Effective Derivation of Similarity Transformations
for Implicit Laplacian Mesh Editing

Hongbo Fu     Oscar Kin-Chung Au    Chiew-Lan Tai    

Computer Graphics Forum (CGF)


Configuration-independent merging. The goal is to merge the Mannequin head model (source) to the Venus model (target). The user only specifies the correspondence between the merging boundaries. In (a) and (b), the Mannequin head model has different positions, orientations and scales. Our configuration-independent merging method produces the same result (c), given the same boundary correspondence. The lines indicate two user-specified key correspondences.

Laplacian coordinates as a local shape descriptor have been employed in mesh editing. As they are encoded in the global coordinate system, they need to be transformed locally to reflect the changed local features of the deformed surface. We present a novel implicit Laplacian editing framework which is linear and effectively captures local rotation information during editing. Directly representing rotation with respect to vertex positions in 3D space leads to a nonlinear system. Instead, we first compute the affine transformations implicitly defined for all the Laplacian coordinates by solving a large sparse linear system, and then extract the rotation and uniform scaling information from each solved affine transformation. Unlike existing differential-based mesh editing techniques, our method produces visually pleasing deformation results under large angle rotations or big-scale translations of handles. Additionally, to demonstrate the advantage of our editing framework, we introduce a new intuitive editing technique, called configuration-independent merging, which produces the same merging result independent of the relative position, orientation, scale of input meshes.

mesh editing, similarity invariant, Laplacian coordinates, configuration-independent, mesh deformation, mesh merging

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An earlier version: Technical report, HKUST-CS05-01, Jan 2005


An example of rotation of local features resulting from translation of handles. The deformation results by [Zayer et al. 2005] (Left), [Lipman et al. 2004] (Middle) and ours (Right) when the handle at the tail of the dinosaur undergoes a big-scale translation.


(a) The original Armadillo model with four handles specified. (b) and (c): Two views of the deformed model by applying several rigid transformations to the handles. (d) The deformation result with the same view as (c) but without shearing removal.


The hind part of the Feline model is deformed and merged to the fore part of the Dinosaur model. (b) is the result of configuration-independent merging with the configuration in (a), and (d) is the result of configuration-dependent merging with the configuration in (c). In the configuration-dependent merging, the two feet are specified as handles (purple), thus remain fixed. The same boundary correspondence is used for both merging.

    author = {Hongbo Fu and Oscar K.-C. Au and Chiew-Lan Tai},
    title = {Effective derivation of similarity transformations for implicit {L}aplacian
    mesh editing},
    journal = {Computer Graphics Forum},
    year = {2007},
    volume = {26},
    pages = {34--45},
    number = {1},