Approximate Star-Shaped Decomposition of Point Set Data

Transcription

1 Eurograhics Symosium on Point-Based Grahics (2007) M. Botsch, R. Pajarola (Editors) Aroimate Star-Shaed Decomosition of Point Set Data Jyh-Ming Lien George Mason University, Fairfa, Virginia, USA Abstract Simlification or decomosition is a common strategy to handle large geometric models, which otherwise require ecessive comutation to rocess. Star-shaed decomosition artitions a model into a set of star-shaed comonents. A model is star shaed if and only if there eists at least one oint which can see all the oints of the model. Due to this interesting roerty, decomosing a model into star-shaed comonents can be used for comuting camera locations to guard a given environment (the art-gallery roblem), skeleton etraction, oint data comression, as well as motion lanning. In this aer, we roose a simle method to artition (or cluster) oint set data (PSD) into aroimately star-shaed comonents. Our method can be alied to both 2D and 3D PSD and can be naturally etended to higher dimensional saces. Our method does not require or comute any connectivity information of the inut oints. The roosed method only requires the osition and the outward normals of oints. Our eerimental results show that the size of the final decomosition is close to otimal. Categories and Subject Descritors (according to ACM CCS): I.3.5 [Comuting Methodologies]: Comuter Grahics[Comutational Geometry and Object Modeling] 1. Introduction Larger and more comle models can be catured chealy nowadays using laser range scanners or stereo cameras. Large geometric models require a lot of comutation resources to rocess. Even with current CPU and GPU advances, simlification is usually required in order to handle such models efficiently. Decomosition is a kind of simlification that artitions a model into simler-shaed and smaller-sized comonents. Star-shaed decomosition artitions a model into a set of star-shaed comonents. A model is star shaed if and only if there eists at least one oint which can see all the oints of the model. Due to this interesting roerty, decomosing a model into star-shaed comonents has many alications. For eamle, star-shaed decomosition can be used to etract shae descritors, e.g., a skeleton. Star-shaed decomosition has been used to comute roadmas and find aths in robotic motion lanning [VM05]. Star-shaed decomosition is also closely related to the art-gallery roblem (see [Chv75]): Finding the fewest cameras to guard a given environment. jmlien@gmu.edu In this aer, we roose a simle method to artition (or cluster) a oint set into aroimately star-shaed comonents. This work rovides an alternative aroach to the eisting star-shaed decomosition methods that are mostly focused on decomosing 2D shaes with continuous boundaries. Our work is insired by more and more techniques roosed to work directly on oint set data (PSD) instead of on reconstructed meshes, e.g., rendering [RL00,ABCO 03], comression, feature etraction [PKG03], and surface analysis [PG01], mesh offsetting [CWRR06], to name just a few. One of the reasons for the oularity of oint-based methods is that the connectivity of the oints is not always easy to comute. An imortant benefit of working directly on oints is that our method can be alied to both 2D and 3D oint sets and can be naturally etended to higher dimensional saces. Our method does not require or comute any connectivity information of the inut oints. The roosed method only requires the ositions and the outward normals of oints, which can usually be comuted efficiently for data obtained from many sources, such as laser scanner or stereo cameras. Another imortant feature of the roosed method is that, as shown in our eerimental results, our decomosition is close to otimal. In all of our eeriments, the size of the final decomosition is only 1.1 to less than five times larger than the otimal size.

2 We begin our discussion by reviewing some of the related work in Section 2. In Section 3, we define the terms and roerties that we will use throughout the aer. The main algorithm of our decomosition method is resented in Sections 4 to 6. Finally, we show eerimental results of the roosed algorithm in Section 7. q s V q s V (P) 2. Related Work Star-Shaed Decomosition. There is little known about 3D star-shaed decomosition. On the contrary, decomosing simle olygons into 2D star-shaed subolygons is well studied; see a survey in [Kei00]. Similar to most of the decomosition roblems, decomosing a olygon with holes into minimum number of star-shaed comonents is NPcomlete [Kei83]. For olygons without holes, the artitioning can be done in O(nlogn) time [AT81] or O(n) [Gho83] time and result in 3 n comonents. Finally, Keil [Kei85] achieved the otimal solution for star-shaed artition with the minimum number criterion in O(n 5 r 2 logn) time, where r is the number of the refle vertices. Star-shaed decomosition is related to guarding an art gallery [Chv75]. A olygon is said to be guarded if it is covered by the visible regions of the guards. The visible region of a guard is a star-shaed comonent. The roblem of finding minimum guards is known to be NP-comlete for olygons with or without holes [OS83, LL86]. Aroimate aroaches also have high comleities, e.g., O(n 5 logn) time [Gho87] and O(n 4 logn) time [AGS88]. In 3D, aroimate aroaches have been roosed to guard terrain (see [BMKM05]). Decomosition of Points. Several methods have been roosed to decomose oint set data into meaningful comonents, e.g., by Dey et al. [DGG03]. Recently, Yamazaki et al. [YNBH06] roosed a decomosition method based on the estimation the centrality of each oint using aroimated geodesic distance. Desite their romising results, one major drawback of this aroach is that the comutation for the centrality is time consuming and requires large memory consumtion. Further simlification of the model and aroimation of the centrality are required to rovide reasonable comutation time. Aroimated Decomosition. The main idea of aroimated decomosition is to take advantage of ignoring detailed features of the model to efficiently roduce smaller decomositions than the eact aroach. Aroimate conve decomosition (ACD) of olygons [LA06] and of olyhedra [LA07] are methods that decomose a model into nearly conve comonents. Lien and Amato have shown that ACD can be generated more efficiently and has more alications than eact conve decomosition methods. As we will show, the aroimate star-shaed decomosition has similar benefits. (a) Figure 1: (a) A olygon. (b) A oint set reresenting the same shae. See Section 3 for detail. 3. Preliminaries: Visibility Before going into details about our decomosition method of oints, let us first review some simle geometric roerties about shae. A shae S is usually reresented by its boundary S and the boundary of a shae is usually reresented by a set of connected oints, i.e., edges and facets. Eamles include 2D olygons and 3D olyhedra. Several geometric roerties can be easily comuted using boundary reresentations. For eamle, given two oints and q, we can check if can directly see q (with resect to S) by checking if the line segment q connecting and q interests the boundary S. That is we can define the visibility of two oints as follows. Definition 3.1. Visibility. Let and q be two oints and let S be a shae. Points and q are visible from each other if and only if q S, where q is the oen set of q. Now, given a oint inside a shae S, we call a set of oints that are visible from the visible region V of (see Figure 1(a)). Definition 3.2. Visible region. Let S be a oint of S. The visible region, denoted as V, of is a set of oints in S that are visible from, i.e., V = {q S q S }. It is easy to see that the visible region V of is a star shae, in which the oint is visible from all the oints of V. Therefore, we can now simly define a star shae and star-shaed decomosition using the concet of visible region. Definition 3.3. Star shae and star-shaed decomosition Shae S is a star shae if and only if there eists a oint S so that V S. A star-shaed decomosition of a shae S is a set of star shaes {S i } whose union is S. Note that, so far, our definitions of visibility and star shae deend on a continuous boundary reresentation of the shae, however, this continuous boundary will not be available in a oint-based reresentation. (b) Point-based reresentation. Now let us return our attention back to oint sets. A oint set data is also a tye of boundary reresentation ecet that these oints are not connected into lines or meshes. Without connectivity, the boundary S of a shae S becomes more ambiguous, thus the definitions and roerties mentioned reviously seem to be not valid anymore. For eamle, how can we check if two oints are visible from each other without the elicit rerec The Eurograhics Association 2007.

3 sentation of the boundary? Fortunately, as we will see, these roerties can be aroimated closely without comuting the connectivity of the oints. We let P be a oint set. We assume that the oint set P is a samle of the boundary S of a shae S and each oint of P is associated with an outward normal direction n. First, let s consider the visibility. Let be an arbitrary oint and be a oint of P. We observe that if the viewing line from to and the outward normal of oint in the oosite direction, must be invisible from. In this case, is called a back oint of. Definition 3.4. Back oint. Given a oint and a oint P. The oint is invisible from if the normal of is ointing in the oosite direction of, i.e., n < 0, ecet the discontinuous oints. Note that Definition 3.4 rovides a necessary condition to identify invisible oints of a guard but not a sufficient condition. In order to find (or aroimate) all visible oints, we have to identify the oints that are occluded by the back oints as well. More recisely, we define the ε-view of a guard as follows. Definition 3.5. ε-view. Given a guard, an ε-view from to a oint is reresented as a cone whose ae is, ae angle is ε and base center is. Any back oint of in s ε-view can occlude the view. Then we can immediately define the occluded oints of a guard. Definition 3.6. Occluded oints. A oint is occluded from a guard if and only if there eists a back oint q in s ε-view and q is closer to than is to. An eamle of the the ε-view is shown in Figure 2. This definition of ε-view hels to identity occluded oints and therefore the visible oints. Using the concet of ε-view, we define ε-visibility. Figure 2: An ε-view of a guard towards a oint. If the oint q is a back oint of, the view is occluded and is not visible by. Definition 3.7. ε-visibility. A oint is ε-visible from a oint if is neither a back oint nor an occluded oint of. Then, a set of ε-visible oints is called the ε-visible region of, denoted as ε-v. ɛ Finally, we say that a oint set P is an ε-aroimate star Points on the boundary of a visible region can be further classified into connected comonents. We call the boundary oints of each comonent discontinuous oints. An eamle is the oint s in Figure 1(b). The discontinuous oints must be refle vertices (whose internal angle is larger than 180 ). The number of discontinuous oints is usually relatively small comaring to the the number of the invisible oints. q shae (or simly ε-a ) if there eists a oint whose ε- visible region is P. Then, as before, an ε-a decomosition of P is simly a set of ε-a s whose union is P. Note that the quality of the aroimation of the true visibility deends on the value of ε. The quality degrades when ε is too large or too small. We will discuss how to comute ε from the oint set in Section 6. In the following section, we will assume that ε is given by the user and start to sketch our method for comuting A decomosition. 4. Aroimate Star-Shaed (A ) Decomosition The framework of the A decomosition is rather straightforward. We first select a oint at random from the oint set P that is not visible from any eisting guards. Then we add as a guard and s ε-visible region as an ε-a to the decomosition. The rocess is iterated until all oints in P are ε-visible by at least a guard. We ick this strategy because of its simlicity and because it has been shown to roduce imressive results for olygons [AMP05]. More imortantly, as we will see in our eerimental results in Section 7, this strategy generates only 1.1 to less than 5 times more guards than any otimal solutions do for some common models in comuter grahics (see Table 1). The main challenge of our A decomosition framework is to aroimate the visible region of a guard. Because of its crucial role in this framework, we will send the entire net section addressing this challenge. 5. ε-visible Region of A Guard In this section, we will discuss methods to comute the ε- visible region of a guard. We first roose a basic method in Section 5.1. Then we will imrove of the efficiency and quality of this basic method in Section 5.2 and in Section 5.3, resectively Basic Aroach This basic aroach is comosed of two stes: Radial artitioning and ε-visible oint identification. Radial artitioning. Given a oint set P and a guard, we use to artition P. To do so, we convert the oints of P to a sherical coordinate system centered at. Then, we artition the sherical coordinate system into 2( π ε )2 equallysized buckets. Each bucket reresents an ε-view of. Finally, each oint of P is assigned to a bucket according to its coordinate. In our imlementation, we use an enclosing bo instead of a shere, and we roject P to the boundary of the bo. An eamle of such a artitioning is shown in Figure 3(a). A benefit of using a bo is its simlicity in etending to high dimensional sace. Find visible oints. We identify ε-visible oints from each bucket. That is, for each bucket, we find the closest back oint of (see the shaded bucket in Figure 3(b)). Then all oints in the bucket that are closer than to are

4 (a) (b) Figure 3: (a) A radial artitioning of the oint set P from a guard. Points in P are rojected to the boundary of a bo defined around. (b) For each bucket, the closest invisible oint is found and all oints that are closer than are considered visible. (c) Darker oints are considered as visible oints from, which collectively form an A of. (c) visible by. Figure 3(c) shows all the ε-visible oints from, which collectively form the A of. Algorithm 5.1 outlines this rocess. Algorithm 5.1: A (P,,ε) comment: Point set P and query oint Assign P into 2( π ε )2 buckets {B i } of s radial artitioning for each bucket B i Comute the closest back oint B i to if eists then d do else d + for each { oint r B i if r < d do then V V r A naïve imlementation of Algorithm 5.1 is of O(n) time comleity for a oint set with n oints. The comutation efficiency can be further imroved if a satial data structure is built on P. Details of this idea will be elaborated in Section 5.2. Moreover, since the guard is selected at random, can have oor visibility. In this case, more guards will be needed to guard the entire oint set. Therefore, in order to reduce the number of guards needed to cover the sace, we find a better from. Details of this method will be discussed in Section 5.3. are artitioned evenly at their center into eight oint sets. (In our eeriment we arbitrarily ick k = 16.) Then a minimum bounding bo is constructed for each oint set as a child of the original bo. This rocess iterates until no boes can be slit. Radial artitioning. Similar to the radial artitioning of the entire oint data, we erform a radial artitioning on the bo tree constructed above. The idea is to traverse the bo tree to-down and find a set of boes that can fit into the buckets. Starting with the bounding bo (the root), we check if all of its vertices belong to the same bucket. If so, we assign the bo to the bucket. Otherwise, we oen the bo and erform the same test for each of the eight child boes and reeat until no unfit boes left. Find visible oints. Similar to the basic method, we find the closest back oint in each bucket. To do so, we first sort the boes from near to far and then start to eamine oints in each bo until we find a back oint. Since checking if a bo fits into a bucket takes only constant time and the number of boes is much smaller than the number of oints, artitioning boes is more efficient than artitioning all oints. Using a bo tree, the comleity of comuting an A becomes outut sensitive, i.e., O(k + m), where k is the number of visible vertices and m is the overhead of building the bo tree and sorting the boes in each bucket Imrove A Efficiency: Bo-tree rerocessing We use a bo tree to re-rocess the inut oint set and to imrove the erformance of the A comutation discussed above. A bo tree is similar to an octree ecet that, in the bo tree, each node is always a smallest (ais-aligned) bounding bo of the enclosed oints. Intuitively, our lan is to lace oints near a guard in smaller boes and lace far away oints in larger boes. With this data structure, we hoe we can find the closest back oint in the nearby small boes without checking the far away large boes. Bo tree construction. The construction of the bo tree starts with the minimum bounding bo of the entire oint set. If there are more than k oints in a bo, these oints 5.3. Imrove A Quality: A Eansion Our goal here is to find a better guard than the randomly selected guard. Here, better means larger visible region. Our strategy is to comute the kernel of the visible oints of a guard and find a guard from the kernel. We define a kernel of a oint set as the following. Definition 5.1. A kernel of a set oints P is another set of oints K. Every oint of K can see all the oints of P. The kernel K of a set of oints V can be comuted as the intersection of half-saces defined by the oints in V (see Figure 4). The key to comute the kernel efficiently is that we can convert these half-saces to the oints V in the

5 K V Figure 4: K is the kernel of the oints V. K is simly the intersection of the half saces (shown as the lines and arrows in the figure) defined by the oints (shown as dark oints in the figure) and oint normals. dual sace, where V is defined as the following: V = { nq q V }. q n q Then the boundary oints in the kernel K are simly dual to the facets of the conve hull of V, which can be comuted efficiently in O(nlogn), where n is the number of oints in V. Lemma 5.2. Let H i = { ( i ) n i < 0} be a half sace defined by a oint i V. The kernel K of V is i H i. Proof. We know that if a oint i V is visible from a oint, the view direction i and the normal of i oint in the same direction. For to be a member of the kernel K, this criterion must hold for all oints of V, i.e., i n i > 0 = ( i ) n i < 0, i, i.e., i H i. Once the kernel is known, we comute a visible region for each verte in the kernel. Since the visible regions of the vertices of the kernel are larger than or equal to V (P) (see Lemma 5.3), our visible region must eand monotonically. This rocess is reeated until no eansion can be gained. The subroutine A -EXPAND is defined in Algorithm 5.2. Algorithm 5.2: A -EXPAND(,V ) Comute the kernel K from V Find k K so that k has the largest visible region V k if V k > V then return (A -EXPAND(k,V k )) else return (,V ) Lemma 5.3. Given a oint and its visible oints V, Algorithm 5.2 must return a guard whose visible region is no smaller than V. Proof. Let K be the kernel of V and let K. Because can see all the oints in V, V V. Therefore V V. It is clear that the time comleity of comuting A - EXPAND is higher than comuting A. One ass of A - EXPAND without the recursive call takes O(n 2 + nlogn) time. In the worst case, only one more oint becomes visible by alying one ass of A -EXPAND. When this haens, the recursion deth is O(n) and the total time comleity for A -EXPAND is O(n 3 ), which is not ractical for most roblems that we are interested in in this work. To reduce the time comleity, we modify Algorithm 5.2 so that only log K oints randomly selected from the kernel K are considered and the recursion stos when the imrovement is less than P /c oints, where c is a user-defined constant. This heuristic is outlined in Algorithm 5.3. Algorithm 5.3: A -EXPAND2(,V ) Comute the kernel K from V Let K contain log K random vertices from K Find k K so that k has the largest visible region V k if V k > V + P c then return (A -EXPAND2(k,V k )) else return (k,v k ) The time comleity of Algorithm 5.3 now becomes O(nlogn). Eerimentally we observe that A -EXPAND and A -EXPAND2 roduce similar results while A -EXPAND2 is much more efficient. For the rest of this aer, A -EXPAND2 is used (if not said otherwise) to comute the A of a given oint. 6. Putting It All Together Algorithm 6.1 shows a fleshed-out version of the A decomosition. Note that in the last ste of Algorithm 6.1 we simle assign each oint to its closest visible guard to roduce the final decomosition. Figure 5 shows an eamle of the A decomositions of a 3-d oint cloud. Algorithm 6.1 has O(knlogn) time comleity, where n and k are the number of oints in P and guards, resectively. Algorithm 6.1: A -DECOMP(P) build a bo tree from P reeat randomly select a oint P invisible by any guards build ε-v of using the bo tree (g,ε-v g) =A -EXPAND2(,ε-V ) add (g,ε-v g) to the decomosition until every oint in P is visible Assigneveryointtoitsclosestvisibleguard Now, the remaining task is to find the value of ε from a given oint set. As we mentioned before, the value of ε has great influence on the quality of the final decomosition. A small ε makes invisible oints visible while a large ε eliminates many visible oints. Our goal is to find the smallest ε that will not classify invisible oints as visible.

6 eternal medial ais q r Figure 6: There must be a samle, such as the oint, in the first intersection of s ε-view and the olygon to ensure that q will not become a visible oint of. Figure 5: A decomositions of the oint data of a horse model (with A -EXPAND2). Each of the 17 large dots in the figure reresents a guard. (This figure is much informative in color. Please refer to the PDF file if you have a black and white rintout.) 6.1. The value of ε The value of ε deends on how dense the oint set P is samled from the boundary of a shae S and the eternal medial ais of S. Lemma 6.1 uses the samling density to comute the value of ε. Lemma 6.1. When ε = ma i {2 arctan( 4γ δ i )}, where δ is the samling density of P and γ i is the distance from a oint i P to its closest oint on the medial ais, Algorithm 6.1 will never classify invisible oints as visible oints. Proof. We show this statement is true by assuming that ε is known and show that if no invisible oint is missed by Algorithm 6.1, then the samling density is 4γ i tan( 2 ε ). Figure 6 shows an eamle of the worst scenario, in which the guard is laced on the boundary of the shae. In addition, s ε-view is aligned with its outward normal and the first segment of the shae blocking the view is erendicular to the view. This make the intersection of the view and the segment shortest. To make Algorithm 6.1 recognize this boundary, there must be a oint samled from the intersection, e.g., the oint in Figure 6. Otherwise, will see through this boundary and classy some invisible oints as visible, e.g., the oint q. Therefore the length of the intersection of the ε-view and the invisible boundary defines the samling density δ. It s easy to see that the length of the intersection is 4γ tan( 2 ε ), where γ is the distance between the eternal medial ais and. Thus, we conclude that if ε = 2 arctan( 4γ δ ) all back oints will be found and no invisible oints are visible by. Both samling density δ and distance to the medial ais γ i can be estimated from the oint set. To estimate, δ, we comute the maimum of the longest distances between each oint and its k nearest neighbors. (We simly set k = 4.) We estimate γ i as the half of the distance between the oint i and its closest back oint. 7. Eerimental Results We imlemented the roosed method in C++. In this section, we show eerimental results from our imlementation using seven common models, which are converted to oint set data and shown in Table 1. We show the the number of guards and comutational efficiency of the roosed methods. We also comare to the estimated lower bound of guards for each oint set. The size of A decomosition is close to the lower bound. We comute the lower bound as the maimum number of guards such that no oints can see more than one guard. Under this definition, we know that no visible regions of any guards can overla and therefore this lower bound must be smaller than the minimum number of guards covering the entire oint set. The estimated lower bound for each oint set is shown in Table 1. Table 1 also shows the averaged number of guards generated by A decomosition over 10 runs. For the cubes, horse, bunny, venus and armadillo models, A decomosition generates just 1.1 to 2.7 times more guards than the lower bound does. Moreover, when the shae of the model is relatively fat (e.g., the cubes, the bunny and the venus models), A decomosition generates only 1.1 to 1.5 times more comaring to the lower bound. On the other hand, A decomosition generates 2.7 times and 4.9 times more guards than the lower bound of the head bone and the david models. The main reason for this is because the lower bound estimation is not accurate enough. This haens when the model has teeth -like or ocket-like regions, e.g., the teeth of the bone model and the ockets around the hair area of the david model. A guard laced at the based of the teeth will rohibit any guards being laced inside the teeth. Figure 7 illustrates this scenario. A eansions reduce decomosition size. Recall that, in Section 5.3, we roosed two A eansions (denoted as full and limited eansions). A eansion is designed to imrove the visibility of a randomly selected guard and to

7 Table 1: Models used in the eeriments. A decomositions of the models are generated with limited eansion. (cubes) (bunny) (venus) (armadillo) (head bone) (david) oint set cubes horse bunny venus armadillo head bone david (above) (Fig. 5) (above) (above) (above) (above) (above) oint size number of A -DECOMP (avg.) guards lower bound Figure 7: In the lower bound estimation, the guard will rohibit us from adding more guards, while the minimum number of guards to cover the environment is seven. Time (sec) no e full e limited e reduce the size of the final decomosition. Figure 8 shows the comutation time and the number of guards. For the limited eansion, we sto eanding when the eansion is not larger than 0.1% of the inut oint size. As you can see from Figure 8, the number of guards generated without any eansion is significantly (2 to 5 times) larger than that with eansions. Moreover, the difference between the guard sizes of the full and the limited eansions is negligible while the limited eansion is always (1.3 to 3 times) faster than the full eansion. Bo tree hels imroving efficiency. The bo tree described in Section 5.2 significantly imroves the efficiency. Figure 9 shows that the imrovement is larger than 100%. Partial coverage. When artial visibility coverage is tolerable, A decomosition can be comuted much more efficiently. In fact, according to Figure 10, if only 90% coverage is needed the number of guards and the comutation time can be halved. Both the size of guards and the comutation time grow eonentially as the coverage ercentage increases. 8. Conclusions In this aer, we roosed a method to decomose a oint set into a set of aroimately star-shaed (A ) comonents. We roosed a method to comute ε-visibility, where ε can be estimated from the inut oint set. Several strategies are investigated to imrove the quality and the efficiency of the A decomosition, namely the bo tree and A eansion. Our eeriments demonstrated these imrovements using seven common models. guards no e full e limited e lower bound cubes horse bunny venus armadillo head bone david Figure 8: Comutation time and guard size with three different A eansions. Each record shows an average of 10 eeriments. Notice that the Y-ais is in logarithmic scale. without bo tree with bo tree Time (sec) 3500 Figure 9: Decomosition time of all seven models with and without bo trees. The major limitation of the roosed method is its efficiency. The comutation can take u to several minutes (4.6 and 3.9 minutes are needed for the bone and the david models, resectively). Fortunately, due to the oularity of the recent multi-core rocessors, we can address this issue by

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