## Close-to-Conformal Deformations of Volumes

Conformal deformations are infinitesimal scale-rotations, which can be parameterized by quaternions. The condition that such a quaternion field gives rise to a conformal deformation is non-linear and in any case only admits Möbius transformations as solutions. We propose a particular decoupling of scaling and rotation which allows us to find near to conformal deformations as minimizers of a quadratic, convex Dirichlet energy. Applied to tetrahedral meshes we find deformations with low quasiconformal distortion as the principal eigenvector of a (quaternionic) Laplace matrix. The resulting algorithms can be implemented with highly optimized standard linear algebra libraries and yield deformations comparable in quality to far more expensive approaches.

The work reported here was supported in part by ONR Award N00014-11-1002, the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics," and a software donation from Side Effects Software. We are grateful to Houman Owhadi and Chiu-Yen Kao for generously sharing their knowledge; Kovalsky and Lipman for providing code from [Kovalsky et al. 2014]; and Paillé for the models used in [Paillé and Poulin 2012]. Last but not least the detailed reviewer feedback helped us greatly improve the paper.

In this work, we are interested in having an efficient algorithm for obtaining deformation of 3D volumetric meshes with low shear distortion, which is, for example, important for preserving quality of 3D texture. The resulting method we arrive at turns out to be related to the quantum field theory (non-abelian gauge theory).

A deformation is recognized as "low distortion" if the deformation is locally mostly just a scale and rotation; that is there is little anisotropic stretching or shearing. The deformation is called "conformal" if it is locally scale and rotation everywhere. Note that there is no 3D conformal deformation within $${\Bbb R}^3$$ other than the Möbius transformations (one of the Theorems named after Liouville, 1850). And a direct optimization over deformation maps for 3D conformality is a numerically challenging non-convex problem.

Here we take a different route. Instead of searching over the deformation maps $${\Bbb R}^3\to{\Bbb R}^3$$ yielding low distortion, we search over a scale-rotation field which almost stitches together into a deformation. The mesh reconstructed from such scale-rotation field is expected to consist mostly scale-rotation, i.e. close to conformal.

Scale-rotation is straightforwardly thought of as a linear map $$aR$$, where $$a\in{\Bbb R}$$ is a scaling factor and $$R\in SO(3)$$ is a rotation matrix. However, a rotation matrix $$R$$, which is subject to the constraint $$R^\intercal R = I$$, is difficult to handle as a variable. An easier way of describing scale-rotation is to use quaternions. If $$q\in{\Bbb H}$$ is a quaternion, then for each $$v\in{\Bbb R}^3\cong {\rm Im}\,{\Bbb H}$$ is an imaginary quaternion (i.e. a 3D vector), $$v\mapsto \overline q v q$$ is a scale-rotation.

We shall look at 3D deformations in the following way. A 3D volumetric shape is a map $$f:M\rightarrow{\Bbb R}^3\cong{\rm Im}\,{\Bbb H}$$, where $$M$$ is a 3-dimensional manifold, and $$f$$ simply says that how the points in this abstract domain lie in $${\Bbb R}^3$$ (a pure-imaginary-valued position attribute). A deformed shape is simply another map $$\tilde f:M\rightarrow{\rm Im}\,{\Bbb H}$$. We say this pair of maps $$f,\tilde f$$ are conformal to each other if $d\tilde f = \overline\lambda df\lambda$ for some quaternion-valued function $$\lambda:M\rightarrow{\Bbb H}$$. That is, the deformed shape $$\tilde f$$ is locally a scale-rotation of $$f$$. As mentioned above, we seek for $$\lambda:M\rightarrow{\Bbb H}$$ so that we can construct $$\tilde f$$ so that $$d\tilde f= \overline\lambda df\lambda$$. We discover that the condition of $$\lambda$$ for there to exist a solution $$\tilde f$$, namely the integrability condition, is a first order linear differential equation $d\lambda+{1\over 2}G df\lambda = 0$ for some parameter $$G:M\rightarrow{\Bbb R}^3$$. We write $$\nabla^G:= d+{1\over 2} Gdf$$, and the integrability condition becomes $\nabla^G\lambda = 0.$ The differential operator $$\nabla^G$$ is recognized as a covariant derivative.

With the vector field $$G$$ being an input parameter, we find $$\lambda$$ by minimizing the Dirichlet energy $E_D(\lambda) = \int_M\left|\nabla^G\lambda\right|^2$ followed by a linear solve of $$d\tilde f = \overline\lambda df\lambda$$ for $$\tilde f$$ in the least-squares sense. In practice, $$E_D(\lambda)$$ can be minimized, with a constraint $$\int_M|\lambda|^2=1$$ preventing trivial solution, by solving the smallest eigenvalue problem of the covariant Laplacian $$\Delta^G = {\nabla^G}^\dagger\nabla^G$$. We also discuss in the paper how to design $$G$$ in order to control the desired output deformation.

In quantum field theory, $$G$$ is a gauge boson interacting with a field $$\lambda$$ representing a fermion.