Handling Degenerate Cases in Exact Geodesic Computation on Triangle Meshes

Transcription

1 The Viual Computer manucript. (will be inerted b the editor) Yong-Jin Liu Qian-Yi Zhou Shi-Min Hu Degenerate Cae in Eact Geodeic Computation on Triangle Mehe Abtract The computation of eact geodeic on triangle mehe i a widel ued operation in computer-aided deign and computer graphic. Practical algorithm for computing uch eact geodeic have been recentl propoed b Surazhk et al (2005). B appling thee geometric algorithm to real-world data, degenerate cae frequentl appear. In thi paper we claif and enumerate all the degenerate cae in a tematic wa. Baed on the claification, we preent olution to handle all the degenerate cae conitentl and correctl. The common uer ma find the preent technique ueful when the implement a robut code of computing eact geodeic path on mehe. Keword Eact geodeic computation Degenerate cae Robutne 1 Introduction An eact geodeic between two point in a 2-manifold meh i a union of line egment within the meh which connect the two point and i locall length-minimized. The computation of eact geodeic path on triangle mehe i a widel ued operation in computer-aided deign and computer graphic. In [5], a practical implementation of the DGP algorithm in [3] i propoed for computing eact geodeic from a ource point to one or all other point efficientl. In the wort cae the DGP algorithm ha compleitie of O(n 2 ) pace and O(n 2 log n) time, while in practice the algorithm i oberved to run in ub-quadratic time. The implementation in [5] can be regarded a a generic algorithm, i.e., it i guaranteed to be correct with generic ituation, while how to handle degenerate cae i not Department of Computer Science and Technolog, Tinghua Univerit, P. R. China Tel.: Fa: {liuongjin, zq, himin}@tinghua.edu.cn Fig. 1 Geodeic computation with a precribed ource point; point on the meh are colored according to the geodeic ditance to the ource point. reported. In thi paper we enumerate all the degenerate cae rien from implementation in [5] and how that in mot cae with arbitraril haped triangle, the degenerate cae are frequentl appear. An eample i illutrated in Fig. 1. The meh ued in Fig. 1 ha 2000 face, 6000 edge and 1028 vertice. The triangle in the meh are arbitraril haped, including both obtue and acute triangle. Given a precribed ource point, there are totall 8807 cae handled, in which 2583 cae are degenerate, about 29.33%. Some degenerate cae are illutrated in Fig. 4. In geometric computation, degenerate cae will increae the intabilit of the generic algorithm. Theoreticall, degenerate cae can be handled b uing the mbolic perturbation cheme [1]. Though it i a powerful tool, thi cheme ma not be applicable in the computation of eact geodeic path. Firt, mbolic perturbation require eact arithmetic, with which man uer are not familiar. Second, uing mbolic perturbation doe not olve the degenerate cae itelf, but an arbitraril-choen nearb general cae. Topolog-oriented implementation i another wa to handle degenerate cae [4]. However, it onl guarantee to output a topolog-conitent olution which ma not be the deired topolog-correct one. In thi paper, to develop a robut and fat eact geodeic algorithm, we preent a tematic olution to

2 2 Yong-Jin Liu et al. σ d 0 d 1 0 b 0 b 1 v Starting from the ource point, the DSP algorithm propagate ditance information in a continuou Dijktralike fahion. When new interval are created, the are placed in a priorit queue orted b minimum ditance back to the ource. When an interval i popped from the queue, interval propagation i performed in one of the three cae howing in Fig. 3. The reader i referred to [5] for full detail of thi algorithm. Fig. 2 6-tuple repreentation (b 0, b 1, d 0, d 1, σ, τ) of the interval. efficientl handle all the degenerate cae with floating point computation [6]. B doing o, geometric predicate are treated conitentl and thu the implemented algorithm i robut. 2 Review of the eact geodeic algorithm We follow the notation in [5] to quick review the DGP algorithm [3]. Shortet path on meh are ra emanating from the ource verte along tangent direction. Interior to a triangle, a hortet path mut be a traight line. When croing an edge, a hortet path mut be a traight line when the previou face i unfolded into the plane containing the net face. The onl vertice (called geodeic vertice below) that a hortet path can pa through are either boundar vertice or the vertice whoe total urrounding angle i larger or equal to 2π. The baic idea of the DSP algorithm i to partition each meh edge into a et of interval. Refer to Fig. 2. Each interval i encoded b a 6-tuple (b 0, b 1, d 0, d 1, σ, τ). b 0, b 1 are parameter meauring ditance along the edge. The unfolded poition of the geodeic verte i encoded b it ditance d 0, d 1 to the interval endpoint. A binar direction τ i ued to pecif the ide of edge on which the ource lie. σ i the length of the path from back to the ource v. Given an interval I on an edge, it ditance field i propagated acro an adjacent face to define new potential interval on the two oppoing edge,. Refer to Fig. 3. Three general cae eit for interval propagation. According to different cae, different new interval are formed on the oppoing edge. If interval alread eit on the oppoing edge, the new interval ma interect ome old one. If two interval interect with a nonempt region δ, a quadratic equation Ap 2 + Bp + C = 0 (1) i olved to determine a new poition p δ uch that the updated range of the two interval I and I are (b 0, p) and (p, b 1), repectivel. 3 Degenerate cae In the eact geodeic algorithm [5], two tpe of degeneracie occur in interval propagation: 1. Degeneracie on geometric interection. Refer to Fig. 3 and 4. Thee degeneracie rie from the determination of interection region between the wedge and the line egment and ; 2. Degeneracie on geodeic dicontinuitie. Due to the numerical error in floating point computation, the olution of equation (1) often generate mall gap or overlap between the new reulted interval; thi give rie to geodeic dicontinuitie along the interval on the edge. 3.1 Degeneracie on geometric interection Baicall, there are 5 degenerate cae in thi cla, a hown in Fig. 4: 1. The poition of lie on edge. Thi cae can happen if the interval i created on in the cae of Fig. 3(c); 2. Three point, b 0, are in a traight line. Thi make the new interval on the edge diappear in the cae of Fig. 3(b); 3. Three point, b 1, are in a traight line. Thi i a mmetric cae of ca; 4. Four point,, b 0, are in a traight line. Thi alo mean that point and b 0. In thi cae, the new interval on the in the cae of Fig. 3(b) mut be treated a the new interval on the in the cae of Fig. 3(c); 5. Four point,, b 1, are in a traight line. Thi i a mmetric cae of cae 4. tice that there are ome degenerate cae compoed of everal baic cae. For eample, referring to Fig. 4, if three point, b 0,, the baic degenerate ca,2,4 occur imultaneoul. Different degenerate cae mut awake different procedure to proce. Treating degenerate cae in random order will reult in catatrophic failure in the algorithm. In Section 4.1, we preent a concie deciion procedure to properl handle all the degenerate cae.

3 degenerate cae in eact geodeic computation 3 d 0 d 1 d 0 d 1 d 1 0 b 0 I b 1 0 b 0 b 1 d 0 0 b 1 v I 2 b 0 e 1 e 2 b (a) 1 (b) (c) Fig. 3 Interval propagation. (a)one new interval created. (b) Two new interval created. (c) One new normal interval and two additional interval (in red) created. b 0 b 1 b 1 b 0 b 1 b 1 v 2 b 0 v 2 b 0 v 2 Cae (1) (2) (3) (4) (5) Fig. 4 Degenerate cae on geometric interection; the haded area indicate the wedge range of b 0 b 1. I i I I.b 0 I.b 1 (a) The new created interval I I j (b) Eiting interval I new i I i I new j I j I inide (c) Preproceing eiting interval I i I j I i 1 I j 1 (d) Finall updated interval I updated Fig. 5 Degeneracie on geodeic dicontinuitie. I updated = {I 0, I 1, } be the et of updated interval on e, four degenerate cae ma occur: 1. Tin interval appear in I; 2. Two conecutive interval in I interect; 3. Two conecutive interval in I eparate b a tin gap; 4. The geodeic ditance at the common endpoint of two conecutive interval are not the ame. Theoreticall, if eact arithmetic i ued, thee cae will not happen or can be regarded a error. However, in practice, when float-point computation i ued and numerical error are unavoidable, thee cae do occur and we regard them a degenerate cae. The olution to handle thee degeneracie i preented in Section Degeneracie on geodeic dicontinuitie After the determination of interection region between wedge b 0 b 1 and edge,, new interval are created. Refer to Fig. 5. Suppoe that a new interval I with range (I.b 0, I.b 1 ) i created on edge e on which there alread eit a et of interval I = {I 0, I 1, } orted b poition on edge, I i 1.b 1 I i.b 0 < I i.b 1 I i+1.b 0. If the interval I and I i I have a nonempt interection region δ = I I i, a quadratic equation need to be olved to determine the minimal ditance for point in δ and update the interval I and I i along edge e. Let 4 degenerate cae In geometric algorithm, teting degenerate cae relie heavil on the incidence deciion uch a whether a point lie on a line or two point [2]. Incidence deciion contribute to geometric predicate. A predicate i a numerical primitive computation whoe value impact the flow of control of an algorithm. To evaluate predicate with float-point computation, we preent a tematic olution in the following ubection. The peudo-code of the overall algorithm i a follow.

4 4 Yong-Jin Liu et al. Begin with b 0 or b 1 Ye new interval on, fig. 4.1), b 0, lie on a ame line Ye interval on, b 0 Ye interval on fig. 4.4) fig. 4.2), b 1, lie on a ame line interval on v 2, b 1 interval on fig. 4.5) Ye Ye fig. 4.3) lie on left ide of line b 0 Ye interval on, b 0 Ye new interval on, fig. 3c) m.to fig. 3a) lie on right ide of line b 1 Ye interval on, b 1 Ye new interval on, m. to fig. 3c) Fig. 6 The flowchart of the deciion tem to handling degeneracie on geometric interection. new interval on, fig. 3a) fig. 3b) Algorithm 1 1. Initialize a priorit queue Q with a given ource point in the meh; 2. while Q i not empt 2.1. pop off the top element q from Q; 2.2. etablih the local tem a hown in fig. 3 baed on q = (b 0, b 1, d 0, d 1, σ, τ); 2.3. find the interection of the wedge b 0 b 1 and, ; handle the degeneracie uing the olution preented in Sec. 4.1; 2.4. update interval on, and Q uing the olution preented in Sec. 4.2; 2.5. if new interval created add them into Q; 4.1 degeneracie on geometric interection Suppoe that we implement the vector operation in a C++ cla. Given a point (or a vector) p, p., p., p.z retrieve it three coordinate. p.length() return the value of the vector length. p q return the value of the inner product of two vector p, q. p q return the vector of the cro product of p, q. ab(c) return the abolute value of c. Denote the machine preciion b ɛ. Refer to Fig. 3. The following rule conit of incidence deciion: If ( b 0 ).length() < ɛ, point and b 0 ; If ( b 1 ).length() < ɛ, point and b 1 ; If ab((( b 0 ) (b 0 )).z) < ɛ, three point, b 0, lie on a traight line; If ((b 1 ) ( b 1 )).z > ɛ, the verte lie right of the wedge and the new interval will be on the. That mean cae (a) in Fig. 3 occur. If ((b 0 ) ( b 0 )).z < ɛ, the verte lie left of the wedge and the new interval will be on the. If ((b 0 ) ( b 0 )).z > ɛ and ((b 1 ) ( b 1 )).z < ɛ, the verte lie inide the wedge formed b two ra b 0 and b 1. That mean cae (b) in Fig. 3 occur; Given the above rule, our goal i to deign a deciion procedure that reduce all poible deciion to a et of predicate a few a poible, which alo guarantee to output a conit and right deciion on chooing the order of different degenerate cae. We preent uch a nontrivial deciion tree in Fig. 6. Given the rule of incidence deciion and the deciion tree a hown in Fig. 6, the code that can robutl and conitentl handle all the degenerate cae in thi cla i readil to build. 4.2 degeneracie on geodeic dicontinuitie Here we preent a robut olution to handling degeneracie on geodeic dicontinuitie. The preented olution ma eem unnecearil complicated at the firt glance. However, it not onl give u a concie wa of programming, but alo it make verification and error etimation poible and ea to realize at each tep b providing determinitic tatu to check. The peudo-code handling

5 degenerate cae in eact geodeic computation 5 degeneracie on geodeic dicontinuitie (ref. the Step 2.3 in Algorithm 1) are a follow. Algorithm 2 1. for all I i I 1.1. let interb0 = ma{i i.b 0, I.b 0 }, and interb1 = min{i i.b 1, I.b 1 }; 1.2. if interb0 < interb if I i.b 0 < interb eparate I i at interb0; let Inew i = (I i.b 0, interb0) and I i = (interb0, I i.b 1 ); inert Inew i into I; if interb1 < I i.b eparate I i at interb1; let Inew i = (I i.b 0, interb1) and I i = (interb1, I i.b 1 ); inert Inew i into I; 2. for all I i I which completel inide I 2.1. update I i and I b olving equation (1); 3. Remove tin interval in I; 4. Sew mall gap in I; 5. In I merge neighbor interval with the ame geodeic verte; 6. (Optional) verification of I if needed. Given the newl created interval I and a et of alread eited interval I = {I 0, I 1, } on edge e, we firt proce all interval in I uch that for each interval in I, it i either completel outide range I or completel inide I. Thi proce i illutrated in Fig. 5c and Step 1 in Algorithm 2 erve thi need. At Step 2 in Algorithm 2, denote the orted ubet b I inide whoe element are completel inide the range of the new interval I. We update interval in I inide in turn. Given I i I inide and I, a quadratic equation i olved. According to the olution, I i = (I i.b 0, I i.b 1 ) ma diappear or hrink into a maller interval I inew = (I inew.b 0, I inew.b 1 ). In the latter cae, we divide interval I = (I.b 0, I.b 1 ) into two part, i.e., I new = (I.b 0, I inew.b 0 ) and I = (I inew.b 1, I.b 1 ), and inert I new into I. Then we continue to proce I i+1 with I until all element in I inide are proceed. Finall, we get an updated interval et I. It i not difficult to check that given the above rule, the element in I cannot be interected to each other. Due to numerical computation, tin interval and mall gap ma occur. Refer to Fig. 5c and Step 3,4,5 in Algorithm 2, the following rule handle thee degeneracie: 1. Detect and remove tin interval. I i I, if I i.b 1 I i.b 0 < ɛ, merge I i with I i 1 or I i+1 ; 2. Detect and ew mall gap. If I i+1.b 0 I i.b 1 < ɛ, let I i+1.b 0 = I i.b 1 be the midpoint of the original I i+1.b 0 and I i.b 1 ; 3. Merge interval with the ame ource point. For an pair I i and I i+1, let the unfolded poition of geodeic verte be i and i+1, repectivel. If ( i i+1 ).length() < ε, merge interval I i and I i face, degenerac rat9.33% 5000 face, degenerac rate 31.13% Fig. 7 An eact geodeic path over the head model with two different reolution mehe. The bigget advantage of Algorithm 2 i the reult of ever tep i predictable and thu code verification i ea to check. 5 Reult B handling all the degenerate cae conitentl and correctl, the implementation of the eact geodeic algorithm [3,5] i ver robut. In thi ection, we preent ome teting eample with the model of variou ditribution of triangle. In each eample, the mall green phere indicate the poition of the precribed ource point with which a ditance field i built b computing the length of geodeic path from the ource to all other point on mehe. B tracing the gradient of the ditance field, a geodeic path from the ource to a detination point on meh i alo hown in each eample. In all eample hown here, the degenerac rate i meaured b the percentage of degenerate cae over all the cae. Tabl ummarize the degeneracie tet on all the eample. In Fig. 7, a head eample with two different reolution model i preented. Both model conit of irregular triangle. On both model, the ource and detination point are the ame and the geodeic path connecting

6 6 Yong-Jin Liu et al. Tabl Degeneracie tet on all the eample; the degenerac rate i meaured b dividing the degenerate cae reulted from geometric interection over all the cae. model face num. all cae degenerac rate fig7a % fig7b % fig % fig9a % fig9b % fig9c % fig9d % fig % them are hown. In Fig. 8, the tet i performed on the maplunck head model. Thi model poee different meh reolution over different region. On thi model, a geodeic path croing region of different reolution i hown. Thee two eample how that (1) more maller the triangle are, more degenerate cae occur; (2) more irregular the triangle ditribution i, more degenerate cae occur. We alo tet the implementation on a diverit of model with arbitrar triangle. Four tpical eample are hown in Fig. 9. Thee eample how that realworld data i likel to contain a large number of degeneracie. B providing a concie and conitent olution to all the degenerate cae, the uer ma find the technique preented in thi paper ueful when he/he implement a robut code to compute eact geodeic over triangle mehe. 6 Concluion Geometric algorithm are enitive to degeneracie rien from pecial poition of everal incident geometric object. Although the general technique [1, 4] eit to handle the degeneracie theoreticall in an geometric algorithm, certain particular application permit much more efficient wa to handle degeneracie. In thi paper we claif and enumerate all the degenerate cae in the computation of eact geodeic on triangle mehe. Baed on the claification, we preent a tematic treatment to handle all the degeneracie conitentl. We alo how b eample that the real-world data i likel to be degenerate. The common uer ma find the preented technique ueful to obtain a robut implementation of the fat eact geodeic algorithm. Reference 1. H. Edelbrunner, E. Mucke. Simulation of implicit: a technique to cope with degenerate cae in geometric algorithm. ACM Tran. Graphic, 1990, 9(1): C. Hoffmann. Robutne in Geometric Computation. Journal of Computing and Information Science in Engineering, 2001, 1(2): Fig. 8 The eact geodeic over the maplunck head model which poee different reolution over different region. The code mut be robut againt large and mall triangle imultaneoul eited on a ingle meh. The degenerac rate of thi model i 33.25%. 3. J. Mitchell, D. Mount, C. Papadimitriou. The dicrete geodeic problem. SIAM J. Comput., 1987, 16(4): K. Sugihara, M. Iri, H. Inagaki, T. Imai. Topologoriented implementation an approach to robut geometric algorithm. Algorithmica, 2000, 27(11): V. Surazhk, T. Surazhk, D. Kiranov, S. Gortler, H. Hoppe. Fat eact and approimate geodeic on mehe. ACM SIGGRAPH 2005, pp J. Zachar. Introduction to Scientific Programming, Santa Clara, CA : TELOS, 1998.

7 degenerate cae in eact geodeic computation 7 a b c d e Fig. 9 The computation of eact geodeic over the divere model with arbitrar triangle.