User-controllable polycube map for manifold spline construction

Abstract

Polycube T-spline has been formulated elegantly that can unify Tsplines and manifold splines to define a new class of shape representations for surfaces of arbitrary topology by using polycube map as its parametric domain. In essense, The data fitting quality using polycube T-splines hinges upon the construction of underlying polycube maps. Yet, existing methods for polycube map construction exhibit some disadvantages. For example, existing approaches for polycube map construction either require projection of points from a 3D surface to its polycube approximation, which is therefore very difficult to handle the cases when two shapes differ significantly; or compute the map by conformally deforming the surfaces and polycubes to the common canonical domain and then construct the map using function composition, which is challenging to control the location of singularities and makes it hard for the data-fitting and hole-filling processes later on. This paper proposes a novel framework of user-controllable polycube maps, which can overcome disadvantages of the conventional methods and is much more efficient and accurate. The current approach allows users to directly select the corner points of the polycubes on the original 3D surfaces, then construct the polycube maps by using the new computational tool of discrete Euclidean Ricci flow. We develop algorithms for computing such polycube maps, and show that the resulting user-controllable polycube map serves as an ideal parametric domain for constructing spline surfaces and other applications. The location of singularities can be interactively placed where no important geometric features exist. Experimen- * Stony Brook University, e-mail: wanghy@cs.sunysb.edu † Stony Brook University, e-mail:mjin@cs.sunysb.edu ‡ Nanyang Technological University, e-mail:yhe@ntu.edu.sg § Stony Brook University, e-mail:gu@cs.sunysb.edu ¶ Stony Brook University, e-mail:qin@cs.sunysb.edu tal results demonstrate that the proposed polycube maps introduce lower area distortion and retain small angle distortion as well, and subsequently make the entire hole-filling process much easier to accomplish.

A divide-and-conquer approach for automatic polycube map construction

“…Automatically Constructed Polycubes. In the existing techniques of constructing polycube maps [1] [4] [5], the polycube maps are constructed manually. Although manual constructions work well for the models with simple topology, it is extremely tedious and time consuming to construct polycube with complicated topology.…”

“…Wang et al presented an intrinsic method to construct the polycube map which avoids the projection of the vertices on a 3D model to the polycube domain [4]. Wang et al presented a technique where the user can interactively control the desired locations and the number of singularities of the polycube map (i.e., the corners in polycubes) which facilitates the manifold spline construction [5]. Lin et al used Reeb graph to segment the surface and then developed an automatic method to construct polycube map [10].…”

“…(The detailed comparisons among different polycube map construction methods are documented in Section 5.) Because of the highly regular structure, polycube maps are very promising in seamless texture mapping and synthesis [11,2], level-of-detail [12], morphing [3], and T-spline construction [4,5].…”

“…Comparison with [5]. Wang et al proposed an interactive method to improve the polycube map for high genus surfaces [5].…”

“…Wang et al proposed an interactive method to improve the polycube map for high genus surfaces [5]. The key difference between this user-controllable method and [4] is that users have full freedom to specify the number and locations of the singularities (the pre-images of polycube corners) on M and their connectivity, i.e., which pair of corners forms a polycube edge, which set of polycube edges form the polycube face, etc.…”