PARTITIONING POSITIONAL AND NORMAL SPACE FOR FAST OCCLUSION DETECTION

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1 Proceedngs of DETC 3 ASME 23 Desgn Engneerng Techncal Conferences and Computers and Informaton n Engneerng Conference Chcago, Illnos USA, September 2-6, 23 DETC23/DAC PARTITIONING POSITIONAL AND NORMAL SPACE FOR FAST OCCLUSION DETECTION Xaopng Qan qan@crd.ge.com Inecton & Manufacturng Technologes GE Global Research P.O. Box 8, Nskayuna, NY 239 Kevn G. Hardng hardng@crd.ge.com Inecton & Manufacturng Technologes GE Global Research P.O. Box 8, Nskayuna, NY 239 ABSTRACT Occluson detecton s a fundamental and mportant problem n optcal sensor necton plannng. Many vewplannng algorthms have been developed for optcal necton, however, few of them explctly develop practcal algorthms for occluson detecton. Ths paper presents a herarchcal ace partton approach that dvdes both postonal and surface normal ace of an object for fast occluson detecton. A k-d tree s used to represent ths partton. A novel concept of δ occluson s ntroduced to detect occluson for objects n an un-organzed pont cloud representaton. Based on the δ occluson concept, several propostons regardng to a range search on a k-d tree have been developed for occluson detecton. Implementaton of ths approach demonstrated that sgnfcant tme can be saved for occluson detecton usng the partton of both postonal and surface normal ace. Keywords: Vsblty, Occluson Detecton, Sphercal Map, K- d Tree ntroducton Optcal sensors have been used n many 3D shape measurement applcatons such as optcal metrology and reverse engneerng. Effectve and effcent necton plannng of these sensor systems s a crtcal task n optcal applcatons. Many plannng systems have been developed for ths purpose. The task of necton plannng s to seek a set of vewponts for the sensor so that all the ponts on the part can be nected optmally from these vewponts. It nvolves the determnaton of extrnsc and ntrnsc parameters for the sensors, optcal llumnaton parameters for the lasers, and optmal placement of lasers and sensors so that part surfaces can be nected wthout occluson or collson. In ths paper, we focus on one aect of necton plannng: developng fast algorthms to ensure occluson-free necton. Camera Laser Measurement volume Part Fgure A vald necton pont for an optcal necton system To ensure a pont on an object s nectable, ths pont must satsfy certan requrements (Fgure ). They nclude ) the pont must le wthn the measurement volume MV, formed by the feld of vew (FOV) and the depth of feld (DOF); 2) the pont must be vsble to both the laser and the camera, 3) the pont must be accessble, that s, there should be no occluson between the pont and the laser or between the pont and the camera. The second condton s often referred to as a vsblty constrant. In ths paper, we refer to a pont as locally vsble when the angle of ncdence of ths pont (angle between the surface normal of a pont and the lne-of-sght formed by the pont and the laser or camera) s smaller than 9 degrees. We refer to a pont as globally vsble when ) the angle of ncdence of ths pont s smaller than 9 degrees, and 2) no pont n the object occludes ths pont. From a gven set of P Copyrght 23 by ASME

2 ponts that are locally vsble to both a laser and a camera, we need to fnd out those ponts that are not occluded by any object. In other words, we need to fnd all the ponts that are globally vsble. These are the ponts that can be nected under a gven vewpont. In a typcal sensor necton plannng system, canddate vews are generated frst. A model coverage test s then conducted to ensure the object s fully covered. Often tmes, the occluson check s not explctly explored. Thus, some locally vsble ponts may not be nected due to occluson. Ths paper presents an effcent approach for fast occluson check based on ace partton. Poston ace partton s often used for collson detecton. However, the occluson detecton nvolves the surface normal check. The approach n our paper s based on a herarchcal parttonng of both poston ace and surface normal ace of an object. The object to be nected s represented n a tessellated model wth each pont havng a poston coordnate and a surface normal. We transform the occluson problem usng the δ occluson concept, so that we can use a k-d tree based ace partton for occluson detecton. For a constructed k-d tree, we use the lne-of-sght to ntersect the 3d bucket. A k-d tree of sx dmensons, consstng of the poston value and the surface normal of each pont, s constructed for the object. For each canddate vew pont, the problem of fndng the ponts from the object fallng nto the measurement volume and satsfyng the vsblty constrants s transformed nto a range search problem n a k-d tree. In the remanng of ths paper, Secton 2 revews pror work on necton plannng and occluson detecton. Secton 3 detals our approach, ncludng δ occluson, a smple algorthm for pont nectablty check and an mproved algorthm based on ace partton. Secton 4 presents the mplementaton of the algorthms and computatonal results. The paper s concluded n Secton 5. 2 Lterature Revew Many sensor plannng systems have been developed for the best placement of sensors and parts to ensure full model coverage, to enhance necton qualty and to decrease buld tme. The plannng methods behnd these systems largely fall nto two categores: ) a generate-and-test approach, and 2) a synthess approach (Tarabans 995). In a generate-and-test approach, the sensor confguratons are generated and then evaluated wth reect to task objectves. In order to lmt the number of canddate sensor confguratons, the confguraton ace s dscretzed. For example, an approach usng known magng sensors and feature-based object model to compute the optmal postons for necton tasks s reported n (Trucco 997). In the synthess approach, the task requrements are characterzed analytcally and the sensor parameter values that satsfy the task constrants are drectly determned from these analytcal relatonshps. For example, a computatonal approach for best sensor setup to mnmze sgnal dynamc range s reported n (Qan 23). In all these necton plannng systems, computng occluson-free vew ponts s a complex task and consumes a lot of computng tme. Most of the necton plannng systems do not provde effcent methods for occluson detecton. For example, a general plannng system s developed for laser scannng, n whch occluson check s done for every vew ponts n the scan plan, but wth no explct mentonng of the algorthm s effcency (Son 22). A cone vsblty based method s used to determne a near optmal decomposton of free form surfaces accordng to surface normal (Elber 998). In the context of NC machnng, hercal geometry based vsblty has been extensvely explored (Chou 992). However, n these approaches, only local vsblty s addressed and global vsblty s not addressed. Methods to calculate the occluson-free loc n terms of boundary representaton and CSG representaton were developed n (Tarabans 996). Exact aect graphs have also been used to analyze CAD models to obtan exact and contnuous vsblty regons coverng the whole ace (Pettjean, 99). However, these methods are rarely appled n practce, n part because of ther computatonal and representatonal complexty (Trucco 997). A concept smlar to occluson detecton s collson detecton. If the geometry of two objects s ntersectng wth each other, there s collson between these two objects. Many algorthms have been developed for effcent collson detecton. Fast collson detecton algorthms typcally employ a dvde-and-conquer scheme by sub-dvdng the object s 3D ace (Ln 998). For example, a tessellated representaton wth a herarchy ace dvson s developed for collson detecton n (Gottschalk 996). In ths approach, they used a Dscrete Orentaton Polytope (DOP). A herarchcal adaptve ace subdvson scheme, the BoxTree, and an assocated dvdeand-conquer traversal algorthm were developed for nteractve vrtual prototypng (Zachmann 997). Typcally herarchcal schemes outperform the non-herarchcal schemes as long as the herarches are not requred to re-buld dynamcally. Occluson check dffers from collson check n that occluson can happen even when there s no collson. Occluson happens when lne-of-sght from a source pont s ntersectng wth any object geometry before t hts the target pont. In collson detecton, typcally only poston ace partton s requred. In the paper, we explore the partton of both postonal and surface normal ace partton for fast occluson detecton. In ths paper, we use a k-d tree (k-dmensonal bnarysearch tree) based representaton for parttonng both postonal and surface normal ace. A k-d tree s a common data structure used to fnd a pont s closest neghbor n a pont set (Fredman 977, Preparata 988). A k-d tree parttons ace nto a set of buckets. A hyper-here s then used to ntersect the bucket to fnd the closest neghbor. The computatonal complexty s O(nlogn) for constructng a k-d tree, and O(logn) for searchng such a tree. 3 Overvew of The Fast Occluson Detecton Approach In ths secton, we frst ntroduce the concept of δ occluson. We then present a smple algorthm for pont nectablty check and an mproved algorthm based on ace partton. 2 Copyrght 23 by ASME

3 3. Proposton for pont cloud based occluson test p x x 2 Lne of sght s p p 2 d p p 3 4 Fgure 2 Assumptons about the object and the tesselaton When a lne-of-sght ntersects wth an object, there would be at least two ntersecton ponts. One of the ntersecton ponts would be nvsble to the source pont. For example, n Fgure 2, the lne-of-sght ntersects wth the objects at the pont x and the pont x 2, both x and x2 are nvsble to sensors. Therefore, we have the followng lemma for the occluson test. Lemma : If among all the ponts n the pont cloud of an object, f we can fnd one pont x, whch meets the followng condtons: ) t s on the lne-of-sght, and 2) ts surface normal forms an angle wth ncdence vector smaller than 9, we then know that pont p s occluded and pont x s an occludng pont. The lne-of-sght n theory has an nfnte-small dameter. The drecton applcaton of an ntersecton test between a lneof-sght and a pont cloud does not always produce the occludng ponts. In order to conduct the occluson test for the pont cloud representaton of an object, we make the followng assumptons about the pont cloud representaton of an object: ) dstance between each pont no larger than δ, 2) mnmal feature sze δ, 2δ 3) curvature radus > π These assumptons ensure that, f there s any ntersecton between a lne-of-sght and an object and the ntersectons ponts are wthn the δ dstance from the lne-of-sght, at least one pont nvsble to the sensor wll be n the pont cloud. Based on these assumptons, we have the followng proposton for occluson check for pont cloud. s p j p j' (a) p Proposton δ occluson Fgure 3 Occluson test Let sensor pont s and canddate pont p from an object pont set P form a lne-of-sght. For any pont p j from the object pont set P, let the dstance between the pont p j and the. s p j p j' (b) p lne-of-sght be d, and let the p j be the closest pont from the to p j. Let the angle between the surface normal n at pont p and the lne-of-sght to be θ. That s, θ cos(. ). If the followng condtons are met, ) d < δ, = a n 2) p j ' p p j ' s <, 3) θ <= 9, then canddate pont p s ether occluded by the surface patch around the pont p j or at most δ dstance away from beng occluded by the surface patch. We refer to ths occluson as δ occluson. Proof: Accordng to the st condton, we know pont p j s wthn δ dstance from the lne-of-sght. From the 3rd condton, we know the pont p j s nvsble to the sensor s. From the 2nd condton, we know the shortest pont between the lne segment and the pont p j les wthn the lne segment. If we move the lne-of-sght by d dstance, so that pont p j les on top of, then we can know the moved lne-of-sght s occluded by pont p j. That s, the pont p j s at most d dstance away from beng occluded by the surface patch around the pont p j. Due to the sample rate of δ dstance, pont p s occluded by the surface patch or at most δ dstance away from beng occluded by the patch. From the above proposton, we know for a gven pont p, f any one pont satsfes the condtons, then the pont p s δ occluded. If no pont from the object set meets the requrements, then the pont p s occluson-free. That s, pont p s globally vsble and accessble for optcal necton. So the complexty of occluson test for any gven canddate pont s O(n) where n s the number of ponts n the pont set. Note that the δ occluson may overshoot the occluson test by dstance δ. However, typcally δ can be made very small when the object s sampled nto a pont cloud. Due to the manufacturng tolerance and the naccuracy durng the part setup, t s often expected the necton pont to be some dstance away from beng occluded. 3.2 Smple algorthm for part nectablty test A part s fully nectable when all the ponts on the part can be nected. In order to check f all the ponts can be nected for a gven necton plan, all the ponts need to be checked aganst the nectablty requrements. That s, each pont has to ) le wthn measurement volume, 2) be locally vsble to both the laser and the camera, 3) be accessble or occluson-free. A smple algorthm for part nectablty check s as follows (Algorthm ). 3 Copyrght 23 by ASME

4 ALGORITHM : A SIMPLE ALGORITHM FOR PART INSPECTABILITY CHECK For each vew pont V n the vew plan. Obtan the measurement volume MV from the vew pont V 2. For each pont p j from the part P If p j MV, then add p j to the wthn pont set W. 3. For each pont p j wthn the measurement volume, p j W If pont p j s vsble to both the laser and camera, then add p j to the local vsble set W For each locally vsble pont p j, p j W2 If pont p j s occluson free, then mark ths pont as nectable. 5. For each pont p j from the part P If p j s not marked nectable, then add p j to the un-nected pont set U. The nput s a set of vew plans. The output s a pont set U that contans all the ponts not covered by any vewpont. It nvolves the followng fve steps.. Obtan the measurement volume for each vew pont. 2. Fnd all the ponts wthn the measurement volume. For a gven vew pont, the measurement volume s typcally a trapezod volume (Fgure ) and t can be represented as a collecton of sx planar half-aces MV = {( p p ) n <= =,6}, where each plane s represented by a pont p and a plane normal n. So whether a pont les wthn the measurement volume can be determned through a set of half-ace calculatons. 3. Check f all the wthn ponts are locally vsble to both the laser and the camera. The local vsblty can be determned through the surface normal comparson. For a canddate pont p wth surface normal n, f n ( s p) >, where s s ether the laser poston or the camera poston, then pont p s vsble to the sensor s. 4. For all the ponts that are locally vsble, we check f they are occluson free. The occluson test s based on the δ-occluson proposton n the last secton. For a gven sensor (ether laser or camera) poston S and a canddate pont p j, we have ths occluson check algorthm. The output of ths algorthm s whether the canddate pont p j s occluded. Algorthm 2: A Smple Algorthm for Occluson Check For each pont p from the part P Project pont p onto the lne p js and obtan the projected pont p Calculate dstance d between p and p, d = ' p p Calculate the angle θ between p js and surface normal n, θ = a cos( n. ) Return False. If d < δ, p ' p p ' s < <= j j, and θ 9 Return True 5. Check f any pont s not covered by all the vewponts. Algorthm 2 has complexty of O(n) where n s the number of ponts n the pont cloud of the part. So the Algorthm (Smple algorthm for part nectablty test) has complexty of O(n 2 ). 3.3 Space partton based algorthm for part nectablty check We mprove the smple algorthm n last secton usng ace partton va a k-d tree. We frst present the constructon of a k-d tree based on postonal ace and normal ace partton. We then transform the smple algorthm nto a range search problem on the k-d tree Parttonng poston and normal ace For many atal problems, ace partton s a common scheme to ncrease computatonal effcency. Snce the nectablty problem nvolves both poston and surface normal constrants, we propose the smultaneous partton of postonal and normal ace to ncrease the algorthm effcency. Constructon of a k-d tree We adopt a classcal k-d tree constructon method (Preparata 988) for our problem. In ths method, a boundary box s needed for ace partton. Dfferent types of boundary boxes have been used for a varety of applcatons. For the smplcty of mplementaton, n ths paper, we use axs-algned boundary box. However, the algorthms presented n ths paper are applcable to other types of boundary boxes. For each pont, three are three postonal values (x, y, z) and three surface normal values (nx, ny, nz). We represent the sx values as a 6-dmensonal pont (x, y, z, nx, ny, nz). Suppose that all the ponts from an object to be nected le n a 6- dmensonal box, whch has some ponts dentfed on ts boundary. We buld the data structure nsde ths box, and defne t recursvely as follows. The set of data ponts s lt nto two parts by lttng the box contanng them nto two chld boxes by a hyper plane, accordng to some lttng rule ecfed by the algorthm; one subset contans the ponts n one chld box, and another subset contans the rest of the ponts. The nformaton about the lttng plane and the boundary values of the ntal box are stored n the root node, and the two subsets are stored recursvely n the two subtrees. When the number of the data ponts contaned n some box falls below a gven threshold, then we call a node assocated wth ths box a leaf node, and we store a lst of coordnates for these data ponts n ths node. (Preparata 988). In ths paper, we lt the boundary box sequentally n x, y, z, nx, ny, and nz order. When the lt happens n the postonal ace, the surface normal boundary of the set may be changed. Lkewse, when the lt happens n the surface normal ace, the boundary box of the postonal ace may also change. In order to have a tght boundary box, durng the 4 Copyrght 23 by ASME

5 constructon process, we calculate and record the new boundary box after each lt. Range search n a k-d tree The range search of a k-d tree s handled dfferently from a standard k-d tree. We have the followng range search algorthm (Algorthm 3). The output s a set of ponts, W, fallng wthn the range R. The algorthm works recursvely. Frst, t checks f the node s a leaf. If t s a leaf, t fnds out all the ponts under the leaf fallng nto the range R and adds then nto W. If t s not a leaf, t checks f the boundary box of the node, extent, completely les wthn the range R. If t does, all the ponts under the node fall nto the range R. It then checks f the boundary box overlaps wth the range. If t does, t recursvely searches ts left sub-tree and rght sub-tree. If not, t returns the pont set W. Algorthm 3 Range Search based on a k-d tree RangeSearch(node N, BoundaryBox Extent, Range R) If (type(n) == leaf) { For each pont p n the leaf node } Else { } If (swthn(p, R) == true), then Axs = node->spltaxs; W p If(sWthn(Extent, R) == true) W all the ponts under N If(sOverlap(Extent, R) == true) { // Left Chld TmpExtent = Extent[2*Axs]; Extent[2*Axs] = N->SpltValue; RangeSearch( N->LeftChld, Extent, R); Extent[2*Axs] = TmpExtent; } // Rght Chld TmpExtent = Extent[2*Axs]; Extent[2*Axs] = N->SpltValue; RangeSearch( N->RghtChld, Extent, R); Extent[2*Axs] = TmpExtent; In the above range search algorthm on a k-d tree, there are three basc Boolean nteracton tests: ) whether a pont p le wthn a range R, swthn(p, R), 2) whether a boundary box extent falls completely wthn the range R, swthn(extent, R), 3) whether the boundary box extent overlaps wth the range R, soverlap(extent, R). The frst test s a basc contanment test. The second and the thrd tests enable the search tme savng on a herarchal tree. For example, n Fgure 4, a pont set s shown n the left fgure and ts ace partton s shown n the rght fgure. In order to fnd all the ponts wthn the range A, the k-d tree wll be selectvely traversed. In ths partcular case, only nodes 6, 3, 2, and 5 wll be checked for Range A. Ths nformaton can be obtaned through the overlap test soverlap(extent, R) snce the boundary boxes of these nodes overlap wth the Range A. In order to search the ponts n Range B, the nodes 6, 3, 2,, wll be checked. Due to the fact that Range B falls completely wthn the boundary box of node s left chld, any ponts n the left leaf of node s automatcally wthn the Range B. Ths nformaton s obtaned through the swthn(extent, R) test. B A (a) Space partton 8 Fgure 4 Interacton tests between boundary box and range Convert a pont nectablty test nto a range search problem In order to take advantage of the computatonal effcency of a range search based on herarchal postonal and normal ace partton, we convert the pont nectablty test nto a range search problem on the k-d tree. We then present several propostons related to a range search for occluson detecton. We dvde the pont nectablty problem nto a two-step problem. Durng the frst step, we seek all the ponts that le wthn the measurement volume and are vsble to both the camera and the laser. We refer to these ponts as locally nectable ponts, meanng these ponts are nectable n a local sense, wthout knowledge whether they meet the global vsblty condtons or not. Durng the second step, we check f any pont n the locally nectable pont set meets the global vsblty condtons,.e. whether t s occluded. Range search for locally nectable ponts The problem of fndng ponts wthn the measurement volume, and meetng vsblty crtera s essentally equvalent to a range search problem. The range s a combnaton of a measurement volume n the postonal ace and a vsblty cone n the gaussan map (Qan 23) n the surface normal ace. The search tme can be saved n two ways when a k-d tree s used. One s, when the boundng box does not overlap wth the measurement volume or the vsblty cone, there s no need for the nectablty test for the ponts wthn the measurement volume. None of these ponts can be nected. The second s, when the boundng box and surface normal hercal convex hull are wthn the measurement volume and the vsblty cone, there s no need for the test for the ponts wthn the measurement volume. All the ponts can be nected. When there s overlap between a node s boundary box and measurement volume (or vsblty cone), then we recursvely conduct a range search for ts left and rght subtrees. In order to use the range search based on a k-d tree, we put forward the follow propostons wth regardng to the nteracton between the range (measurement volume and vsblty cone) and the postonal and normal ace boundary box. Proposton : If a postonal boundary box does not overlap wth measurement volume, or the boundary box of surface normal does not overlap wth vsblty cone of the sensor, none of the ponts wthn the boundary box are vald ponts for measurement. (Completely Out) (b) K-d tree structure 5 Copyrght 23 by ASME

6 Proposton 2: If the boundng box s wthn the measurement volume, and all the eght surface normal of the normal boundary box are wthn the vsblty cone, then all the ponts under ths k-d tree node are locally effectve measurement ponts. (Completely Wthn) The two propostons reectvely correond to the range test, soverlap(extent, R) and swthn(extent, R), n the k-d tree. The frst proposton reveals that the node does not overlap wth the range. The second proposton shows that all the ponts n ths node are completely wthn the range. If we substtute these two functons nto Algorthm 3 based on the two propostons, we get an mproved algorthm for fndng the locally nectable ponts. Range Search for Occluson-free Ponts Once we have the locally vsble ponts, we need to verfy f each pont s occluson-free. The smple algorthm (Algorthm 2) compares each locally nectable pont aganst the whole pont cloud and has the complexty of O(n). We mprove ths smple algorthm by comparng the locally nectable ponts aganst the potentally occludng ponts. The potentally occludng ponts are those ponts fallng nto the convex hull formed by the boundary box of the locally vsble ponts and the sensor. As shown n Fgure 5, for a set of locally nectable ponts (left fgure), we can form the boundary box B. We then form a convex hull C between the sensor and boundary box B. Any pont fallng nto the convex hull other than the locally nectable ponts, f ths pont s not vsble to the sensor, s a potentally occludng pont. Note, accordng to δ-occluson proposton, we only use the pont that s not vsble to the sensor as a bass for occluson detecton. Therefore, the ssue of fndng potentally occludng ponts s agan a range search problem. Here, the range s a combnaton of convex hull n the postonal ace and the vsblty cone n the normal ace. Substtutng ths range nto Algorthm 3 leads to mproved algorthm for fndng potentally occludng ponts. In ths mproved algorthm, we use the followng proposton for range search functon swthn(extent, R). Proposton 3: If there s no pont wthn the convex hull formed by the boundary box of the locally nectable ponts and the sensor, then all the ponts under ths k-d tree node are globally effectve measurement ponts,.e. occluson-free. (Completely Wthn) B Sensor Locally nectable ponts (a) Fnd the boundary box of locally nectable ponts C Occludng ponts Sensor (a) Form the convex hull and fnd the potentally occludng ponts Fgure 5 Seekng potentally occludng ponts Once we fnd potentally occludng ponts, we compare each locally nectable pont aganst the potentally occludng pont set. Such a comparson s also a range search problem. In the postonal ace, the range s a cylnder. The cylnder s centerlne s a lne wth end ponts at the locally nectable pont and the sensor. The radus s δ. In the surface normal ace, the range s the whole ace. For each pont wthn the cylnder, we then compare ts normal to see f the angle between the normal and the lne-of-sght (formed between the sensor and the pont) s larger than 9 degrees (the second condton of Lemma ) Range and Boundng box nteracton test We convert the pont nectablty problem, ncludng occluson, nto a set of range search problem on a k-d tree. It nvolves a set of geometrc nteracton test between the boundary boxes (of postonal and normal ace) and the ranges. The boundary boxes are axs-algned boxes. In postonal ace, the ranges nclude measurement volumes for fndng locally nectable ponts, convex hulls for fndng potentally occludng ponts, and cylnders for determnng f a pont s occluded by the potentally occludng pont set. In surface normal ace, the ranges nclude the vsblty cones of a set of vertces n the boundary boxes of the surface normal. Due to the page lmt, we wll not go nto detals about the nteracton test between the boundary boxes and the ranges. The basc dea behnd our nteracton tests s usng a dmenson reducton method to convert a 3D geometry model and boundng box overlap test nto a lower dmensonal geometry nteracton test. For example, we can convert a 3D cylnder and 3D boundary box nteracton test nto a problem of a pont-lne dstance calculaton and a lne-plane ntersecton test. Specfcally, we frst check the extreme cases. They nclude: ) All the boundng box vertces le wthn the cylnder. Ths can be tested usng pont-lne dstance calculaton. 2) The cylnder centerlne les wthn the boundng box. Ths can be easly known by checkng f end ponts of the centerlne le wthn the box. 3) The two end ponts of the cylnder center lnes le at least r dstance away from the boundary box. After the ecal cases processng, we ntersect the centerlne wth the box. For each boundary plane of the box, we offset the centerlne by dstance of cylnder radus and ntersect the offset lne wth the plane. If the end ponts of both the centerlne and the offset centerlne le outsde of boundary plane, there s no overlap. If the ntersecton pont of offset lne wth the boundary plane les beyond the end ponts of the offset lne, there s overlap. If ether ntersecton pont (from center lne or from offset lnes) les wthn the boundary box, there s overlap. Otherwse, there s no overlap. 4 Implementaton and results The algorthms have been mplemented on an HP-UX. U9/785 machne. We compare the computatonal results of the algorthms based on one benchmark part (Fgure 6). The tessellated model and ts gaussan map are also ncluded n the fgure. The tessellated model ncludes 6553 ponts and surface normal. A snapshot of the frst 2 boundary boxes durng the k-d tree constructon process s shown n Fgure 7 and Fgure 8, reectvely llustratng the parttonng of postonal ace and surface normal ace. Fgure 7 shows one pcture n whch the 6 Copyrght 23 by ASME

7 tessellated model les wthn the boundary box, and one pcture wthout the tessellated model. Fgure 8 shows a wre-frame and a shaded model of the surface normal ace partton. Dfferent colors represent the boundary box at dfferent levels n the k-d tree. The tme for constructng the k-d tree of 6553 ponts s about.355 second. outperformed the method usng parttonng postonal ace only. Fgure shows the tme comparson of these two methods for dfferent feld of vews. The largest sze of the feld of vew and depth of feld s 24 *24 *8. We then tred to set the feld sze to one half, one fourth, and one sxth of ts orgnal sze. The reectve results are shown n Fgure. Agan, the method usng the parttonng of both postonal and surface normal ace consstently outperformed the method usng the parttonng of postonal ace only regardless of the sze of the measurement volume Angle=55 (a) Inecton part (b) Tessellated necton surfaces Fgure 6 A benchmark part (c) Surface normal of necton surfaces Tme (sec) Postonal ace Angle=65 Angle=75 Angle=85 Angle=55 Angle=65 Angle=75.5 Angle=85 Postonal & normal ace Vew Fgure 9 Tme comparson for dfferent sensor vew angles Fgure 7 Parttonng postonal ace Tme (sec) Partton poston ace Partton poston and normal ace Tme (sec) Partton poston ace Partton poston and normal ace Vew Vew (a) /6 FOV (b) /4 FOV 2 Partton poston ace Partton poston and normal ace.4.2 Tme (sec).2.8 Tme (sec).8.6 Partton poston ace Parttonal poston and normal ace Vew Vew (c) /2 FOV (d) FOV Fgure Tme comparson for dfferent feld of vews Fgure 8 Parttonng surface normal ace In order to compare the computatonal tme durng pont nectablty check for dfferent algorthms, we select seven necton vews, under whch we calculate ) the ponts that are wthn the measurement volumes and are locally vsble, and 2) the ponts that are globally vsble,.e. no occluson. We frst compare tme ent n search of vald ponts wthn measurement volume and locally vsble to both the camera and the laser usng ether parttonng postonal ace only or parttonng both postonal and surface normal ace. Fgure 9 shows the tme comparson of these two methods for dfferent sensor vew angles. As the sensor vew angle constrants changes from 55 to 85 degrees, the method usng parttonng both postonal and surface normal ace consstently Fnally, we compare the tme consumed n search of both locally vsble ponts and the occluson-free ponts for three methods under the same sensor vew angle and the same measurement volume constrants. The results are shown n Table. As the table shows, the ace partton based approach, ether postonal ace partton only or the postonal and normal ace partton, outperformed the smple algorthm. The method of parttonng both postonal and surface normal ace consumed the least amount of tme n search of the locally vsble ponts. Table also llustrated the tme consumed n search of occluson-free ponts for seven vewponts. The smple algorthm consumed much more tme than the k-d tree based method, where a set of potentally occludng ponts were frst sought wthn the convex hull composed to the boundary box of locally vsble ponts and the sensor pont. In the table, there are four vews, at whch no ponts le wthn the 7 Copyrght 23 by ASME

8 measurement volume. Therefore, the tme for occluson-free pont search s zero. Table Tme comparson for dfferent methods (Tme n second) Wthn measurement volume & locally vsble Occluson free Smple Algorthm K-d tree wth poston ace partton K-d tree wth postonal and normal ace partton Smple Algorthm K-d tree based Concluson Ths paper presents a fast approach for occluson detecton n optcal necton. We frst ntroduced a novel δ occluson concept for occluson check for tessellated part model. We then transformed the pont nectablty problem nto a range search problem n a k-d tree. We developed several propostons based on the herarchcal postonal and normal ace partton. The results demonstrated that the algorthm usng the partton of postonal ace of the target objects outperformed the smple algorthm that does not nvolve ace partton. The method usng both postonal ace and normal ace parttonng outperformed the method usng the postonal ace partton only. The contrbutons of ths paper nclude a novel δ occluson concept for occluson check for pont cloud, and the method of herarchcally parttonng both postonal ace and surface normal ace for occluson detecton. The algorthms can be enhanced n a few ways n the future. We wll nvestgate the sequence of lt axs nvolvng both poston and surface normal. Currently, a fxed sequence s used and ths may not be the best lt sequence. We wll nvestgate the optmal sze of the ponts n the leaf node of the k-d tree. We wll nvestgate the use of dfferent boundary box representaton, such as dscrete orentaton polytope, nstead of axal-parallel boundary box. REFERENCES Chou, C. Y., Chen, L. L., and Woo, T. C., Separatng and ntersectng hercal polygons: computng machnablty on three, four and fve-axs numercally controlled machnes, ACM Trans. Graphcs 2 (4), , 993. Elber, G., and Zussman, E., Cone vsblty decomposton of freeform surfaces, Computer-Aded Desgn 3 (998) Fredman, J. H., Bentley, J. L., and Fnkel, R. A., An algorthm for fndng best matches n logarthmc expected tme, ACM Transactons on Mathematcal Software, Vol. 3, No. 3, Sep 977, pp Gonzalez-Banos, H.H. and Latombe, J.C., A Randomzed Art-Gallery Algorthm for Sensor Placement. Proc. 7th ACM Symp. on Computatonal Geometry (SoCG'), pp , 2. Gottschalk S, Ln M, Manocha D, OBB tree: a herarchy structure for rapd nterference detecton, SIGRAPPH 96 Vsual Proceedngs, New Orleans, LA, USA 4-9 Aug 996. Ln, M.C, and Gottschalk, S., Collson detecton between geometrc models: a survey, Proceedngs of IMA Conference on Mathematcs of Surfaces, 998. Maver, J. and Bajcsy, R., Occlusons as a Gude for Plannng the Next Vew, IEEE Transactons on Pattern Analyss and Machne Intellgence, Vol. 5, No. 5, May 993, pp Nene, S. A., and Nayar, S. K., A Smple Algorthm for Nearest Neghbor Search n Hgh Dmensons, Techncal Report, Department of Computer Scence, Columba Unversty, 995. Pettjean, S., Ponce J., and Kregman, D. J., Computng exact aect graphs of curved objects: Algebrac surfaces, Internatonal Journal of Computer Vson, 992, pp. 9. Preparata, F. P. and Shamos, M. I., Computatonal Geometry: An Introducton, Sprnger-Verlag, 988. Qan, X. and Hardng, K. G., A Computatonal Approach for Optmal Sensor Setup, SPIE Journal Optcal Engneerng, May 23. Son, S., Park, H. and Lee, K. H., Automated laser scannng system for reverse engneerng and necton, Internatonal Journal of Machne Tools & Manufacture, Vol. 42, 22, pp Tarabans, K., Allen, P., and Tsa, R., A survey of sensor plannng n computer vson, IEEE Trans. Robot. Automat, vol., pp. 86 4, 995. Tarabans, K., Tsa, R. Y., and Kaul, A., Computng Occluson-Free Vewponts, IEEE Transactons on Pattern Analyss and Machne Intellgence, Vol. 8, No. 3, March 996, pp Tarbox, G. H. and Gottschlch, S. N., Plannng for Complete Sensor Coverage n Inecton, Computer Vson and Image Understandng, Vol. 6, No., Jan, pp. 84, 995. Trucco, E., Umasuthan, M., Wallace, A. M., and Roberto, V., Modelng-based Plannng of Optmal Placements for Inecton, IEEE Transactons on Robotcs and Automaton, Vol. 3, No. 2, 997, pp Zachmann, G., Real-tme and exact collson detecton for nteractve vrtual prototypng, 997 ASME Desgn Engneerng Techncal Conferences, Sacramento, CA. 8 Copyrght 23 by ASME