Central European Multimedia and Virtual Reality Conference(2006)C. Sik Lányi, B.Oláh (Editors)

Global Mesh Partitioning for Surgical Planning

Philipp Fürnstahl1, Bernhard Reitinger†1, Reinhard Beichel1, Dieter Schmalstieg1

1Institute for Computer Graphics and VisionGraz University of Technology, Austria

AbstractWe present a set of partitioning tools that classify a tetrahedral mesh into different regions of interest while pre-serving mesh consistency. These regions can then be individually visualized, repositioned, or combined for furtheranalysis or processing. A partitioning operation, either defined analytically(by a formula) or geometrically (by asurface mesh), is applied globally to the model. A hierarchical data structure is used to store region informationand consecutive partitioning operations: it ensures consistency betweenthe specified regions of the volumetricmesh and the visualized surface mesh. Similar to volumetric cutting, subdivision is used to split the initial modelinto regions. Subdivision of tetrahedra that contain multiple intersection pointsper edge is a non-trivial task. Anextension to existing subdivision methods is presented which handles the subdivision of such tetrahedra in an iter-ative way. Since the partitioning of a volumetric mesh is an important task in surgical planning, this paper finallyshows that the presented algorithms can be successfully integrated in a virtual reality planning system.

Categories and Subject Descriptors(according to ACM CCS): I.3.5 [Computer Graphics]: I.3.8 Applications

1. Introduction

Interactive tools, which partition the underlying volumetricmodel into regions of interest, give new opportunities in vi-sualization and planning systems. These tools can be eitherapplied progressively to the model (e.g. a scalpel cut spec-ifies the partitioning surface) or globally, by interactivelydefined partitioning shapes. We aim at the second method:a partitioning operation is used to divide the entire meshinto disjoint regions. Consecutive operations can be used formore complex partitioning.

In our concept, the shape of the tool, which can be modi-fied interactively, defines how the affected parts of the meshwill be separated into regions. Each specified region can thenbe treated as an individual sub-model. This improves inter-active planning since quantitative analysis can be performedon particular regions (or a set of regions). Moreover, par-ticular regions can be visualized, hidden, or modified formore accurate intervention planning. Regions, created dur-ing consecutive partitioning operations, are stored in a hier-archically organized data model based on a binary tree. The

† contact: reitinger@tugraz.at

binary tree approach has the benefit that existing informa-tion is preserved and previously specified regions are stillavailable. This feature is very important for planning envi-ronments but usually not considered by conventional cuttingsimulations where old information is discarded. The binarytree is also used to consistently combine regions for visual-ization (interface extraction) or analysis without node or facedoubling.

In this paper, we introduce a new approach of global tetra-hedral mesh partitioning (especially focusing on very largemeshes) on the basis of three algorithms. In contrast to otherapproaches, the same scheme can be used for both, analyt-ically and geometrically defined partitioning tools. More-over, complex partitioning shapes (e.g. non-planar) can beused. Analytical partitioning is restricted to objects whichcan be described by a formula so it can be carried out effi-ciently. Geometrical partitioning shapes are represented bytriangulated surface meshes and are used for more flexi-ble partitioning. The developed partitioning algorithms op-erate on tetrahedral meshes. In contrast to other approaches,we designed the algorithms to work with high-resolutiontetrahedral meshes. We accept higher computation times but

P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

achieve highest quality for both meshes, the tetrahedral meshand the visualized surface mesh.

Partitioning is especially useful for surgical environmentswhere surgeons need to interactively classify an organ intohealthy (benign) and diseased (malignant) tissue. For ourtarget application, the intervention of a liver tumor resec-tion must be planned using partitioning tools [Rei05]. Fig. 1sketches an example of a liver model which will be parti-tioned into healthy and diseased regions. The partitioningprocedure starts by locating a tumor in the initial mesh. Next,the dataset is partitioned using a plane in order to uncoverthe tumor. Two new regions are created, colored yellow andorange. Subsequently, the left part of the liver (yellow re-gion) is hidden. A suitable shape is then used to apply a morespecific partitioning operation for removing the residual. Fi-nally, the binary tree is used to visualize and measure vol-umes of the healthy and diseased regions in order to obtainimportant quantitative indices.

tumor

liver

(a) initial model

plane

(b) plane partition-ing is applied

shape

(c) shape partitioning isapplied

Figure 1: A medical example showing the classification of atetrahedral mesh into disjoint regions.

2. Related Work

The partitioning operations presented in this paper are con-ceptually related to constructive solid geometry and virtualsculpting but share many technical aspects with existing vol-umetric cutting techniques.

Ganovelli et al. [GCMS01] presented a technique to en-able cuts with a shape on multi-resolution representations.A lookup table is used to determine which basic subdivi-sion case is necessary. Simplicial complex properties are al-ways preserved. In contrast to our approach, this method isrestricted to planar cut shapes but supports shape faces pok-ing inside tetrahedra.

Another approach for specifying virtual resections inliver surgery planning is described by Konrad-Verse etal. [KVPL04]. A deformable cutting plane is used to definethe cut shape. Our algorithms are not limited to planes or de-formable planes and can use other complex shapes as well.In addition, Konrad-Verse’s algorithm is based on vertex dis-placement instead of subdivision yielding more approximateresults. Unfortunately, neither the performance nor the accu-racy of the algorithm are discussed.

A progressive cutting algorithm, using a state machine,was recently published by Bielser et al. [BGTG04]. Theyoutlined a hierarchical subdivision method where alreadysubdivided tetrahedra can further be split, based on the tran-sition of the state machine. In addition, example applicationsare described where scalpel cutting is used to partition andvisualize regions of interest.

Based on this method, Steinemann et al. [SHGS06] intro-duced a real-time cutting approach where the low-resolutionmechanical model is decoupled from the high-resolution vi-sualized model. This allows efficient real-time cutting, how-ever, the visualized surface mesh is only an approximationof the tetrahedral mesh boundary.

Nienhuys and van der Stappen presented methods forcutting in deformable objects using the finite elementmethod [NvdS00,NvdS02]. Their goal was to enable interac-tive cutting in volumetric meshes while preserving the finiteelement mesh model. To avoid the generation of degenera-cies, a node snapping technique was described as well.

Several other techniques like volume sculpting or con-structive volume geometry are related to our approach.Sculpting is done by moving voxel-based tools within themodel (e.g. [FCG00]). In constructive volume geometry, in-troduced by Chen and Tucker [CT00], a set of algebraic op-erations is applied to scalar fields. However, direct volumerendering techniques rather than more efficient polygonalrendering techniques are used for visualization.

3. Data Representation and Visualization

For the proposed partitioning tools, we developed a two-level data model consisting of atetrahedral meshand abi-nary tree.

We call a tetrahedral meshconsistentif its set of tetra-hedra is a 3-dimensional simplicial complex [Ede01]. Theproperties of simplicial complexes enables us to define re-gions (simplicial sub-complexes) and to consistently extractboundaries for visualization (the mesh boundary is inher-ently available).

In addition to the tetrahedral mesh, a binary tree is usedfor storing the actual partitioning results (regions). This sec-tion gives a basic overview of the binary tree representation,interfaces (region boundaries), and finally interface extrac-tion and visualization.

Per definition, each partitioning operation splits the under-lying mesh into two new, disjoint parts. Therefore, a binarytree can be used to store the relation of tetrahedra to certainregions. Each leaf node in the tree represents a region andstores the indices of associated tetrahedra (see Fig.2). Ini-tially, the root of the binary tree stores all tetrahedron indicesof the mesh as one region. If partitioning operations are ap-plied, additional tree levels are created by splitting each leafnode into two child nodes. Thus the path from the root to a

P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

given leaf node can be encoded into an index that representsa sequence of partitioning operations (similar to a locationcode). A binary representation, implemented as a bit vector,is used for encoding: each bit corresponds to either a left orright link in the tree.

To visualize a set of regionsS of a tetrahedral meshM,an interfacecan be defined which represents the boundaryto the remaining regionsM\S. By using the properties ofsimplicial complexes, each interface consists of a set of tetra-hedron faces which can be always represented by a prop-erly connected triangular mesh. These triangles are inher-ently available in the tetrahedral mesh: no additional geom-etry is generated, because the geometry of the parent modelis used. This avoids duplicate vertices or faces and guaran-tees an overall consistent connection between the volumetricmodel and its visualized surface mesh.

The basic idea of the extraction algorithm is to traverse thebinary tree for identifying the list of regions which are targetfor rendering (see Fig.2). A rendering bit vectorr specifiesall regions which should be visualized. For each indexi of atraversed node a bitwiseXORcompare withr is performed(trimming i andr to equal length). Ifr ⊗ i = 0, then the childnodes are rendering candidates and recursively processed.Finally, all visited leaf nodes and their lists of associatedtetrahedra are examined for retrieving the boundary faces.A tetrahedron face is extracted if the neighboring tetrahe-dron does not belong to a region which should be rendered(defined byr).

shape

plane

(a) liver model

target for rendering

1

10

100 101 110 111

11

(b) e.g. render bit vectorr = 11

Figure 2: A plane and shape partitioning operation is ap-plied to the tetrahedral mesh in Figure (a). Figure (b) showsthe corresponding binary tree. The root node consists of oneregion. After plane partitioning, two child nodes are created,partitioning the domain into two pieces. If applying anotheroperation, 4 new leaf nodes, storing the regions, are created.Interfaces can be extracted to visualize arbitrary regions.

Since vertices of the mesh are shared by adjacent meshprimitives, a concept is required which allows the assign-ment of attributes to vertices. For each tetrahedron, belong-ing to a certain region, a unique set of attributes must be

available (e.g. to render each region in a different color).Therefore, thewedge conceptproposed by Hoppe [Hop98]is used. A wedge acts as a wrapper for vertices. It stores avertex index and additional local parameters but is associ-ated with exactly one tetrahedron. This has the effect of al-locating multiple wedges each of which is indexing a similarvertex but storing different attributes.

4. Partitioning Algorithms

Global modification of tetrahedral meshes are computation-ally expensive and acceptable computation times require thedesign of efficient algorithms. We introduce a hierarchy ofthree partitioning algorithms which differ in the complexityof the partitioning shape and consequently in computationtimes. The first algorithm uses an analytically defined planeand is typically applied in order to perform coarse partition-ing operations. Due to its simple shape, several algorithmparts can be specialized and optimized (e.g. subdivision).The second algorithm uses a generalized analytical shape.Contrary, the partitioning shape of last algorithm is specifiedby a triangular surface mesh for most flexible partitioning.In general, a partitioning algorithm consists of four essentialparts:

1. Collision detection: tetrahedra which are intersected bythe partitioning tool are detected.

2. Intersection point calculations: after testing if an inter-section point is not yet calculated, it is stored in a hashtable.

3. Subdivision: intersected tetrahedra must be split accord-ing to the intersection points for a unique region assign-ment.

4. Assignment to regions and binary tree update: alltetrahedra can now be assigned to one of the newly cre-ated regions. A new level is added to the binary tree andtetrahedron indices are stored in the corresponding leafnodes.

According to these four steps, the following partitioningalgorithms were developed.

4.1. Analytical Plane Partitioning

The plane partitioning algorithm starts with a traversal ofthe tetrahedral mesh and determines for each tetrahedronwhether it is intersected by the plane. Tetrahedra which arenot split, are immediately assigned to a region (correspond-ing to the side of the plane). A tetrahedron is intersected bythe plane if not all of its vertices lie on the same side of theplane. This verification is done by calculating the signed dis-tance between all tetrahedron vertices and the plane. The dis-tance is computed by using robust orientation tests to avoidfloating point inconsistency [She97]. If calculated distanceshave different signs, then the corresponding edges containintersection points and the tetrahedron is target for subdi-vision. According to the signed distances of the vertices, a

P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

lookup table is used to determine the appropriate subdivisioncase (eight cases, one possible intersection point per edge).

Before subdivision, the intersection points are calculatedusing basic vector algebra. In order to keep the mesh consis-tent, shared intersection points of adjacent tetrahedra mustbe stored uniquely in the vertex buffer. Therefore, a hash ta-ble is used for efficient propagation of intersection points toadjacent tetrahedra. The index of each newly calculated in-tersection point is stored in the hash table. The correspond-ing tetrahedron edge uniquely defines the hash table’s keyvalue.

In case of plane partitioning only one intersection pointper tetrahedron edge can exist. Therefore, an ordinary sub-division technique can be applied. We chose a subdivisionmethod by Dompierre et al. [DLVC99] who splits quadri-lateral faces by choosing the diagonal which contains thesmallest vertex index in the face. This method is efficient,does not depend on floating point calculations, and guaran-tees that tetrahedra faces are always split uniquely.

Before adding newly created tetrahedra to the mesh, in-valid orientations are detected and fixed. In addition, the bi-nary tree is updated by adding the tetrahedron index to thecorresponding node in tree (representing the correct region).

4.2. Analytical Shape Partitioning

Shape partitioning provides a more flexible way than planepartitioning, because more general shapes can be used tospecify partitions. The method requires shapes, which canbe defined by a formula but leads to an efficient algorithm.Since a partitioning operation must completely split eachtetrahedron (required by the binary tree concept), partiallysplit tetrahedra are not considered. In these cases the shapepartitioning methods are approximative, but subsequentlygenerate less tetrahedra and vertices. This has a positive ef-fect on the mesh complexity. Since the partitioning shapecan be specified analytically, the algorithm is efficient evenfor non-planar shapes.

An analytically defined partitioning shapeP has to pro-vide two functions. An intersection functionI(et) that cal-culates the intersection points betweenP and a tetrahedronedgeet, and a distance functionD(pt) that determines thelocation of a pointpt relative to the partitioning shape. Asan example, the distance function of a sphere is defined asD(p) = (px −mx)

2 + (py −my)2 + (pz−mz)

2 − r2, wherecenter pointm and radiusr specify the partitioning sphere.I(et) is then an ordinary sphere – line segment intersectiontest.

At first the distance function is calculated for each ver-tex p of the mesh by traversing the vertex buffer. After-wards the mesh tetrahedra are traversed and the result of theD(p) is stored in the corresponding wedges of each tetrahe-dron (see Section3). Three states represent this result:inside

(D(p) < 0), intersection(D(p) = 0), andoutside(D(p) > 0).Since partially split tetrahedra are ignored, intersected tetra-hedra can be identified corresponding to the sign ofD: ifD(pi) > 0 andD(p j ) < 0 for tetrahedron verticespi andp j ,then this tetrahedron is target for subdivision. Otherwise itcan be assigned immediately to the inside or outside region(according to the sign of the distance function).

Subsequent to the identification of an intersected tetrahe-dront, the exact intersection points are calculated. Similar toSection4.1, a hash table stores the intersection information.In order to uniquely calculate the intersection points betweent andP, the hash table is examined for intersection points:for each edge of the current tetrahedron, existing intersec-tion points are retrieved. If no entry for an edge is found, allits intersection points are calculated (and stored in the hashtable) using the intersection function.

Intersecting tetrahedra with non-planer shapes can resultin multiple intersection points per tetrahedron edge (e.g. asmall sphere can intersect a tetrahedron edge twice). Theiterative subdivision (described in the next paragraph) sub-divides such tetrahedra according to the order of stored in-tersection points. Moreover, the used hash table approachmust be extended. An ordinary hash table with the abilityto store objects with equivalent key values is organized inbuckets (usually arranged in a linked-list). These buckets areextended to so-calledsorted bucketsby storing all intersec-tion points of a tetrahedron edge in a specific order. In ourcase, all intersection points of a tetrahedron edge are sortedbased on the distance to the edge’s start-point. The order rep-resents the correct cut sequence and is required later in thesubdivision process.

After all intersection points are calculated, each inter-sected tetrahedront must be subdivided for a unique assign-ment to the corresponding region. Basically, 10 symmetri-cal different subdivision cases are required to split a tetra-hedron into two parts (see Fig.3). However, the subdivisionof a tetrahedron that contains edges with multiple intersec-tion points is a non-trivial task. We introduce an iterative ap-proach: In the first iteration, the currently active intersectionpoint of each tetrahedron edge is determined by choosingthe first stored point in the corresponding sorted buckets ofthe hash table (see Fig.4(a)). Subsequently, a lookup table isused to determine how to rotate or flip the vertices to fit thedefault orientation in order to perform the basic subdivisionby Dompierre [DLVC99]. Active intersection points of thenext iteration are retrieved by incrementing the position inthe sorted buckets of the edges (Fig.4(b)). Finally, the algo-rithm is iteratively applied to subdivided tetrahedra until allintersection points are processed (shown in Fig.4(c)).

In the last step of the partitioning algorithm each subdi-vided tetrahedron is assigned to the corresponding region.It contains only vertices of the original tetrahedron or in-tersection points (labeled withintersectionstate). Thus thevertex states, which are stored in the wedge data structures,

P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

(a) 2 tets (b) 1 tet, 1pyramid

(c) 1 tet, 1prism

(d) 2 prisms (e) 4 tets

(f) 4 tets (g) 1 tet, 2pyramids

(h) 2 tets, 2pyramids

(i) 2 tets, 1pyramid, 1prism

(j) 4 tets, 2pyramids

Figure 3: Partitioning cases of a tetrahedron and an arbi-trary partitioning shape with subdivision into primitives. In-tersection points are colored red; red-black points denotetetrahedron vertices lying on the shape. Red colored facesdenote configurations which are required definitely. Eitheryellow or green colored faces are required in 4 cases to guar-antee, that the tetrahedron is split completely. Yellow-blackor green-black points must lie on the partitioning shape re-spectively.

p1

p2

p3

(a) initial state

p1

i2

i3p2

p3

i1

(b) 1st iteration

i4

i5

p1

i1

i2

i3p2

p3

(c) 2nd iteration

Figure 4: Iterative tetrahedra subdivision with multiple in-tersection points per edge (simplified shown as a trianglesubdivision). In each iteration an ordinary subdivision isperformed, using active intersection points (red arrows).

are then used to assign those tetrahedra to the correct region.If a tetrahedron does not contain a vertex with stateoutside,it is assigned to the inside region. Respectively, if a tetrahe-dron does not contain aninsidevertex, it is assigned to theoutside region. Tetrahedra, where no unique assignment ispossible, are assigned to the outside.

4.3. Geometrical Shape Partitioning

More flexible partitioning can be achieved by using a tri-angulated surface mesh to define the partitioning shape. Inorder to divide the tetrahedral mesh into disjoint regions, thepartitioning shape must be a closed mesh or an object home-omorphic to an open disc.

All tetrahedra, which are intersected by partitioning shapefaces, must be identified. In addition, remaining tetrahedrahave to be assigned to the inside or outside region as well.Both steps can be handled efficiently by using an octree, gen-

erated from the partitioning shape. Since the shape is usuallymodified interactively, the octree generation must be doneon-the-fly.

For each vertex of the tetrahedral mesh, it is determined ifa point lies inside or outside of the partitioning shape. Tetra-hedron vertices, lying on a partitioning shape face, are iden-tified later during exact intersection point calculations. A rayshooting approach, which can handle open surfaces, is used:

• A ray is shot, starting from each vertex of the volumetricmesh. The ray direction is specified by the barycenter ofa certain partitioning shape face (e.g. primitive with in-dex 0). This guarantees that the ray always intersects thepartitioning shape.

• The intersected partitioning shape face with the minimaldistance to the ray origin must be identified. Starting at theroot of the octree, child nodes, which are hit by the ray, aretraversed recursively in the order of the smallest distanceto the ray origin. Once a leaf node with stored primitivesis visited, an exact ray – triangle intersection test is per-formed for each stored primitive [MT97]. Subsequently,the algorithm terminates and returns the primitive indexwith the minimum distance.

• If the supporting plane’s normal vector of the chosenprimitive points in the same direction as the ray (com-puted in step one), then the tested vertex is located insidethe partitioning shape.

In the next step, the tetrahedra of the mesh are traversed.According to the results of the ray shooting, theinside oroutsidestate is set in the wedges of each tetrahedron. If allfour vertices have the same state, the tetrahedron is imme-diately assigned to the corresponding region. Otherwise, itis intersected by the partitioning shape and the exact inter-section points must be calculated. Each tetrahedron is testedagainst the octree (bounding box tests) to retrieve the listof overlapping partitioning shape faces. For each of theseshape faces, the exact intersection points between the tetra-hedron and the supporting plane of the shape face are calcu-lated (similar to Section4.1). Subsequently, it must be testedwhether a calculated intersection point lies inside the parti-tioning shape face by projecting the point and the triangleinto 2D: a robust 2D orientation test [She97] finally deter-mines if the intersection point is located in the triangle andcan be stored.

Analogue to analytical shape partitioning, the hash tableapproach is used for the propagation of intersection points.Since partitioning shape faces are tested in arbitrary order,the verification whether an intersection point is already cal-culated, is more complex. The following information mustbe stored in the hash table for each intersection point:

• The index of the intersection pointpk.• The index of the partitioning shape facef which intersects

the tetrahedron edge inpk.• The vertex indices of the edge off if pk lies on this edge.

P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

• The vertex index of a point off if pk coincides with thispoint.

To ensure mesh consistency and avoid degeneracy, each in-tersection point must be inserted in the hash table only once.Three possible cases can arise:

1. If an intersection pointpk lies inside a single partitioningshape facef, thenpk is only shared by adjacent tetrahedrafaces. A comparison of the identifier off with stored hashtable values is sufficient to verify thatpk was already cal-culated.

2. If pk lies on an edge shared by two partitioning shapefaces, then additionally the vertex indices of the affectedpartitioning shape edge must be compared with corre-sponding hash table values.

3. Otherwise, ifpk coincides with a vertex off, then thisvertex index is compared with stored hash table values aswell.

Finally, the actual subdivision and the assignment of subdi-vided tetrahedra is performed according to Section4.2.

5. Results

We evaluated the algorithms in 45 test cases for perfor-mance and quality measurements. In this test environment,high-quality tetrahedral meshes with sizes ranging between10,000 and 280,000 tetrahedra were used. In case of geomet-rical shape partitioning, partitioning shapes were specifiedby 800−2,000 triangular primitives.

The performance measurements were done on a Pen-tium 4 with 2 GHz. Experiments showed that in a typicalpartitioning operation 1,000−5,000 tetrahedra were inter-sected by the partitioning tool. The graph of Figure5 givesa runtime comparison of all algorithms. On average, a com-plete partitioning operation, applied to a 50,000 tetrahedramesh, took 75 ms, 320 ms, and 685 ms for plane, analyti-cal shape, and geometrical plane partitioning. We carried outparticular runtime measurements for collision detection, in-tersection point calculation, subdivision, and region assign-ment.

Plane partitioning can be performed very efficiently. Col-lision detection and region assignment was performed in7 ms per 10,000 tetrahedra. In the remaining time, intersec-tion points are consistently calculated and the subdivision isperformed. For 1,000 tetrahedra, these operations togethertook 28 ms on average.

Analytical shape partitioning takes longer. Collision de-tection, intersection point calculations, and region assign-ment only depend on the complexity of the distance andintersection function. In case of a sphere, those steps tookabout 17 ms for 10,000 tetrahedra. In addition, iterative sub-division, which is more complex and thus has more over-head, contributed with 100 ms per 1,000 tetrahedra.

The additional runtime increase if using geometricalshapes is primarily caused by the collision detection (octree)and region assignment (ray shooting). We used an on-the-fly generated octree with depth 5 (100−220 ms generationtime in experiments). The described ray shooting approachtook about 40 ms for 1,000 mesh vertices. Finally, the con-sistent propagation of calculated intersection points is moreruntime expensive since triangular faces are processed in-stead of evaluating analytical functions.

We also analyzed the quality of all tetrahedra, createdafter a partitioning operation, since degenerated tetrahe-dra are a known problem of subdivision. In our exper-iments, 4− 7.1% of subdivided tetrahedra were degen-erated according to a quality ratio proposed by Nien-huys [NvdS02]. Node snapping was applied on-the-fly dur-ing subdivision [NvdS02], reducing the number of degenera-cies by 52% on average.

Peer-to-peer comparisons to state-of-the-art techniquesare difficult since most of them aim at local real-time cuttingwhere small-sized tetrahedral meshes are used (< 5,000).Most related toglobal partitioningis a technique of Ganov-elli et al. [GCMS01] where planar cut shapes are used tocut multi-resolution meshes with 37,000 tetrahedra. The run-time, given in their paper, consists of tetrahedra subdivision(60− 530 ms) and data structure updates (150− 344 ms).Ganovelli achieves a total runtime of 219− 875 ms (Pen-tium 2, 400 MHz).

0 0.5 1 1.5 2 2.5 3

x 105

0

500

1000

1500

2000

2500

3000

tim

e [

ms]

mesh size [tetras]

planeanalytical shapegeometrical shape

Figure 5: Average runtime of the proposed partitioning al-gorithms in relation to the initial mesh size (P4, 2 GHz).

5.1. Application

Several 3D applications require the ability to define and alterarbitrary regions of a volumetric mesh. In our case, the par-titioning algorithms are integrated into a virtual liver surgeryplanning system [Rei05]. This system provides the possibil-ity for surgeons to simulate atypical (non-segment oriented)tumor resections.

P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

Figure7 shows a medical scenario where partitioning isused to plan a surgical intervention. The partitioning con-cept is helpful in supporting physicians, who need to care-fully consider a limited number of alternatives for the mostpromising outcome of an intervention. In this example, twotumors are target for resection (a). Inside the semitransparentliver the vessel structure can be seen. After investigation, thesurgeon has several different possibilities for removing thediseased parts. In strategy 1, the dataset is partitioned usinga plane in order to uncover the tumors (b). The left part ofthe liver is hidden after the partitioning operation (c). Subse-quently, two options must be considered. In strategy 1a ((e)to (h)), the surgeon plans to remove (hide) each tumor sepa-rately. Therefore, two spheres are positioned for applying ananalytically defined shape partitioning operation. In contrast,both tumors are removed at once with a single bigger spherein strategy 1b ((i) and (j)). Another strategy simulates thetumor resection by removing a single tissue part: a geomet-rically defined deformable grid is applied in strategy 2 ((k)and (l)).

Since the hierarchically organized data structure of Sec-tion 3 is used, the different intervention strategies can bestored in a single data model. Moreover, it also enables toswitch between those strategies (or combine them) in real-time, e.g. for measuring the volume of removed tissue.

Figure 6 shows two different scenarios of the virtualliver surgery planning where the presented partitioning toolsare interactively used in a virtual reality setup. By using atracked pencil and panel, partitioning tools can be specifiedand partitioning can be initiated intuitively.

An evaluation with a surgeon has shown that the intro-duced partitioning tools can be used successfully withina complete planning process (evaluations are detailedin [Rei05]). Surgeons can save time by using the presentedatypical planning methods compared to traditional interven-tion planning. In addition, the performance of the partition-ing algorithms was stated reasonable, since they are initiatedin a separate thread without disturbing interactive rendering.

6. Conclusion and Future Work

We have presented an entire concept for implementing par-titioning tools into interactive visualization or planning en-vironments. Based on the required complexity of the tool,one of three algorithms can be chosen. Evaluations haveshown that the provided algorithms are feasible for surgicalplanning in interactive systems. The presented methods arenot limited to surgical environments, thus partitioning has abroad field of application in volumetric visualization.

Nevertheless, several improvements are reasonable andaddressed for future work. The algorithms can be furtheroptimized, however, experiments indicate that the runtimeis sufficient for interactive planning. In addition, situationsmay arise where a unique assignment of tetrahedra to the

correct region is not possible. In this case, the methods arecurrently approximative. We intend to overcome this prob-lem by performing on-the-fly mesh refinement.

Acknowledgments

This work was funded by the Austrian Science FundFWFunder contract no. P17066.

(a) The right part of the liverincludes a tumor. An analyticalshape partitioning is initialized,choosing a sphere primitive.

(b) The sphere is interactivelyscaled and positioned to coverthe tumor completely.

(c) After partitioning, each re-gion can be modified separatelywhile the information is storedin one model. In this case, theleft region is set to transparent.

(d) Another strategy is chosenwhere geometrical shape par-titioning, using a deformablegrid, is initialized.

(e) The grid is interactively de-formed in such a way that thetumor is completely containedin the left liver part.

(f) Two regions are created af-ter the operation. The left oneis hidden in order to verify thatthe tumor was successfully re-sected.

Figure 6: Partitioning methods embedded into a virtualreality-based liver surgery planning setup.

P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

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P. Fürnstahl et al. / Global Mesh Partitioning for Surgical Planning

(a) (k) (l)

(b) (c) (d)

(e) (f)

(g) (h) (j)

(i)

Strategy 2: deformable grid

Strategy 1a: two separate spheresStrategy 1b:

single sphere

plane partitioning to uncover tumors

initial liver dataset

Figure 7: Medical scenario showing different possibilities for the removal of liver tumors. Either plane partitioning and an-alytical shape partitioning (strategy 1, 1a, 1b), or geometrical shape partitioning (strategy 2) are used two simulate differentintervention strategies.