Growing Self-Organizing Maps for Surface Reconstruction from Unstructured Point Clouds

Transcription

1 Growing Self-Organizing Map for Surface Recontruction from Untructured Point Cloud Renata L. M. E. do Rêgo, Aluizio F. R. Araújo, and Fernando B.de Lima Neto Abtract Thi work introduce a new method for urface recontruction baed on Growing Self-organizing Map, which learn 3D coordinate of each vertex in a meh a well a they learn the topology of the input data et. Each map grow incrementally producing mehe of different reolution, according to the application need. Another highlight of the preented algorithm refer to the recontruction time, which i independent from the ize of the input data. Experimental reult how that the propoed method can produce model that approximate the hape of an object, including it concave region and hole, if any. T I. INTRODUCTION HE problem of recontructing the urface of object from a point cloud i quite common in many area uch a CAD deign, virtual reality and medicine. The urface recontruction method aim to create a model to reproduce an object hape given a et of point of the urface of thi object. The large number of point to be proceed contitute an ubiquitou difficulty, i.e. a challenge that recontruction method mut face. Depending on their purpoe, ome application of urface recontruction require accurate and detailed model while other require model with le vertice and polygon to be effective. Then, a good recontruction method mut be able to produce model according to different reolution of the target object, depending on the problem; to avoid further proceing uch a implification or refinement procedure. The ample point ued for recontruction can be claified a tructured or untructured depending on the exitence or abence of connectivity information, repectively. One of the mot challenging tage of urface recontruction method applied to untructured point cloud i to obtain the correct connectivity about the ample point. Some recontruction method uch a Deformable Model [1], etablih the meh topology beforehand. The main pitfall of thi approach i the topologie of the produced mehe which are equivalent to the pre-defined one; ometime, producing poor reult. For intance, a pre-defined topology with no hole can produce only mehe without hole. Renata Rêgo and Aluizio F. R. Araújo* are with Center of Informatic, Federal Univerity of Pernambuco, Recife/Brazil. *Contact phone: #4321 and aluizioa@cin.ufpe.br. Fernando Buarque de Lima Neto (IEEE M 2006) i with Department of Computing Sytem at Polytechnic School of Engineering - Pernambuco State Univerity, Recife/Brail Many well etablihed technique, propoe olution to the urface recontruction problem from a geometric point of view. Thee algorithm require long proceing time for the input point cloud and do not cale up properly with very large point cloud [2], [3]. A different perpective for urface recontruction i to look at it a a learning problem. Under thi perpective ome method were propoed baed on elf-organization [4], [5], [6], [7]. Thee method can learn the coordinate of the meh vertice and can handle very large input point cloud. The method draw the point cloud and ue the ample a training data. During the learning proce, the vertice of the meh have their poition adapted to fit the input data. In thi particular ene, learning-baed urface recontruction method are equivalent to deformable model once they change the hape of an initial model toward the hape of a target object. Previouly propoed recontruction method baed on elf-organizing map include Kohonen SOM [4], [5], Growing Cell Structure [6], and Topology Repreenting Network [7]. However, they can not imultaneouly learn the input data topology and grow incrementally to produce mehe with different reolution a Growing Neural Ga (GNG) [8] can do. We propoe a urface recontruction method baed on GNG to learn topology through elf-organization. Furthermore, thi alternative can produce mehe with different reolution becaue the meh grow incrementally during the learning proce. The tandard GNG learning algorithm, however, produce only wireframe, o the propoed method include ome change in thi algorithm in order to produce triangular mehe. Thee change include an extenion of the Competitive Hebbian Learning, the ue of GCS vertex inertion operation and a different operation to remove edge from the mehe. Experimental reult how that the propoed method learn vertice coordinate and the connectivity among thee vertice building mehe to reproduce the hape of target object. The learning proce ucceively generate mehe with increaing number of node and thu different reolution. Thi paper i organized a follow: Section II preent ome previou work on urface recontruction: elforganizing map and reaon why they are uitable for olving urface recontruction problem, and their actual ue in recontruction method. Section III decribe the propoed

2 method. The experiment and reult are preented and dicued in Section IV. Thi paper concluion, including achievement and limitation of the propoed method, i in Section V. II. PREVIOUS WORK An active reearch topic i the problem of urface recontruction from untructured point cloud. A number of approache handle thi problem imply by geometric perpective, an example of that i the approach propoed by Hoppe [2]. Another relevant approach to urface recontruction i baed on deformable model [9] that are lowly rehaped toward the target object. Such a recontruction employ energy evaluation or force function to ae each produced hape. The proce contruct mehe to approximate the hape to the point cloud. Example of thi approach can be found in the review work of Montagnat et al.[1]. A third perpective to urface recontruction i to undertand it a a learning problem [10].According to thi approach all neceary information to recontruct a urface can be learned from the available data. Different elforganizing map are ued a learning method to olve a urface recontruction problem. A. Self-organizing Map Self-organizing map (SOM) were originally propoed by Kohonen [11] for the viualization and abtraction of highdimenional data. SOM baic tructure i formed of an input and an output layer. Uually the topology of the output layer conit of a two-dimenional grid of node. Each output unit i connected with all input node through a weight vector. The learning algorithm of SOM i compoed by three main tep: competition to identify a winner, cooperation to correlate activation of a winner and it neighbor, and adaptation to update the weight vector. The ability of SOM to contruct topological map from an input data ditribution can be employed in olving the urface recontruction problem. Depite thi ability, predefined number of node and the connection between them contraint the accuracy of the topological map produced by SOM [12]. Model propoed to overcome thi limitation can be ued for urface recontruction. The Topological Repreenting Network (TRN) [12] ha the network ize pre-defined to contruct topology preerving map. TRN i een a a combination of Neural Ga (NG), placing the node according to the probability ditribution of input data, with Competitive Hebbian Learning (CHL), building a topology with thee node, i.e. to perform topology learning [13]. Such ability i a uitable feature to be ued a a olution for the problem of topology detection of a point cloud in urface recontruction method. Depite the favorable characteritic, SOM and TRN hare a relevant limitation: the previou choice of the number of node for the map. A olution to thi drawback wa to build map growing incrementally a thoe contructed by Growing Cell Structure (GCS) [14] and Growing Neural Ga (GNG) [8]. The former generate map coniting only of baic building block, hypertetrahedron of dimenionality k choen in advance. Unlike the GCS, a map generated by GNG may have node with different connectivity and the topology may have different dimenionalitie in different part of the map. A the TRN, the GNG i able to learn the topology of input data through Competitive Hebbian Learning [13]. Hence, GNG can be een a a GCS variant without it topological retriction or a a growing verion of TRN. The ue of SOM, TRN and GCS in the urface recontruction problem ha already been tudied, a number of achievement were obtained, however ome limitation till perit a they are dicued in the next ub-ection. B. Self-organizing Map for Surface Recontruction Self-organizing map are ued in recontruction method becaue they can generate a map that cloely matche the hape of a target object which ha it hape repreented by a point cloud. The elf-organization proce demand only the patial coordinate of the input point in order to execute the recontruction. The recontruction proce of method baed on elforganizing map ue elected point within the input point et a training data for the learning algorithm. When elforganizing map are ued for recontruction, the meh vertice correpond to the output node of the map. The weight vector of the network determine the vertice poition in the meh wherea the connection between the node correpond to the edge of the meh. So hereafter the term map and meh, node and vertex, and connection and edge, will be ued interchangeably. Surface recontruction method baed on Kohonen SOM ability to ditribute the vertice in a polygonal meh are preented in variou work [4], [5], where (i) the topology of the meh i predetermined and (ii) Kohonen learning algorithm i carried out to obtain correct 3D coordinate of every vertex of the meh. Depite all achievement, SOM preented difficultie to recontruct concave region [4], [5]. That i, after the learning proce ome vertice or triangle may be untable, dangling among region denely populated by the data. To avoid uch problem thee method employ ome meh operation: edge wap, edge collape, vertex plit and triangle ubdiviion. Furthermore, both work ue meh refinement algorithm to obtain a new meh with more element, once SOM doe not inert new node during the learning proce. Ivriimtzi et al. [6] ha propoed a recontruction method, called Neural Mehe, baed on GCS. A traight forward conequence i that Neural Mehe pre-define neither the number of node nor the exact topology; however, the generated mehe are alway topologically

3 equivalent to the initial meh, uually a tetrahedron. To overcome thi limitation, a topology learning tep in Neural Mehe algorithm wa introduced [15] to enable the formation of handle and boundarie of a urface. Boundarie are then created by removing triangle and handle are created by merging thee boundarie. The Neural Mehe grow incrementally, hence, uch a method generate mehe of different reolution at different tage of the learning proce. So, differently from other SOM-baed method, Neural Mehe do not need meh refinement algorithm for generating multireolution mehe. The Extended Neural Ga (ENG) [7] ue an extenion of Competitive Hebbian Learning to dicover the topology of the input point cloud. Thi extenion allow ENG to generate not only the connection between node, a the tandard CHL doe, but alo the triangular face of a triangular meh. ENG create a meh that i not a 2-manifold urface [7] becaue the topology learning proce doe not prohibit the exitence of ome anomalie uch a the occurrence of edge hared by more than two face, of edge diconnected from any face, and of face oriented toward different direction. To olve thee anomalie, in ENG there i a manifold creation algorithm which i applied to the non-manifold mehe produced by ENG. A a TRN-baed method, the number of node i fixed in ENG, o that the meh reolution of the recontruction generated with thi method mut be previouly defined. The promiing reult obtained with the neuralrecontruction method how that elf-organizing map are uitable for the tak of generating mehe to repreent the hape of a particular object given only a et of patial coordinate of point belonging to the object urface. However, the mentioned method till have limitation, uch a (i) to require previouly defined number of node and (ii) incapacity to learn topology due retriction for connection change from the initial configuration. III. PROPOSED SOLUTION In thi ection, we introduce a new method to detect the topology of the point cloud, a ENG doe, and to generate mehe with different reolution during the learning proce, a Neural Mehe doe. The neural olution propoed here i an extenion of the Growing Neural Ga and imultaneouly ue the vertex inertion operation employed by Growing Cell Structure. We found that thi combination of feature in addition to other modification carried on the tandard GNG i able to generate polygonal mehe, compoed by vertice, edge and face, intead of the wireframe compoed only by vertice and edge. We refer to the propoed method a Growing Self- Recontruction Mehe (GSRM). The peudo-code of GSRM learning algorithm and parameter are introduced and dicued below. The learning parameter of GSRM are ε n (winner learning rate), ε b (winner neighbor learning rate), λ (number of adaptation tep until a new node inertion), β (decreaing rate for all error counter), α (decreaing rate for the node between which a new node have been inerted), age max (maximal age for an edge to be conidered valid), nv (top criterion number of node in the map). The main proceing tep of GSRM are detailed below: 1. Start the map with a et A of three node with weight vector randomly choen from the input point et; 2. Draw an input point ξ from the point cloud P; 3. Find the firt ( w ), the econd ( w ( w ), and the third ) bet matching node (i.e. winner) of the map, that i, the three vertice (weight vector) of the map with hortet ditance from ξ ; w ξ w ξ A (1) 1 i i w ξ w ξ i A { 1} (2) i 2 w ξ w ξ i A { 1, 2} (3) i 3 4. Create (or reinforce) connection between thee node, and triangular face of the meh, according to the extended Competitive Hebbian Learning ; 5. Update the winner error counter according to Equation (4): 2 E = w ξ 1 (4) 1 6. Move 1 and it topological neighbor toward ξ with learning rate ε b e ε n, repectively: w w 1 i where N = ε ( ξ w b = ε ( ξ w n 1 1 i ) ) N i 1 i the et formed by the neighbor of 1 (5), (6) 7. Update the age of all edge emanating from 1 according to Equation (7): age = age +1 (7) 8. Remove invalid edge, i.e. thoe edge with age greater than age max, and their incident face; 9. If the number of input ignal preented o far to the network i an integer multiple of the parameter λ, then inert a new node. A. Decreae the error variable of q e f : E = αe E = αe (8), (9) q q f B. Interpolate the error variable of r from q e f : f

4 E = 0.5( e + E ) (10) r q f where: q i the node with the highet error counter, f i the q neighbor with highet error counter and r i the newly inerted node. 10. Decreae the error variable of all node, according to Equation (11): E = β E ( A) (11) 11. If a topping criterion (e.g. net ize or ome performance meaure) i not fulfilled, continue from tep 2. GSRM conider the number of node in the map a topping criterion a well. The main difference between GSRM and the tandard GNG are found in tep 4, 8, and 9. Such modification involve extenion of the Competitive Hebbian Learning, the procedure for edge removal and the vertex inertion operator. Thee difference are further dicued below. A. Extended Competitive Hebbian Learning (ECHL) In the Growing Self Recontruction Mehe, CHL i extended to create the topological face. The ECHL i decribed a: 1) For each ample preented to the map, three winner node are determined, i.e., node whoe weight vector are nearer to the ample then all the other node; 2) If there i not a ynaptic connection between each pair of thee node, thi connection i then created. If any of thee connection previouly exited thi fact i reinforced by the etting of it age to zero; 3) If the created or reinforced connection do not form a triangular face, uch a face i then created. B. Edge and Incident Face Removal During tep of weight vector adaptation, ome of the edge generated in the network may become invalid and are removed. A in GNG, an edge aging cheme i ued here. Edge whoe age become higher than a given threhold are automatically excluded from the meh. In tandard GNG thee edge are imply removed and o are the vertice without incident edge. In GSRM the elimination of edge implie in the removal of their incident face a well. Thi becaue thee face can not exit without any of the incident edge. The excluion occur a follow: 1) Remove face incident to the invalid edge, i.e. with age larger then age max. 2) Remove edge without incident face. The invalid edge are removed here becaue their entire face have been removed in the previou tep. Some other edge without incident face can be removed a well. 3) Remove vertice without incident edge. C. Vertice Inertion In the tandard GNG, a new vertex ( r ) i inerted between two exiting vertice ( q and f ). The original edge connecting q and f i removed and two new edge, connecting q and f to r are then created. Thi operation doe not create new triangular face when a new vertex i inerted. To overcome thi retriction, GSRM inert a new vertex in a imilar way to GCS. Figure 1 illutrate thi operation. The inertion of a new vertex in the meh happen a follow: 1) Inert a new vertex ( r ) and initialize it weight vector with the average of q and f. 2) Create edge connecting r to q and to f and to their common neighbor (hatched edge in Figure 1(b)). 3) Replace each face incident to the edge between q and f by two new face, one incident to the edge between r and q and the other incident to edge between r and f. 4) Remove the original edge connecting r e f (hatched edge in Figure 1(a). (a) (b) Figure 1 - (a) Original Meh (b) Meh after a new vertex inertion. IV. EXPERIMENTS Thi ection preent the reult of urface recontruction carried out by the propoed neural recontruction method applied to three ynthetic object: the Hand, the Max-Planck face, gently granted by Ionnai Ivrimitzi, and the Stanford Bunny, available on the internet (i.e. at Stanford repoitory 1 ). The point cloud contain only the patial coordinate of vertice in the original meh, no additional informational about the connectivity information among thee vertice are ued for the recontruction that are preented bellow. In all experiment preented here the mehe have five thouand vertice. The experiment carried out in thi work aim at howing the three major achievement of our new method: (1) The propoed learning algorithm for recontruction can learn (a) vertice coordinate and (b) the connectivity among the vertice, a GNG doe. Thi in addition to producing triangular mehe, a GCS doe, while tandard GNG only produce wireframe and GCS impoe retriction on the meh topology; (2) The propoed method i able to produce mehe with different reolution throughout learning proce; (3) The recontruction time of the propoed approach doe 1 The Stanford 3D Scanning Repoitory can be reached on the internet at acceed on October, 2006.

5 not depend on the point cloud ize. The Haudorff ditance [16] wa ued to compare the ditance between original and generated mehe with: GSRM, GCS (tandard), and Neural Mehe. The rationale i: the maller the ditance, the bet a recontructed meh repreent the original one. Mehe produce with ENG could not be compared becaue the comparion metric ued wa unavailable. A GNG doe not produced mehe (only wireframe), o the Haudorff ditance could not be calculated either. Figure 2 how the wireframe produced by GNG that approximate the hape of the recontructed object but are not polygonal mehe. Figure 3 illutrate the topological retriction of the GCS. Figure 3(a) how that thi map can not recontruct concave region (ee fale bridge between the finger) and Figure 3(b) how that GCS can not recontruct hole, ee the counter-example at the bottom of the hand. (a) (c) (b) (d) Figure 2 - Wireframe (5k vertice) produced with tandard GNG (e) (f) Figure 4 Hand, Max Plunk, and Bunny recontruction obtained with GSRM. (a) (b) Figure 3 Hand recontructed with tandard GCS [14]. Compare the two previou figure with reult preented in Figure 4, which are recontruction produced by GSRM. Note the remarkable learn ability of the preented method that can be evoked through the recontruction of concave region, in chart (a), (c) and (e), and, of hole a een in the chart (b), (d) and (f) thee three latter are bottom view of image preented in chart (a), (c) and (e), repectively. Due to it fractal growing, GSRM can generate mehe of different reolution at ditinct training tage. So it need not necearily to ue any refinement or implification method. Figure 5 how an example of thi feature. The learning time i independent of the number of point from the point cloud, once ampling i the only tep in which the point cloud i proceed and it i not amount depend. Table 1 how the Haudorff Ditance between the original mehe and thoe recontructed with GCS, GSRM (calculated with Metro Tool [16]), and Neural Mehe (taken from [10]). (a) (b) (c) Figure 5 - GSRM Fractal Growing with different reolution at ditinct learning tage: (a) 20 vertice, (b) 100 vertice, (c) 500 vertice. TABLE 1 HAUSDORFF DISTANCES BETWEEN THE ORIGINAL MODELS AND THE RECONSTRUCTION OBTAINED WITH GSRM AND NEURAL MESH. GCS GSRM Neural Meh Max Planck Hand Bunny The reult confirm that GSRM produce better reult then tandard GCS. Thi i becaue point filling concave region and hole in GCS mehe do not have correpondence with point in the original mehe. Although the overall number preent better reult for the recontruction of the Neural Meh, there are ome important remark: 1. The number of vertice ued in the Neural Meh model wa 20 thouand veru mere five thouand for the GCS,

6 GSRM model. 20k model wan t generated for the GSRM due to time retriction, a will be dicued later; 2. The reult preented for the Neural Meh doe not relate to the original method propoed by Ivriimtzi in 2003 [6]. It i related to another method alo propoed by Ivriimtzi in 2004 [15], when a topology learning tep wa introduced. Notice that the original Neural Meh i baed in GSC and, a aid before, thi network can not learn topology. Although the propoed method could recontruct the hape of target object, the mehe generated have ome limitation that could be olved by pot-proceing thee mehe in a future work. The limitation concern to mehe that are not 2 manifold, and undeirable hole, that can be eaily ditinguihed from the urface hole. The manifold meh creation algorithm propoed by the author of ENG [7], could be applied to olve the limitation of GSRM mehe. There are alo other algorithm that create manifold from non-manifold polygonal mehe, a the one propoed in [17]. Once the meh become a manifold, the Haudorff ditance between the original meh and the one generated with GSRM (hown in Table 1) hould decreae due the elimination of undeirable face whoe point doe not have correpondent point in the original meh. Another limitation of the method i that the training time increae with the current number of vertice in the meh. Thi i becaue of the winner earch at tep 4 of peudocode, the earch for the highet error counter node at tep 11 and the updating of each node error counter at tep 12. The winner earch time could be reduced with an octree baed earch [6]. Step 11 and 12 could be fater, refer to [10]. V. CONCLUSION AND FUTURE WORK The neural approach put forward in thi article ucceeded in recontructing urface model of the 3D object from point cloud repreenting their hape. The highlight of the propoed method are (i) topology learning, which i a challenging feature for recontruction method, (ii) time of recontruction independent of the ize of the point cloud and (iii) model generated at different reolution. The main problem detect for the method o-far are: (i) non 2-manifold mehe, (ii) mall amount of undeirable hole and (iii) prolonged learning time when dealing with many thouand of vertice. At thi point we have devied poible olution for the firt two problem by mean of pot-proceing of the meh with manifold creation algorithm a propoed at [7] and [17]. A for the ometime elevated learning time problem, it could be olved with algorithm that make the winner earch fater, a the octreebaed earch [6] and olution that could faten other tep of the algorithm, refer to Saleem [10]. VI. REFERENCES [1] J. Montagnat, H. Delingette, and N. Ayache. 'A review of deformable urface: topology. geometry and deformation," Image and Viion Computing, vol.19, [2] H. Hoppe, T. 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