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系統識別號 U0026-1501201922350900
論文名稱(中文) 最小二乘法在影像處理上的應用
論文名稱(英文) An Application of Least Squares Method for Image Process
校院名稱 成功大學
系所名稱(中) 數學系應用數學碩博士班
系所名稱(英) Department of Mathematics
學年度 107
學期 1
出版年 108
研究生(中文) 于世偉
研究生(英文) Shih-Wei Yu
學號 L16054025
學位類別 碩士
語文別 中文
論文頁數 57頁
口試委員 指導教授-沈士育
口試委員-廖炳松
口試委員-王進猷
中文關鍵字 影像處理  最小二乘法 
英文關鍵字 Image processing  Least squares method 
學科別分類
中文摘要 本論文要探討的是如何從兩張相似的影像,利用最小二乘法及方向導數將影像進行數學分析計算後得到該仿射轉換關係,再進行影像比對處理,比對的方式是將其中一張影像透過求得的仿射轉換關係轉換後的影像與另一張影像進行相減,相減的方式是取影像的每個像素點的RGB值,從影像的第一個點到最後一個點依序相減。
在第二章的部分,討論兩張影像間有3個以上相對應的共軛對時,利用這些共軛對找到兩張影像間的轉換關係,最後證明其實就是仿射轉換的最小二乘法。但若是沒有這些共軛對依然能夠將轉換關係給求得,所以接下來進行的實驗比對中不需要共軛對,而是分別在兩張影像的某個區域內做比對。
第三章的實驗過程是先將一張標準影像給一個仿射轉換得另一張影像,在此仿射轉換分別設定為平移與旋轉兩個狀況下做討論,也就是在已知的轉換關係下去求證;而第四章的實際比對中,是真實取兩張影像,且對於該轉換關係是未知的情形下做討論,結果在本文有詳細說明。
英文摘要 This thesis is to explore how to obtain the affine transformation relationship from two similar images by using the least squares method and the directional derivative to calculate the image, and then compare the images. The comparison method is one of them. The image is subtracted from the other image by the obtained affine transformation relationship. The subtraction is performed by taking the RGB value of each pixel of the image from the first point to the last point of the image. Subtract in order.
In the second chapter, when there are more than three corresponding conjugate pairs between two images, the conjugate pairs are used to find the conversion relationship between the two images, and finally the least squares method of affine transformation is proved. However, if these conjugate pairs are still able to obtain the conversion relationship, the next experimental comparison does not require a conjugate pair, but is compared in a certain area of the two images.
In the third chapter, the experimental process is to convert a standard image into an affine and convert it into another image. In this case, the affine transformation is set to be discussed in the case of translation and rotation, that is, in the known conversion relationship. In the actual comparison of the fourth chapter, two images are actually taken, and the discussion is made in the case where the conversion relationship is unknown, and the results are described in detail in this paper.
論文目次 目錄
中文摘要---------------------------------------------Ⅰ
英文延伸摘要-----------------------------------------Ⅱ
誌謝-------------------------------------------------Ⅴ
目錄-------------------------------------------------Ⅵ
圖目錄-----------------------------------------------Ⅷ
第一章- 緒論
1.1前言----------------------------------------------1
1.2影像處理------------------------------------------2
1.3數位影像------------------------------------------4
1.4自動光學檢查--------------------------------------6
1.5研究目的及方法------------------------------------8
1.6其他章節簡介-------------------------------------10
第二章- 數學模型
2.1影像變形-----------------------------------------12
2.2最小二乘法---------------------------------------14
2.3仿射變形之最小二乘法證明-------------------------16

2.4偏微分方向導數-----------------------------------21
2.4.1平移誤差之方向導數--------------------------25
2.4.2旋轉誤差之微分------------------------------26
第三章- 最小二乘法之應用
3.1平移誤差-----------------------------------------28
3.2旋轉誤差-----------------------------------------37
第四章- 實際比對
4.1影像前置處理-------------------------------------43
4.2平移誤差之比對-----------------------------------45
4.3旋轉誤差之比對-----------------------------------48
第五章- 延伸討論與結論
5.1延伸討論-----------------------------------------52
5.2結論---------------------------------------------55
參考文獻----------------------------------------------56
參考文獻 [1]Steger, Carsten; Markus Ulrich; Christian Wiedemann ,2018. Machine Vision Algorithms and Applications (2nd ed.). Weinheim: Wiley-VCH.
[2]Maria Petrou, Costas Petrou,2010. "Image Processing:The Fundamentals, Second Edition," Chichester, U.K.,Wiley.[3]Azriel Rosenfeld,1969, Picture Processing by Computer, New York: Academic Press.
[4]Matt Pharr and Greg Humphreys,July 2004. Physically Based Rendering: From Theory to Implementation, Morgan Kaufmann.[5]http://apisc.com/aoi-introduction.htm
[6]http://www.skin168.net/2013/04/medical-ultrasound.html
[7]https://www.kmuh.org.tw/www/kmcj/data/10108/11.htm
[8]Wang Guorong; Wei Yimin; Qiao SanZheng,2004. Equation Solving Generalized Inverses. Generalized Inverses:Theory and Computations. Beijing: Science Press. : 第6頁.
[9]Bretscher, Otto ,1995. Linear Algebra With Applications (3rd ed.). Upper Saddle River, NJ: Prentice Hall.
[10]Stigler, Stephen M. ,1981. "Gauss and the Invention of Least Squares". Ann. Stat.
[11]Kaplan, W,1991. "The Directional Derivative." §2.14 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 135-138.
[12]Morse, P. M. and Feshbach, H, 1953. "Directional Derivatives." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 32-33.
[13]Rodney Coleman,2012. Calculus on Normed Vector Spaces. Springer. .
[14]Jain, R., Kasturi, R. and Schunck, B.G,1995. “Machine Vision” McGraw-Hill International Edition
[15]Berger, Marcel ,1987. Geometry I, Berlin: Springer
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