累積機率曲線理論推導與廣角監視器鏡頭設計之公差分析探討

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姓名

?廷政(Ting-Cheng Wen) 查詢紙本館藏

畢業系所

光電科學與工程學系

論文名稱

累積機率曲線理論推導與廣角監視器鏡頭設計之公差分析探討

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至系統瀏覽論文 (2030-1-14以後開放)

摘要(中)

廣角鏡頭因其大視角特性，廣泛應用於監視器系統、虛擬實境（Virtual Reality, VR）、擴增實境（Augmented Reality, AR）、醫學成像及自動駕駛等領域。然而，與小視場角鏡頭相比，廣角鏡頭在設計與製造過程中面臨更大的挑戰，包括嚴重的畸變、邊緣模糊、色差等成像問題，以及由製造誤差引起的成像品質下降。本研究針對上述問題，提出以累積機率曲線（Cumulative Probability Curve, CPC）理論為基礎，結合光學設計軟體 Code V的公差分析，建立一套完整的公差分析方法，並使鏡頭設計者更精確的預測鏡頭實際製造後的成像品質。
從歷史研究結果可看出，累積機率曲線理論被廣泛應用在社會學、經濟、建築、工業製造等行業的統計模擬中，它能量化有限的資料，並做出符合自然法則的模擬，給出有效的結果。應用累積機率曲線理論量化各項公差對成像品質的影響，可以幫助鏡頭設計者在設計階段最佳化公差的分配，進而減少畸變等像差，提高成像品質。
半視角在90時，近軸像高為無限大，但真實像高不可能為無限大，因此，在半視角75至90之間的影像會重疊在一起，此為廣角鏡頭一大缺點。本研究設計的80廣角鏡頭，為了避免此缺點採用F-theta畸變來計算理想像高，並以此判斷成像面扭曲程度。
本研究也針對所設計的80廣角鏡頭，以敏感度分析、反敏感度分析兩種方法，分析了各種常見公差項，如曲率半徑、厚度、位移、傾斜等，提出了適當的修改策略。此方法不僅能縮短設計到製造的週期，還能提高廣角鏡頭的生產一致性與穩定性。
本研究不僅為廣角鏡頭設計提供了實用的指導，也以累積機率曲線理論模擬鏡頭製造與組裝之公差分析，對學術界與產業界都具有很好的價值。未來，相關技術可進一步應用於高端光學系統的設計與優化，助成像光學產業更進一步的創新與發展。

摘要(英)

Wide-angle lenses, known for their large field-of-view characteristics, are widely used in various fields, including surveillance systems, Virtual Reality (VR), Augmented Reality (AR), medical imaging, and autonomous driving. However, compared to lenses with smaller fields of view, wide-angle lenses face greater challenges in design and manufacturing. These challenges include severe distortion, edge blurring, chromatic aberration, and degradation of imaging quality caused by manufacturing errors.
To address these issues, this study proposes a comprehensive tolerance analysis method based on the Cumulative Probability Curve (CPC) theory, combined with the tolerance analysis functionality of the optical design software Code V. This approach enables lens designers to more accurately predict the imaging quality of lenses after manufacturing.
Historical studies reveal that CPC theory has been widely applied in statistical simulations across fields such as sociology, economics, architecture, and industrial manufacturing. It quantifies limited data to perform simulations that align with natural laws, yielding effective results. By applying CPC theory to quantify the impact of various tolerances on imaging quality, lens designers can optimize tolerance allocations during the design phase, thereby reducing aberrations such as distortion and improving imaging quality.
When the half field-of-view reaches 90°, the paraxial image height becomes infinite. However, the actual image height cannot be infinite, leading to image overlap in the range of 75° to 90° half field-of-view, a significant drawback of wide-angle lenses. For the 80° wide-angle lens designed in this study, F-theta distortion was applied to calculate the ideal image height, which was then used to evaluate the extent of image plane distortion.
Additionally, this study analyzed the designed 80° wide-angle lens using two methods: sensitivity analysis and desensitization analysis. Common tolerance factors such as curvature radius, thickness, displacement, and tilt were examined, and appropriate modification strategies were proposed. This approach not only shortens the design-to-manufacturing cycle but also enhances the production consistency and stability of wide-angle lenses.
This research provides practical guidance for wide-angle lens design and uses CPC theory to simulate tolerance analysis during lens manufacturing and assembly. It holds significant value for both academia and industry. In the future, these related technologies can be further applied to the design and optimization of high-end optical systems, driving innovation and development in the imaging optics industry.

關鍵字(中)

★ 監視器鏡頭
★ 廣角鏡頭設計
★ 累積機率分佈
★ 鏡頭製造與組裝公差

關鍵字(英)

★ surveillance lens
★ wide-angle lens design
★ cumulative probability distribution
★ lens manufacturing and assembly tolerance

論文目次

摘要 i
英文摘要 iii
致謝 v
目錄 vi
圖目錄 x
表目錄 xv
1 第一章緒論 1
1-1 研究動機 1
1-2 文獻回顧 1
1-2-1 鏡頭與公差分析發展 1
1-2-2 廣角鏡頭的發展 3
1-3 論文架構 7
2 第二章設計理論 8
2-1 成像高度 8
2-2 有效焦距 9
2-3 解析度 10
2-4 繞射極限 11
2-5 畸變 13
2-6 F-theta畸變 14
2-7 累積機率分佈函數定義 16
2-7-1 期望值定義 16
2-7-2 標準差定義 16
2-8 機率密度函數（常態分佈） 17
2-8-1 機率密度函數定義 17
2-8-2 常態分佈機率密度函數的常數項定義 18
2-9 累積機率分佈函數 19
2-9-1 機率密度函數h(x)的積分 19
2-9-2 3-Sigma法則 20
2-9-3 累積機率分佈函數f(x)定義 22
2-9-4 累積機率分佈函數f(σ)計算 22
2-10 CODE V公差分析之累積機率分佈函數應用 24
2-10-1 Code V公差分析與標準差關係 24
2-10-2 Code V公差分析之高斯分佈MTF分佈情形 24
2-10-3 0視場之MTF變化量的標準差計算 25
2-11 製造與組裝公差 27
2-11-1 中心公差 28
2-11-2 偏心公差 35
2-11-3 可製造之公差範圍 41
3 第三章設計過程 42
3-1 設計規格 42
3-2 設計目標 42
3-3 設計參考起始值 44
3-4 最佳化過程 46
3-5 鏡頭設計數據 47
3-6 設計結果分析 49
3-6-1 成像品質MTF分析 50
3-6-2 橫向色差分析 51
3-6-3 畸變量評估 52
3-6-4 像方主光線角度 53
3-6-5 相對照度 54
4 第四章公差分析探討 55
4-1 敏感度分析過程 55
4-2 Code V的累積機率分佈圖 60
4-2-1 0視場在180 lp/mm下MTF變化量的累積機率分佈圖之標準差及期望值計算 61
4-3 敏感度分析結果 63
4-4 Excel畫公差分析的累積機率分佈圖 80
4-4-1 Code V曲線圖的誤差 80
4-4-2 累積機率分佈圖MTF的最小值與最大值 81
4-4-3 Excel面積圖畫高斯分佈 83
4-4-4 各視場的高斯分佈圖與累積機率分佈圖 84
4-4-5 修正Code V的累積機率分佈 90
4-4-6 修正後累積機率分佈圖與Code V累積機率分佈圖比較 92
4-5 反敏感度分析過程 98
4-5-1 反敏感度分析計算 99
4-5-2 反敏感度分析公差限制範圍 101
4-6 反敏感度分析結果比較 102
4-7 敏感度分析與反敏感度分析三種MTF變化量最大值比較 118
5 第五章結論未來展望 121
5-1 結論 121
5-1-1 累積機率曲線理論的應用價值 121
5-1-2 廣角鏡頭公差敏感度分析與反敏感度分析 121
5-1-3 設計與製造的實際應用建議 122
5-2 未來展望 122
5-2-1 多場景應用的擴展 122
5-2-2 鏡頭補償器的應用 123
5-2-3 進一步優化累積機率曲線理論模型 123
6 參考文獻 124

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指導教授

孫文信(Wen-Shing Sun)

審核日期

2025-1-20

推文