Area-preserving Parameterization Research Articles

Area-Preserving Parameterization with Tutte Regularization

Area-Preserving Parameterization with Tutte Regularization

Adaptive area-preserving parameterization of open and closed anatomical surfaces

The parameterization of open and closed anatomical surfaces is of fundamental importance in many biomedical applications. Spherical harmonics, a set of basis functions defined on the unit sphere, are widely used for anatomical shape description. However, establishing a one-to-one correspondence between the object surface and the entire unit sphere may induce a large geometric distortion in case the shape of the surface is too different from a perfect sphere. In this work, we propose adaptive area-preserving parameterization methods for simply-connected open and closed surfaces with the target of the parameterization being a spherical cap. Our methods optimize the shape of the parameter domain along with the mapping from the object surface to the parameter domain. The object surface will be globally mapped to an optimal spherical cap region of the unit sphere in an area-preserving manner while also exhibiting low conformal distortion. We further develop a set of spherical harmonics-like basis functions defined over the adaptive spherical cap domain, which we call the adaptive harmonics. Experimental results show that the proposed parameterization methods outperform the existing methods for both open and closed anatomical surfaces in terms of area and angle distortion. Surface description of the object surfaces can be effectively achieved using a novel combination of the adaptive parameterization and the adaptive harmonics. Our work provides a novel way of mapping anatomical surfaces with improved accuracy and greater flexibility. More broadly, the idea of using an adaptive parameter domain allows easy handling of a wide range of biomedical shapes.

Open and closed anatomical surface description via hemispherical area-preserving map

Spherical harmonics, one of the most widely used basis functions for shape description, rely heavily on a spherical parameterization of the given surface. Additionally, spherical harmonics based 3D modeling works well only for closed surfaces, while many anatomical structures are hemisphere-like open objects. Therefore, it is more natural to have a hemisphere-based approach for their shape description. In this work, we propose a novel framework for the shape description of open and closed hemisphere-like surfaces. We first develop two hemispherical area-preserving parameterization methods for simply-connected open and closed surfaces respectively, and then utilize the hemispherical harmonics basis functions to yield an accurate representation of hemisphere-like anatomical surfaces. We assess the performance of the proposed framework for the shape description of human head. In particular, 60 hemispherical anatomical surfaces (20 closed brain surfaces, 20 closed skull surfaces, and 20 open scalp surfaces) constructed from human head MRI scans are utilized for this purpose. For the three types of surfaces, our framework achieves a significant improvement in the surface reconstruction accuracy by 75%, 80% and 50% respectively when compared to the spherical harmonics based approach. This suggests that our new shape description framework can facilitate the biomedical analysis of hemisphere-like anatomical objects.

Optimized surface parameterizations with applications to Chinese virtual broadcasting

Surface parameterizations have been widely applied in computer-aided design for the geometric processing tasks of surface registration, remeshing, texture mapping, and so on. In this paper, we present an efficient balanced energy minimization algorithm for the computation of simply connected open surface parameterizations with balanced angle and area distortions. The existence of a nontrivial accumulation function of the proposed algorithm is guaranteed under some mild conditions, and the limiting function is shown to be one-to-one. Comparisons of the proposed algorithm with angle- and area-preserving parameterizations show that the angular distortion is close to that of an angle-preserving parameterization while the area distortion is significantly improved. An application of the proposed algorithm involving surface remeshing, registration, and morphing to the Chinese virtual broadcasting technique is demonstrated.

Curvature adaptive surface remeshing by sampling normal cycle

Surface meshing plays a fundamental important role in Visualization and Computer Graphics, which produces discrete meshes to approximate a smooth surface. Many geometric processing tasks heavily depend on the qualities of the meshes, especially the convergence in terms of topology, position, Riemannian metric, differential operators and curvature measures.Normal cycle theory points out that in order to guarantee the convergence of curvature measures, the discrete meshes are required to approximate not only the smooth surface itself, but also the normal cycle of the surface. This theory inspires the development of the remeshing method based on conformal parameterization and planar Delaunay refinement, which uniformly samples the smooth surface, and produces Delaunay triangulations with bounded minimal corner angles. This method ensures the Hausdorff distances between the normal cycles of the resulting meshes and the smooth normal cycle converges to 0, the discrete Gaussian curvature and mean curvature measures of the resulting meshes converge to their counter parts on the smooth surface.In the current work, the conformal parameterization based remeshing algorithm is further improved to speed up the curvature convergence. Instead of uniformly sampling the surface itself, the novel algorithm samples the normal cycle of the surface. The algorithm pipeline is as follows: first, two parameterizations are constructed, one is the surface conformal parameterization based on dynamic Ricci flow, the other is the normal cycle area-preserving parameterization based on optimal mass transportation; second, the normal cycle parameterization is uniformly sampled; third, the Delaunay refinement mesh generation is carried out on the surface conformal parameterization. The produced meshes can be proven to converge to the smooth surface in terms of curvature measures.Experimental results demonstrate the efficiency and efficacy of proposed algorithm, the convergence speeds of the curvatures are prominently faster than those of conventional methods.

Area-Preserving Parameterizations for Spherical Ellipses

Presentamos nuevos métodos para la estimación eficiente de la radiancia incidente proveniente de fuentes de luz con forma de disco. Por ello, introducimos tres nuevos mapeos conformes por área de la proyección en ángulo sólido del disco. Nuestros métodos mejoran significativamente el trabajo preexistente, y han sido implementado en un motor de render convencional orientado a producción.

Topological Disk Mesh Morphing Based on Area-Preserving Parameterization

Mesh morphing is a technique which gradually deforms a mesh into another one. Mesh parameterization, a powerful tool adopted to establish the one-to-one correspondence map between different meshes, is of great importance in 3D mesh morphing. However, current parameterization methods used in mesh morphing induce large area distortion, resulting in geometric information loss. In this paper, we propose a new morphing approach for topological disk meshes based on area-preserving parameterization. Conformal mapping and Mobius transformation are computed firstly as rough alignment. Then area preserving parameterization is computed via the discrete optimal mass transport map. Features are exactly aligned through radial basis functions. A surface remeshing scheme via Delaunay refinement algorithm is developed to create a new mesh connectivity. Experimental results demonstrate that the proposed method performs well and generates high-quality morphs.

Density-Equalizing Maps for Simply Connected Open Surfaces

In this paper, we are concerned with the problem of creating flattening maps of simply connected open surfaces in $\mathbb{R}^3$. Using a natural principle of density diffusion in physics, we propose an effective algorithm for computing density-equalizing maps with any prescribed density distribution. By varying the initial density distribution, a large variety of flattening maps with different properties can be achieved. For instance, area-preserving parameterizations of simply connected open surfaces can be easily computed. Experimental results are presented to demonstrate the effectiveness of our proposed method. Applications to data visualization and surface remeshing are explored.

Area-preserving mesh parameterization for poly-annulus surfaces based on optimal mass transportation

This work proposes a novel method for computing area-preserving parameterization for genus zero surfaces with multiple boundaries (poly-annuli), which is based on discrete optimal mass transportation and surface Ricci Flow. We first begin with a conformal mapping (which may greatly distort area) by Ricci Flow and then correct the area distortion using the mass transport procedure via a convex optimization. The method is intrinsic and stable, and the resulting parameterization preserves area element and minimizes angle distortion. Comparing with existing algorithms, our method is more general and flexible. It can handle surfaces with more complicated topology, and gives users full control of the target measure, such as the areas of the holes. We have tested the method for applications in various fields. Our experimental results demonstrate the efficiency and efficacy of the proposed method.

Optimal mass transport for geometric modeling based on variational principles in convex geometry

In geometric modeling, surface parameterization plays an important role for converting triangle meshes to spline surfaces. Parameterization will introduce distortions. Conventional parameterization methods emphasize on angle-preservation, which may induce huge area distortions and cause large spline fitting errors and trigger numerical instabilities.To overcome this difficulty, this work proposes a novel area-preserving parameterization method, which is based on an optimal mass transport theory and convex geometry. Optimal mass transport mapping is measure-preserving and minimizes the transportation cost. According to Brenier's theorem, for quadratic distance transportation costs, the optimal mass transport map is the gradient of a convex function. The graph of the convex function is a convex polyhedron with prescribed normal and areas. The existence and the uniqueness of such a polyhedron have been proved by the Minkowski-Alexandrov theorem in convex geometry. This work gives an explicit method to construct such a polyhedron based on the variational principle, and formulates the solution to the optimal transport map as the unique optimum of a convex energy. In practice, the energy optimization can be carried out using Newton's method, and each iteration constructs a power Voronoi diagram dynamically. We tested the proposal algorithms on 3D surfaces scanned from real life. Experimental results demonstrate the efficiency and efficacy of the proposed variational approach for the optimal transport map.