User-controllable polycube map for manifold spline construction

Polycube T-spline has been formulated elegantly that can unify T-splines 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. Experimental 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.