Non-Distorted Texture Mapping Using Angle Based Flattening

Transcription

1 Non-Distorted Texture Mapping Using Angle Based Flattening A. Sheffer and E. de Sturler Abstract: This article introduces a new method for surface parameterization for texture mapping. In the first step of the algorithm we compute a planar triangulation of the faceted surface, using an Angle Based Flattening (ABF) formulation. This provides a mapping function from a three-dimensional surface to the plane. The ABF formulation defines the flattening problem as a constrained minimization problem in terms of angles (only). After generating the flat triangulation we compute a mapping from the flat triangulation to a rectangular region. The mapping minimizes the distortion of distances and areas in the parameterization with respect to the original three-dimensional surface. The texture mapping function is generated by combining the two mapping functions. The method is more general and robust than previous methods and is guaranteed to generate a valid solution. It can handle any surface for which a connected, valid, two-dimensional parameterization exists, including surfaces with large curvature gradients. It does not require the boundary of the flat two-dimensional domain to be predefined or convex. The flattening step of the algorithm uses the necessary and sufficient constraints for a valid two-dimensional triangulation. As a result, the existence of a theoretical solution to the flattening procedure is guaranteed. The second step of the algorithm aims at the preservation of distances in the combined mapping, resulting in a minimal distortion texture mapping. Keywords: texture mapping, parameterization, triangulation, angle based flattening (ABF), constrained optimization. 1. INTRODUCTION Texture is an essential component of computer generated images. To create a texture on a threedimensional surface model, a texture mapping procedure is used. Texture mapping provides a projection function from a three-dimensional surface to a two-dimensional texture pattern. Such mapping corresponds to a two-dimensional parameterization of the three-dimensional surface. To minimize the distortion of the texture, the parameterization function has to preserves the surface metric structures as much as possible. Parameterization of three-dimensional surfaces has many other applications as well, including finiteelement surface meshing [Sheffer and de Sturler 2000a; Marcum and Gaiter 1999], surface reconstruction [Floater 1997], multiresolutional analysis[eck et al. 1995], formation of ship hulls, generation of clothing patterns [McCartney et al. 1999], and metal forming. When the surface is represented using an analytic description, this description can often be used to provide the parameterization. However, many computer graphics models are represented by a triangular tessellation and the analytic representation of the surface is often not available. A triangulated mesh is the natural description of surfaces constructed from scattered data such as input from laser range scanners or sampling on a regular three-dimensional grid. An algorithm for two-dimensional parameterization or flattening of tessellated surfaces first constructs a two-dimensional mesh with a similar connectivity to the three-dimensional surface. Then, a parametric function is defined between the two-dimensional mesh facets and their three-dimensional counterparts. ACM Journal Name, Vol. V, No. N, Month 20YY, Pages 1 0??.

2 2 Several approaches for parameterization of tessellated surfaces have been suggested. In [McCartney et al. 1999] the authors suggest a heuristic approach for triangulation flattening. The method is based on optimal local positioning of projected nodes, based on a sequential addition of the nodes. Similar ideas were suggested in [Samek et al. 1986] and [Bennis et al. 1991]. The methods are efficient and produce good results for nearly planar surfaces. However, they do not guarantee the preservation of the metric structures in the two-dimensional mesh or even the validity of the mesh. Eck et al.[eck et al. 1995] suggest the use of harmonic maps to generate the two-dimensional projection of the three-dimensional mesh. The algorithm produces approximations of good quality, and provides an accurate mapping function. A major disadvantage of the method is that it requires the boundary of the two-dimensional mesh to be predefined and convex. Another drawback is that the method does not guarantee the validity of the resulting flat mesh (i.e. it can generate inverted elements). Floater [Floater 1997] describes a parameterization method which computes the positions of the nodes in the flat mesh using the solution of a linear system based on convex combinations. Floater was the first, to our knowledge, to formally define what types of surfaces can be parameterized and to provide a parameterization method which is provably correct for any such surface. The method also guarantees that the resulting flat mesh is valid. However, similarly to the algorithm in [Eck et al. 1995] the method requires the boundary of the two-dimensional mesh domain to be predefined and convex. Marcum [Marcum and Gaiter 1999] introduces the use of finite-element techniques to compute the locations of the flat mesh nodes. The method computes the boundary as part of the solution, using an iterative procedure that alternatingly computes the boundary while keeping the interior fixed and vice versa. However, no validity guarantees are given for this method. A lot of research on computing parameterizations of tessellated surfaces has been done in the context of computer graphics, since parameterization is required to generate non-distorted texture mappings [Foley et al. 1992]. Some papers like [Haker et al. 2000] involve a mapping to a sphere, which provides good parameterization for closed surfaces, but cannot handle surfaces which have boundaries. Zigelman et al. [Zigelman et al. 2000] provide a method for flattening surfaces by using multi-dimensional scaling. It computes the two-dimensional domain boundaries as part of the solution. The method does not guarantee the validity of the resulting flat mesh. Levy and Mallet [Levi and Mallet 1998] suggest a method which provides user control on the spread of the distortion across the flattened mesh. The method uses some of the formulations introduced in [Floater 1997], and as a result has similar limitations. Contribution In [Sheffer and de Sturler 2000a] we presented a new method for the parameterization of tessellated three-dimensional surfaces via a mapping to a two-dimensional domain. The method was developed in the context of finite-element surface meshing. The suggested procedure is based on the observation that a planar triangular mesh is defined by the angles within each triangle, up to global scaling, rotation or translation. Based on this observation, our method formulates the flattening problem solely in terms of the planar triangulation angles. The algorithm minimizes the relative distortion of the planar angles with respect to their counterparts in the three-dimensional surface, while satisfying a set of constraints that ensure the validity of the flat mesh. This formulation yields a constrained minimization problem defined entirely in terms of flat mesh angles; the locations of the flat mesh nodes do not play a role. The solution of this problem provides the set of angles which determine the flat triangulation. We refer to this new method as the Angle Based Flattening or ABF algorithm. The ABF formulation does not require the two-dimensional boundary to be predefined and does not place any restrictions on the shape of the boundary or the surface curvature. The solution method based on this formulation is provably correct; see [Sheffer and de Sturler 2000b; 2000a]. The result of the procedure is a valid two-dimensional mesh, which maintains the original surface connectivity and minimizes the distortion of the mesh angles

3 3 resulting from the mapping to a plane. This minimizes the shape distortion of the mapping. In this work we adopt the proposed method for use in texture mapping. The ABF procedure is used as the first step in generating a parameterization suitable for texture mapping. We need a second step to ensure the preservation of lengths, which we describe below. In this paper we provide several examples of texture mapping using the developed algorithm. The main advantages of our method are: (1) The method provably provides a valid parameterization for any surface for which a parameterization exists. (2) The method minimizes the distortion of both shape and length caused by the parameterization. (4) We compute the boundary of the two-dimensional domain as part of the parameterization algorithm; we do not need to define the boundary in advance. (5) The two-dimensional domain can have any shape; specifically, it does not have to be convex. (6) The robustness of the method is not affected by the input mesh quality due to the direct use of angles in the formulation of the flattening algorithm (part 1 of our parameterization for texture mapping). (7) The ABF formulation avoids the problem of scaling to work on meshes with different feature sizes, i.e. the tolerances and error measurements do not need to be modified when different mesh element sizes are used. 2. ALGORITHM OVERVIEW In order to preserve the surface metric structures during parameterization both angle and length distortion have to be minimized. For a general surface there is no mapping to the plane which fully preserves distances or areas [Ahlfors and Sario 1960]. Neither is there a mapping to the plane that for a faceted surface with non-zero curvature fully preserves angles. Hence, we suggest a method which strives to minimize both types of distortion. The first step of the algorithm minimizes the angular distortion and the second minimizes the distances distortion. Based on Riemann s theorem [Ahlfors and Sario 1960], for a smooth surface there exists a mapping to the plane which is conformal, i.e., it preserves the surface angles. Such a mapping is also guaranteed to locally preserve the shape and the metric structures. In general, however, a conformal mapping does not exist for a faceted (non smooth). Hence, the first part of our algorithm computes a mapping from the three-dimensional surface to the plane which approximates a conformal mapping. The method finds the optimal two-dimensional boundary for the mapping based on the shape preservation (angle minimization) criteria. As observed earlier, the angles in a two-dimensional triangulated mesh fully define the mesh metric structures (up to a global scaling factor). Hence we can formulate the mesh flattening problem using the flat mesh angles as the variables. In the first step of the algorithm the mapping to the plane is computed through a constrained minimization problem based on the angle based flattening formulation (Section 3). The constraints are the necessary and sufficient conditions for a valid two-dimensional mesh, and we minimize the (relative) deviation of the angles from their optimal two-dimensional projections. This step is guaranteed to provide a valid parameterization which minimizes the angular distortion. Since the preservation of lengths is also important for texture mapping, a second mapping which minimizes length distortion has to be applied. After providing a flat parameterization using the ABF algorithm we compute a mapping from the bounding box of the flat projection to a rectangular region in the plain. The mapping is designed to minimize the distortion of the distances and areas with respect to the original three-dimensional surface. The computation is based on applying a mesh smoothing procedure to a Cartesian grid. The smoothing uses a sizing function which is based on the ratios between the lengths of edges in the three-dimensional surface and their counterparts in the flat surface. It is described in detail in Section 4. The texture mapping function is defined by the product of the two mapping functions. We will now discuss each parameterization stage in detail.

4 4 3. ANGLE BASED FLATTENING This section describes the angle based flattening algorithm, which provides the first component of the texture mapping function. First, it formulates the flattening procedure in terms of angles, defining a constrained minimization problem. Then, it discusses the numerical solution of the problem. Finally the locations of the flat mesh nodes are computed based on the angles. 3.1 Definition of the Constrained Minimization Problem We use the following notation to define the objective function to be minimized and the constraints. The index i always indicates faces, the index j indicates angles inside a face, and the index k indicates nodes. f i, i = 1... P, are the triangulation faces (either in the flat mesh or in the original mesh, as will be clear from the context). α j i, i = 1... P, j = 1, 2, 3, are the flat mesh angles; the angles in a face are numbered counterclockwise in increasing order. The vector of all angles is denoted α. β j i, i = 1... P, j = 1, 2, 3, are the corresponding original mesh angles. N k, k = 1... M, are the mesh nodes, where k = 1... M int (< M) indexes the interior nodes. α j(k) i is the angle in face f i at node N k. is the angle in the original mesh corresponding to α j(k) i. β j(k) i In the following, we implicitly assume the obvious relations among the indices that derive from the connectivity of the mesh. So when we write i βj(k) i, the index i runs only over those faces that contain node N k. The sum is therefore over all angles adjacent to node N k Objective Function. The objective function (to be minimized) is defined by F (α) = N i=1 j=1 3 (α j i φj i )2 w j i (1) where φ j i is the optimal angle for αj i in the two-dimensional mesh, and the wj i > 0 are weights. Our standard initial choice for the weights is w j i = (φj i ) 2. This makes the objective function measure the relative distortion of the angles rather than the absolute distortion. We derive the optimal angles φ j i from the angles β j i by computing a scaling factor per node: { j(k) β φ j i (k) = i β j(k) 2π i βj(k) i, N k is an interior node, i, N k is a boundary node, where N k is the mesh node to which the face f i is attached at the corner j. Since the input mesh is supposed to be valid, we assume that where ε 1 is an arbitrarily small (input) parameter. β j i ε 1 > 0, (3) Constraints. The following constraints are necessary and sufficient to ensure that the resulting mesh is valid. (1) g (1) i,j αj i ε 2 > 0, for i = 1... P, j = , and some ε 2 such that ε 1 ε 2 > 0 ; (2) g (2) i αi 1 + α2 i + α3 i π = 0, for i = 1... P ; (3) g (3) k i αj(k) i 2π = 0, for N k : k = 1... M int (interior nodes); (2)

5 (4) g (4) k if j(k) = 1. 5 Πi sin(αj(k)+1 i ) Π i sin(α j(k) 1 i ) 1 = 0, for N k: k = 1... M int, where j(k) + 1 = 1 if j(k) = 3 and j(k) 1 = 3 The first two constraints deal with the validity of individual faces. Constraint (1) maintains the orientation of a face (up or down) with respect to the mesh, and (2) ensures that each face is valid. Constraints (3) and (4) are necessary to ensure the topological validity of the mesh, because the connectivity of the mesh is not an explicit constraint. To explain this we introduce the following terminology. We define the wheel associated with an interior node as the set of all faces that share this interior node. We refer to the edges in the wheel connected to the interior node as the spokes. Constraint (3) guarantees that after fixing the position of an interior node and the direction of one spoke, going over all spokes in counterclockwise order around the wheel, all shared edges (spokes) of neighboring faces coincide. Constraint (4) guarantees that after fixing the length of one (arbitrary) spoke, going over all spokes in counterclockwise order around the wheel, that the length of the last spoke (coinciding with the first) agrees with the length of the first spoke. Note that, since we only compute angles, the lengths of the intermediate spokes can be made to agree by stretching or shrinking the faces appropriately. So we only need one constraint. Going over all interior nodes, it is easy to see that the interior of the mesh is valid. The necessity of the third constraint is obvious. The necessity of the fourth constraint is best illustrated by Figure 1, where the constraint is violated. Since the connectivity of the mesh is not explicit in the computation of the angles of all faces, we must make sure that the (arbitrarily chosen) first face in a wheel and the last face agree on the length of the shared edge. The fourth constraint is derived as follows. Consider an interior node as shown in Figure 1, and assume we fix the edge length l 1. Now from the angles α 1 1 and α 2 1 in face f 1 we can compute l 2 using the relation l 1 = sin α2 1 l 2 sin α1 1. (4) Using l 2 and the angles α 1 2 and α 2 2 in face f 2 we can compute l 3 to obtain an equation analogous to (4) Combining (4) and (5) we can relate l 3 to l 1 l 2 = sin α2 2 l 3 sin α2 1. (5) l 1 = l 1 l 2 = sin α2 1 sin α2 2 l 3 l 2 l 3 sin α1 1 sin α2 1. (6) Applying (4) for each face as we go in counterclockwise direction around the wheel, we can compute l 4, l 5,..., l 7. However, for the mesh to be valid we need l 7 = l 1. Using an equation analogous to (6) to relate l 7 to l 1 we get l 1 = l 1 l2 l6 = sin α2 1 l 7 l 2 l 3 l 7 sin α1 1 sin α2 2 sin α2 1 sin α2 6 sin α6 1 = 1, (7) which demonstrates the necessity of our fourth constraint. Going over the mesh faces and interior nodes, it is easy to see that if the constraints are satisfied the interior of the mesh is valid. Note that our constraints do not avoid potential crossing of the boundary edges. We discuss this concern in Section Solution of the Constrained Minimization Problem We solve the constrained minimization problem as follows. As argued in Section 5, for any valid input to our algorithm a valid planar mesh exists. Since the optimal angles φ j i are strictly positive and our objective

6 6 l 5 l 4 l 3 l 6 l 2 α j l 7 α i l 1 α α 1 2 l 1 l 2... sin( ) = sin( ) l 1 α 1 l 2 α 2 l 7 sin( α1 ) α =... i l1 sin( α 2 ) α j sin( ) sin( ) Fig. 1. Incompatibility of edge length in a wheel. function measures the distance between the angles α j i and the optimal angles φj i, we may prevent the algorithm from making angles too small by increasing the weight in the objective function for those angles rather than explicitly preventing the algorithm from moving outside the feasible set. This corresponds to a change in the norm that measures the distance to our optimal angles. Moreover, because of their simple form, the inequality constraints are very easy to check. Therefore, we formulate the optimization problem without explicitly taking the inequality constraints into account. If the optimization algorithm makes a certain angle too small we reject the iterate, adjust the weight of that angle, and continue. We use a standard Lagrange multiplier formulation for the optimization problem with the equality constraints. The auxiliary objective function then becomes F (α) + N i=1 M int λ i g (2) i (α) + k=1 µ k g (3) k Mint (α) + k=1 ν k g (4) k (α). (8) We use Newton s method to solve for a stationary point of the auxiliary function that satisfies the equality constraints. We modify Newton s method as indicated above to make sure we satisfy the inequality constraints. In most of our examples Newton s method converges in a few iterations and we do not need to adapt the weights. The sparse linear systems of equations that arise in Newton s method are solved using a sparse direct solver from the SuperLU package [Demmel et al. 1999; Demmel et al. 1999]. 3.3 Node Placement After the algorithm computes the angles of the projected mesh, we compute the placement of the mesh nodes on the z = 0 plane. As stated earlier the angles of a two-dimensional mesh define the mesh up to a linear transformation (translation, rotation or scaling). Hence, by placing the end nodes of a single mesh edge in the plane, we fully define the placement of the other nodes, based on the mesh angles. The procedure is done as follows. Choose a mesh edge e 1 = (N 1 a, N 1 b ). Map N 1 a to (0, 0, 0). Map N 1 b to ( e1, 0, 0). Push e 1 on the stack S. While S not empty, pop an edge e = (N a, N b ). For each face f i = (N a, N b, N c ) containing e: If f i is marked as set continue.

7 7 If N c is not yet projected to the z = 0 plane, compute it s position based on N a, N b and the face angles α 1 i, α2 i and α3 i. Mark f i as set, push (N a, N c ) and (N b, N c ) on the stack. The node placement gives us a parameterization function t 1 from the three-dimensional surface to a flat domain. Figure 2 shows two examples of applying the flattening procedure to three-dimensional surfaces. The generated textures are shown in Figure 7. (a) (b) (c) (d) Fig. 2. Flat triangulation examples. (a)(b) Folded plane (nearly developable surface). (a) Original model. (b) Flat triangulation. (c)(d) Human face. (c) Original model. (d) Flat triangulation. 3.4 Boundary Intersections The set of constraints defined in Section 3.1 guarantees the validity of the flat mesh at any interior node. However it does not prevent the flat surface from having self-intersections at the boundary. Boundary intersections make the mapping from the surface in R 3 to R 2 a many to one mapping instead of one to one. This makes the inverse mapping invalid, and hence can cause difficulties for some applications. However for texture mapping only the R 3 to R 2 mapping is used, providing the texture value for each point on the three-dimensional surface. The mapping in case of boundary overlaps remains continuous and smooth and therefore it preserves the texture patterns. Hence, for texture mapping purposes the intersections can simply be ignored. A texture for a model with intersecting two-dimensional projection is shown in Figure 3. Note that this is not true when the mapping folds over on the interior, as this causes folding of the texture pattern as well. For other applications, such as surface meshing where the mapping has to be one to one, an intersection removal procedure [Sheffer and de Sturler 2000a] can be used. The suggested approach handles selfintersections as a post-processing step. First, the method generates a flat surface mesh by solving the

8 8 (a) (b) (c) Fig. 3. Texture mapping for a spiral surface with non unique two-dimensional mapping. (a) Original surface (110 elements). (b) Flat parameterization containing boundary intersections. (c) Final texture. constrained minimization problem (Equation 8). Then, the mesh is tested for intersections of boundary edges. If an intersection exists additional constraints are added to the system of equations above. The method then solves the new constrained system of equations using the previous solution as the initial guess for the solver. The constraints are computed so as to remove the intersection between the edges by increasing the angles on the interior loop generated by the intersection (Figure 4). θ 1 θ θ 2 3 θ 1 θ 2 θ 3 e 1 e 2 e 1 e 2 Fig. 4. Removing a boundary intersection. The intersecting edges are pulled apart by increasing the size of the boundary angles in the interior loop created by the intersection. 4. LENGTH PRESERVING MAPPING IN R 2 After the flat triangulation is generated, the mapping t 1 from the three-dimensional surface to R 2 can be computed. For every point in the three-dimensional surface we define its image under t 1 as the point with the same barycentric coordinates in the flat mesh triangle as the original had in the corresponding surface triangle. This mapping minimizes the angular distortion, but it may not preserve lengths or areas accurately. To improve length (distance) preservation we compute an additional mapping t 2 from R 2 to R 2 as explained below. 4.1 Sizing Function To compute the mapping t 2, we first compute a sizing function representing the (isotropic) distortion of length, i.e., linear distortion, at any point on the flat mesh. The sizing function S is computed as follows. For each edge e = (N k, N l ), compute the distortion ratio as R k,l = 2D e 3D e, the ratio of the edge length in the flat mesh to the edge length in the original surface.

9 9 For each N k, k = 1... M, compute the distortion ratio at the node, R k, as the average of the distortions computed over the edges containing N k. For each point p we locate the mesh triangle f i = (N a, N b, N c ) it belongs to. We compute the linear distortion S(p) using the barycentric coordinates of p in f i. For p = N a u + N b v + N c w, u + v + w = 1 we set S(p) = R a u + R b v + R c w. 4.2 Mapping A texture mapping function T defines a mapping from the three-dimensional surface to a rectangular domain in R 2. If t 1 is used as this function, then the linear distortion of the mapping will be approximately S. To reduce the distortion, an additional mapping t 2 from R 2 to R 2 needs to be introduced with linear distortion roughly equal to S 1. The combined mapping will then have the desired property of near zero linear distortion. To compute such t 2 we consider the inverse problem of finding t 1 2 and inverting it. To achieve this the following procedure was developed. Let F be the flat triangulation. 1. Compute the bounding box B of F. 2. Generate a uniform Cartesian grid G 1 for the bounding box. 3. Generate a copy G 2 of the grid G 1, and embed F in G Adapt G 2 by grid smoothing using the sizing function S defined on F as explained below. (The mapping from G 1 to G 2 gives us the desired t 1 2, i.e. a mapping with linear distortion S.) 5. Define t 2 : R 2 R 2 by mapping any point on G 2 to a point on the corresponding triangle in G 1 at the same barycentric coordinates. The steps of the procedure are shown in Figure 5. The flat triangulation, Figure 5-(b), is embedded in the smoothed grid, Figure 5-(e). We define the texture mapping function T as: T t 2 t 1 : R 3 R 2 (9) Because of the use of the additional sizing based mapping t 2, the final texture mapping function preserves lengths fairly accurately. 4.3 Smoothing Mesh smoothing is a procedure which changes a mesh specifications (size/element shape) by changing the locations of the mesh nodes, without changing the mesh connectivity [Owen 1998]. In our application we use the smoothing procedure to modify the mesh sizing. We start with an inital uniform grid and modify it to conform to the sizing function defined above. There are several methods [Canann et al. 1998] for smoothing two-dimensional meshes. We use Laplacian smoothing [Field 1988] for its simplicity. The procedure to modify G 2 to conform to the sizing function S is as follows. Note that the smoothing algorithm uses only the Cartesian grid edges and ignores the mesh diagonals. 1. For each edge e = (N a, N b ) G 2 compute the desired edge length (up to a global scaling factor): (Outside of F, S is set to 1). l(e) = S(N a) + S(N b ) 2 2. Apply Laplacian smoothing to adjust the edge lengths to the sizing function: (10)

10 10 (a) (b) (c) (d) (e) (f) Fig. 5. Texture mapping for a cat head model. (a) The head (256 elements). (b) Flat parameterization. (c) Texture using only t 1 (without sizing function). (d) Uniform grid G 1 of the bounding box. (e) Smoothed grid G 2. (f) Final texture using T. 2.1 For each interior node N G 2 in turn, 2.2 Repeat 2.1 while d > ɛ. compute N = 1 e=(n,n ) l(e) N 1 e=(n,n ) l(e) compute d = max(d, N N ), and set N = N 3. Recompute the desired lengths and repeat the smoothing (goto 1.), until the lengths l(e) or the node locations no longer change. As discussed in the next section the procedure converges to a mesh which conforms to the sizing function. For the texture mapping we use the mapping function from the smoothed grid G 2 to the original uniform, Cartesian grid G 1.,

11 5. ALGORITHM VALIDITY 11 To show that the presented algorithm is valid we need to prove that both parts of the algorithm provide valid parameterization functions t 1 and t 2. We will only outline such a proof here. In [Sheffer and de Sturler 2000a] we prove the correctness of the ABF procedure, which provides the first step of the texture mapping. It shows that given valid input the algorithm is guaranteed to converge to a valid flat triangulation that minimizes angular distortion. This section briefly repeats the main points of the proof and the definition of a valid input to the flattening procedure. 5.1 Valid Input Definition To define the input to the algorithm and prove the existence of a solution to the constrained minimization problem defined earlier, we use the following standard definitions from graph theory [Marshall 1971]. A (simple) graph is defined as G = (V, E) where V = {i : i = 1... N} are the graph nodes and E the graph edges. E = {(i, j), i j} is a subset of all unordered pairs of nodes. A planar embedding of a graph G is defined as follows: each node i is mapped to a point in R 2, each edge (i, j) is mapped to a curve whose endpoints are i and j, the only intersections between curves are at the common endpoints. A graph G is said to be planar if it has a planar embedding. A valid input to the algorithm is a straight line embedding in R 3 of a triangulated planar graph. By definition, if a triangulated surface is not an embedding of a planar graph it does not have a valid twodimensional parameterization, since a parameterization is a planar embedding. Intuitively, for a surface to be an embedding of a planar graph it has to be a manifold, have a single boundary loop and no tunnels. Figure 6 shows a surface mesh which is not an embedding of a planar graph. To test if an input set is valid we use the planarity testing algorithm by Hopcroft and Tarjan [Hopcroft and Tarjan 1974], which runs in linear time (in the mesh size). Fig. 6. A manifold surface with a single boundary loop, which is not an embedding of a planar graph. In practice, we do allow input meshes which have more than one boundary loop, allowing a single outside boundary and multiple interior boundary loops. In such a case, the interior loops are triangulated by connecting their boundary nodes prior to starting the minimization procedure (e.g. Figure 10(b)). 5.2 Validity of the ABF Algorithm As described above the constraints defined in Section 3.1, are sufficient to guarantee the validity of the flat triangulation. Hence, if the minimization algorithm converges to a solution, the resulting mesh is guaranteed to be valid. Therefore all that remains to show is that the solution space for the minimization procedure is not empty and that a robust Newton type algorithm [J. E. Dennis and Schnabel 1996] will

12 12 converge to a minimum in the feasible space. Fary s Theorem [Fary 1948] states that every planar graph has an embedding in R 2 with straight edges. This means that for every valid input to our algorithm a valid flat triangulation exists. Since our constraints for a valid triangulation are necessary, the solution space for the minimization procedure is not empty. A compactness argument shows that our objective function has at least one local minimum on the closure of the constrained set. In that case, if the nonlinear function is smooth, globally convergent extensions of Newton s method are guaranteed to find a solution [J. E. Dennis and Schnabel 1996]. A demonstration of the smoothness properties provided in [Sheffer and de Sturler 2000a] concludes our discussion. 5.3 Validity of Texture Mapping The second part of the texture mapping algorithm computes the function t 2 from a smoothed grid to a uniform one. Given a fixed domain boundary, Laplacian smoothing is guaranteed to converge [Field 1988], and since the sizing function is continuous the iterative smoothing is guaranteed to converge as well. For a convex domain Laplacian smoothing is also guaranteed to provide valid meshes (with no inverted elements) [Field 1988]. Hence, the second part of the algorithm is guaranteed to converge to a valid mesh. Concluding, both steps of the algorithm are guaranteed to converge to valid solutions and combining the two mappings together provides a mapping which is valid. The new mapping appears to minimize both angular and area/length distortions. It is a classical result in differential geometry [Ahlfors and Sario 1960], that it is not possible to map a general surface to a plane without some distortion of the distances. It is also impossible to map a faceted surface with non-zero local curvature to a plane without some angular distortion. Hence at best a compromise can be made between length preservation and conformality of the map, minimizing the two types of distortion. This is what the presented method tries to achieve. 6. EXAMPLES (a) (b) Fig. 7. Textures generated for the models in Figure 2 Throughout the paper the texture mapping procedure is demonstrated on several examples of varying complexity. The properties of our examples are summarized in Table I. The number of iterations in the second stage (the computation of t 2 ) measures the outer iterations, involving the computation of the edge lengths (steps (1) (3)). We measure both angular and length distortion of the flat surface with regard to the original mesh. We use F (α) 3P (Equation 1) with the weights wj i w j i = (φj i ) 2 to measure angular distortion. We use normalized edge length ratio 1 {e} set to the initial choice of e ( 2D e 3D e 1)2,

13 13 where the node locations in two dimensions are computed using T. In both cases the division by number of angles or edges is done to normalize the error, making it independent of mesh size. Model Num. Runtime Num. t 1 Num. t 2 Distortion Distortion faces (sec.) iterations iterations (angle) (length) Folded Plane e-7 2.5e-6 Face Cat Head Balls Dome Clown Full Cat Cat with Seams Table I. Flattening examples data. Figure 7 shows the texture mapping for the models in Figure 2. Figure 7(a) shows the texture generated for a nearly developable surface. It demonstrates the distortion minimization property where the resulting flat surface has near zero distortion compared to the original three-dimensional surface. Figure 7(b) shows the texture for the face model in 2(c). The checkered pattern emphasizes the preservation of both angles and distances in the mapping. In Figure 5 we show a model of a cat s head (cut around the neck). The model is quite complex and has a large overall curvature, hence the distortion is relatively high. The figure shows the difference between using the result of the first step directly for the texture mapping and using the combined mapping. Adding the smoothing step dramatically reduces the distance and area distortion without affecting the angular deformation. Figure 8 shows another example emphasizing the need for the smoothing step. The dome shaped surface also demonstrates the robustness of our method in handling badly shaped triangles. The original surface contains multiple sliver, triangles which are likely to cause problems for flattening methods based on node locations, but they are not a problem for our method which uses angles as variables. Figure 9 displays a surface build from three spheres positioned at 120 around a joint axis. The spheres are cut at about quarter way up from the base, to create a surface which can be parameterized. The surface mesh is highly non-smooth with high local curvature changes and with multiple sliver triangles, but, despite this, the parameterization converges in a small number of iterations and gives good results. Note that despite the increase in model size (compared to the cat and dome models), the number of iterations required to obtain a solution does not increase. In Figure 10 we demonstrate the method s ability to handle surfaces with interior holes, and significant gradation in element sizes. An earlier example (Figure 3) shows the texture mapping for a surface, where the first step of the parameterization resulted in boundary intersections and non unique mapping to the plane. It demonstrates that this is not an obstacle for texture mapping which is concerned with having a valid mapping from three dimensions to a plane rather than the other way around. Finally, Figure 11 shows a complicated model of an entire cat (without the planar base). This example demonstrates the robustness of our method and its ability to handle highly complicated surfaces. For any flattening algorithm the distortion increases with the increase in the surface area to perimeter ratio as well with the increase in local and global curvature. As a result the texture distortion for a model as complicated as the entire cat tends to be high (Figure 11(c)). The main alternative for reducing the

14 14 (a) (b) (c) (d) (e) Fig. 8. Texture mapping for a dome shape. (a) Original surface (256 elements). (b) Flat parameterization (providing t 1 ). (c) Texture using only t 1 (without sizing function). (d) Smoothed grid G 2. (e) Final texture using T. distortion seems to be the introduction of seams in the model as shown in Figure 11(d). This reduces the texture distortion but introduces seams into the texture (Figure 11(f)). The seams were added using a fully automatic procedure which we are currently testing. 7. SUMMARY We have proposed a new method for texture mapping based on the combination of a recently proposed algorithm for flattening three-dimensional surfaces and an algorithm for mesh smoothing that takes into account a given sizing function. We have outlined the main properties and underlying theory of our algorithm. Our method can be applied to very general problems and has few restrictions. Specifically, it does not require the boundary to be predefined or convex. The surface flattening algorithm is based on minimizing the (relative) distortion of the mesh angles in each face subject to the necessary and sufficient conditions of a valid two-dimensional mesh. The mesh smoothing algorithm takes the sizing function that results from the flattening algorithm into account to reduce distortion of length and area. In our examples we demonstrate that the proposed method generates good texture maps for complex surfaces in a modest amount of time. Although we must solve a nonlinear system of equations, in practice the work involved is modest. For all our test cases convergence seemed to be very fast. The convergence seems loosely dependent on the

15 15 (a) (b) (c) Fig. 9. Texture mapping for 3 balls (non-smooth surface). (a) Original surface (1032 elements). (b) Flat parameterization (providing t 1 ). (c) Final texture using T. (a) (b) (c) (d) Fig. 10. Texture generation for a surface with holes. (a) The original model. (b) Model with the additional triangles filling the holes. (c) Flat triangulation of the model. (d) Texture.

16 16 (a) (b) (c) (d) (e) (f) Fig. 11. Texture mapping for a full cat model (without the planar base). (a) The model (671 elements). (b) Flat parameterization. (c) Generated texture. (d) The model with seams (in red). (e) Flat parameterization containing seams. (f) Final texture. curvature and level of complication of the three-dimensional mesh, which is to be expected. An extension of the algorithm we currently investigate is the automatic introduction of seams and surface subdivision as shown in Figure 11. Adding seams significantly reduces the texture distortion but introduces seam lines into the texture. The current research concentrates on two questions: how to introduce the shortest seams possible, while reducing the distortion to an acceptable level; and how to glue textures continuously across the seams, so they will not show. Another area of future research is the parameterization of very large models. For models with hundreds of thousands of elements, solving the flattening problem directly will likely be very expensive. We may significantly reduce the computation time for such surfaces by using a mesh simplification procedure to produce a mesh with fewer elements. The parameterization obtained for the simplified mesh, can then be adjusted to handle the original.

17 8. ACKNOWLEDGMENTS 17 The work reported here was supported, in part, by the Center for Process Simulation and Design (NSF DMS ), the Center for Simulation of Advanced Rockets (DOE LLNL B341494), and Sandia National Laboratories (SNL 16806). The authors would like to thank S. Har-Peled for his contribution to the development of the surface cutting procedure and John Hart for useful discussions and pointers to the literature. REFERENCES Ahlfors, L. V. and Sario, L Riemann Surfaces. Princeton University Press, Princeton, New Jersey. Bennis, C., Vézien, J. M., and Iglésias, G Piecewise surfaces flattening for non-distorted texture mappinng. Proc. SIGGRAPH 91 25, Canann, S. A., Tristano, J. R., and Staten, M. L An approach to combined laplacian and optimization-based smoothing for triangular, quadrilateral, and quad-dominant meshes. 7th International Meshing Roundtable, Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S., and Liu, J. W. H A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Appl. 20(3), Demmel, J. W., Gilbert, J. R., and Li, X. S SuperLU user s guide. available from xiaoye/superlu/index.html. Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., and Stuetzle, W Multiresolution analysis of arbitrary meshes. Computer Graphics (Annual Conference Series, SIGGRAPH 95), Fary, I On the straight line representations of planar graphs. Acta Sci. Math. 11, Field, D. A Laplacian smoothing and delaunay triangulations. Communications in Applied Numerical Methods 4, Floater, M. S Parametrization and smooth approximation of surface triangulations. Computer Aided Geometric Design 14, Foley, J., van Dam, A., Feiner, S., and Hughes, J Computer Graphics. Addison-Wesley Publishing, Massachusetts. Haker, S., Angenent, S., Tannenbaum, A., Kikinis, R., Sapiro, G., and Halle, M Conformal surface parameterization for texture mapping. IEEE Transactions on Visualization and Computer Graphics 6 (2), Hopcroft, J. and Tarjan, R Efficient planarity testing. Journal of ACM 21(4), J. E. Dennis, J. and Schnabel, R. E Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAMpub, Philadelphia, PA , USA. Levi, B. and Mallet, J Non-distorted texture mapping for sheared triangulated meshes. Proc. SIGGRAPH 1998, Marcum, D. L. and Gaiter, J. A Unstructured surface grid generation using global mapping and physical space approximation. 8th International Meshing Roundtable, Marshall, C. W Applied Graph Theory. Wiley, New-York. McCartney, J., Hinds, B. K., and Seow, B. L The flattening of triangulated surfaces incorporating darts and gussets. Computer-Aided Design (CAD) 31, Owen, S. J A survey of unstructured mesh generation technology. 7th International Meshing Roundtable, Samek, M., Slean, C., and Weghorst, H Texture mapping and distortions in digital graphics. The Visual Computer 2(5), Sheffer, A. and de Sturler, E. 2000b. Surface parameterization for meshing by triangulation flattening. Proc. 9th International Meshing Roundtable, Sheffer, A. and de Sturler, E. 2000a. Surface parameterization for meshing using angle-based flattening. To appear in Engineering with Computers. Zigelman, G., Kimmel, R., and Kiryati, N Texture mapping using surface flattening via multi-dimensional scaling. CIS report CIS , submitted to IEEE Trans. on Visualization and Computer Graphics.