974 Y.Guo et al.Computers Graphics 29 (2005)972-979 1.2.Overview of our new approach its application in constrained texture mapping.Section 4 describes the details of an adaptive approach for local Our proposed method consists also of three steps.It texture coordinate refinement.Experiment data are first parameterizes a 3D mesh model using the angle- given in Section 5.In the last section,we summarize based-flattening (ABF)method [9]:it then specifies the the proposed method and highlight the future work. corresponding feature points on the 3D model and the texture plane by user interaction and calculates the texture coordinates of all other sample points on the 3D 2.ABF parametrization mesh by harmonic mapping;finally it performs an "adaptive local mapping refinement"to adjust the Our texture mapping method first parameterizes the texture coordinates of some points within the mis- 3D mesh onto the planar region.The chosen parame- matched area.Compared with previous methods,our trization method should minimize the angular or metric method has the following advantages: distortion of the embedded 2D mesh,while still guaranteeing its validity.The ABF method [9]intro- Distortion minimization.Due to the energy minimiza- duced by Sheffer et al.satisfies these conditions.This tion property of the harmonic mapping,the metric method defines the flattening problem entirely in terms distortion of our constrained texture mapping is of angles and minimizes the shape distortion of the small,as is testified by the results. embedded triangles by maintaining a consistent angular Analytical solution.It is an analytically accurate distribution of incident edges around each vertex on the solution to the constrained texture mapping problem, parametric plane.After the angular distributions have while other methods [13,15]are based on iterative been computed by solving an energy equation,the flat optimization and provide only approximate solutions. mesh can be fully determined once we fix the position of High performance.Our method can deal with a large an initial interior vertex and the length and direction of amount of data.Since it involves solving only sparse an initial edge incident to that vertex. linear symmetric equations,our algorithm is much As the ABF method can ensure the validity of more efficient and much faster than others. parametrization and minimize the angular distortion,it Real-time refinement.With our "adaptive local is employed in our approach to parameterize the 3D mapping refinement"technique,the result of the mesh.Fig.3 shows the parametrization results of two mapping can be refined in real-time. 3D mesh models. Concise process.Our method need only specify a few correspondence constraints then solves the texture coordinates using a harmonic map while the method 3.The harmonic map provided by Kraevoy et al.needs to segment the mesh and texture into several patches with Steiner We apply the harmonic map technique to establish the vertices [15]. correspondence between the parameterized 2D mesh and the texture plane.Harmonic maps have been The remainder of this paper is organized as follows. studied intensively by many researchers over a long Section 2 gives a brief review of ABF parametrization. period of time.It is an extreme of the energy density Section 3 introduces the concept of harmonic map and that satisfies the corresponding Euler-Lagrange (a) (b) (c) (d) Fig.3.3D models and corresponding results of ABF parametrization.(a),(c)the face model (1344 triangles and 690 vertices)and the cow head model (1896 triangles,972 vertices);(b),(d)the parametrization results of (a)and (c).1.2. Overview of our new approach Our proposed method consists also of three steps. It first parameterizes a 3D mesh model using the angle￾based-flattening (ABF) method [9]; it then specifies the corresponding feature points on the 3D model and the texture plane by user interaction and calculates the texture coordinates of all other sample points on the 3D mesh by harmonic mapping; finally it performs an ‘‘adaptive local mapping refinement’’ to adjust the texture coordinates of some points within the mis￾matched area. Compared with previous methods, our method has the following advantages:
Distortion minimization. Due to the energy minimiza￾tion property of the harmonic mapping, the metric distortion of our constrained texture mapping is small, as is testified by the results.
Analytical solution. It is an analytically accurate solution to the constrained texture mapping problem, while other methods [13,15] are based on iterative optimization and provide only approximate solutions.
High performance. Our method can deal with a large amount of data. Since it involves solving only sparse linear symmetric equations, our algorithm is much more efficient and much faster than others.
Real-time refinement. With our ‘‘adaptive local mapping refinement’’ technique, the result of the mapping can be refined in real-time.
Concise process. Our method need only specify a few correspondence constraints then solves the texture coordinates using a harmonic map while the method provided by Kraevoy et al. needs to segment the mesh and texture into several patches with Steiner vertices [15]. The remainder of this paper is organized as follows. Section 2 gives a brief review of ABF parametrization. Section 3 introduces the concept of harmonic map and its application in constrained texture mapping. Section 4 describes the details of an adaptive approach for local texture coordinate refinement. Experiment data are given in Section 5. In the last section, we summarize the proposed method and highlight the future work. 2. ABF parametrization Our texture mapping method first parameterizes the 3D mesh onto the planar region. The chosen parame￾trization method should minimize the angular or metric distortion of the embedded 2D mesh, while still guaranteeing its validity. The ABF method [9] intro￾duced by Sheffer et al. satisfies these conditions. This method defines the flattening problem entirely in terms of angles and minimizes the shape distortion of the embedded triangles by maintaining a consistent angular distribution of incident edges around each vertex on the parametric plane. After the angular distributions have been computed by solving an energy equation, the flat mesh can be fully determined once we fix the position of an initial interior vertex and the length and direction of an initial edge incident to that vertex. As the ABF method can ensure the validity of parametrization and minimize the angular distortion, it is employed in our approach to parameterize the 3D mesh. Fig. 3 shows the parametrization results of two 3D mesh models. 3. The harmonic map We apply the harmonic map technique to establish the correspondence between the parameterized 2D mesh and the texture plane. Harmonic maps have been studied intensively by many researchers over a long period of time. It is an extreme of the energy density that satisfies the corresponding Euler–Lagrange ARTICLE IN PRESS Fig. 3. 3D models and corresponding results of ABF parametrization. (a), (c) the face model (1344 triangles and 690 vertices) and the cow head model (1896 triangles, 972 vertices); (b), (d) the parametrization results of (a) and (c). 974 Y. Guo et al. / Computers & Graphics 29 (2005) 972–979