doi:10.1016/j.cag.2005.09.013 
Copyright © 2005 Elsevier Ltd All rights reserved.
Technical Section
A novel constrained texture mapping method based on harmonic map
Yanwen Guoa, b, 
, 
, Jin Wanga, Hanqiu Sunc, Xiufen Cuia and Qunsheng Penga, b
aState Key Lab of CAD&CG, Zhejiang University, Hangzhou 310027, China
bDepartment of Mathematics, Zhejiang University, Hangzhou 310027, China
cDepartment of Computer Science and Engineering, CUHK, Hong Kong
Available online 6 October 2005.
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Abstract
In this paper, we present a novel constrained texture mapping method based on the harmonic map. We first project the surface of a 3D model on a planar domain by an angle-based-flattening technique and perform a parametrization. The user then specifies interactively the constraints between the selected feature points on the parametric domain of the 3D model and the corresponding pixels on the texture image; the texture coordinates of other sample points on the 3D model are determined based on harmonic mapping between the parametric domain of the model and the texture image; finally we apply an adaptive local mapping refinement to improve the rendering result in real-time. Compared with other interactive methods, our method provides an analytically accurate solution to the problem, and the energy minimization characteristic of the harmonic map reduces the potential distortion that may result in the constrained texture mapping. Experimental data demonstrate good rendering effects generated by the presented algorithm.
Keywords: Texture mapping; Parametrization; Harmonic map
Fig. 1. Constrained texture mapping. S(x,y,z) defines a one-to-one correspondence between surface A of R3 and a subset D of R2 with some user-defined constraints. For example, the point P on the canthus of the 3D model must correspond with the pixel P| on the canthus of Hepburn. 
is the inverse of 
.
Fig. 2. A cow head mapped with the image of a leopard, using the algorithm presented in this paper.
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).
Fig. 4. (a) On the 3D mesh model, the green vertices belong to 1-ring of vi; the blue are 2-ring of vi. (b) The green vertices are 2-ring region of vi. (c) On the texture plane, ti is the texture point corresponding to vi. Δ1, Δ2 of vi is the sum of displacement of the green points, blue points, respectively, during each adjustment of ti on the texture plane.
Fig. 5. Mapping a facial image (b) to a 3D face model (a), we get (c). However, the quality of the eyelid (region in red rectangle of (c)) is less satisfactory and needs be refined. (d) is a close-up of the eyelid. (f) is the result of “adaptive local mapping refinement” with two iterations after adjusting pi and (g) is a close-up of the refined eyelid. (e) is the zoomed region surrounded by the red rectangle in (a); the vertices within the 2-ring region of vi are showed in green; while the vertices included in the 3-ring of vi are shown in yellow. As the 2-ring region of vi contains rare vertices, solving the local mapping is real-time.
Fig. 6. A face model is mapped with different textures: (a), (c) are original textures with constrained feature points; (b), (d) are textured results.
Fig. 7. Texturing of a body model: (a) 3D body model; (b) the texture image with constrained feature points (red dots); (c) textured model.
Fig. 8. Texturing of a cow head: (a) the texture image with constrained feature points (red dots); (b), (c) textured model at different views.
Table 1.
Performance results
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