Efficient Constraint Evaluation Algorithms for Hierarchical Next-Best-View Planning

Transcription

1 Efficient Constraint Evaluation Algoriths for Hierarchical Next-Best-View Planning Ko-Li Low National University of Singapore Anselo Lastra University of North Carolina at Chapel Hill (a) 4 (b) (c) Figure. (a) The eight 3D views in a coputed view plan. The views ay be at different height. (b)&(c) Two views of the polygonal odel constructed fro the eight range scans. Abstract We recently proposed a new and efficient next-bestview algorith for 3D reconstruction of indoor scenes using active range sensing. We overcoe the coputation difficulty of evaluating the view etric function by using an adaptive hierarchical approach to exploit the various spatial coherences inherent in the acquisition constraints and quality requireents. The ipressive speedups have allowed our NBV algorith to becoe the first to be able to exhaustively evaluate a large set of 3D views with respect to a large set of surfaces, and to include any practical acquisition constraints and quality requireents. The success of the algorith is greatly dependent on the ipleentation efficiency of the constraint and quality evaluations. In this paper, we describe the algorithic details of the hierarchical view evaluation, and present efficient algoriths that evaluate sensing constraints and surface sapling densities between a view volue and a surface patch instead of siply between a single view point and a surface point. The presentation here provides exaples for the design of efficient algoriths for new sensing constraints.. Introduction With the advent of affordable active range sensing devices, reconstructing detailed 3D digital odels of realworld objects and environents has becoe ore coon. A typical reconstruction would require ultiple range scans ade fro different scanning locations. The set of scanning locations ust be chosen carefully so that each location satisfies a set of acquisition constraints and the reconstructed 3D digital odel can eet a set of quality requireents. This tas is nown as view planning. For 3D reconstruction, a priori nowledge of the scene geoetry is not available to the autoatic view planner. The first scan is ade fro a view selected by a huan operator, and for each subsequent scan, the planner ust deterine its best view based on the inforation collected fro the previous scans. This is often called the next-best-view (NBV) proble. The NBV proble is inherently a local optiization proble since global geoetric inforation is unnown. It is NP-hard, and since it can be reduced to the setcovering proble, it is often solved approxiately using a greedy approxiation algorith. A greedy NBV algorith selects the view that axiizes a view etric as the best view for the next scan. The ajor challenge to a practical NBV solution is an efficient ethod to evaluate the view etric for a large set of views, using inforation provided by a partial odel of the scene. Each evaluation can be coputationally very expensive, since a large aount of inforation of the partial odel ay be involved, and visibility coputations and other constraint evaluations are expensive. This apparent coputation difficulty has liited any previous NBV algoriths to siple and sall objects, incoplete search space, incoplete set of acquisition constraints and reconstruction quality requireents, and low-quality acquisition. Soe early algoriths even ignore self-occlusion of the objects [2].

2 However, an efficient algorith to evaluate the view etric is actually possible. Recently, we proposed a hierarchical approach [5] to adaptively exploit the various spatial coherences inherent in the acquisition constraints and quality requireents. Results show that the hierarchical approach can speed up view evaluation by one to two orders of agnitude over the straightforward ethod used in the previous NBV algoriths. These ipressive speedups have allowed our NBV algorith to becoe the first to be able to exhaustively evaluate a large set of 3D views with respect to a large set of surfaces, and to include any practical acquisition constraints and quality requireents. Our proposed hierarchical view evaluation algorith was inspired by the hierarchical radiosity algorith [4], which can be generalized to evaluate pair-wise interactions between extended objects. The success of the hierarchical view evaluation is due to the evaluation of the sensing constraints and surface sapling densities between a view volue and a surface patch, instead of between a single view point and a single surface point at a tie. The straightforward, inefficient, single-view-pointto-single-surface-point approach is used in alost every previous view planning algoriths [2, 3, 6, 7, 8]. The purpose of this paper is to describe the algorithic details of the hierarchical approach, and present efficient algoriths that evaluate sensing constraints and surface sapling densities between a view volue and a surface patch. The presentation here provides exaples for the design of efficient algoriths for new acquisition constraints. Section 2 presents a suary of the aterial in [5]. The reader is encouraged to refer to the paper for ore details. In Section 3, we present the details of the hierarchical view evaluation algorith, and the efficient ipleentation of the individual constraint and sapling density evaluation algoriths. Section 4 presents soe results of our hierarchical view evaluation. We conclude the paper in Section The Hierarchical NBV Algorith For our NBV algorith, we have forulated a view etric and used an adaptive hierarchical ethod to efficiently evaluate the view etric for a large set of views. The view etric incorporates the reconstruction quality requireents and acquisition constraints. Our view etric taes into account the following two reconstruction quality requireents. () Copleteness. The NBV algorith tries to axiize the aount of surface area acquired so that the reconstructed odel can be ore coplete. (2) Surface sapling quality. Our algorith tries to axiize the surface area that reaches a required surface sapling density. Several acquisition constraints ust be observed when planning a view of the scanner. Each constraint can be classified as one of the following types. () Positioning constraints. The physical construction of the scanner and the capability of the positioning device (a) (b) (c) (d) Figure 2. Different types of surfaces in a partial odel. (a) True surfaces. (b) Occlusion surfaces (red). (c) Hole-boundary surfaces (blue). (d) Iage-boundary surfaces (green). can constrain the scanner s physical position. A view that satisfies all the positioning constraints is called a feasible view. (2) Sensing constraints. These constraints deterine whether a surface point in the scene can be easured fro a view. For exaple, a surface point cannot be easured by the scanner if it is not visible fro the range sensor. (3) Registration constraint. Due to positioning error of the scanner, each new scan has to be explicitly aligned to the previous scans. However, this registration is not guaranteed to be successful. Our view planning algorith can ensure that the new scan to be acquired fro the planned view can be successfully registered with the previous ones. However, we will not discuss the forulation and evaluation of the registration constraint in this paper. 2.. Partial Model The partial odel consists of the acquired surfaces (called true surfaces) and three types of false surfaces: () occlusion surfaces, (2) hole-boundary surfaces, and (3) iage-boundary surfaces. They are shown in Figure 2. These false surfaces are added to connect holes caused by occlusions, issing saples, and range iage boundaries, respectively. These surfaces enclose a volue of nown epty space. The false surfaces provide clues to how the unnown volues can be resolved by subsequent scans. One partial odel can be erged to another of the sae environent by perforing the union of their nown epty volues and the union of their true surfaces. In our ipleentation, the partial odel is represented using an octree. All surface types and epty space are represented View Metric Our view etric is shown in Eq. (), ( v) score of view v. h ( v) f ( v) r( v) c( v, p) w( p) t( v, p)dp p S h is the = () S is the set of all surface points in the current partial scene odel; it includes all true and false surfaces; f ( v) is if view v is a feasible view, otherwise f ( v) is 0; r ( v) is if the registration constraint is satisfied at r v is 0; view v, otherwise ( )

3 ( v, p) c is if all the sensing constraints between view v and surface point p are satisfied, otherwise c ( v, p) is 0; w ( p) is the weight or iportance value assigned to the surface type (false or true surface) of p; t ( v, p) is the iproveent to the recorded sapling density at p if a scan is ade fro view v. We use the following definition for t ( v, p). t( v, p) = ax ( 0, in( s( v, p), D) q( p) ) (2) s ( v, p) is the sapling density at p if it is scanned fro view v; this is referred to as the new scan sapling density; D is the sapling density requireent for all surfaces; ( p) q is the axiu sapling density at which p has been scanned previously; this is referred to as the recorded sapling density; if p is on a false surface, q p is 0; then ( ) 2.3. Algorith Overview Our strategy to evaluate h () v for all the views is to evaluate Eq. () in pieces, fro least to ost expensive to copute. Figure 3 shows the ajor steps in the evaluation of h () v. We first evaluate f () v for all views to eliinate the infeasible views. Next, we use our hierarchical view evaluation ethod to evaluate the integral part of Eq. () for all the feasible views. The feasible views are then raned by their current scores. Starting fro the highestscore view, the registration constraint function, r ( v), is evaluated. The first view found to satisfy the constraint is output as the next best view. To support the hierarchical view evaluation, surface voxels in the partial octree scene odel are grouped into planar patches. The planar patches are then raned in descending order of iportance, so that the ost iportant ones can be used to evaluate the views first. Coputing feasible views. An octree is used to represent the feasible view volues. This feasible view octree contains a subset of the epty space in the cuulative partial odel, and it is carved out using the positioning constraints. new view (F) chec registration constraint (E) ran views (A) copute feasible views (D) evaluate views partial scene odel (B) extract planar patches (C) ran patches terinate Figure 3. The ajor steps in the NBV algorith. Extracting planar patches. Surface voxels that have not reached the required sapling density are grouped into planar patches. Each patch has the following attributes: () a rectangle, (2) an approxiate surface area, (3) the average recorded sapling density, and (4) the sapling deficit. The sapling deficit is defined as the nuber of saples needed to ae the average recorded sapling density equal to the sapling density requireent D. Raning patches. The ost iportant patch should have the greatest potential ipact on the value of the view etric in Eq. (). This leads to the following: the patch iportance value of P = w ( P) sapling deficit of P, P is the patch, and w ( P) is the weight assigned to the surface type of P, which is the sae as the weight w ( p) in Eq. (). The patches are then sorted in descending order on their patch iportance values Hierarchical View Evaluation Let g ( v) be the integral part in Eq. (), i.e. g( v) = c( v, p) w( p) t( v, p)dp (3) p S The next step of the NBV algorith is to evaluate g ( v) for all the feasible views. Due to the potentially large area of surfaces in the partial scene odel, a brute-force approach would be ipractical. However, the aount of coputation can actually be reduced by exploiting the spatial coherences in the sensing constraints and the sapling quality function. The idea is that if a constraint is satisfied between a view v and a surface point p on the partial odel, very liely the sae constraint is also satisfied between another view v and p, provided v is near to v. The sae constraint is also very liely to be satisfied between v and another surface point p that is near p. We exploit these spatial coherences using a hierarchical approach. Neighboring views are first grouped into view volues, and neighboring surface points are grouped into surface patches. The constraint is evaluated between each view volue V and a patch P. If it is entirely satisfied or entirely not satisfied between V and P, then the constraint evaluation is considered copleted between every view in V and every surface point i. If the constraint is partially satisfied between V and P, then we subdivide either V or P, and continue the evaluation on the children Forulation Suppose all the false surfaces and under-sapled true surfaces in the partial odel have been partitioned into N i =, K,N, then Eq. (3) can be rewritten as patches P i g N () v = g( v,p i ) i= (4)

4 ( v,p) c( v, p) w( p) t( v, p)dp g = (5) p P Now, we will focus on evaluating views with respect to a patch P, instead of with all the surface area in the partial odel. Suppose the values of c ( v, p) and t ( v, p) reain constant between a view volue V and a patch P, v V and p P, then g ( v,p) can be coputed as g( v,p) = g( V,P) = c( V,P) w( P) t( V,P) a( P) (6) t( V,P) = ax ( 0, in( s( V,P), D) q( P) ) (7) and c ( V,P), w ( P) and s ( V,P) are siilarly defined as c ( v, p), w ( p) and s ( v, p) ; a ( P) is the patch area of P, and q ( P) is the average recorded sapling density of P. In actual fact, the value of s ( v, p) does not stay constant between V and P. However, if every s ( v, p) between V and P is bounded within a sall interval, then we consider it approxiately constant. The value of s ( v, p) between V and P is considered approxiately constant if sax ( V,P) sin ( V,P) ε (8) s sax ( V,P) s in ( V,P) and s ax ( V,P) are the iniu and axiu s ( v, p) between V and P, respectively. We have chosen to let s ( V,P)= s in ( V,P), and copute g ( V,P) using Eq. (6). If any sensing constraint is found entirely not satisfied between V and P, then s ( V,P) need not be coputed and g ( V,P) = 0. If c ( v, p) is not constant or s ( v, p) is not approxiately constant between V and P, then we cannot copute g ( V,P) using Eq. (6). We can subdivide either V or P, and apply Eq. (6) on the sub-volues or the sub-patches. If patch P is subdivided, then ( V,P) = g( V,P ) + L g( V,P ) g + (9) are the sub-patches of patch P. If view P, K,P volue V is subdivided, then ( V,P) (,P), K,g( V,P) g V, g is replaced with V, K,V i g g( V,P) if v V. c is constant and s is approxiately constant between the view volue and the patch. are the subvolues of V. In this case, ( v,p) = The subdivision stops when ( v, p) ( v, p) 3. Hierarchical View Evaluation Algorith The following describes the algorithic details and ipleentation of the hierarchical view evaluation to evaluate the integral part of the view etric. It is assued that the range sensor is onostatic, and all the saples in a range iage are easured fro a single i center of projection or viewpoint. This assuption is true for any coercial id-range and long-range laser scanners that use tie-of-flight range sensing. Many of such scanners also have 360 horizontal FOV but liited vertical FOV. Since we assue that the scanner is always in the upright orientation, each view of the scanner is effectively only a 3D position. We use the feasible view octree to represent the 3D view volues. In our ipleentation, c ( v, p) consists of four separate sensing constraints: () The axiu-range constraint, represented by c 0 ( v, p). If the distance between view v and surface point p is ore than the axiu effective range of the range sensor, then c 0 ( v, p) = 0, otherwise c 0 ( v, p) =. (2) The vertical-field-of-view constraint, represented by c ( v, p). If the surface point p is outside the vertical field of view of the scanner at view v, then c ( v, p) = 0, otherwise c ( v, p) =. (3) The angle-of-incidence constraint, represented by c 2 ( v, p). If the angle between the surface noral vector at p and the direction vector fro p to v is greater than a threshold angle, then c 2 ( v, p) = 0, otherwise c 2 ( v, p) =. (4) The visibility constraint, represented by c 3 ( v, p). If the line of sight fro v to p is occluded, then c 3 ( v, p) = 0, otherwise c 3 ( v, p) =. Consequently, the binary function c ( v, p) is defined as c ( v, p) = c 0 ( v, p) c ( v, p) c 2 ( v, p) c 3 ( v, p). Siilarly, c ( V,P) is ade up of c 0 ( V,P), c ( V,P), c 2 ( V,P), and c 3 ( V,P), c i ( V,P) = if c i ( v, p) = for all v V and p P, or c i ( V,P) = 0 if c i ( v, p) = 0 for all v V and p P, otherwise c i ( V,P) is undefined. Intuitively, when c i ( V,P) is undefined, it eans that the corresponding constraint is only partially satisfied between V and P. In Figure 4 is a siplified C-lie procedure to evaluate g ( V,P). Here, input viewcell V is a feasible view volue. Each input Boolean eleent C_in[i] is true if c i ( V,P) is already nown to be, otherwise C_in[i] is false to indicate that c i ( V,P) is unnown. The input Boolean arguent S_const is true if the relative error of s ( V,P) is nown to be bounded by ε s. The input arguent S is ( V,P) s if S_const is true. Initially, the procedure EvaluateView() is called with all C_in[i]=false and S_const=false. The function EvaluateConstraint(i,V,P) evaluates ( V,P) c i ( V,P) = 0 or c i ( V,P) = other integer values to indicate ( V,P) function c i and returns 0 or to indicate, respectively, or returns any c i is undefined. The EvaluateSaplingDensity(V,P,

5 &SMin,&SMax) evaluates the iniu and axiu new scan sapling densities between V and P. The details of both functions are described in the subsections below. In our ipleentation, a viewcell is subdivided into eight equal sub-viewcells, as a patch is subdivided into four sub-patches by splitting its rectangle into four equal parts. A viewcell is not subdivided if it has reached the iniu viewcell size. Siilarly, a patch is not subdivided if it has reached the iniu patch size. When either a viewcell or a patch is to be subdivided, we subdivide the patch if its longer side is larger than the viewcell s width, otherwise the viewcell is chosen. It is iportant to note that when c i ( V,P) =, it is also true that c i ( V, P ) = and ci ( V,P) = for all sub-patches P of P and all sub-viewcells V of V. Therefore, when c i ( V,P) =, and EvaluateView() is called with the sub-patches P or the sub-viewcells V, there is no need to recopute c i ( V, P ) and ci ( V,P). This is siilar for s ( V,P), when its relative error has been deterined to be bounded by ε s. This iportant observation can eliinate a large aount of coputation since once a constraint is deterined to be 0 or for V and P, it needs not be evaluated anyore for their descendents. After all patches have been evaluated (or the allotted view evaluation tie is up), a viewcell s score is not yet propagated down to its children. Since each child viewcell EvaluateView( Viewcell *V, Patch *P, bool C_in[4], bool S_const, float S ) bool C[4] = C_in[0], C_in[], C_in[2], C_in[3] ; for ( int i = 0; i < 4; i++ ) if (!C[i] ) int t = EvaluateConstraint( i, V, P ); if ( t == 0 ) return; if ( t == ) C[i] = true; if (!S_const ) float SMin, SMax; EvaluateSaplingDensity( V, P, &SMin, &SMax ); if ( ( SMax - SMin ) / SMax <= epsilon_s ) S_const = true; S = SMin; if ( MIN( S, D ) - q(p) <= 0 ) return; if ( C[0] && C[] && C[2] && C[3] && S_const ) V->score += w(p) * ( MIN( S, D ) - q(p) ) * a(p); else if ( ToSubdividePatchFirst( V, P ) ) SubdividePatch( P ); for ( int = 0; < P->nuChildren; ++ ) EvaluateView( V, P->child[], C, S_const, S ); else SubdivideViewcell( V ); for ( int = 0; < V->nuChildren; ++ ) EvaluateView( V->child[], P, C, S_const, S ); Figure 4. A procedure to evaluate g(v, P). contains part of the view volue of its parent viewcell, the scores in the children should include the parent s score. Therefore, the score of each viewcell should be updated by adding to it the scores of its ancestors. The center point of each leaf node of the feasible view octree is a candidate view, to be raned and tested for the registration constraint. The first candidate view that satisfies the registration constraint is chosen as the best view for the next scan. 3.. Constraint and Sapling Density Evaluations This section describes the ipleentation of EvaluateConstraint() and EvaluateSapling Density() to evaluate c i ( V,P) and s ( V,P), respectively. The success of the hierarchical view evaluation depends on how efficiently they can be evaluated. In actual fact, EvaluateConstraint(i,V,P) need not evaluate c i ( V,P) precisely, in the sense that when EvaluateConstraint(i,V,P) returns or 0, it iplies that c i ( V,P) = or c i ( V,P) = 0, respectively, but the inverse iplication need not be true. When c i ( V,P) = or c i ( V,P) = 0, EvaluateConstraint (i,v,p) ay return undefined. This is preferred when it is expensive to precisely deterine whether c i ( V,P) = or c i ( V,P) = 0. By returning undefined, the precise evaluation of the constraint is left to the sub-patches of P or the sub-viewcells of V, and because of their saller sizes, they are ore liely to belong to one of the easy cases. Of course, when c i ( V,P) is undefined, Evaluate Constraint(i,V,P) ust return undefined. For the sae purpose, EvaluateSapling Density(V,P,&SMin,&SMax) need not return the precise iniu and axiu new scan sapling densities between V and P. It is allowed to underestiate the iniu new scan sapling density and overestiate the axiu new scan sapling density. The following sections describe the algoriths. There are certainly soe other efficient ways to accoplish these operations, but these provide exaples for the ipleentation of new constraints. When V is indivisible or P is indivisible, they are treated as points, the algoriths are generally trivial, so they are not described here Maxiu-Range Constraint Let the axiu effective range of the range sensor be R. When ax c 0 ( V,P) =, the distance between any point in patch P and any view in V is equal to or less than R. ax To deterine this, four iaginary spheres of radius R ax are centered at the four corners of the patch s rectangle. If the entire viewcell V is inside all the four spheres, then EvaluateConstraint(0,V,P) returns. The viewcell can be approxiated with a sphere to speed up the coputation. If the viewcell (or its

6 sphere) intersects or is inside soe but not all the four spheres, undefined is returned. When c 0 ( V,P) = 0, the distance between any point in patch P and any view in V is greater than R. This can ax be deterined as follows. Let C be the iaginary convex hull of the four spheres of radius R that are centered at ax the four corners of the patch s rectangle. If the viewcell is entirely outside the convex hull C, then c 0 ( V,P) = 0. By approxiating the viewcell with a sphere, it is not hard to efficiently deterine whether the sphere is outside the convex hull. When the viewcell (or its sphere) has been deterined to be outside the convex hull, EvaluateConstraint(0,V,P) returns 0. For all other cases, EvaluateConstraint (0,V,P) returns undefined to indicate that the actual value of c 0 ( V,P) is still uncertain, or that the constraint is satisfied by only soe, but not all, pairs of views and patch points Vertical-Field-of-View Constraint The scanner is assued to have a 360 horizontal FOV, but a liited vertical FOV, as shown in Figure 5. To deterine whether a surface point is within the vertical FOV, we copute the angle between the y-axis (vertical axis) and the vector fro the view position to the surface point. If the angle is less than θ top or ore than 80 θ bot, then the surface point is outside the vertical FOV. When c ( V,P) = 0, every point in patch P is outside the FOV of every view in V. Figure 6 illustrates a ethod to deterine whether c ( V,P) = 0. If the directions of all the four directed lines in Figure 6(a) are in the botto outside region of the vertical FOV, then EvaluateConstraint(,V,P) returns 0. Siilarly, in Figure 6(b), if the directions of all the four directed lines are in the top outside region, EvaluateConstraint(,V,P) also returns 0. If soe of these directed lines are inside and soe are outside the vertical FOV, then Evaluate Constraint(,V,P) returns undefined. Figure 7 illustrate how to deterine whether c ( V,P) =. If all the four corners of the patch s rectangle are on the positive sides of both planes A and B, then the patch is entirely inside the vertical FOV of every view in the viewcell and the value to be returned inside θ top θ top θ bot θ bot outside y outside outside inside outside view position Figure 5. The vertical field of view of the scanner. by EvaluateConstraint(,V,P) is. Otherwise, EvaluateConstraint(,V,P) returns undefined Angle-of-Incidence Constraint The angle of incidence, φ, of a surface point p fro a view position v is the angle between the surface noral vector at p and the direction vector fro p to v. If this angle is greater than a threshold angle φ ax, then c 2 ( v, p) = 0, otherwise c 2 ( v, p) =. To deterine whether c 2 ( V,P) =, four open-ended cones are set up at the four corners of the patch s rectangle as shown in Figure 8. The base of each cone extends infinitely in the direction of the patch s noral vector, and the half angle at the apex of each cone is φ. If the viewcell V (or its sphere) is ax entirely inside all four cones, then c 2 ( V,P) =, and EvaluateConstraint(2,V,P) returns. If the sphere sphere viewcell viewcell patch s rectangle (a) (b) Figure 6. (a) The four directed lines are tangent to the sphere and point at the four corners of the patch s rectangle. Each directed line touches the sphere at the lowest point it is still tangent to the sphere. If the angles between the y-axis and all the directed lines are larger than 80 θ bot, then the patch is entirely in the botto outside region of the vertical FOV of the viewcell. (b) Each of the four directed lines touches the sphere at the highest point it is still tangent to the sphere. If the angles between the y-axis and all the directed lines are less than θ top, then the patch is entirely in the top outside region of the vertical FOV of the viewcell. y θ top θ bot patch s rectangle Figure 7. Deterining whether a patch is entirely inside the vertical FOV of every view in the viewcell. Planes A and B are both tangent to the sphere of the viewcell. The planes noral vectors n A and n B are coplanar with the noral vector of the patch. If all the four corners of the patch s rectangle are on the positive side (the side the noral vector is pointing) of both planes A and B, then the patch is entirely inside the vertical FOV of every view in the viewcell. n A n B plane A plane B

7 viewcell intersects or is inside soe but not all four cones, then EvaluateConstraint(2,V,P) returns undefined. To deterine whether c 2 ( V,P) = 0, the viewcell V (or its sphere) ust be entirely outside the openended convex hull that encloses all the four cones in Figure 8. In this case, EvaluateConstraint (2,V,P) returns 0, otherwise it returns undefined Visibility Constraint Here, we are testing the visibility between a viewcell and a rectangle-bounded patch. When c 3 ( V,P) =, it iplies the viewcell and the patch are totally visible to each other. To deterine that, one has to ensure that there is no occluder in the shaft between the viewcell and the patch, which is the 3D volue occupied by the line segents connecting every point in the viewcell to every point on the patch. To deterine c 3 ( V,P) =, planes are constructed to enclose the shaft between the viewcell and the patch s rectangle. The non-epty-space voxels in the partial octree odel are then tested, in a top-down traversal, against all the planes to find out if any of the voxels intersects the volue bounded by the planes. If not, EvaluateConstraint (3,V,P) returns. When c 3 ( V,P) = 0, the viewcell and the patch are totally invisible to, or totally occluded fro, each other. Deterining total invisibility or total occlusion between two extended objects is a difficult proble because the total occlusion ay be caused by ultiple occluders that are not connected to each other []. In the absence of an efficient algorith, we have chosen to use a probabilistic approach to estiate total occlusion. The ethod is illustrated in Figure 9. On the viewcell s sphere, the great circle parallel to the patch is first identified. Then, an equal nuber of rando points is generated in each quadrant of the disc bounded by the great circle, and in each quadrant of the patch s rectangle. Rays are shot fro the rando points on the disc to the points on the patch s rectangle. 6 rando rays are generated in this way. If all the rando rays are occluded, then it is estiated that the patch is totally occluded fro the viewcell, and EvaluateConstraint(3,V,P) returns 0. Otherwise, it is assued that the patch is partially visible Figure 8. The four open-ended cones set up at the four corners of the patch s rectangle. is the noral vector of the patch. patch s rectangle great circle on the sphere that is parallel to the patch rays Figure 9. Generating rando rays fro the viewcell s sphere to the patch s rectangle to estiate total occlusion. fro the viewcell, and EvaluateConstraint (3,V,P) returns undefined. The hierarchical structure of the partial octree odel is exploited to accelerate the deterination of whether a ray intersects any non-eptyspace voxels. There is no ill-effect when total occlusion is erroneously declared as partial visibility, except that it ay cause an unnecessary subdivision of the viewcell or the patch. On the other hand, it ay be undesirable when partial visibility is erroneously declared as total occlusion, since the patch will be disregarded. However, since the ethod is probabilistic, the issed patch ight still be reconsidered in a later acquisition cycles New Scan Sapling Density patch s rectangle The function EvaluateSaplingDensity(V,P, &smin,&smax) outputs the iniu and the axiu new scan sapling densities between V and P. Let α be the angle interval between two successive saples acquired by the range scanner, and r be the distance fro the view position v to the surface point p. The surface sapling density around point p is s( v, p) = d = cosφ αr φ is the angle of incidence of the laser bea at surface point p. When the values of α and d are fixed, the locus of the view position is the surface of a sphere with radius /(2αd), and the sphere is tangent to the surface at p. All points inside the sphere have sapling densities greater than d, and points outside have sapling densities less than d. We call it the sapling density sphere of p. Since the function EvaluateSaplingDensity() is allowed to under-estiate the iniu sapling density, it is sufficient to construct, for each corner of the patch s rectangle, the sallest sapling density sphere that encloses the viewcell, and let S be the largest of the four spheres. The iniu sapling density fro the viewcell V to the patch P is estiated as /(2αR), R is the radius of S. The viewcell ay be approxiated by a sphere. The function EvaluateSaplingDensity() is allowed to over-estiate the axiu sapling density. Let p be the point on the patch s rectangle that is closest to the viewcell s sphere. Then, let S be the largest sapling density sphere of p that touches a point of the viewcell s sphere but does not enclose the sphere. The axiu sapling density fro the viewcell V to the patch P is estiated as

8 (a) (b) planning syste is robust for real-world applications. Figure 0. (a) The feasible view volues to be evaluated. (b) The results of evaluating the patch (agenta) with the feasible view volues. The best 500 viewcells are shown. /(2αR), R is the radius of S. 4. Results Figure 0(a) shows the feasible view volues coputed for a partial odel of a large living roo. These feasible view volues are then evaluated with a patch shown in agenta color in Figure 0(b). The resulting best 500 viewcells are shown. The iniu viewcell size used is inch 3, and the iniu patch size is 2 2 inch 2. A brute-force ethod too seconds to evaluate the feasible views with the patch, while our hierarchical algorith too just.9 seconds a difference of ore than 20 ties. Typical speedups for indoor scenes are between 0 to 00 ties. Generally, larger and sipler scenes, and saller iniu viewcell and iniu patch sizes result in larger relative speedups. We have tested our NBV planning syste in siulations and on real scenes. The scanners used in both cases have only 3D translational poses, and have full horizontal FOVs but liited vertical FOVs. The siulated scanner has pose errors, and produces range data with range errors, drop-outs and outliers. Figure (a) shows the coputed view plan for acquiring scans of a synthetic living roo. The acquisition was anually terinated after the eighth scans. Figure (b)&(c) show the polygonal odels reconstructed fro the eight range iages. Every cycle, the hierarchical view evaluation was able to evaluate alost all the patches with all the feasible views within the allotted two inutes. Figure shows the acquisition of a real building interior using a DeltaSphere-3000 laser scanner. The acquisition process was anually terinated after the fifth scan. Figure (a) shows the view plan. Every cycle, ore than 75% of the patch areas can be evaluated with all the feasible views. This experient shows that our NBV (a) (b) 5. Conclusion Our hierarchical view evaluation ethod has ade exhaustive 3D view evaluation for greedy NBV planning practical. This is ainly due to the evaluation of the sensing constraints and surface sapling densities between a view volue and a surface patch, unlie previous NBV algoriths, which siply evaluate between a single view point and a single surface point at a tie. We have presented efficient algoriths to evaluate the individual sensing constraints and sapling quality between view volues and surface patches. The descriptions also serve as exaples for the design of efficient algoriths for new acquisition constraints. Acnowledgeents We than John Thoas, Heran Towles, Lars Nyland, and 3rdTech Inc. for their technical help, and than Andy Wilson and the UNC Walthrough Group for the beautiful house odel. This wor is supported by NSF grant nuber ACI References [] Daniel Cohen-Or, Yiorgos Chrysanthou, Claudio Silva, and Frédo Durand. A Survey of Visibility for Walthrough Applications. IEEE Transaction on Visualization and Coputer Graphics, 9(3):42 43, July [2] C. I. Connolly. The Deterination of Next Best Views. Proceedings of IEEE International Conference on Robotics and Autoation, pp , 985. [3] H. González-Baños, E. Mao, J.-C. Latobe, T. M. Murali and A. Efrat. Planning Robot Motion Strategies for Efficient Model Construction. Proceedings of International Syposiu on Robotics Research, pp , 999. [4] Pat Hanrahan, David Salzan, Larry Aupperle. A Rapid Hierarchical Radiosity Algorith. ACM SIGGRAPH Coputer Graphics (Proceedings of SIGGRAPH '9), 25(4):97 206, July 99. [5] Ko-Li Low and Anselo Lastra. An Adaptive Hierarchical Next-Best-View Algorith for 3D Reconstruction of Indoor Scenes. Technical Report TR06-003, Departent of Coputer Science, University of North Carolina at Chapel Hill, January [6] N. A. Massios and R. B. Fisher. A Best Next View Selection Algorith Incorporating a Quality Criterion. Proceedings of British Machine Vision Conference, 998. [7] R. Pito. A Sensor Based Solution to the Next Best View Proble. Proceedings of IEEE International Conference on Pattern Recognition, pp , 996. [8] J. M. Sanchiz and R. B. Fisher. A Next-Best-View Algorith for 3D Scene Recovery with 5 Degrees of Freedo. Proceedings of British Machine Vision Conference, Figure. (a) The view plan coputed for a real scene. (b) The final partial odel and the feasible view volue.