muhammad hussain - semantic scholar...e-mail: [email protected] or ... bc de db ce ddb cee e...

Muhammad Hussain محمد حسين

From: "Chen Yen-Wei" <[email protected]>To: <[email protected]>Cc: <[email protected]>Sent: Tuesday, January 07, 2003 7:08 PMSubject: Acceptance letter (IJIG-013)

Page 1 of 1

IJIG-013 Fast, Simple, Feature Preserving and Memory Effidcient Sinmplification of Triangle Meshes Authors: M.Hussain et al. Dear Authors, We are pleased to inform you that your above paper has been accepted for publication in International Journal of Image and Graphics (IJIG). Please prepare your final paper according to following process: 1. Please prepare your final manuscript in IJIG format. You can get the format from IJIG website: http://www.worldscinet.com/ijig/ijig.shtml. 2. A biographical sketch and photograph of each author should be included in the manuscript. 3. Please sign and fill in the copyright form, which can download from IJIG website also. 4. Please send me two hardcopies and one softcopy (PDF format) for each paper, plus the signed copyright form by January 25, 2003. If you have any questions, I look forward to hearing from you. Thank you fro your contribution to our special issue. Best regards Yen-Wei CHEN Guest Editor of IJIG Spec. Iss. Faculty of Eng., Univ. of the Ryukyus, Okinawa 903-0213, JAPAN Phone: +81-98-895-8703; FAX:+81-98-895-8708 E-mail: [email protected] or [email protected] __________________________________________________ Do You Yahoo!? Yahoo! BB is Broadband by Yahoo! http://bb.yahoo.co.jp/

September 23, 2003 18:22 WSPC/164-IJIG 00124

International Journal of Image and GraphicsVol. 3, No. 4 (2003) 653–670c© World Scientific Publishing Company

FAST, SIMPLE, FEATURE PRESERVING AND MEMORYEFFICIENT SIMPLIFICATION OF TRIANGLE MESHES

MUHAMMAD HUSSAIN∗, YOSHIHIRO OKADA† and KOICHI NIIJIMA‡

Graduate School of Information Science and Electrical Engineering,Kyushu University, 6-1, Kasuga Koen, Kasuga, Fukuoka 816-8580, Japan

∗[email protected]†[email protected]‡[email protected]

Received 1 July 2002Revised 1 December 2002Accepted 7 January 2003

We propose a new iterative algorithm for the automatic geometric simplification of atriangle mesh based on edge collapse transformations. The way how geometric deviationresulted from an edge collapse transformation is measured plays a basic role in inducinga priority ordering on the set of edge collapse transformations to control the simpli-fication process. We introduce a new technique to measure this deviation based onlyon current simplified mesh and the new concept of accumulating the cost of collapse,which are easy to implement and involve simple computations making the algorithmcomputationally efficient. No geometric history is needed to be retained, so the resultingalgorithm is also memory efficient. Results and numerical comparisons show that ouralgorithm generates simplified meshes of good visual fidelity, which compares well withthose by other methods in terms of maximum and mean geometric error and it preservesthe visually important features of the original mesh.

Keywords: Polygonal surface reduction; level of details; edge collapse.

1. Introduction

Automatic simplification of large triangle meshes is an important problem in com-puter graphics. Triangle meshes are commonly used to represent 3D surfaces andthey serve as the de facto standard for fast interactive visualization. Latest ad-vances in CAD systems and scanning devices have given rise to very complex andhighly detailed triangular meshes; surface reconstruction and isosurface extractionmethods also result in very densely sampled meshes; meshes consisting of millionsof triangular faces are commonplace. Such meshes are usually huge and mostlyuniformly sampled; the density of the model is not adapted to the detail actuallyneeded to represent the local geometry. The growing size and complexity of triangu-lar meshes have surpassed the development in rendering systems and transmissioncapabilities, so it is hard to navigate and render such meshes at interactive frame

653


654 M. Hussain, Y. Okada & K. Niijima

rates due to the sheer number of triangles. Using triangle meshes with varying de-tails in different application contexts is the only way to deal with this problem.The strategy to achieve this goal is based on multiresolution modeling,1,27 thatallows processing geometry at multiple levels of detail. In addition simplificationalgorithms are at the heart of constructing multiresolution representation from theinitial surface geometry.

The importance of simplification techniques has motivated an intense researchin this area. During the past years, many simplification algorithms have appeared inthe graphics literature2–14,25 which reduce the number of triangles to a particulardesired triangle count or until a particular error threshold is met. These methods,based on different topological operators and various heuristics to measure geometricerror, address the problem from different angles and provide sub-optimal solutionsthat contemplate different practical trade-offs. They mainly target on geometricsimplification, topological simplification or view-dependent simplification.

We focus only on geometric simplification. Most algorithms for geometric sim-plification are based on iterative approach, where small local geometric change isintroduced according to some optimality criterion. This optimality criteria is usuallybased on one of the two approaches to measure the approximation errors: local andglobal error. Those algorithms which follow the local approach compare the currentmesh with the previous simplified version; the existing methods11,13,25,28 based onthis approach are fast and computationally efficient but yield poor approximations.The algorithms2,4,7,8,12,15,16 are based on global measure of error although theyproduce high quality simplifications, but they make comparisons with the originalgeometric model and thus require that the geometric history is carried along thepartly simplified mesh, making the iterative process memory consuming and slow.Memoryless Simplification10,26 is the only algorithm that is memory efficient andstill produces high quality results but computationally it is not so fast. None of thesehave used the concept of accumulating the cost of collapse; this concept in a wayis an implicit comparison with the original mesh, having the process of reductionwithout retaining geometric history and consuming extra memory. We measure thegeometric deviation locally and then accumulate it as the edge collapse transforma-tions are applied. Our proposed measure of geometric fidelity is intuitive and doesnot involve complex computations. That is why our algorithm is not only memoryefficient but is also faster than almost all those iterative methods which use geo-metric history to execute optimality criteria and even Memoryless Simplification,and it yields good quality results. Due to the involvement of a dihedral angle in thedefinition of our error metric, it automatically preserves the surface discontinuitiesand prevents surface artifacts such as folds.

The remainder of this paper is organized as follows. In Sec. 2, we give an overviewof some of the related iterative simplification algorithms that have appeared in theliterature. Section 3 outlines our simplification algorithm. The main components ofour algorithm have been discussed in Sec. 4. In Sec. 5, the algorithm is validatedby showing and discussing some examples, and comparing them with published


Memory Efficient Simplification of Triangle Meshes 655

simplification schemes. Section 6 concludes the paper and presents some possibledirections for future research.

2. Background and Related Work

Most of the iterative simplification methods can be classified into three categorieswith respect to the topological operator that they adopt: vertex decimation, edgecollapse, and face decimation. Face decimation methods17–19 constrict a face andimmerge its adjacent faces; they usually result in poor approximations. Vertexdecimation2,4,13,15,16 algorithms decimate a vertex and retriangulate the hole; whilethese algorithms produce good approximations, the need for an algorithm to retri-angulate the hole makes the algorithm slow. Although edge collapse operation is aspecial case of vertex decimation, it does not require any triangulation algorithm,so edge collapse algorithms are faster and they result in good approximations. Ouralgorithm is also based on edge collapse operation, and so we will center our atten-tion on edge collapse algorithms. For a thorough survey of simplification algorithms,consult the papers of Refs. 20 to 22.

An arbitrary edge collapse transformation merges the two vertices of an edge intoa single vertex, thus removing two triangles from the mesh (see Fig. 1(a)). To reducea mesh, edge collapse transformations are applied iteratively in a greedy fashion

eru v

t1

t2

v

(a)

v

d e dd ee

bd ec

e

v1v0

b c d e

db ce ddb cee

e

ev

v v

v

e

e

e

e

e

(b)

Fig. 1. (a) Edge collapse transformation. (b) Simplex operators.



until a mesh of required triangle count or when the tolerance is obtained. So we haveto make two decisions: (1) how to prioritize edge collapse transformations whichlead to sub-optimal solution and (2) how to choose the substitute vertex, which isthe vertex to which the end vertices of the collapsing edge will merge. As far as thechoice of a substitute vertex is concerned, there are two main approaches in commonuse: subset placement (half-edge collapse) and optimal placement. Subset placementcauses one of the end vertices of the collapsed edge to be chosen as a substitutevertex, and it is the simplest strategy one can adopt. In optimal placement, thesubstitute vertex is not necessarily a vertex of the original mesh and it can floatfreely in space in order to minimize some error measure; it results in resampledvertices.

The sequence of edge collapse transformations is automatically determined byan error measure which reflects the local geometric deviation of the mesh resultedfrom an edge collapse transformation. The way in which error is measured is thebasic differentiating factor between different algorithms of this class. Various sim-plification approaches measure the approximation error in many different mannersmostly based on any one of the two policies: local evaluation and global evaluationof approximation error.

Hoppe’s algorithm8 for constructing progressive meshes is the pioneer of theclass of edge collapse-based algorithms. It uses an error measure that is defined asthe average distance from the proposed new triangles in the mesh to a set of samplepoints on the original model which are carried along as a geometric history. Whilethis algorithm produces high quality approximations, several distance-to-surfacemeasurements make it quite slow. Gueziec7 defines a tolerance volume as a convexcombination of spheres located at each vertex of the simplification. He selects edgesbased on the shortest edge length and then chooses a new vertex position suchthat the original surface is guaranteed to lie within that volume. This algorithmalso produces good quality results, and appears to be somewhat slow, but it isfaster than Progressive Meshes. The algorithm of Kobelt et al.9 maintains linksbetween points on the original mesh and the corresponding neighborhood on theapproximation, and the distances between these points and the associated facesdefine the approximation error; this algorithm is relatively fast but it is still slowand memory consuming.

Ronfard and Rossignac12 assign to each vertex the set of planes associatedwith its incident triangles for geometric history. As a result of one edge collapsetransformation, two vertices are merged into one and the new vertex inherits theplanes of the merged vertices. The maximum distance from the new vertex to itssupporting planes is used as an error metric to measure the edge collapse cost. Gar-land and Heckbert5 used this work as the starting point of their own simplificationalgorithm QSlim. Instead of maintaining a list of planes, they measured the squareddistance from the collection of planes associated with triangles incident on a vertexand stored them as a symmetric 4 × 4 matrix, one matrix per vertex. While their



approach is fast and gives high quality approximations, it is not memory efficient;for each vertex it stores ten floats and for a polygonal model consisting of somemillion polygons a very large amount of memory is consumed to store this infor-mation. The memoryless algorithm proposed by Lindstrom and Turk10 uses linearconstraints, based primarily on the conservation of volume, in order to decide theedge collapse sequence and the position of the new vertex. The most interestingaspect of this algorithm is that it makes decisions based purely on the current ap-proximation alone. It produces good quality simplifications and is fairly efficient,particularly in memory consumption but it is rather slow as compared to QSlim. Allthese algorithms except the memoryless simplifications algorithm retain geometrichistory while the decimation process is carried out, and no one uses the idea ofaccumulating the cost of collapse.

3. Overview of our Algorithm

In this section we outline our algorithm and briefly describe its main characteristics.First of all, to fix the ideas a brief description of terminology and notation is in place.

3.1. Terminology and notations

Triangles are the most popular drawing primitive in Computer Graphics, and so a3D surface model of any physical object is usually represented by a triangular mesh.A triangular mesh is specified by a pair (P, K), where P is a set of n point positionsP = {vi ∈ R3|1 ≤ i ≤ n} and K is an abstract simplicial complex which containsall the topological information. In other words, P and K describe the geometry andtopology of a triangle mesh. The complex K is the set of subsets of {1, 2, 3, . . . , n},which are known as simplices; vertex is a 0-simplex, edge is a 1-simplex and face is a2-simplex. We represent the vertex (0-simplex) by v with its geometric counterpartas a 3D vector v. An edge (1-simplex) e is a subset {ve

0, ve1}. An oriented edge is

represented by an order pair (ve0, v

e1) and is denoted by �e. A triangle (2-simplex) t

is a set of oriented edges i.e. t = {→e0,

→e1,

→e2} or t = {(ve0

0 , ve01 ), (ve1

0 , ve11 ), (ve2

0 , ve21 )}

or simply t = (v0, v1, v2).According to the definition of simplex operators � � and � � as adopted in Ref. 10,

�v�, ��v��, ��v�� and �e� stand for edges incident on v, triangles incident on v,neighboring vertices of v, and vertices of e, respectively. ��e�� and ��e�� representthe edges and the triangles respectively that are incident upon the end vertices ofe as shown in Fig. 1(b).

3.2. Outline of the algorithm

Our simplification method, like most related algorithms is a simple greedy proce-dure. It is based on the half-edge collapse and it uses the criterion for the evaluationof approximation error which has been detailed in Sec. 4 to prioritize the half-edge



collapse transformations. It takes the original mesh as input and yields progressivemesh representation as an output. It involves the following steps:

• Compute the cost of collapse for each half-edge in the original triangle mesh usingour proposed error metric and put the half-edges in priority ordering.

• Choose the half-edge �e = (v0, v1) with minimum cost of simplification and sub-stitute it with v1. During this operation triangles �e� become singular and arediscarded. The remaining edges ��e�� − {e} and triangles ��e�� − �e� incidentupon v0 and v1 are updated so that all occurrences of v0 are replaced with v1.

• Re-evaluate the cost of collapse for the edges ��e�� − {e} after the collapse ofedge e. Add the cost of collapse of the edge �e = (v0, v1) to those of the half-edgeswhich start from the vertex v1 and update the priority queue of edge collapsetransformations.

The main characteristics of our algorithm are as follows:

• Our algorithm automatically preserves visually important features of a sur-face model in a better way as compared to most of the existing simplificationalgorithms.

• Memory consumption is one of the important factors which effects the efficiencyof an algorithm. Our algorithm accumulates errors and it needs not store anykind of geometric history, so it is memory efficient and can manage huge trianglemeshes.

• Our algorithm is computationally faster than almost all existing iterative edgecollapse algorithms except QSlim.

• It automatically prevents the occurrence of surface artifacts such as folds.• Simplified versions resulted from our algorithm are comparable with those by

published methods in terms of maximum and mean geometric error and theybear good visual resemblance with the original meshes.

4. Main Components

4.1. Topological operator

For an iterative simplification algorithm, the choice of a particular topological op-erator has no significant effect on the results; what matters is the way on how tomeasure the geometric deviation from the original shape.9 As such we have de-cided to use subset-placement or half-edge collapse as a topological operator inour algorithm. The vertices of the simplified mesh resulted from half-edge collapsetransformations always form a proper subset of the original vertices; this makes pro-gressive transmission of meshes more effective and it is crucial for the integratedlevel of detail extraction. Moreover there are many applications where optimal re-placement is not allowed or feasible, e.g. in the case of datasets where the samplingof a scalar/vector field is associated with the mesh vertices and in the re-sampledlocations where the field values cannot be computed safely.



4.2. Error metric

We base our criterion for the evaluation of the cost of an edge collapse transforma-tion on an intuitive observation. When an arbitrary half-edge collapse transforma-tion �er (v0, v) �→ v takes place, (see Fig. 2(a)) two of the edges �v0� will degenerateand the remaining will be displaced. Similarly the tow of the triangles ��v0�� willdegenerate and the remaining will be displaced. This displacement of edges and tri-angles is responsible for geometric deviation. We measure the geometric distortioncaused by the displacement of each of the triangles ��v0�� − �er� as the multipleof the dihedral angle turned through by the triangle and the area swept out by anadjacent edge. To be precise, consider the triangle t = (v0, v1, v2) ∈ ��v0�� − �er�,see Fig. 2(b). The half-edge collapse transformation �er (v0, v) �→ v will cause thistriangle to turn through the angle θ and the edge (v0, v1) to sweep out an areaequal to the area of triangle t′′ = (v0, v, v1). Therefore the error that will be partlyintroduced because of triangle t = (v0, v1, v2) is given by

Qt = Aθ ,

v1

2

v0

v1

v2

v0v v

t t t er

er

v

,

(a)

v1

v2

v0

t

er

�nt

nt’

tv,

(b)

Fig. 2. (a) Collapse of the edge. er = {v0, v} will map the triangle t = (v0, v1, v2) onto thetriangle t′ = (v, v1, v2) and will cause the edge {v0, v1} of the triangle t = (v0, v1, v2) to describethe triangle t′′ = (v, v1, v0). (b) First the triangle t = (v0, v1, v2) will rotate through the angle θand then it will be scaled and sheared.



where

A =12‖a× b‖ , a = v0 − v1 , b = v1 − v ,

and θ is the dihedral angle between the triangles t and t′ = (v1, v2, v); it is measuredas the angle between the normal vectors to the two triangles and it assumes thevalues from 0◦ to 180◦. The computation of θ will involve the evaluation of trigono-metric functions and so it will render the process slow. Scaling by 1/90 maps therange of values of θ onto [0, 2]. The range of values of 1 − nt · nt′ , where nt andnt′ are unit normals to the triangular faces t and t′, is also [0, 2], so to reduce thecomputational cost, we approximate θ by 1 − nt · nt′ . Although this is not a goodapproximation of θ, it serves our purpose. Our goal is to compare the approximationerrors caused by different edge collapses, and for practically feasible edge collapses,the value of θ is much less than 90◦, so the proposed approximation of θ will affectthe two error values in the same manner, and the overall comparison result will notbe affected. We can also consider the area swept out by the edge (v2, v0), but inour experiments we found that it makes no difference.

The cost of collapse of the edge �er will be the sum of errors contributed by eachof the triangles ��v0��−�er�, where v0 is the starting vertex of �er (see Fig. 2(a)), i.e.

Cost(�er) =∑

t∈��v0��−�er�Qt .

Differential geometry tells us that the behavior of a surface is characterized wellby the first and second fundamental forms. The first fundamental form reflects thelocal distortion and the second fundamental form provides the complete informationabout local curvature. In discrete setting, for the sake of efficiency, they can beestimated by geometric analogies.9 In our criterion for approximation error thequantity A can be considered to account for the local distortion of the surface andthe dihedral angle θ measures the local change in curvature; so our criterion in away is associated with local fairness of first and second order.

4.3. Boundary simplification

Boundary edges need special treatment to preserve the boundary of an open surfacemodel; on the boundary, vertices must be prevented from sliding into the innerregion of the surface. One simple solution is to penalize the boundary edges withedge length, but this will preserve the boundary at the cost of a large numberof triangles along the boundary and may result in a very large number of slivertriangles along the boundary. We propose a simple and more effective solution todeal with this problem. We can categorize the half-edges along the boundary intotwo main types: (1) the half-edge that has either staring or terminating vertex onthe boundary e.g. �e = (v, v1) and (2) the half-edge that has both end vertices onthe boundary e.g. �e = (v1, v2), see Fig. 3(a). We deal with each case separately.

The half-edge collapse transformation �e(vs, vt) �→ vt substitutes the half-edgewith terminating vertex, so the collapse of the half-edge having terminating vertex



v3

v2v

1

v0

�

e=( , )’

v1v

2

�0

v

1

2e

e

e

e1

2

,

,

e=( , )’

v1

v2

(a)

v0

v1

e

t

v2

v

t,

(b)

Fig. 3. (a) Edge e1 = {v0, v1} will turn through an angle of φ if the half-edge �e = (v1, v2) iscollapsed whereas the collapse of the half-edge �e = (v2, v1) will make the edge e2 = {v2, v3}turn through an angle of φ0, φ0 < φ. (b) As a result of the collapse of edge e = {v0, v}, trianglet = (v0, v1, v2) will fold over the mesh.

on the boundary does not need special handling. However if the starting vertex ison the boundary, then the collapse of such an half-edge will deform the boundary,so we have to prevent the collapse of such an half-edge.

The half-edge having both end vertices on the boundary must be dealt tactfully.This will obviously collapse to a vertex along the boundary. Now the problem ishow to guide the greedy approach so that it is not entrapped in a local minimum. Soto help the algorithm get out of this problem, we use some heuristics. We assume asimilar part of the mesh on the exterior side of the edge as that on the interior sideto bring it in line with the interior edges. Note in Fig. 3(a), edge e = {v0, v1} maybe collapsed either to v1 or v2, but to achieve better results e must be collapsedto v1. To achieve this we penalize the cost of collapse of half-edges �e = (v2, v1)and �e = (v1, v2) with edge and length scaled by φ and φ0 respectively; φ and φ0

are angles between edges e1 and e′1 as well as e2 and e′2 respectively as shown inFig. 3(a). So the cost of collapse of the half edge �e = (v1, v2) whose both verticesare on the boundary will be

Cost(�e) = λφ‖v1 − v2‖ + 2∑

t∈��v1��−�e�Qt ,



where φ = 1 − u1 · u2. u1 and u2 are the unit vectors along the edges e1 = (v0, v1)and e′1 = (v0, v2) as shown in Fig. 3(a). Here λ is a parameter used to controlthe quality of boundary preservation. It can assume values greater than 1. In ourexperiments we have found that practically feasible results can be obtained usingthe value of λ in the range of 0 < λ ≤ 50. A user can choose the value of λ as near50 as tightly boundary is needed to be preserved.

Instead of φ, curvature κ at vertex v1 can also be used. We estimate the curvatureat v1 by the curvature of a circle passing through the three vertices v0, v1, and v2

23

as follows:

κ =‖(v0 − v1) × (v2 − v1)‖

‖v0 − v1‖ ‖v2 − v1‖ ‖v0 − v2‖ .

In this case λ ∈ (0, 1] and a user can choose a value of λ according to his/her needs.

5. Experimental Results and Discussions

We tried implementating the FMS of our algorithm on several public domain largetriangular meshes and have achieved good results. Our method can simplify verylarge models consisting of millions of triangular faces in a fairly short time andthe simplified models bear good visual resemblance with the originals. To validatethe asserted precision and efficiency of our method, we make comparisons withQSlim,5 Memoryless Simplification (MS),10 JADE2 and Simplification Envelopes(SE)4 among the published simplification algorithms. We choose the Stanford bunnyand hand models as test models because of their complex structures.

Table 1 lists the computation time taken by QSlim and FMS to simplify variousmodels shown in Figs. 6 and 7. Notice that our algorithm is almost twice as slowthan QSlim. We run both the algorithms on a 800 MHz Intel PentiumIII machinewith 384 MB of main memory. From the results reported in Ref. 10 (see Table 1),it is obvious that MS is about 5 times, JADE about 10 times and SE about 17times slower than QSlim. Therefore we can safely conclude that our method is thefastest one after QSlim. As far as QSlim is concerned, FMS consumes at least 40bytes per vertex less memory than QSlim. We use Version 2.5 of Metro tool24 toevaluate the quality of simplified meshes.a Graphs shown in Figs. 4 and 5 illustratethe mean geometric and maximum geometric errors between the original and thesimplified models created by FMS and other algorithms. We plotted 1000 times theratio of the error and the bounding box of the original model along logarithmicy-axis and the number of faces along logarithmic x-axis. It is apparent that ouralgorithm compares favorably with all the algorithms except MS in terms of mean

aWe have evaluated the meshes with the options: Metro Original.ply Simplified.ply -s -t.



Table 1. Time taken in seconds to reduce to one face.

Model Model Size (#faces) FMS (s) QSlim (s)

Fandisk 12,946 2.9 1.5

Bunny 69,451 9.0 4.0

Crater 199,114 26.2 12.4

Hand 654,666 86.7 39.9

Blade 1,765,388 266.2 125.1

0.01

0.1

1

10

100 1000 10000 100000

Memoryless SimplificationQSlimFMSJade

Simplification Envelopes

Me

an

ge

om

etr

ice

rro

r

Model size (faces)

0.1

1

10

100

100 1000 10000 100000

Memoryless Simplification.

QSlim

FMS

Jade


Maxim

um

ge

om

etr

ice

rro

r

Model size (faces)

Fig. 4. Mean and maximum geometric error for bunny model.



0.01

0.1

1

10

1000 10000 100000 1000000

FMS

Memoryless Simplification

QSlim

Jade


Me

an

ge

om

etr

ice

rro

r

Model Size (faces)

0.01

0.1

1

10

100

1000 10000 100000 1000000

Jade

Memoryless Simplification

FMS

QSlim


Maxi

mu

mg

eom

etr

ice

rro

r

Model size (faces)

Fig. 5. Mean and maximum geometric error for hand model.

geometric errors. The reason of this difference is that the models are simplified byMS with the option of optimal placement whereas FMS uses subset placement. Themodels simplified by MS are the courtesy of Perter Lindstrom. It also compareswell with all algorithms except JADE in terms of the maximum geometric error.

Now we highlight the other aspects of our algorithm and concentrate on thosefeatures of a model which are geometrically and visually important.

5.1. Folds in the surface

Folds may appear when an edge to be collapsed is surrounded by a very concavepolygon. Some algorithms e.g. QSlim5 use special heuristics to deal with the prob-lem of folds. Our proposed error measure intuitively prevents folds in the mesh. Note



Fig. 6. (a) Original bunny model, faces: 69451. Bunny models simplified by (b) FMS, faces 1394,(c) JADE, faces: 1396, (d) Simplification Envelopes, faces: 1907 and (e) QSlim, faces: 1395. (f)Original blade model, faces: 1765388. Blade model simplified by (i) FMS, faces: 15526. (g) Originalhypersheet model (upper), faces: 19416 and simplified version (lower) by FMS, faces: 533, λ = 25.(h) Original teapot model (upper) faces: 17712 and reduced version (lower) by FMS, faces:1000.



Fig. 7. (a) Original hand model, faces: 654666 faces. Hand models reduced by (b) FMS, faces4956 (c) JADE, faces: 4958 (d) Memoryless simplification, faces: 5000 and (e) QSlim, faces:4958. (f) Original crater model (upper), faces:199114 and simplified version by FMS, faces: 904.(g) Original fandisk model, faces: 12946. Reduced fandisk by (h) FMS, faces: 396 and (i) QSlim.



in Fig. 3(b), when edge e = {v0, v} collapses and v0 coincides with v, trianglet = (v0, v1, v2) will fold over, thus creating a fold in the mesh. In this case, theangle between the triangles t = (v0, v1, v2) and t′ = (v1, v2, v) will bear greatervalue and our error measure will cause greater values to be added to the cost ofedge collapse thereby preventing this edge collapse. Consider Figs. 7(g)–7(i) ouralgorithm automatically prevents folds whereas QSlim creates folds and it needsextra heuristic to prevent these folds.

5.2. Preserving boundary

In the case of open surface models, the boundary is one of the main geometricfeatures which have effect on the visual appearance of a model, and it must bepreserved properly for good visual fidelity. Our algorithm is capable of preservingthe boundary with varying degrees of tightness. It has a practically important anduseful characteristic whereby it provides some control to the user to preserve theboundary according to his/her needs. The model shown in Fig. 6(g) is a hyper-sheet which demonstrates how beautifully FMS can preserve the boundary of thesimplified model. One can see well shaped triangles along the boundary.

5.3. Preserving feature lines

Feature lines are sharp edges whose two adjacent faces have a dihedral angle ofless than some threshold. These lines reflect the overall geometric appearance of amodel and are visually very important. Since the definition of our error measureinvolves the dihedral angle, it automatically preserves the feature lines withoutany additional aids. Note that Fig. 7(h) is a simplified fandisk model consistingof 396 faces in which 3% of the original, features lines are preserved, albeit highlysimplified.

5.4. Preserving high frequency detail

High frequency details resolve the visually important features of a model. Again, asthe dihedral angle is an important factor of our measure of accuracy, our algorithmalso preserves high frequency detail automatically. Observing Fig. 7(f), the simpli-fied crater model shows that in spite of a 99.5% reduction, bumps and creases areapparent.

5.5. Adaptivity

Curvature is one of the most important geometric quantity. A good simplificationscheme is assumed to preserve the high curvature regions in a mesh for good visualeffects. As our proposed error measure is based on dihedral angle, it automaticallyachieves adaptive simplification. Consider Fig. 6(h), the high curvature regions havedense and small triangles elongated along the direction of high curvature whereasthe relatively flat regions have large triangles.



5.6. Large models

Our algorithm can efficiently simplify very large models. The model shown inFig. 6(f) is a turbine blade consisting of about 1.8 million triangular faces. Simpli-fication of this model is a challenging task because of its sheer size, complicatedtopology with a large number of tiny holes, and complex geometry with many sharpedges. Our method spent just 4 minutes and 22 seconds to simplify this model. Thereduced version shown in Fig. 6(i) consists of 15 526 faces (0.08% of the original)demonstrating that in spite of drastic simplifications, all important details of themodel is preserved and it bears good visual resemblance with the original model.

6. Summary and Future Work

We have proposed a new polygonal simplification method based on a new wayof measuring the approximation error and the new idea of accumulating the costof collapse. Our method has very good trade off between memory consumption,computation time and accuracy. It is the fastest method after QSlim and is mem-ory efficient like the Memoryless Simplification. It compares favorably with thepublished methods either in terms of maximum geometric error or mean geomet-ric error. It can simplify huge models consisting of millions of triangular faces ina relatively short time. It preserves the essential features of an object even aftersignificant reductions. It can be used for applications which require visual fidelitybut not tight error bound, as well as applications which need the vertices of thesimplified mesh to be a proper subset of original vertices. We intend to extend itto include surface attributes.

Acknowledgments

We thank the anonymous referees for their valuable comments, which helped im-prove the quality of the paper.

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Muhammad Hussain received his MSc degree in AppliedMathematics and the MPhil degree in Computational Mathe-matics specializing in CAGD from the University of the Punjab,Lahore, Pakistan, in 1990 and 1993 respectively.

In 1995, he joined the Department of Mathematics, Pun-jab University, Lahore, Pakistan as a lecturer where in additionto teaching graduate courses, he participated in a project onCAGD.

In 1999, he received the Monbusho Scholarship and proceeded to Japan for hisdoctoral studies. He is currently at the Graduate School of Information Science andElectrical Engineering, Kyushu University, Fukuoka, Japan and is pursuing a doc-toral degree in Computer Science. His current research interests include geometricmodelling, physical-based modelling and virtual reality.

Yoshihiro Okada received his BSc, MSc and doctoral degreesin Engineering from the Electrical Engineering Department, Fac-ulty of Engineering, Hokkaido University, Japan, in 1988, 1990and 1993 respectively.

He was an Assistant Researcher at the Faculty of Engineer-ing, Hokkaido University from 1993 to 1998. He was an AssociateProfessor at the Computing Center, Kyushu University, Japanfrom 1999 to 2000. He has been an Associate Professor in the

Department of Informatics, Graduate School of Information Science and ElectricalEngineering, Kyushu University, Japan since April, 2000. His current research inter-ests include 3D graphics, software architecture, network collaboration technologyand human-computer interface.

Koichi Niijima received the BS degree, the MSc degree, andthe PhD degree in Mathematics from the Kyushu University,Japan, in 1967, 1969 and 1977, respectively.

In 1971, he was a Research Assistant at the Kyushu Uni-versity. From 1974 to 1981, he was an Associate Professor atthe Fukuoka Women’s University. He worked from 1981 to 1988as an Associate Professor of Mathematics at the Kyushu Uni-versity, and from 1988 to 1996, he was a Professor of Control

Engineering at the Kyushu Institute of Technology. Since 1996, he has been withthe Kyushu University, where he is currently a Professor in the Department of In-formatics, Graduate School of Information Science and Electrical Engineering. Hiscurrent research interests include wavelets, in particular lifting wavelets, and theirapplications to image analysis and signal processing and geometric modeling.

muhammad hussain - semantic scholar...e-mail: [email protected] or ... bc de db ce ddb cee e...

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