從珍貴罕見的材料中，儘可能分割出最大矩形塊(Maximum rectangular block, MRB)可增加材料之使用率。自1984 年開始討論矩形分割問題以來，由於此問題簡單的描述與定義，其應用一直被限制在較小的範圍。為了擴展其應用領域，在本論文中，擬將矩形分割問題延伸到一般更廣泛之範圍。首先，將原始材料之矩形邊界由一個任意封閉的區域來取代。由於是一般任意輪廓， 其處理範圍可以包含更多相關之材料。當最大矩形塊被分割取得時，材料的剩餘封閉空間(Remaining closed space, RCS)可以再被分割。我們提出一種盲目搜尋法(Blind search algorithm, BSA)，可全域性沿著邊界逐點搜尋最大之矩形塊，且此方法能從母材中連續地從大到小分割出最大矩形塊，直到所設定的門檻值為止。
雖然在一個任意封閉區域分割出最大矩形塊之問題可成功獲得處理，但仍有二個限制尚待解決：第一個限制是最大矩形塊的兩個邊緣必須與影像軸平行；第二個限制則是母材內部必須是均勻一致的，即在材料中沒有瑕疵。為解除這兩個限制，我們發展出相關演算法，並確認這些演算技術對皮革材料的應用成果。儘管所提出的演算法已有效的解決分割問題，但仍遺留一個可以改良的空間。即如何提高最大矩形塊獲得之搜尋效率。因此，我們將提出基因演算法來取得材料之最大矩形塊，並將其結果與盲目搜尋法比較，確認基因演算法確實能達到近於最佳化的分割性能，且能提昇運算之效率。儘管本論文的研究對象為皮革材料，但所提出的方法可以很容易地延伸到其它工業材料，特別是運用在那些昂貴的材料之上。

To determine the maximum rectangular block (MRB) from a rare material as larger as possible indicates to increase of the rate of material usage. The cutting problem has been addressed since 1984. But its applications were strongly restricted due to simple definition of the cutting problem. In order to expand the area of applications, in this dissertation, a general cutting problem will be considered. At first, the rectangular boundary of the original material is replaced by an arbitrary closed region. Due to the general material profile, many other materials can be involved. When the maximum rectangular block has been obtained, the remaining closed space (RCS) of the material can be divided again. A blind search algorithm (BSA), which globally searches the MRB point-by-point from the boundary points of the contour, will be developed. The BSA is able to acquire the MRB from mother material continuously from larger areas to smaller ones until a predefined threshold value is reached.
Although the MRB in an arbitrary closed region can be successfully resolved, two problems are still unsolved. The first limitation is that both edges of the MRB must be parallel with image axes. The second limitation is that the mother material needs to be uniform, i.e., no defects inside the material. In order to release these two assumptions, some algorithms will be presented. Applications of those techniques to the leather material will be demonstrated. In spite of resolving the cutting problem by the presented algorithms, a possible improvement is needed for larger MRBs. The challenge about larger MRBs is that how to make the searching process more efficiently. Therefore, two new methods of GA to obtain the MRB are proposed. By comparing the results using the BSA, the GA approaches are verified to be able to reach the near-optimal performance. Even though only leather material is focused in this research, the proposed methods can be easily extended to other industrial materials, especially for those expensive materials.

Acknowledgments.……………………………………………….. i
Contents.……………………………………………………….…. ii
List of Figure Captions.……………………………………….…. v
List of Table Captions.…………………………………………... viii
Notations.…………………………………………………………. ix
Chinese Abstract.…………………………………………….....… xi
English Abstract.…………………………………………………. xii
Chapter 1 Introduction.……………………………………….... 1
1.1 Goals and Motivation.………………………………….... 1
1.2 Background and Paper Review.…………………………. 2
1.3 Dissertation Outline.…………………………………….. 7
Chapter 2 MRB in Uniform Material.………………………... 10
2.1 Definitions and Explanation.………………………….… 10
2.2 Methods and Algorithms.……………………………….. 14
2.2.1 Quadrant judgment.………………………………... 14
2.2.2 Determination of the MRB.………………………... 17
2.2.3 Construction of the remaining closed space.………. 18
2.2.4 Automatic division of global region.………………. 20
2.2.5 Dead zone.…………………………………………. 21
2.3 Computation Improvement……………………………... 22
2.4 Experiments and Application.………………..…………. 23
Chapter 3 Optimization of GMRB Cutting for a Non-Uniform
Arbitrary Closed Region.………………………... 39
3.1 Release of the First Limitation.………………………. 39
3.1.1 The global MRB.………………………………... 40
3.1.2 Reduction of the working space.………………... 40
3.1.3 Contour modification after rotation.……………. 41
3.1.4 Contour reconstruction.…………………………. 44
3.1.5 Successive cutting.………………………………. 45
3.2 Release of the Second Limitation.…………………….. 47
3.2.1 Establishment of total boundary points.…………. 47
3.2.2 Single GMRB.………………………………….. 48
3.2.3 Successive GMRB.……………………………… 49
3.3 Experiments and Discussion.…………………………. 50
Chapter 4 GA-based Optimization for the Maximum Rectangular Block.……………………………….. 59
4.1 The Genetic Algorithm.……. ……………………...... 59
4.2 GA for the MRB Problem.……..…………………….. 60
4.2.1 Coding the work volume.……………………….. 60
4.2.2 Fitness and selection.…………………………… 61
4.2.3 Standard genetic algorithm (SGA).….………….. 63
4.2.4 Tuning using the Hooke method.……………….. 65
4.2.5 Adaptive genetic algorithm (AGA).…………….. 65
4.3 Experimental Verification.……………………………. 67
4.4 Discussions.…………………………………………... 68
Chapter 5 Conclusions.………………………………………. 77
5.1 Contributions………………………………………….. 77
5.2 Future Research Topics.………………………………. 79
Bibliography……………………………………………………. 81

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