# Detail-Preserving Shape Unfolding
## 1 Introduction  
The canonical form of 3D shape is very useful for computer graphics applications, such as shape retrieval [1; 2; 3], shape correspondence [4; 5; 6] and texture mapping [7; 8], as it largely reduces the complexity of 3D shapes caused by various poses. Currently, a popular and commonly used method to obtain canonical forms of 3D shapes is MDS, which comes in classical [9; 10], least-squares [11; 12; 13], and landmark forms [14; 15; 16; 17]. The basic principle of MDS is to maintain the geodesic distances of vertex pairs for the shapes under different poses. To achieve this, it minimizes the sum of squared distance differences between geodesic and Euclidean distances of all vertices pairs on a 3D mesh. However, only satisfying such distance constraints may suffer from serious distortions, leading to a poor accuracy for context-based shape retrieval [3]. A possible way to reduce such distortions is to preserve the local structures of 3D shapes during shape unfolding, that is to make the shape unfolding as rigid as possible [18; 19]. In order to preserve as many geometric structures as possible, some researchers tried to obtain feature preserved canonical forms. Most of them first utilized the standard MDS [11] of a shape as guidance, then shape analysis techniques, such as shape deformation based registration [5] or skeleton [20], are applied to prevent distortions with the help of mesh part segmentation [21; 22; 23]. Recently, [PERSON] and [PERSON] [24] used mass-spring system with non-linear volume constraints to preserve geometric details. Later, [PERSON] et al. [25] introduced an automatic mesh unfolding algorithm by solving a semidefinite programming. The basic idea is to maximize the total variance of the vertex set for a given 3D mesh, while preserving the details by minimizing locally linear reconstruction errors. However, existing shape unfolding methods have several limitations which hamper their applicability. First, many of them cause significant geometric distortions, as they are the variants of MDS [11], such as the fast MDS [14], non-metric MDS [26] and accelerated MDS [27]. Second, the geodesic distances between all pairs of vertices or landmarks need to be calculated, which is highly time-consuming. Therefore, it is not applicable for high resolution meshes which have hundreds of thousands of vertices. Third, some methods [3; 28] have tedious operation processes and their performances highly rely on the accurate execution of each step. Finally, several proposed methods [24; 25] need to solve complex non-linear optimization problems which are always time-consuming. In addition, it is difficult to guarantee the robustness of these methods. Motivated by the above issues, we present a novel mesh unfolding method which has better ability of structure-preserving and ease of use. Our method is based on shape deformation, a technique that can explicitly control the geometric details of meshes. In short, we revisit the mass-spring system model proposed by [PERSON] and [PERSON] [24] and reformulate it in terms of rigid transform energy [19]. Our proposed formulation brings more benefits. Firstly, our model is very simple which only needs to solve linear systems during alternate iteration process. Secondly, it can well preserve the rigidity of the original shape and reduce the geometric distortions as rigid transform energy [19] is considered. Furthermore, our algorithm can be accelerated by using parallel computation and cascade strategy further enhances the robustness of algorithm. In addition, the proposed approach doesn't need to calculate the pairwise geodesic distances and also has no tedious operation processes, which are timing-consuming and difficult to control. In summary, the main contributions of our paper are as follows:  
* A novel shape unfolding method is proposed for non-rigid 3D mesh based on shape deformation technique. It makes the local deformation be approximately rigid and more details can be preserved.