方向資訊對於許多幾何塑模與處理演算法是否成功扮演了很重要的角色，即使現今已發展了許多重建演算法，從未定向之樣本點重建三維物體表面仍是一件很具挑戰性的工作。在本論文中，我們提出一名為方向推論法(Orientation inference)之共作式演算法，以及一名為二向樹(Binary orientation tree)之空間分割結構，它們可以很容易地整合進一個典型的表面重建流程，以幫助現有之演算法處理未定向之樣本點。

方向推論法之架構起始於建置包含一組未定向區域表面的逼近物體表面階層架構，而區域表面表示為輻射基底函數(Radial basis function)的線性組合。藉由將區域表面視為節點(Node)，我們將決定全域一致方向的問題定義為一圖模型最佳化的問題。藉由檢查鄰近區域表面之間的符號一致性，一能量函數被定義來對不一致的方向變化施加較高的能量。藉由在圖模型上對該能量函數進行最小化，可以得到每個節點的最佳標示，亦即全域一致的區域表面方向。區域推論的結果透過一波前傳遞(Front-propagation)的方式得到全域解。重建之表面透過一簡單有效的檢測程序以找出錯誤的區域表面並加以修補。此外，我們也提出一新的漸進式重建演算法，它是在表面逼近的過程中，迭代地增加更多已定向的點以增進逼近的準確度。

給定一完整之未定向點集合，我們建立二向樹以將空間大致地區分為輸入點集合的內外部，並利用於容積與表面重建的問題。二向樹的建立利用了傳統的八元樹(Octree)分割技術，同時每個節點的端點(Corner)都被賦予一標籤，以指明該端點相對於輸入點集合的內外關係。由根節點起始，透過一區域成長的過程，相連的空節點之端點可以迅速地決定其標籤。其餘的端點則透過隱藏點消除(Hidden point removal)運算子檢查其可視性來決定其標籤。在二向樹建立的過程
中，伴隨著輸入點集合的離群點(Outliers)可以很有效地被偵測，在離群點被移除，並且將二向樹端點的內外標籤都決定之後，二向樹能夠支援任何的容積與表面表示技術。為重建物體表面，我們利用了二向樹提出了一能夠從未定向之輸入點群重建隱式表面的 MPU 演算法。

Orientation information plays an essential role for the success of many geometric modeling and processing algorithms. Despite of the development of many existing algorithms, it remains challenging to reconstruct 3D object surfaces from unoriented sample points. In this thesis, we propose a cooperative algorithm called orientation inference and a space partitioning structure called binary orientation trees (BOTs), which can be easily integrated into a typical surface reconstruction pipeline to enable the existing algorithms to deal with unoriented data sets.

The orientation inference framework starts from building a surface approximation hierarchy comprising of a set of unoriented local surfaces, which are represented as a weighted combination of radial basis functions. We formulate the determination of the globally consistent orientation as a graph optimization problem by treating the local implicit patches as nodes. An energy function is defined to penalize inconsistent orientation changes by checking the sign consistency between neighboring local surfaces. An optimal labeling of the graph nodes indicating the orientation of each local surface can thus be obtained by minimizing the total energy defined on the graph. The local inference results are propagated over the model in a front-propagation fashion to obtain the global solution. The reconstructed surfaces are consolidated by a simple and effective inspection procedure to locate the erroneously fitted local surfaces. A progressive reconstruction algorithm that iteratively includes more oriented points to improve the fitting accuracy and efficiently updates the RBF coefficients is proposed.

Given a complete unoriented point set, we propose the binary orientation tree (BOT) for volume and surface representation, which roughly splits the space into the interior and exterior regions with respect to the input point set. The BOTs are constructed by performing a traditional octree subdivision technique while the corners of each cell are associated with a tag indicating the in/out relationship with respect to the input point set. Starting from the root cell, a growing stage is performed to efficiently assign tags to the connected empty sub-cells. The unresolved tags of the remaining cell corners are determined by examining their visibility via the hidden point removal operator. We show that the outliers accompanying the input point set can be effectively detected during the construction of the BOTs. After removing the outliers and resolving the in/out tags, the BOTs are ready to support any volume or surface representation techniques. To represent the surfaces, we also present a modified MPU implicits algorithm enabled to reconstruct surfaces from the input unoriented point clouds by taking advantage of the BOTs.

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