دانلود رایگان مقاله بازسازی سطح ضمنی فشرده از طریق تقریبی تانسور

Abstract

Implicit representations have gained an increasing popularity in geometric modeling and computer graphics due to their ability to represent shapes with complicated geometry and topology. However, the storage requirement, e.g. memory or disk usage, for implicit representations of complex models is relatively large. In this paper, we propose a compact representation for multilevel rational algebraic spline (MRAS) surfaces using low-rank tensor approximation technique, and exploit its applications in surface reconstruction. Given a set of 3D points equipped with oriented normals, we first fit them with an algebraic spline surface defined on a box that bounds the point cloud. We split the bounding box into eight sub-cells if the fitting error is greater than a given threshold. Then for each sub-cell over which the fitting error is greater than the threshold, an offset function represented by an algebraic spline function of low rank is computed by locally solving a convex optimization problem. An algorithm is presented to solve the optimization problem based on the alternating direction method of multipliers (ADMM) and the CANDECOMP/PARAFAC (CP) decomposition of tensors. The procedure is recursively performed until a certain accuracy is achieved. To ensure the global continuity of the MRAS surface, quadratic B-spline weight functions are used to blend the offset functions. Numerous experiments show that our approach can greatly reduce the storage of the reconstructed implicit surface while preserve the fitting accuracy compared with the state-of-the-art methods. Furthermore, our method has good adaptability and is able to produce reconstruction results with high quality.

7. Conclusions and future work

In this paper, we have developed an adaptive surface reconstruction method based on a new implicit representation— Multilevel Rational Algebraic Splines. To generate a compact representation in order to reduce the storage requirement, we propose a local fitting model by introducing a low-rank regularization term. We then convert this model into a convex optimization problem, which can be solved by the ADMM algorithm efficiently. We obtain the compact representation of the MRAS surface using the low-rank tensor approximation technique based on CP decomposition. A number of experimental results have shown that our approach not only produces very compact representations, but also achieves comparable results with some state-of-the-art surface reconstruction methods. The capability to handle the non-uniform sampling data, the noisy data and the incomplete data has also been evidenced by the numerical examples. Regarding future work, our method should greatly benefit from the parallelization of our approach on GPU for acceleration, since the calculations of the MRAS representation in all the cells of each level are independent and can be carried out simultaneously. Another interesting direction for acceleration is to develop specialized fast algorithms for solving our local fitting model and performing the CP decomposition. We believe that our method can be useful in many applications, such as mesh compression, level of details and progressive compression and transmission, where the ability to produce compact and multilevel representation is crucial. Applications of tensors in other geometric modeling problems are also worthy of further study.