本論文發展了一套以Variational Implicit Surface為基礎，由緻密、精確三維座標點重建物體表面模型之快速演算法。物體表面模型重建乃科學與工程範疇中一重要之研究課題，並在諸如電腦圖學、視覺、CAD、醫學影像等領域中被廣泛應用。
傳統上，Variational Implicit Surface藉由建立單一的radial basis function並以其值域描述欲重建之物體表面，但其指數成長之複雜度使其不適用於處理日益龐大的資料量。本論文提出的片段式隱式表面(piecewise implicit surface)表示方法提供了一演算方法上的改進，並且能夠輕易處理包含數十萬以上資料點之模型。本論文乃基於divide-and-conquer之精神，將輸入資料適當地分割□partitioning)為一組叢集(cluster)，使得傳統方法得以輕易地施行於單一叢集，由此產生之區域表面最後組合成一完整之表面模型。
本論文提出之重建演算法主要包含兩部分：1) 分割隱式表面模型2) 漸進式重建(progressive reconstruction)演算法。本論文中證實了Variational Implicit Surface的可分割性，意謂在輸入資料經過適當分割之前提下，叢集數目並不影響重建之表面模型，分割隱式表面模型之用意僅在於降低複雜度。而在漸進式重建演算法中之關鍵元素，同時亦為本論文之另一主要貢獻，在於Schur complement formula的引入，憑藉其快速更新radial basis function係數之助益，片段式隱式表面模型乃成為一多重解析度(multi-resolution)之表示方法。論文最末亦展示了本物體表面重建演算法之效能與各式重建實例。

This thesis describes an efficient method for automatic reconstruction of closed, piecewise smooth and seamless surfaces from accurate and dense 3D points based on variational implicit surface. The problem of surface reconstruction arises in a diversity of applications in scientific and engineering domains such as computer graphics, animation, CAD, scientific visualization and medical imaging etc.
Traditionally, a variational implicit surface uses a single implicit function formulated as a sum of weighted radial basis functions to describe the unknown surface. However, the global support nature and exponential growth of the complexity of RBFs makes it infeasible for modeling data sets with more than only a few thousand points. The piecewise representation of implicit surfaces outlined in this thesis provides an algorithmic solution to the difficulties in handling large data sets faced by traditional approaches. The basic principle of our method is to partition the input data points into a set of clusters and apply the traditional method to them separately. The decomposed local implicit patches are then joined together to form the complete surface.
The reconstruction method has two major parts: 1) partitioning of the implicit surfaces and 2) progressive reconstruction. A key component in part 2 and another main contribution of this thesis is the introduction of a novel iterative refinement algorithm based on the Schur complement formula. The effectiveness of the proposed method is demonstrated by a number of experimental results of reconstructed surfaces from real-world scanning data.

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