隨著科技發展迄今，影像處理已有數十餘年。在許多研究中發現以正六角形為結構呈現數位影像，可以比傳統正方形結構呈現數位影像更貼近真實影像。除了可以更加刺激人類視覺感觀，在電腦視覺方面也大大提升了辨識率。但目前以正六角形為結構的顯示器鮮為少見，使得目前諸多以正六角形為結構的相關研究環境在建置上相當不便。本論文提出可以在目前既有的傳統正方形結構顯示器上快速、少量記憶體虛擬正六角形結構影像，並且可在正方形結構影像與虛擬正六角形結構影像之間正、逆轉換，以加速正六角形結構的相關學術研究。在未來科技發展至以正六角形結構密鋪而成的顯示器之日，勢必需要在正方形結構顯示器與正六角形結構顯示器之間做一轉換。本論文以影像放大方式將正方形結構顯示器與正六角形結構顯示器之間做轉換，除了更換正六角形結構顯示器之外，其餘硬體不需更換。而正六角形結構具有影像每60度無失真旋轉的特性，但影像旋轉後需要一空間範圍容納正六角形結構影像，本論文以數據歸納法精確的求出影像每60度無失真旋轉所需的空間範圍，成功解決每60度無失真旋轉所需的空間範圍難題。

Image processing technologies has been developed for several decades. In many researches, it is found that digital images expressed by hexagonal structure is more close to real image than the images expressed by traditional square structure. It not only provides more exciting human being vision impression but also increases the identification rate of computer vision. But it is difficult to find a hexagonal structure monitor now and the difficulty makes it not easy to build a research environment for the study of hexagonal structure. The paper proposes a method to simulate a hexagonal structure image in a traditional square monitor fast and with a little memory, and can execute the transformation between square structure images and simulated hexagonal structure images. The research is expected to speed up the related studies about hexagonal structure. The transformation from a square monitor to a hexagonal monitor is necessary for the future technologies covered by hexagonal monitors. The study transforms a square monitor to a hexagonal monitor by enlarging an image without changing the other hardware except for changing the hexagonal monitor. A hexagonal structure has a feature that keeps an image without distortion for every 60 degrees rotation. But there must be a space to accommodate a hexagonal structure image for each image rotation. The study uses data induction to calculate the necessary space for accommodating the image without distortion obtained in every 60 degrees rotation and solves the problem of space needed for 60 degrees rotation without distortion successfully.

目錄

第一章　緒論.............................................................1
1.1 研究背景.....................................................1
1.2 研究動機.....................................................2
1.3 研究目的.....................................................4
第二章　相關研究.........................................................5
2.1 正六角形結構影像相關研究.....................................5
2.2 虛擬正六角形結構.............................................6
2.3 正六角形結構座標.............................................7
2.4 虛擬正六角形結構影像.........................................9
2.5 正六角形結構影像無失真旋轉..................................11
2.6 正六角形結構影像無失真旋轉空間範圍..........................13
第三章　研究方法........................................................14
3.1 直覺式正六角形結構座標......................................14
3.2 虛擬正六角形結構影像快速轉換演算法..........................15
3.2.1 影像正轉換............................................15
3.2.2 影像逆轉換............................................17
3.3 實體正六角形結構影像轉換演算法..............................18
3.3.1 以高為基準............................................19
3.3.2 以寬為基準............................................20
3.4 正六角形結構影像無失真旋轉空間範圍計算......................21
3.4.1 正六角形結構影像旋轉60度空間範圍公式推導..............22
3.4.2 正六角形結構影像旋轉120度空間範圍公式推導.............25
3.4.3 正六角形結構影像旋轉180度空間範圍公式推導.............28
3.4.4 正六角形結構影像每60度無失真旋轉空間範圍公式使用......30
第四章　實驗與結果......................................................32
4.1 虛擬正六角形結構影像轉換....................................32
4.1.1 影像正轉換............................................32
4.1.2 影像逆轉換............................................33
4.2 虛擬正六角形結構影像每60度無失真旋轉........................34
4.2.1 無失真旋轉60度........................................34
4.2.2 無失真旋轉120度.......................................36
4.2.3 無失真旋轉180度.......................................38
4.2.4 無失真旋轉240度.......................................40
4.2.5 無失真旋轉300度.......................................42
4.3 虛擬正六角形結構影像每60度無失真旋轉實作....................44
4.4 比較........................................................47
第五章　結論與未來研究方向..............................................51
5.1 結論........................................................51
5.2 未來研究方向................................................52
參考文獻................................................................53
誌謝....................................................................56
簡歷....................................................................57


圖目錄

圖1.1 花朵的旋轉對稱....................................................2
圖1.2 正六角形的平移對稱................................................2
圖1.3 正六角形結構排列組合形成三條垂直線................................3
圖1.4 正方形結構排列組合形成兩條垂直線..................................3
圖2.1 電子視網模視覺技術................................................5
圖2.2 近似模擬正六角形..................................................6
圖2.3 虛擬正六角形像素..................................................6
圖2.4 虛擬正六角形結構..................................................7
圖2.5 正方形結構座標....................................................7
圖2.6 蜂巢式網路座標....................................................8
圖2.7 螺旋式正六角形結構座標............................................8
圖2.8 正方形結構影像放大14倍後套入虛擬正六角形結構二分之一縮小圖........9
圖2.9 正方形結構影像放大14倍後套入虛擬正六角形結構局部放大圖...........10
圖2.10 虛擬正六角形結構影像週圍轉換不完全...............................10
圖2.11 正方形結構無失真旋轉90度.........................................11
圖2.12 正方形結構影像無失真旋轉90度.....................................11
圖2.13 正六角形結構三條垂直線劃分6個區塊................................12
圖2.14 正六角形結構三條垂直線劃分6個區域................................12
圖2.15 正六角形結構影像無失真旋轉60度...................................13
圖3.1 直覺式正六角形結構座標...........................................14
圖3.2 直接以對映方式套入虛擬正六角形結構...............................15
圖3.3 7x8正方形像素....................................................16
圖3.4 虛擬正六角形結構影像直接對映逆轉換...............................17
圖3.5 正六角形結構轉換為方形面積示意圖.................................18
圖3.6 正六角形結構影像無失真旋轉後空間範圍數據歸納.....................21
圖3.7 正六角形結構影像旋轉60度空間範圍示意圖...........................22
圖3.8 正六角形結構影像旋轉60度空間範圍區塊一公式推導...................22
圖3.9 正六角形結構影像旋轉60度空間範圍區塊二公式推導...................23
圖3.10 正六角形結構影像旋轉60度空間範圍區塊三公式推導...................23
圖3.11 正六角形結構影像旋轉120度空間範圍示意圖..........................25
圖3.12 正六角形結構影像旋轉120度空間範圍區塊一公式推導..................25
圖3.13 正六角形結構影像旋轉120度空間範圍區塊二公式推導..................26
圖3.14 正六角形結構影像旋轉120度空間範圍區塊三公式推導..................26
圖3.15 正六角形結構影像旋轉180度空間範圍示意圖..........................28
圖3.16 正六角形結構影像旋轉180度空間範圍公式推導........................28
圖3.17 正六角形結構旋轉60度與240度......................................30
圖3.18 正六角形結構旋轉120度與300度.....................................30
圖4.1 正轉換之64x64正方形結構影像......................................32
圖4.2 正轉換之73x64正方形結構影像......................................32
圖4.3 正轉換之73x64虛擬正六角形結構影像二分之一縮小圖..................33
圖4.4 逆轉換之73x64虛擬正六角形結構影像二分之一縮小圖..................33
圖4.5 逆轉換之73x64正方形結構影像......................................34
圖4.6 逆轉換之64x64正方形結構影像......................................34
圖4.7 5x5正六角形結構影像無失旋轉0度至60度.............................34
圖4.8 5x5正六角形結構影像無失真旋轉60度空間範圍與X、Y軸數值示意圖......35
圖4.9 5x5正六角形結構影像無失真旋轉60度像素搬移........................36
圖4.10 5x5正六角形結構影像無失真旋轉0度至120度..........................36
圖4.11 5x5正六角形結構影像無失真旋轉120度空間範圍與X、Y軸數值示意圖.....37
圖4.12 5x5正六角形結構影像無失真旋轉120度像素搬移.......................38
圖4.13 5x5正六角形結構影像無失真旋轉0度至180度..........................38
圖4.14 5x5正六角形結構影像無失真旋轉180度空間範圍與X、Y軸數值示意圖.....39
圖4.15 5x5正六角形結構影像無失真旋轉180度像素搬移.......................40
圖4.16 5x5正六角形結構影像無失真旋轉0度至240度..........................40
圖4.17 5x5正六角形結構影像無失真旋轉240度空間範圍與X、Y軸數值示意圖.....41
圖4.18 5x5正六角形結構影像無失真旋轉240度像素搬移.......................42
圖4.19 5x5正六角形結構影像無失真旋轉0度至300度..........................42
圖4.20 5x5正六角形結構影像無失真旋轉300度空間範圍與X、Y軸數值示意圖.....43
圖4.21 5x5正六角形結構影像無失真旋轉120度像素搬移.......................44
圖4.22 73x64正六角形結構影像無失真旋轉0度二分之一縮小圖.................44
圖4.23 100x87正六角形結構影像無失真旋轉60度二分之一縮小圖...............45
圖4.24 100x86正六角形結構影像無失真旋轉120度二分之一縮小圖..............45
圖4.25 73x65正六角形結構影像無失真旋轉180度二分之一縮小圖...............46
圖4.26 100x87正六角形結構影像無失真旋轉240度二分之一縮小圖..............46
圖4.27 100x86正六角形結構影像無失真旋轉300度二分之一縮小圖..............47
圖4.28 放大14倍後之正方形像素...........................................47
圖4.29 奇數與偶數行相差二分之一像素.....................................48
圖4.30 虛擬正六角形結構局部影像.........................................49


表目錄

表3.1 正六角形結構影像每60度無失真旋轉空間範圍公式......................31
表3.2 正六角形結構影像每60度無失真旋轉後與X和Y軸交點位置公式............31
表4.1 正逆轉換運算時間與記憶體使用量比較表..............................49
表4.2 影像處理常用影像之正逆轉換運算時間與記憶體使用量比較表............50

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