本文為美國明尼蘇達州立大學（作者：William Joseph Beksi）的博士論文，共126頁。

由於低成本感測器的可用性，三維點雲資料集變得越來越普遍。鐳射探測和測距（LIDAR）、立體光、結構光和飛行時間（ToF）都是捕捉環境三維表示的感測器例子。這些感測器越來越多地出現在智慧手機、平板電腦、機器人和自動駕駛汽車等移動裝置和機器中。隨著硬體技術的發展，需要演算法和資料結構來以創新和有意義的方式處理這些感測器生成的資料。

本文發展並應用代數拓撲方法處理三維點雲資料集。近年來，拓撲資料分析（TDA）領域已經成熟，使得研究人員能夠使用考慮資料“形狀”的技術來分析點雲資料集。這包括拓撲特徵，例如連線的元件、孔、空洞和高維的類似物體。這些思想已經成功地應用於自然嵌入到度量空間（如歐幾里德空間）中的資料集，在度量空間中，點之間的距離可以用來形成一個引數化的空間序列。通過研究這個序列不斷變化的拓撲結構，我們獲得了有關底層資料的資訊。

在論文的第一部分，我們提出了一個快速的方法來建立一個三維VietorisRips複合體，使我們能夠近似一個點雲的拓撲結構。該複合體的構造分為三個並行階段：最近鄰搜尋、邊列表生成和三角形列表生成。邊和三角形列表可以用於持久同調計算。

在論文的第二部分，我們提出了利用持久同調理論的思想來分割三維點雲資料的方法。提出的演算法首先生成點雲資料集的簡單復表示，然後計算對應於連通分量個數的復表示的第零同調群。最後，我們提取資料集中每個連線分量的簇。我們的研究表明，這些方法在存在噪聲和取樣條件差的情況下能夠穩定地分割點雲資料，從而比現有的分割方法具有優勢。

在論文的第三部分，我們通過引入一個近似線性時間演算法來解決計算拓撲中的一個開放問題，該演算法用於增量計算拓撲持久1-cycle。此外，我們發展了第二個演演算法，利用第一個演演算法的輸出來產生一個生成樹，在這個生成樹上可以計算出非有界的最小1-cycle。然後使用這些非有界最小1-cycle來定位和填充資料集中的孔。實驗結果表明，我們的演算法可以有效地重建由噪聲感測器資料產生的三維點雲表面。

在論文的第四部分，我們開發了一個全域性特徵描述符，稱為拓撲持久點簽名（STPP），它對三維點雲資料的拓撲不變數（第零個和第一個同調群）進行編碼。與現有技術相比，STPP是一種具有競爭力的三維點雲描述符，並且對噪聲感測器資料具有彈性。我們的實驗證明，STPP可以作為一個獨特的特徵，從而允許三維點雲處理任務，如目標檢測和分類。

本論文的研究主要是在兩個方向上實現有效、高效、可擴充套件的三維點雲拓撲處理方法。我們提出了這些演算法，並分析了它們的理論效能和正確性證明。我們還通過使用公開資料集的實驗來證明結果的可行性和適用性。

3D point cloud datasets are becoming morecommon due to the availability of low-cost sensors. Light detection and ranging(LIDAR), stereo, structured light, and time-of-flight (ToF) are examples ofsensors that capture a 3D representation of the environment. These sensors areincreasingly found in mobile devices and machines such as smartphones, tablets,robots, and autonomous vehicles. As hardware technology advances, algorithmsand data structures are needed to process the data generated by these sensorsin innovative and meaningful ways.

This dissertation develops and appliesalgebraic topological methods for processing 3D point cloud datasets. The areaof topological data analysis (TDA) has matured in recent years allowingresearchers to analyze point cloud datasets using techniques that take intoaccount the ‘shape’ of the data. This includes topological features such asconnected components, holes, voids, and higher dimensional analogs. These ideashave been successfully applied to datasets which are naturally embedded in ametric space (such as Euclidean space) where distances between points can beused to form a parameterized sequence of spaces. By studying the changingtopology of this sequence we gain information about the underlying data.

In the first part of the thesis, we presenta fast approach to build a 3D Vietoris-Rips complex which allows us to approximatethe topology of a point cloud. The construction of the complex is done in threeparallelized phases: nearest neighbors search, edge list generation, andtriangle list generation. The edge and triangle lists can then be used forpersistent homology computations.

In the second part of the thesis, wepresent approaches to segment 3D point cloud data using ideas from persistenthomology theory. The proposed algorithms first generate a simplicial complexrepresentation of the point cloud dataset. Then, the zeroth homology group ofthe complex which corresponds to the number of connected components iscomputed. Finally, we extract the clusters of each connected component in the dataset.We show that these methods provide a stable segmentation of point cloud data underthe presence of noise and poor sampling conditions, thus providing advantages overcontemporary segmentation procedures.

In the third part of the thesis, we addressan open problem in computational topology by introducing a nearly linear time algorithmfor incrementally computing topologically persistent 1-cycles. Further, wedevelop a second algorithm that utilizes the output of the first to generate aspanning tree upon which non-bounding minimal 1-cycles can be computed. Thesenon-bounding minimal 1-cycles are then used to locate and fill holes in adataset. Experimental results show the efficacy of our algorithms forreconstructing the surface of 3D point clouds produced by noisy sensor data.

In the fourth part of the thesis, wedevelop a global feature descriptor termed Signature of TopologicallyPersistent Points (STPP) that encodes topological invariants (zeroth and firsthomology groups) of 3D point cloud data. STPP is a competitive 3D point clouddescriptor when compared to the state of art and is resilient to noisy sensordata. We demonstrate experimentally that STPP can be used as a distinctive signature,thus allowing for 3D point cloud processing tasks such as object detection andclassification.

This dissertation makes progress towardseffective, efficient, and scalable topological methods for 3D point cloudprocessing along two directions. We present algorithms with an analysis oftheir theoretical performance and proof of correctness. We also demonstrate thefeasibility and applicability of our results with experiments using publicly availabledatasets.

1. 引言
2. 數學背景
3. R3 Vietoris-Rips復形式的快速重建
4. 點雲分割
5. 點雲空洞邊界檢測
6. 點雲簽名
7. 結論與討論

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