應用線掃描結構光重建表面傾斜與三維形貌之研究

Abstract

本研究提出了一種適用於線掃描三角架構的測量物體表面傾斜的方法。此方法引用了計算機圖學和投影幾何學的概念，建立表面傾斜角度與其他物理量之間的關係。核心概念是分析相機拍攝到的線條紋之線寬與光強度的變化，再透過設計適當的正向模型來解釋傾斜角和特徵變化之間的關係，進而得知表面傾斜的大小。利用三角函數和微積分的概念即可建立傾斜角度對線寬的影響的模型。傾斜角度對光強響應的模型則是引用蘭柏特餘弦定律來建立。在量測場景中，傾斜角為逆問題的解。反向模型可被重新表述為有約束條件的優化問題，因此傾斜角可以透過尋找目標函數的最小值來獲得。驗證過程分為兩大部分，正向模型與反向模型的驗證。經由驗證結果得知，線寬正向模型與實驗觀測值匹配，$R^2$統計量約為0.644。與三次元量床的量測數據相比，本研究提出的量測方法在關鍵尺寸的量測上仍然有次毫米等級的量測誤差。其誤差來源推測是三維點座標重建演算法的缺陷，以及實際觀測之光條紋與理論假設產生背離所致。

In this research, a methodology for measuring surface tilting angle of an object from a triangulation line scanner is proposed. The method implemented concepts from computer graphics and projective geometry. The idea is to detect the changes of linewidth and intensity in the line pattern, the surface tilting angle and the position of the surface can be reconstructed. Appropriate mathematical model explaining the relationship between tilting angles and the features are designed. The rationale behind the concept is also proven theoretically. The model for tilting angle to linewidth can be established using trigonometry and calculus, while the model for tilting angle to intensity is modeled by Lambert''s cosine law. In measuring scenario, the tilting angles are the solution to the inverse problem. A reformulation of the inverse problem yields a more convenient way for finding the solution. Finding the tilting angles are thought of as a constrained optimization problem. The validation of the methodology is separated into two parts, one is the validation of forward models and the second is the validation of the inverse problem, i.e. the overall performance. It is found that the linewidth forward model matches observation with R-squared value around 0.644. The validation of overall methodology shows that there are still rooms for improvements. It is speculated that sub-millimeter bias is caused by geometrical constraint of the measuring system, failure of presumed condition and inappropriate reconstruction algorithm for the 3D coordinates.