 2 – 4 for a total of runs each.
Note that for this benchmark an Amilo notebook by Fujitsu Siemens Computers has been used. Its hardware consists of a Pentium®Dual-Core CPU T4200 @ , and RAM. The disk operating system of this computer is an Ubuntu 10.04 LTS Linux distribution.
iii.1 Processing of CT-Scan Data of the Stanford Terra-Cotta Bunny
As a first application, we present in Fig. 26 a VESTA isosurface rendering
for CT-scan data of the Stanford Terra-Cotta Bunny STAN01 .
Note, that this particular rendering uses a flat shading (cf.,
Ref. FOLE93 ) for the -valued isosurface, which has been
generated while using VESTA in a high resolution (“H”) “mixed” mode
(the latter uses gray-level information to determine locally, whether
a “disconnect” or a “connect” mode should apply).
In Table 2, all benchmark information is listed. In particular, all three considered versions of codes yield the same results with respect to the numbers of surface cycles. The bunnies’ surface is apparently smooth enough, so that -cycles up to length six suffice for an adequate result. Note that the MCA produces the largest amount of redundant information, i.e., three support points for each created surface triangle, whereas the original VESTA produces always the minimum of the required support points. The marching VESTA avoids in our particular implementation redundant information only within each given 3D data scanning -cell.
Postprocessing for the removal of redundant support point information has not been performed by us, since we intended to make only minimal adaptations to the source code of the MCA implementation by Bourke et al. BOUR94 . Therefore, this according effort has not been put into the marching VESTA implementation either, in order to be able to provide benchmark results that allow for a better comparison among the codes. As a result, the here used MCA and marching VESTA codes execute somewhat faster than they normally would, if redundant support points were removed. Note that the original VESTA executes the fastest for this particular data set (cf., the bold faced “L”-times in Table 2).
|Technique||original VESTA||marching VESTA||extended MCA|
|L: Time [s]|
|H: Time [s]|
iii.2 Isosurface Rendering for X-Ray Microtomographic Data
In the field of x-ray microtomography STOC08 , like
tomography HERM09 , one creates cross-sections during scans of a 3D object.
These cross-sections are 2D images with pixel sizes in the micrometer range.
The stacking of the 2D images results in a 3D image (data set).
Such a 3D image represents a virtual model of the original object.
Hence, x-ray microtomography provides a way to create a virtual model from an
object without destroying it.
Let us begin with the sequence of 2D gray-level images with pixels, which is shown in Fig. 27. After their stacking, they represent a 3D image. The pixels here assume gray values in the range from to . These bounds represent black and white, respectively. In the following, we shall apply VESTA to this 3D image in order to render an isosurface with isovalue . First, all voxels that have a gray value larger or equal to are selected. The cluster of selected voxels is shown in Fig. 28.a, together with the total volume of the voxels. Note that this initial 3D shape of selected voxels is already enclosed with a surface that has no holes at all; namely the union of all boundary faces that enclose all active voxels.
Here we shall use gray-level information to automate the (consistent) local choosing of the “disconnect” and the “connect” mode, respectively. If two voxels that ought to be enclosed have only one single voxel edge in common, we determine the (here linearly interpolated) gray value of corresponding point of ambiguity. This gray value is evaluated as the average of the gray-levels of the four voxels, which share this common edge.
After application of VESTA, but without support point displacement, one obtains – as intermediate steps – the results as illustrated in Fig.s 28.b – 28.d. The wire frame in Fig. 28.d represents the union of all VESTA -cycles where . In Fig. 28.e, a much denser wire frame is shown, which is the result of the break down of the -cycles () into -cycles while making use of additional average points as described in the theoretical section (cf., Fig.s 9 and 14). In a next step, the VESTA -cycles are depicted as solid triangles (cf., Fig. 28.f). Note that the shade of gray of each triangle is determined through the evaluation of its normal vector.
The preliminary VESTA surface as shown in Fig. 28.f looks somewhat bulky. However, we should stress that all VESTA surfaces are at this stage of the processing perfect in the sense, that they are non-degenerate, i.e., they always enclose a volume that is larger than zero, and they never self-intersect and/or overlap each other. Furthermore, the information of the interior/exterior of the enclosed shapes is propagated correctly at all times.
In a final step, we generate an isosurface from the VESTA surface. This is done by displacement of all voxel centers within the bounds defined by their range vectors, , (cf., Fig. 2) and by subsequent reevaluation of the average points, which have been used for the -cycle breakup. In particular, the gray-levels of the two voxels, which define a given range vector, , are interpolated linearly here (note that other techniques may be applied BOEH07 ; SALO05 ).
The final isosurface representing an isovalue of for the given 3D image data is shown in Fig. 28.g. And in Fig. 28.h, this isosurface is superimposed with normal vectors of the visible triangles. Note that this isosurface is quite smooth considering the rather coarse granularity of the underlying 3D voxel space. Furthermore, the directions of the normal vectors of the surface elements, i.e., triangles, are not limited to the six directions of the normals of the initial boundary faces.
While using the processing steps as outlined in this subsection, in Fig. 29, the rendering of a VESTA isosurface with isovalue is shown for 3D x-ray microtomographic image data of a metallic foam SEID03 . In order to better perceive the depth of the data, the rendered surface has been color encoded prior to its transformation into shades of gray. The corresponding benchmark results are presented in Table 3. Note that the original VESTA needs a little more computing time than the marching VESTA and the MCA codes, however it generates significantly lesser surface support points.
iii.3 Freezeout Hypersurface Extraction for Expanding Fireballs
One of the foremost objectives within the field of heavy-ion physics is
the exploration of the equation of state (EOS) of nuclear
matter (cf., Ref. FRIM10 and references therein).
In heavy-ion physics experiments so-called “fireballs” are created, which
are very hot and dense zones of nuclear matter.
The theoretical description of these fireballs includes – but is not
limited to – a relativistic hydrodynamic component, in which an EOS has
to be explicitly postulated (cf., e.g., Ref.s STRO86 ; CSER94 ).
The resulting space-time evolution of the fireball is driven by the EOS,
and hence, is expected to manifest itself in calculated multi-particle
production spectra that will eventually be compared to experimental data.
In some theoretical descriptions (cf., e.g.,
Ref.s COOP75 ; CHEN10 ),
the space-time evolution of the (ultra-relativistic) fluids is in part
represented through a so-called “freezeout hypersurface” (FOHS), which is an
isosurface within space-time, e.g., with respect to the temperature of the fluid.
The FOHS has to be extracted from the hydrodynamic simulation
data so that further (numerical) calculations can be performed.
Let us consider here the dimensional relativistic hydrodynamic simulation
code HYLANDER-C BRS97 ; BRS99 that can be used to study radially
symmetric, so-called “central”, heavy-ion collisions.
The numerical HYLANDER-C simulations are performed on a cartesian grid in the two spatial (2D) variables, and , and in the temporal (+ 1D) variable . In particular, denotes the radius of the system, the longitudinal direction, and the time, respectively. A solution of the hydrodynamic equations may be viewed as 3D image data where each voxel contains continuous values for each of the various physical field quantities under consideration, such as temperature, , energy density, , components of the fluid velocity , etc. Before we treat 3D simulation data, we shall first take a look at a temporal sequence of spatial 2D data.
In Fig. 30, a temporal sequence of temperature fields is shown as a function of the longitudinal direction, , and the radius, . The time steps remain the same between two successive frames, which are numbered from to . The hydrodynamic grid has originally a much higher resolution than the here shown down-sampled 2D images. Each 2D frame in the figure has a resolution of pixels, but eventually we shall process a slightly larger 3D data set (cf., Table 4). Note that this lower resolution provides sufficient numerical accuracy for the desired FOHS in this given example. Black pixels correspond to a fluid temperature of , whereas white pixels refer to a fluid temperature of and/or higher.
For all frames DICONEX isocontours with isovalue (“freezeout temperature”, ) have been extracted (white contours). The last frame, no. , does not contain any isocontour, since all of its pixels have temperatures below . Note that unphysical line segments have been removed where (at least) one end point has a radial value that is smaller than zero (cf., also Ref. BRS09 on the extraction of a FOHS in 2D).
If one stacks up the DICONEX freezeout contours of Fig. 30 in 3D with the intent to construct a surface (cf., Fig. 31.a), one requires both orientations of the contours. The latter is a consequence of the fact that in 3D one can look upon a 2D pixel in two ways, e.g., from “above” and from “below”. However, if we consider in Fig. 30 the transition from frame no. to frame no. , we observe a correspondence problem. Apparently, a single isocontour has to be transformed into two isocontours, but it is unclear, how it can be accomplished within this approach.
Fig.s 31.b and 31.c offer the two alternate ways to build stacks of isocontours from the other possible projections within the 3D space under consideration. Again, unphysical line segments have been removed in the figures where (at least) one end point has a radial and/or a temporal value that is smaller than zero. For our particular example, no correspondence problems occur this time, but it cannot be ensured for different scenarios. Note that before the invention of VESTA in the year 2002 BRS03 , FOHS construction within HYLANDER-C was initiated while using stacks of isocontours from the planes at fixed values as shown in Fig. 31.b.
|Technique||original VESTA||marching VESTA||extended MCA|
|L: Time [s]|
|H: Time [s]|
In order to avoid any correspondence problems within 3D, the D
relativistic hydrodynamic simulation code HYLANDER-C starts nowadays from the
consideration of the full 3D stack of voxels, which each represent a
temperature that is equal to or higher than (cf., Fig. 31.d).
The application of VESTA (cf., Fig. 32), while using average points
for the breakup of the -cycles into -cycles as explained above, yields the
isosurface for the isovalue , which is depicted in
Fig.s 33.b and 33.c.
During the process of FOHS construction, all unphysical triangles have been
discarded, i.e., those, which have (at least) one corner with a radial and/or a
temporal value that is smaller than zero.
Note that the union of all isocontour edges, which are shown in Fig.s 31.a – 31.c, is also given through the set of triangular VESTA surface mesh edges as shown in Fig. 32.c (without the unphysical ones) and vice versa. The gray shading of each triangle in Fig.s 33.a – 33.c is determined from the knowledge of the normal vector of each of the triangles of the FOHS. Benchmark comparisons between the various VESTA and MCA codes are provided in Table 4. Although the extended MCA is here the fastest, it also creates the most redundant information, i.e., support points instead of only .
Since the number of normal vectors can be quite large for a generated FOHS, it may be desirable to reduce this number for the speed up of subsequent, further calculations. In Fig. 33.d, a single decomposed VESTA -cycle is shown. To each triangle, its normal vector (gray), (), has been attached to the average (or “center of mass”) point of each the three triangle corner points. Note that the orientations of these normal vectors have been inherited from the initial orientation of the VESTA -cycle.
In order to prevent further averaging of other field quantities, which are potentially present within the initial 3D voxel data set, and because we intend to speed up subsequent calculations, it is adequate to construct the total normal vector, , as the sum of the triangle normals,
where is associated with the average (or “center of mass”)
point of each support point of the VESTA -cycle under consideration.
Note that in the case of VESTA -cycles,
In this case, is associated with the center of mass point of
the corresponding single triangle.
For further considerations of normal vector construction, we would like to
refer the reader to Ref. CHEN10 .
In summary, all the details that are necessary for a successful implementation of
the volume-enclosing surface extraction algorithm, which has been named VESTA, have
been explained here for the very first time.
VESTA surfaces are always perfect in the sense, that they are non-degenerate, i.e.,
they always fully enclose a volume that is larger than zero, and they never
self-intersect and/or overlap each other
(i.e., prior to a possible move of the initial boundary face centers within the
bounds of the corresponding range vectors).
We would like to stress that holes can never occur within a VESTA surface. Hence, VESTA surfaces do not require any kind of repair since potential ambiguities are correctly resolved. Furthermore, the information of the interior/exterior of the enclosed shapes is propagated also correctly at all times. In particular, the VESTA surfaces may be viewed as the 3D analog of the DICONEX contours in 2D. Note that VESTA could be extended to adaptive and/or unstructured 3D grids as well. Furthermore, it is fully compatible with its 4D counterpart, STEVE BRS04 , which is capable of the processing of time-varying voxel data. In fact, the ideas which have been presented in this paper can be applied to -dimensional spaces PAT11 .
The fact that VESTA surface cycles are confined to voxel neighborhoods is a result, but not a prerequisite as it is the case in Refs. LORE87 ; BLOO94 ; HO05 . Unlike the mesh generators presented in Ref.s LORE87 ; HO05 , VESTA is not template-based. Instead, VESTA uses the single building block, which is shown in Fig. 2.b. VESTA will find all of the required surface segments, which are determined with a fast and 100% robust construction technique.
We have demonstrated the usage of VESTA for several rather diverse applications, namely for the creation of isosurfaces based on 3D image data in the fields computer tomography and x-ray microtomography, respectively, as well as for the creation of a freezeout hypersurface from a set of 3D numerical simulation data in the field of relativistic heavy-ion physics. Various benchmarks have shown that VESTA can easily compete with the Marching Cubes algorithm, e.g., as far as computing speed is concerned. In addition, VESTA can produce six different types of surface outputs.
This work has been supported by the Department of Energy under contract W-7405-ENG-36. I am indebted to Dr. Dan Strottman for the extended support after my return to Germany.
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