Pu Huang Intersection-Free and Topologically Faithful...

Pu HuangDepartment of Mechanical and Automation

Engineering,

The Chinese University of Hong Kong,

Hong Kong, PRC

Charlie C. L. Wang1

Department of Mechanical and Automation

Engineering,

The Chinese University of Hong Kong,

Hong Kong, PRC;

Epstein Department of Industrial and Systems

Engineering,

University of Southern California,

Los Angeles, CA 90089

e-mail: [email protected]

Yong ChenEpstein Department of Industrial and Systems

Engineering,

University of Southern California,

Los Angeles, CA 90089

Intersection-Free andTopologically FaithfulSlicing of Implicit SolidWe present a robust and efficient approach to directly slicing implicit solids. Differentfrom prior slicing techniques that reconstruct contours on the slicing plane by tracing thetopology of intersected line segments, which is actually not robust, we generate contoursby a topology guaranteed contour extraction on binary images sampled from given solidsand a subsequent contour simplification algorithm which has the topology preserved andthe geometric error controlled. The resultant contours are free of self-intersection, topo-logically faithful to the given r-regular solids and with shape error bounded. Therefore,correct objects can be fabricated from them by rapid prototyping. Moreover, since we donot need to generate the tessellated B-rep of given solids, the memory cost our approachis low—only the binary image and the finest contours on one particular slicing planeneed to be stored in-core. Our method is general and can be applied to any implicit rep-resentations of solids. [DOI: 10.1115/1.4024067]

Keywords: direct slicing, solid, implicit representation, self-intersection free, topologi-cally faithful

1 Introduction

Slicing CAD models is a crucial operation for generating toolpaths in rapid prototyping. Prior slicing algorithms focus on com-puting the intersection curves between a model represented by po-lygonal meshes and a sequence of parallel planes, which becomesan unstable step for the whole procedures of rapid prototyping ifthe triangular meshes are self-intersected (or overlapped). In addi-tion, more and more modeling approaches represent objects withcomplex structures by implicit solids since such representationsare mathematically compact and robust. When conventional slic-ing methods are used, the implicit solids must be first tessellatedinto triangular meshes and then intersected by slicing planes.However, generating a self-intersection free and topologicallyfaithful polygonal model from an implicit solid is not easy (seeRefs. [1,2]. for detailed discussions). Specifically, the triangularmodels produced can have problems like gaps, degenerated trian-gles, overlapped facets, nonmanifold entities and self-intersections. Using conventional slicing techniques to generatecontours for rapid prototyping from such problematic triangularmeshes will result in an incorrect object. For example, as shownin Figs. 1 and 2, unexpected gaps are produced on the Buddhamodel fabricated by fused deposition modeling (FDM). This gapis caused by the self-intersected polygonal model, which brings ininverse in/out membership classifications to some planar contours(see the contours on the plane with height¼ 1.79 in. in Fig. 2).

The topology of a fabricated model is usually required to behomeomorphic to the given solid. This is very important to applica-tions such as biomedical engineering—e.g., a fabricated modelwith an incorrect topology may merge two tubes that should be sep-arated into one, which is very dangerous for medical treatments(see Fig. 3). Meanwhile, shape approximation errors between theextracted contours and the exact ones defined by intersectingthe given solid with the slicing plane must also be controlled. Tofabricate an object with high accuracy in a conventional way of

tessellation, a massive number of triangles may be generated andstoring them in-core will use up the memory of a computer system.This motivates our research to develop a direct slicing algorithm togenerate self-intersection free and topologically faithful contoursfrom a general implicit solid.

Problem Definition. For a given implicit solid H ¼ fpjf ðpÞ� 0;8p2<3g and a slicing plane P, a contour C ¼ M\P isdefined as a topologically faithful contour when M is a surface r-homeomorphic to the exact surface boundary, @H, of H. In thispaper, we compute contours which are

(1) topologically faithful when the given solid H is r-regular(2) self-intersection free, and(3) with the shape approximation error minimized

Our approach consists of three major steps. Firstly, a binaryimage B of an r-regular solid H is sampled on the slicing plane P,where an appropriately selected sampling distance r0 (with itsbound relating to the value r) ensures that the contour �C0 gener-ated from B is homeomorphic to C. Secondly, �C0 is iterativelysmoothed into �Cm by a constrained Laplacian operator that pre-vents topological changes and self-intersections. Lastly, a con-strained contour simplification is applied to simplify �Cm into acontour ~C, which has fewer line segments and ~C satisfies the threerequirements previously given in the problem definition. Proofsfor the correctness of ~C are also given in this paper. A flow chartof our direct slicing approach is presented in Fig. 4.

The technical contribution of our approach is two-fold:

• For an r-regular solid represented by the implicit indicatorfunction (i.e., � for inside and þ for outside), the contoursgenerated by our approach are topologically faithful, self-intersection free, and with the shape approximation error con-trolled. Therefore, the topology and shape of the final modelfabricated by using these contours are preserved. Rigorousproofs of these good properties are given.

• As a direct slicing approach, it is efficient in computation andmemory-usage since only the information on a particular slic-ing plane is involved, which is different from those techni-ques which first polygonize a given solid into B-reps and thengenerate contours from the B-reps (Ref. [6]).

1Corresponding author.Contributed by the Computers and Information Division of ASME for publication

in the JOURNAL OF COMPUTING AND INFORMATION SCIENCE IN ENGINEERING. Manuscriptreceived July 5, 2011; final manuscript received March 6, 2013; published onlineApril 26, 2013. Editor: Bahram Ravani.

Journal of Computing and Information Science in Engineering JUNE 2013, Vol. 13 / 021009-1Copyright VC 2013 by ASME

Downloaded From: http://computingengineering.asmedigitalcollection.asme.org/ on 03/19/2014 Terms of Use: http://asme.org/terms

Fig. 1 Incorrect contours generated by slicing a given model will produce a model with unwanted gaps (and/or membranes) inrapid prototyping: (a) the given Buddha model in implicit representation (actually Layered Depth-Normal Images (LDNI) [3]), (b)the correct model fabricated from contours generated by our approach, (c) the problematic object fabricated by slicing thelocally self-intersected polygonal model extracted from (a) using a variation [4] of dual-contouring [5], and (d) a zoom-view ofthe incorrect layer. The models are fabricated by FDM.

Fig. 2 Tool path of five consecutive layers generated in InsightTM version 7.0 by slicing a polygonal model extracted from theimplicit Buddha solid given in Fig. 1(a) by [4]. Each layer is in thickness of 0.01 in., and the regions of the Buddha model and thesupporting structures are displayed in green and cyan respectively. Pay attention that an incorrect in/out membership classifi-cation is given on the layer with height 5 1.79 in., which is caused by a local self-intersection on the polygonal model. As aresult, the inside of the Buddha model is filled with supporting structure by mistake.

Fig. 3 A example of tubes in biomedical engineering: (left) theexpected tubes configuration, (right) the manufactured modelwith incorrect topology merging two tubes that should be sepa-rated—this is very dangerous for medical treatments Fig. 4 A flow chart of our direct slicing approach

021009-2 / Vol. 13, JUNE 2013 Transactions of the ASME


Solids represented by several implicit representations, includinglayered depth-normal images (LDNI) [3,4], binary space partition(BSP) [7] and adaptively supported radial basis functions (RBF)[8], are tested in this paper to demonstrate the functionality of ourapproach.

The remainder of the paper is organized as follows. Afterreviewing the related work in Secs. 2 and 3 presents the method togenerate the first topologically faithful contours. The smoothingtechnique presented in Sec. 4 is applied to the contours to furtherimprove their fairness while preventing self-intersections. A varia-tional shape approximation algorithm is introduced in Sec. 5 togenerate topology-preserved and error-bounded contours whichhave fewer line segments from the smoothed ones. The experi-mental results are discussed in Sec. 6 and our paper ends with theconclusion section.

2 Related Work

The basic problem for directly slicing an implicit solid is tocompute the intersection between the solid and a plane. Accordingto a review in Ref. [9], the methods can be classified into two cat-egories: analytical and numerical. Analytical methods find preciseintersection points by solving polynomial equations derivedfrom implicitization [10]. However, these methods can only beapplied to algebraic surfaces and the computing speed is generallyslow. Numerical strategies like subdivision [11] and marching[12,13] do not require precise analytical representation ofsurface boundary. Subdivision methods intersect a tessellatedpiecewise linear approximation of the given implicit surface witha plane, which have the problem that some small intersectionloops will be missed if the subdivision stops at an improperlevel. Marching based methods (e.g. Ref. [14]) always startfrom an initial point, and then proceed to march along a curve,but they suffer from robustness problem at critical points ofcontours (i.e., the point where two loops join into one). Tracingbased contour generation algorithms like Ref. [14] may also misssome small loops. In a follow-up work in Ref. [15], the authorsdivide the MLS surfaces into several slabs with each slab havingthe same topology, then use a tracing strategy to generate contoursfor each slab separately. However, unlike our approach, theirmethod is specialized for moving least-square surfaces (MLS)and they do not provide rigorous proof of the self-intersectionfree property and the topological faithfulness as our approach.Recently, their work is further extended to use point set surface(PSS) to compute spline NC paths for high-speed machining(Ref. [16]).

In the computer graphics community, divide-and-conquer algo-rithms like marching cubes (MC) [17,18] and dual contouring(DC) [5] have been developed to polygonize implicitly definedscalar fields. To generate a smooth surface, real values are definedon grid nodes. In Ref. [19], authors introduced a compact methodfor extracting smooth isosurfaces on grids with only binary valuesstored on nodes, which inspires the constrained smoothing step inour approach (Sec. 4). The theoretical work presented in Ref. [20]provides the criteria of topological equivalence between a 3Dobject surface and the model reconstructed from discrete binarysamples stored on grid nodes. We apply these criteria to ourapproach to govern the steps of binary image sampling and con-tour extraction (see Sec. 3).

The last step of our algorithm needs to simplify a contour into anew one with fewer line segments. This relates to mesh simplifica-tion work in literature, where some approaches collapse mesh ele-ments greedily (e.g., Refs. [21–24]) and others cluster faces intoseveral proxies like planes or quadratic surface patches (e.g.,Refs. [25–33]). Among these methods, the variational shapeapproximation (VSA) method in Ref. [33] has the global distor-tion error minimized. Our contour simplification algorithm fol-lows the strategy of VSA and is modified to ensure that thesimplified contour is intersection-free and homeomorphic to theexact contour.

3 Sampling and Contour Generation

We start to analyze the appropriate sampling rate to guaranteethe extraction of topologically faithful contours by briefly review-ing the relevant definitions and theorems given in Ref. [20].

Definition 1. A solid H � <3 is called r-regular if, for eachpoint p2 @ H, there exist two osculating open balls of radius r to@H at p such that one lies entirely in H and the other lies entirelyout.

Theorem 1. For an r-regular solid H, the boundary surface, M,generated by a topology preserving method on cubic grids is r-homeomorphic to the exact surface boundary, @H, if the cubewidth r0 of grids satisfies

ffiffiffi3p

r0 < r.The above definition and theorem are derived from Definition 1

and Theorem 16 of Ref. [20], which is the foundation of our bi-nary image sampling and topologically faithful contouring. Notethat r0 here is different from r0 used in Ref. [20]. Details about thetopology preserving methods can also be found in Ref. [20].

3.1 Sampling. To generate topologically faithful contours sothat a model homeomorphic to the exact boundary of H can befabricated from them, we sample the 2D solid, H \ P, into a bi-nary image I and then generate a contour �C0 from I. The followingtheorem is derived to guarantee that �C0 is topologically faithful.

Proposition 1. For an r-regular solid H, the contour generatedby a topology preserving method on square grids is topologicallyfaithful if the width r0 of the squares satisfies

ffiffiffi3p

r0 < r.PROOF. Without loss of generality, we can assume that the slic-

ing plane P overlaps the boundary of a layer of 3D grids used inTheorem 1. Moreover, the square grids are the boundary of these3D grids. Therefore, when

ffiffiffi3p

r0 < r is given on the planar grids,the criterion given in Theorem 1 is also satisfied.

Following this proposition, we sample a binary image I from �Hon the slicing plane P with the pixel distance r0. We first rotate thegiven solid H into the coordinate system with x–o–y plane parallelto the slicing plane P. Then, the dimension of the binary imagecan be determined by the intersection between P and the boundingbox of H. The resolution is defined according to the value of r0.Figure 5 shows an example of the binary image sampled from asolid of Dragon model. Note that the sampling of binary image isnot limited to the planes perpendicular to the major axes. As longas the in/out membership tests can be efficiently conducted on thegiven solid H, we can generate the binary image on a slicing planein any orientation. Figure 6 demonstrates the binary images gener-ated by the slicing planes in different orientations.

3.2 Topologically Faithful Contouring. After obtaining abinary image, the marching square method introduced in Ref. [17]is used to generate an approximate contour C0 that is topologicallyfaithful. Defining the edge on a square grid with different in/outstatus on its two endpoints as a stick, the contour C0 can beformed by the line segments linking the middle points of sticks inall grids. Note that, in the rest of this paper, endpoints are notincluded when we refer to a stick (i.e., it is defined on an openinterval). Figure 7 shows the lookup table we used in marchingsquare method to construct contour edges.

Proposition 2. The contour generated by using the lookup tableshown in Fig. 7 is topologically faithful.

PROOF. The proof is straightforward and quite similar to that ofProposition 1. As the binary image I is sampled at a rate accordingto Proposition 1 and the planar square grids are considered as theboundary of 3D grids, the only topologically ambiguous configu-ration shown in the lookup table (i.e., Fig. 7(d)) is derived by thetopology preserving contouring method (e.g., Ball Union in Ref.[20]). Note that, ambiguity of the configuration shown in Fig. 7(d)comes from that the contours in this square could be either 1) twoedges E1 ¼ ðv1; v2Þ and E2 ¼ ðv3; v4Þ or 2) another two edgeE1 ¼ ðv1; v3Þ and E2 ¼ ðv2; v4Þ.

Specifically, the contour reconstructed in this way on a slicingplane P can also be considered as using the plane P to intersect a

Journal of Computing and Information Science in Engineering JUNE 2013, Vol. 13 / 021009-3


discrete surface of @H generated by the Ball Union method pre-sented in Ref. [20]. Its topology in each square is the same aswhat we list in Fig. 7.

3.3 r-Regularity and Accuracy in Rapid Prototyping. Thecontour generated by the above method is ensured to be on a sur-

face homeomorphic to the boundary of an r-regular solid H.Although not all implicit solids are r-regular, this assumption isreasonable to the models to be fabricated by rapid prototyping(RP). The value of r actually relates to the smallest componentthat can be fabricated by an RP machine (e.g., the diameter ofplastic filaments on an FDM machine as well as the finest positionthat can be provided by motion controller). Most commercial x–ypositioning systems used in rapid prototyping can achieve the pre-cision of 10�3 in. The diameter of plastic filaments is usuallygreater than 10�2. Moreover, for an implicit solid represented byLDNI [3,4], the accuracy of the solid is also limited by the resolu-tion of LDNI. It is meaningless to make the grid width of a binaryimage smaller than that of LDNI. Therefore, take the Donnamodel shown in Fig. 8 as an example, we choose r0�3 to generatethe binary images, slightly larger than that of LDNI(3:93� 10�3). Usually, selecting r0 with a value no larger than5� 10�3 is good enough for models fabricated by FDM. For mod-els with very small values of r, our algorithm needs a grid withvery high resolution which could be a problem of computer mem-ory. In this case, solid models need to be processed to r-regularwith a larger r by using the techniques like morphologicaloperators.

4 Constrained Smoothing

Since line segments on the contour �C0 are generated by linkingthe middle point of sticks, there is no self-intersection on �C0.However, the shape of �C0 is not smooth (e.g., the contour shownin Fig. 10(a)); as such, we apply a Laplacian operator based

Fig. 5 A binary image sampled from a Dragon solid: (a) the solid H is intersectedby a slicing plane P , (b) the region of �H 5 H \ P on the binary image, and (c) azoom-view of the binary image, where the nodes inside �H are displayed in soliddots while outside nodes are shown in hollow

Fig. 6 Binary image generation of slicing in different orienta-tions: (left) with slicing direction ð0; 1; 0Þ and (right) slicingalong ð1; 1; 1Þ. The bottom row shows the binary images gener-ated on example slicing planes.

Fig. 7 Lookup table for the marching square method with topology preserved. Sticks are inyellow. Sampling nodes inside the solid H are shown in black while the outside nodes aredisplayed in white. The contour edges linking sticks are labeled as E .



smoothing technique to improve it. The advantage of Laplaciansmoothing is its efficiency and stability, but the major drawback isthat the unwanted shrinkage always occurs when it is iterativelyapplied to a closed shape (in 2D or 3D). A constrained Laplaciansmoothing is developed here, which intrinsically solves theshrinkage problem since we guarantee to generate topologicallyfaithful contours. In other words, the smoothed contours cannotviolate the in/out status of any sampling node on the binary imageI. To ensure that, a good strategy is to constrain each vertex vi onthe contour so that it must stay on the stick on which it initiallylies. In addition, by applying this ‘sliding-on-stick’ strategy, wecan guarantee that the resultant contours are self-intersection free.

Proposition 3. When moving the vertices on a contour, no intersec-tion occurs if the vertices are only moved on the sticks holding them.

PROOF. For a vertex vi that is generated from the stick ti, slidingit on ti can only bring itself to a new position belonging to thepoint set fpjp ¼ aps

i þ ð1� aÞpei Þ; a2 ð0; 1Þg, where ps

i and pei

are the two endpoints of ti, respectively. For any contour edge E,if we allow its two vertices vi and vj to slide only on their respec-tive sticks ti and tj, E can sweep out a range U, which containsany of its possible occurrence position.

For the contour edges with vi and vj located on two adjacentsticks ti and tj, the region U is a triangular region formed by theendpoints of ti and tj excluding the boundaries not overlappedwith ti or tj (e.g., p1p3 and p4 in Fig. 9(a)). For the contour edgeswith vi and vj on two opposite sticks (e.g., Fig. 9(b)), U is a squareexcluding the boundaries not overlapped with ti or tj (e.g., p1p2

and p3p4 in Fig. 9(b)). Since the sweeping envelopes of the edgeson the contour do not have any overlap with each other, the propo-sition is thus proved.

Proposition 4. Deforming a topologically faithful contour bymoving its vertices only on the sticks holding them will generate acontour which is still topologically faithful.

PROOF. The proof of this proposition is straightforward. First,deforming a contour in this way will not change the “in”–“out”

status of any samples on the binary image I since the contour isnot moved across any of the samples. Second, self-intersectionswill not be generated during such a kind of deformation (see Prop-osition 3).

The constrained smoothing technique introduced in Ref. [19]ensures every vertex sliding on its stick by projecting the displacedvertex back to the nearest point of its stick. However, some vertexmay eventually have orthogonal displacement to its stick and thecontour is stuck in an sub-optimal shape (see Fig. 10(e)). Weobserved that for each contour vertex vi, its two adjacent verticesvi�1 and viþ1 have only five configurations for their position on dif-ferent sticks with considering rotational symmetry (see Fig. 11).For all these configurations, the line segment connecting vi�1 andviþ1 intersect with either the stick holding vi (configurations inFigs. 11(b), 11(c), 11(d), and 11(e)) or an ending point of this stick(configuration in Fig. 11(a)). Based on this observation, we investi-gate a sliding-based constrained smoothing without using projec-tion. On each vertex vi, it is performed in two steps.

(1) Calculating the intersection point vint between the line seg-ment connecting vi�1 and viþ1 and the stick griping vi withits two ending points counted.

(2) Moving vi in the ratio of s towards vint

vnewi ¼ vi þ sðvint � viÞ

where s ¼ 0:4 is selected to balance the efficiency and the stabil-ity of computation.

The contour �C0 can be smoothed into �Cm by repeatedly applyingthese two steps to all vertices until the average movement of verti-ces in an iteration is less than 10�3 of r0. Figures 10(b), 10(c), and10(d) give a comparison of the smoothing results between sliding-based, projection-based constrained smoothing and ordinary Lapla-cian smoothing. Besides, it is obvious that �Cm is homeomorphic to�C0 since no intersection occurs during this contour evolution.

Fig. 8 An example model fabricated from the contours generated by our method with r 0�3: (a)the given Donna model in the LDNI representation, (b) the sliced contours of respective layersat 2:30, 3:00, and 3:66 in. heights, and (c) the resultant model fabricated by FDM

Fig. 9 The sweeping envelopes for contour edges in grids with different configurations. The shadow regionsrepresent the sweeping envelops. Note that some vertices and edges are excluded from the sweeping envelops.These are p1, p3, p4, edge(p3,p4) for configuration 1; p1, p2, p3, p4, edge(p1,p2), edge(p3,p4) for configuration 2;p1, p3, p4, edge(p1, p3) for configuration 3; and p1, p2, p3, p4, edge(p2, p4) for configuration.



5 Contour Simplification

The smoothed contour, �Cm, usually provides a very good shapeapproximation of the exact contour generated by @H \ P. How-ever, to ensure the topological faithfulness, a relative small valueof r0 may be selected for a model with large dimensions. Thisleads to contours with a lot of very short line segments, which sig-nificantly increase the memory cost. The situation becomes moreserious if the software controlling the RP machine does not run inan out-of-core manner (i.e., loading the contours for all layers

from the contour file at the same time). Moreover, using too manysmall line segments to represent the contours will dramaticallydecrease the efficiency of subsequent processing steps in RP likegeneration of supporting structure and tool-path planning. Basedon our observation, in the smoothed contours, there are alwaysseveral successive edges lying almost in the same straight line,which implies these edges can be simplified into one single edgewith little distortion error introduced. Therefore, a contour simpli-fication algorithm preserving topology and shape approximationerror is investigated in this section to further improve the topolog-ically faithful contours for slicing implicit solids.

5.1 Variational Segmentation. The variational shapeapproximation approach [33] imitates the well-known Lloyd’salgorithm [34] to cluster mesh entities into several regions, andfor each region, it uses a planar proxy to approximate the wholeregion. The Lloyd’s algorithm-based relaxation procedure isemployed to minimize the global shape distortion error. We adopttheir basic idea and develop our constrained simplification algo-rithm for 2D contours.

For any contour, at the very beginning, we randomly select nedges fEig as seeds which will be used to start growing a cluster.The number of clusters, n, can be selected to be proportional tothe total number of edges on this contour (1

a with a positive integera as clustering ratio). Here, each proxy is a line defined by a pointxi on the line and a normal vector ni perpendicular to the directionof the line. The proxies, Qi, are initialized by the seed edges.

We need to grow the proxies on the contour simultaneously andbuild n clusters minimizing the shape approximation error. Foreach seed edge Ei, we insert its two adjacent edges, Ej and Ek,into a minimal queue, !. The queue is keyed by the distortionerror presented on the edges according to a particular proxy (e.g.,the edges Ej and Ek inserted according to the proxy Qi have theweights, DðEj;QiÞ and DðEk;QiÞ, in !). Note that it is possible tohave an edge inserted in the queue more than once (i.e., by differ-ent proxies adjacent to the edge). Here, the distortion error ismeasured by L2 norm. Generally, the L2 error metric for anyregion R and its proxy Q is defined as

L2ðR;QÞ ¼ð ð

x2R

k x�PQðxÞ k2 dx (1)

where PQðxÞ means the orthogonal projection of the argument onthe proxy Q. The L2 distortion error DðE;QÞ between an edge Eand the linear proxy Q can be evaluated by

DðE;QÞ ¼ 1

3ðd2

0 þ d21 þ d0d1Þ k E k (2)

Fig. 10 A comparison among different smoothing strategieson the contour generated for the binary image region shown inFig. 5(c): (a) the contour reconstructed by the topology preserv-ing marching square method, (b) the shrinking contour after or-dinary Laplacian smoothing, (c) the resultant contour afterprojection-based constrained smoothing, (d) the resultant con-tour after sliding-based constrained smoothing, and (e) thezoom-in view of contour vertices stuck in sub-optimal shape

Fig. 11 All the five configurations for the position of vi�1 and viþ1 on different sticks with con-sidering rotational symmetry



where d0 and d1 are the orthogonal distance from two endpointsof E to the line defined on Q and k E k represents the length ofedge E.

The growing of clusters is performed by repeatedly removingthe edge from the top of ! (i.e., the edge with the smallest distor-tion error). For each edge Et removed from !, we check if it hasbeen assigned to a proxy. If not, we assign it to the proxy Qp

which is used to evaluate its distortion error, DðEt;QpÞ, in !. Oth-erwise, no operation is given according to Et. After the edge Et isassigned to a proxy Qp, the two edges El and Er adjacent to Et areinserted into ! according to the weights, DðEl;QpÞ and DðEr;QpÞ,if they have not been assigned to any proxy. The removing andinserting operations will not stop until ! becomes empty, i.e.,when every edge has been assigned to a proxy. This relaxationbased clustering process always provides connected and nonover-lapped segmentations on a contour.

After obtaining an n-clustering result from a set of seed edges,we need to update each proxy Qi in order to minimize the distor-tion error between Qi and its corresponding region Ri (the cluster).We update the linear proxy Qi by recomputing its normal direc-tion ni and the site point xi where the line passes through. xi issimply the barycenter of its corresponding region Ri, which isgiven by

xi ¼

XE2Ri

k E k ðvs þ veÞ

2XE2Ri

k E k(3)

where vs and ve are the two endpoints of an edge E. ni could bedetermined by computing the eigenvector corresponding to thesmallest eigenvalue of the covariance matrix of Ri. The covari-ance matrix Mi of Ri can be calculated by

Mi ¼XE2Ri

k E k ðACAT þ vsvTs þ vsv

Tb þ vbvT

s Þ � xixTi

XE2Ri

k E k

(4)

where

A ¼ ve � vs 0½ �;C ¼1

30

0 0

" #and vb ¼

1

2ðve � vsÞ

After updating all n proxies, we search a new seed edge Ei in eachregion Ri which has the smallest distortion error according to thenew proxy. Then, a new segmentation is computed starting fromthese new seed edges.

After applying this growing-updating process for several itera-tions (e.g., 20 iterations always can provide satisfactory results),the whole contour has been successfully segmented into n regionsand the overall distortion error has been minimized (see Fig. 12(a)for an example).

5.2 Topology and Distortion Verification. After performingthe segmentation of a smoothed contour �Cm, the simplest way togenerate a simplified contour is to replace the edges on �Cm by nedges where each links the starting and the ending points of aregion Ri. However, simplifying contours in this way will makesome sample points on the binary image I which are originally‘inside’ the region, H \ P, become ‘outside,’ or vice versa—i.e.,topological faithfulness is not preserved. Moreover, as shown inFig. 13, intersections and degenerate contours can be generated onthe contours which are intersection-free before the simplification.Another important issue of concern is the bound of distortion errorintroduced by simplification. As shown below, we investigate anovel verification procedure to solve both the topology faithful-ness and distortion error bound problems together. We notice thatevery vertex is guaranteed to be still on its corresponding stick

after smoothing. In other words, the variational clustering actuallystarts from a topologically faithful and intersection-free contour.The verification procedure is based on this assumption of the inputcontour.

Fig. 12 An illustration of contour simplification for thesmoothed contour shown in Fig. 10(d): (a) the variational clus-tering result on the contour with different line type representingdifferent regions, and (b) the final simplified contour after topol-ogy and distortion verification

Fig. 13 A contour that is originally intersection-free couldbecome intersected or degenerate by replacing the curvedregion with line segments



Proposition 5. For a region on the contour defined by asequence of connected vertices fv1; v2; � � � ; vng, it can be simpli-fied into a single edge �E connecting v1 and vn by sliding itsðn� 2Þ internal vertices fv2; v3; � � � ; vn�1g on their respectivesticks ft2; t3; � � � ; tn�1g if and only if the resultant edge �E inter-sects all of the sticks: ft2; t3; � � � ; tn�1g.

PROOF. See Fig. 14 for an illustration.

(1) Sufficiency: Suppose the resultant edge �E intersects all of thesticks ft2; t3; � � � ; tn�1g; the intersection point v0k between �Eand tk is on both �E and tk. Since both vk and v0k is on tk, it isobvious that vk can move to v0k just by sliding on tk. Hence,the whole contour region can be simplified into �E by slidingthe n� 2 internal vertices fv2; v3; � � � ; vn�1g on their respec-tive sticks ft2; t3; � � � ; tn�1g.

(2) Necessity: Suppose that the contour region can be simpli-fied into a single edge �E by sliding the n� 2 internal verti-ces fv2; v3; � � � ; vn�1g on ft2; t3; � � � ; tn�1g; each vk on tk

(k2 ½2; n� 1�) can have a corresponding point v0k projectedon the simplified segment �E by sliding vk on tk. v0k is onboth tk and �E. Thus, tk and �E intersect each other. h

This proposition is thus proved.

Deforming an intersection-free and topologically faithful con-tour by sliding the vertices on the sticks holding them will notchange the properties of topological faithfulness and intersection-free (see Propositions 3 and 4). Because of this, we develop theverification procedure below.

For each segmented region Ri, we first estimate its simplifiededge �Ei by connecting its two ending vertices. We verify whether�Ei can be obtained from sliding the internal vertices on their corre-sponding sticks by testing if �Ei intersects all these sticks (seeProposition 5 for the correctness of such a test). If �Ei intersects allthe tested sticks, we go to the distortion error test. The distortionerror between Ri and �Ei can be evaluated by

D0ðRi; �EiÞ ¼XE2Ri

DðE; Lð �EiÞÞ (5)

where Lð �EiÞ gives the line equation of �Ei. The simplification on Ri

can pass this test only if D0ðRi; �EiÞ <¼ �, where � is a user speci-fied distortion tolerance. If either of these two tests fails, theregion Ri is further separated into two regions which are deter-mined by a local variational clustering conducted only in Ri. Thistrial-and-error procedure is recursively performed until all the seg-mented regions on the contour satisfy both of the two verifica-tions. Then, each segmented region is converted into a simplifiededge of the contour. Note that this trial-and-error procedure isguaranteed to stop since in the worst case, each region is a singleedge on the smoothed contour �Cm. Still, the topologically faithfulcondition is satisfied and the distortion error for every such regionis simply zero. The effectiveness of our verification technique isdemonstrated in Fig. 12(b).

6 Results and Discussion

We have implemented the proposed approach in a Cþþ pro-gram. The examples shown in this paper are all tested on a PCwith Intel Core 2 Quad CPU Q6600 2.4 GHz.

The two engineering models, shown in Figs. 15 and 16, give acomparison between prior slicing algorithm (in commercial soft-ware) and our approach. Due to the self-intersection in the tessel-lated triangular meshes from implicit solids, a prior slicingalgorithm may produce incorrect contours, and consequently thetool path of part material will also have defects (see Figs. 15(b)and 16(b)). Unwanted films will be produced for both of the twoexamples as a result. In addition, the filigree and truss models

Fig. 14 Sliding the six vertices on their respective sticks inorder to form a single edge connecting the starting and endingvertices of this region

Fig. 15 An example of slicing the Filigree model: (a) a mesh tessellated from implicit Filigree model, (b) the contours and theircorresponding tool path generated by InsightTM version 7.0 on the layer with 0.77 in. height, (c) the contours generated by ourapproach for the same layer and their corresponding tool path, (d) the rendered slicing contours generated by our approach



demonstrate that our approach can easily handle the solids withcomplex topology.

In order to verify the effectiveness of our topology verificationtechnique, we analyzed the resultant contours for Buddha andTruss models which are shown in Figs. 17 and 18. For the layer at2.90 in. height of Buddha, self-intersection will be removed if weuse the topology verification (see Figs. 17(c) and 17(d)). Mean-while, the topology verification technique can successfully pre-vent degenerate contours (see Figs. 18(c) and 18(d)) caused bynarrow intersection regions.

Three biomedical models, Donna, Hand-complex and Spine areshown in Figs. 8, 21, and 22, respectively. From the rendered slic-ing contours shown in Figs. 21(b) and 22(b) and the fabricatedmodel by FDM in Fig. 8(c), we can see that the resultant contoursprovide very good shape approximation of the original solids. For

Hand-complex and Spine, we conduct a study on the effect ofclustering ratio a (see Sec. 5) to the performance of our approach.As can be seen from Figs. 21(c) and 22(c), the maximum regionaldistortion error is generally increasing as a increases. However,the rate of error increasing becomes slow as a becomes largerbecause the approach has to perform more topology anddistortion-error verification operations under this situation. Wealso compare the number of resultant contour edges and the timeconsumption for different values of a. The resultant contour edgenumber first decreases as expected when a < 10, while itincreases slowly after a � 10. This is because the contours requireat least a certain number of edges to guarantee topologically faith-ful and distortion-error bounded properties. Once the variationalsegmentation does not provide enough regions to satisfy these twoproperties, more regions will be generated through subdividing

Fig. 16 An example of slicing the Truss model: (a) a mesh tessellated from implicit Truss model, (b) the contours and their cor-responding tool path generated by InsightTM version 7.0 on the layer with 2.44 in. height, (c) the contours generated by ourapproach for the same layer and their corresponding tool path, (d) the rendered slicing contours generated by our approach

Fig. 17 A comparison between resultant contours with and without topology verification: (a)the given Buddha model in the LDNI representation, (b) a binary image sampled from the layerwith 2.90 in. height, (c) the resultant contour without topology verification, (d) the resultant con-tour with topology verification



regions locally in verification stage. Since the subdivided regionscannot freely move to optimally fit the contour shape as what theycan do in the variational segmentation stage, the more subdivisionare applied, the more contour edges tend to be generated on theresults. Time consumption increases quickly as a becomes toolarge because we need to perform more verification operations,which repeatedly check whether two line segments intersect eachother to detect violation of topologically faithful property. Basedon our experimental tests, selecting a between 10 and 15 shows agood trade-off. All testing results presented in this paper are gen-erated by using a ¼ 10.

Our direct slicing approach is general for any implicit represen-tations. Figures 19 and 20 give demonstration of our approach on

BSP and RBF solids, respectively. In our prototype implementa-tion of slicing RBF solids, we use the adaptively supported RBFgenerator which is available on Ohtake’s software website2. Sta-tistics of experimental tests are shown in Tables 1–3 for LDNI,BSP, and RBF representations, respectively. For all the tests onLDNI, the grid width r0 of binary images is set to be no less thanthat of the input LDNI solid. Hence, the resolution of LDNIshould be large enough to make sure the value of r0 is in the orderof 10�3. The column, maxðD0ðRi; �EiÞÞ, reports the maximum re-gional distortion error defined by Eq. (5) on simplified contours.

Fig. 18 A comparison between resultant contours with and without topology veri-fication: (a) the given Truss model in the LDNI representation, (b) a binary imagesampled from the layer with 0.30 in. height, (c) the resultant contour without topol-ogy verification, and (d) the resultant contour with topology verification

Fig. 19 An example of slicing the Hand-complex model: (a) the original Hand-complex model, (b) the rendered slicing contours, (c) the chart of maximum re-gional distortion error versus the clustering ratio a, (d) the chart of simplified con-tour edge number versus a, and (e) the chart of time consumption versus a

2http://www.den.rcast.u-tokyo.ac.jp/yu-ohtake



http://www.den.rcast.u-tokyo.ac.jp/ yu-ohtake

http://www.den.rcast.u-tokyo.ac.jp/ yu-ohtake

Fig. 20 An example of slicing the Spine model: (a) the original Spine model, (b) the renderedslicing contours, (c) the chart of maximum regional distortion error versus the clustering ratioa, (d) the chart of simplified contour edge number versus a, and (e) the chart of time consump-tion versus a

Fig. 21 A demonstration of our approach on BSP solid: (a) the given Rocker-arm model in BSP representation and the result-ant rendered contours, (b) the binary image and corresponding contours for the layer with 1.66 in. height, and (c) the binaryimage and corresponding contours for the layer with 0.85 in. height

Fig. 22 A demonstration of our approach on RBF solid: (a) the given Armadillo model in adaptively supported RBF representa-tion and the resultant rendered contours, (b) the binary image and corresponding contours for the layer with 2.90 in. height, and(c) the binary image and corresponding contours for the layer with 1.30 in. height



In our tests, the tolerance of regional distortion error is actuallyset as square of r0. Therefore, it is easy to find that our constrainedshape simplification can successfully bound the distortion errorwith respect to the tolerance. We use the thickness of 0:01 in. forall the slicing tests, and the number of slicing layers are listed inthe last column of the tables.

7 Conclusion

In this paper, we present a direct slicing approach for implicitsolids. We investigate two main techniques, constrained Laplaciansmoothing and contour simplification, which can produce self-intersection free and topologically faithful contours. In addition,we provide the proofs for the correctness of our approach. Theapproach presented in this paper also allows good distortion errorcontrol on the generated contours and has been shown to be veryefficient.

Even though uniform binary image works well, the wholeapproach could be more efficient if we can make it in an adaptiveresolution. We consider this as our near future work. The chal-lenge is to retain the self-intersection free and topologically faith-ful properties on the contours after moving the computation intoan adaptive sampling strategy. Meanwhile, the distortion errorintroduced between layers will also be considered and modeled to

further improve the quality of models fabricated from the contoursgenerated by our approach.

Acknowledgment

The research conducted in this paper is supported by HongKong RGC/GRF Grant Nos. (CUHK/417109 and CUHK/417508). The third author is supported by the National ScienceFoundation Grant No. CMMI-0927397. The authors would like tothank Yuen-Shan Leung and Allan Mok for generating the LDNIsolid and the FDM object shown at the beginning of this paper,and Ms. Siu Ping Mok for proof reading the manuscript.

References[1] Ju, T., and Udeshi, T., 2006, “Intersection-Free Contouring on an Octree Grid,”

Proceedings of the 14th Pacific Conference on Computer Graphics andApplications.

[2] Varadhan, R., Krishnan, S., Zhang, L., and Manocha, D., 2006, “ReliableImplicit Surface Polygonization Using Visibility Mapping,” Proceedings of theSymposium on Geometry Processing.

[3] Chen, Y., and Wang, C. C. L., 2008, “Layered Depth-Normal Images for Com-plex Geometries—Part One: Accurate Sampling and Adaptive Modeling,” Pro-ceedings of the ASME IDETC/CIE Conference, 28th Computers andInformation in Engineering Conference, New York, Paper No. DETC2008-49432.

Table 1 Statistics of experimental tests on LDNI

ExamplesModel size(L�W� H)

LDNIresolution

LDNIgrid width r0

Timea

(in min) maxðD0ðRi; �EiÞÞ

Contour edges no.before versus

after simplificationLayer

no.

Dragon 0:90� 2:00 � 1:41 1500 1.44� 10�3 1.50� 10�3 2.10 (0.48) 3.46� 10�9 458 k/79 k 414Spine 10:30� 1:72 � 2:18 3000 3.77� 10�3 4.00� 10�3 7.68 (2.61) 1.49� 10�7 1305 k/267 k 218Hand 4:65� 6:63 � 2:27 3500 2.08� 10�3 2.50� 10�3 8.53 (3.01) 1.66� 10�8 1338 k/255 k 227Donna 7:14� 6:10 � 4:41 2000 3.93� 10�3 4.00� 10�3 12.55 (3.10) 6.02� 10�8 2468 k/368 k 441Truss 2:40� 5:82 � 2:54 1300 4.93� 10�3 5.00� 10�3 20.18 (0.88) 2.32� 10�7 3973 k/1027 k 254Filigree 8:00� 8:00 � 1:21 4000 2.20� 10�3 2.50� 10�3 43.13 (2.78) 3.96� 10�8 3110 k/545 k 121

aThe reported time includes binary image sampling (parallelized using multi-thread processing library–OpenMP), contour reconstruction, constrainedLaplacian smoothing and contour simplification. The value in brackets represents time for binary image sampling.

Table 2 Statistics of experimental tests on BSP

ExamplesModel size

(L�W � H)BSP

tree sizeBSP tree

complexitya r0Timeb



after simplification Layer no.

Oct-flower 3:50� 3:50 � 2:49 151,309 21 2.00� 10�3 4.89 (2.77) 3.13� 10�9 834 k/90 k 249Rocker-arm 5:00� 1:52 � 2:57 156,813 22 2.00� 10�3 5.25 (1.95) 3.13� 10�9 1359 k/166 k 257B-Torus 5:05� 5:05 � 3:86 225,319 23 2.50� 10�3 18.41 (5.93) 1.40� 10�8 3823 k/428 k 386Gear 6:35� 6:35 � 3:90 500,99 23 3.00� 10�3 20.8 (6.26) 2.49� 10�8 3897 k/420 k 390

aThe complexity of BSP tree is defined as the average depth of its leaf nodes.bThe reported time includes binary image sampling (parallelized using multi-thread processing library–OpenMP), contour reconstruction, constrainedLaplacian smoothing and contour simplification. The value in brackets represents time for binary image sampling.

Table 3 Statistics of experimental tests on adaptively supported RBF

ExamplesModel size

(L�W � H)Points no. forsurface fitting

Basic functionsnumber r0

Timea



after simplificationLayer

no.

Armadillo 3:17� 2:88 � 3:78 80,000 72,787 3.00� 10�3 9.16 (4.73) 3.01� 10�8 1548 k/348 k 387S-chair 3:50� 6:32 � 3:35 16,113 14,010 2.00� 10�3 14.98 (11.45) 1.01� 10�8 1366 k/197 k 335Horse 5:02� 2:30 � 4:17 19,851 9797 2.00� 10�3 26.33 (20.61) 1.27� 10�8 2094 k/294 k 417Bunny 6:23� 4:83 � 6:17 34,834 21,068 3.00� 10�3 29.85 (18.68) 3.59� 10�8 3528 k/588 k 617

aThe reported time includes binary image sampling (parallelized using multi-thread processing library–OpenMP), contour reconstruction, constrainedLaplacian smoothing and contour simplification. The value in brackets represents time for binary image sampling.



[4] Wang, C. C. L., and Chen, Y., 2008, “Layered Depth-Normal Images for Com-plex Geometries C Part Two: Manifold-Preserved Adaptive Contouring,” Pro-ceedings of the ASME International Design Engineering TechnicalConferences and Computers and Information in Engineering Conferences, NewYork, Aug. 3–6, Paper No. DETC2008-49576.

[5] Ju, T., Losasso, F., Schaefer, S., and Warren, J., 2002, “Dual Contouring ofHermite Data,” ACM Trans. Graphics, 21(3), pp. 339–346.

[6] Chua, C. K., Leong, K. F., and Lim, C. S., 2003, Rapid Prototyping: Principlesand Applications, World Scientific, Singapore.

[7] Fuchs, H., Kedem, Z. M., and Nalor, B. F., 1980, “On Visible Surface Genera-tion by a Priori Tree Structures,” Proceedings of the ACM SIGGRAPH 1980,pp. 124–133.

[8] Ohtake, Y., Belyaev, A., and Seidel, H. P., 2004, “3D Scattered Data Approxi-mation With Adaptive Compactly Supported Radial Basis Functions,” Proceed-ings of the Shape Modeling International 2004, pp. 31–39.

[9] Luo, R. C., and Ma, Y., 1995, “A Slicing Algorithm for Rapid Prototyping andManufacturing,” Proceedings of the IEEE International Conference on Roboticsand Automation, 1995, pp. 2841–2846.

[10] Farouki, R. T., 1986, “The Characterization of Parametric Surface Sections,”Comput. Vis. Graph. Image Process., 33, pp. 209–236.

[11] Lee, R. B., and Fredericks, D. A., 1981, “Intersection of Parametric Surfacesand a Plane,” IEEE Comput. Graph. Appl., 4(8), pp. 112–117.

[12] Barnhill, R. E., and Kersey, S. N., 1990, “A Marching Method for Parametric Sur-face/Surface Intersection,” Comput. Aided Geom. Des., 7, pp. 257–280.

[13] Barnhill, R. E., Farin, G. E., Jordan, M., and Piper, B. R., 1987, “Surface/Sur-face Intersection,” Comput. Aided Geom. Des., 4, pp. 3–16.

[14] Yang, P., and Qian, X., 2008, “Adaptive Slicing of Moving Least SquaresSurfaces: Toward Direct Manufacturing of Point Set Surfaces,” ASME J. Com-put. Inf. Sci. Eng., 8(3), p. 031003.

[15] Qiu, Y., Zhou, X., and Qian, X., 2010, “Direct Slicing of Cloud Data WithGuaranteed Topology for Rapid Prototyping,” Int. J. Adv. Manuf. Technol.,(accepted).

[16] Liu, Y., Xia, S., and Qian, X., 2012, “Direct Numerical Control (NC) Path Gen-eration: From Discrete Points to Continuous Spline Paths,” ASME J. Comput.Inf. Sci. Eng., 12(3), p. 031002.

[17] Lorensen, W. E., and Cline, H. E., 1987, “Marching Cubes: A High Resolution3D Surface Construction Algorithm,” Comput. Graph., 21(4), pp. 163–169.

[18] Kobbelt, L. P., Botsch, M., Schwanecke, U., and Seidel, H., 2001, “FeatureSensitive Surface Extraction From Volume Data,” Proceedings of the ACMSIGGRAPH 2001, pp. 57–66.

[19] Chica, A., Willliams, J., Andujar, C., Brunet, P., Navazo, I., Rossignac, J., andVinacua, A., 2008, “Pressing: Smooth Isosurfaces With Flats From BinaryGrids,” Comput. Graph. Forum, 27(1), pp. 36–46.

[20] Stelldinger, P., Latecki L. J., and Siqueira, M., 2007, “Topological EquivalenceBetween a 3D Object and the Reconstruction of Its Digital Image,” IEEE Trans.Pattern Anal. Mach. Intell., 29(1), pp. 126–140.

[21] Hoppe, H., 1996, “Progressive Meshes,” Proceedings of the ACM SIGGRAPH1996, pp. 99–108.

[22] Klein, R., Liebich, G., and Strasser, W., 1996, “Mesh Reduction With ErrorControl,” Proceedings of the IEEE Visualization, pp. 311–318.

[23] Garland, M., and Heckbert, P. S., 1998, “Simplifying Surfaces With Color andTexture Using Quadric Error Metrics,” Proceedings of the IEEE Visualization,pp. 263–269.

[24] Lindstrom, P., and Turk, G., 1998, “Fast and Memory Efficient Polygonal Sim-plification,” Proceedings of the IEEE Visualization, pp. 279–286.

[25] Maillot, J., Yahia, H., and Verroust, A., 1993, “Interactive Texture Mapping,”Proceedings of the ACM SIGGRAPH 1993, pp. 27–34.

[26] Kalvin, A. D., and Taylor, R. H., 1996, “Superfaces: Polygonal MeshSimplification With Bounded Error,” IEEE Comput. Graph. Appl., 16(3), pp.64–77.

[27] Inoue, K., Itoh, T., Yamada, A., Furuhata, T., and Shimada, K., 1999,“Clustering a Large Number of Faces for 2-Dimensional Mesh Generation,”Proceedings of the Eighth International Meshing Roundtable, pp. 281–292.

[28] Sheffer, A., 2000, “Model Simplification for Meshing Using Face Clustering,”Comput.-Aided Des., 33, pp. 925–934.

[29] Sander, P., Snyder, J., Gortler, S., and Hoppe, H., 2001, “Texture Mapping Pro-gressive Meshes,” Proceedings of the ACM SIGGRAPH 2001, pp. 409–416.

[30] Garland, M., Willmott, A., and Heckbert, P. S., 2001, “Hierarchical Face Clus-tering on Polygonal Surfaces,” Proceedings of the 2001 Symposium on Interac-tive 3D Graphics.

[31] Grinspun, E., and Schroder, P., 2001, “Normal Bounds for Subdivision-SurfaceInterference Detection,” Proceedings of the IEEE Scientific Visualization, pp.333–340.

[32] Levy, B., Petitjean, S., Ray, N., and Maillot, J., 2002, “Least Squares Confor-mal Maps for Automatic Texture Atlas Generation,” Proceedings of the ACMSIGGRAPH 2002, pp. 362–371.

[33] Cohen-Steiner, D., Alliez, P., and Desbrun, M., 2004, “Variational ShapeApproximation,” Proceedings of the ACM SIGGRAPH 2004, pp. 905–914.

[34] Lloyd, A. P., 1982, “Least Square. Quantization in PCM,” IEEE Trans. Inf.Theory, 28(2), pp. 129–137.



http://dx.doi.org/10.1145/566654.566586

http://dx.doi.org/10.1016/0734-189X(86)90115-5

http://dx.doi.org/10.1016/0167-8396(90)90035-P

http://dx.doi.org/10.1016/0167-8396(87)90020-3

http://dx.doi.org/10.1115/1.2955481

http://dx.doi.org/10.1115/1.2955481

http://dx.doi.org/10.1115/1.4006463

http://dx.doi.org/10.1115/1.4006463

http://dx.doi.org/10.1145/37402.37422

http://dx.doi.org/10.1111/j.1467-8659.2007.01039.x

http://dx.doi.org/10.1109/TPAMI.2007.250604

http://dx.doi.org/10.1109/TPAMI.2007.250604

http://dx.doi.org/10.1109/38.491187

http://dx.doi.org/10.1016/S0010-4485(00)00116-0

http://dx.doi.org/10.1109/TIT.1982.1056489

http://dx.doi.org/10.1109/TIT.1982.1056489

Pu Huang Intersection-Free and Topologically Faithful...

Documents

Transcript of Pu Huang Intersection-Free and Topologically Faithful...