形態影像處理可以透過結構元素對物體作出有效率的修補，近年來形態處理已成功應用於工業檢測以及醫療影像處理，在本篇論文中，我們將用兩種方式來研究形態影像處理的效率：四分樹法以及平行處理法。
在四分樹演算法中，任何二元影像皆可轉換成以固定比例為2的次方的黑色與白色方塊，使得整張影像的擴張可透過分別方塊的擴張而完成。另外，任何的二元影像可以位元（bit）的方式表現，使得我們可以在個人電腦上使用平行處理的概念去增加形態影像處理的速度。
實驗的結果都顯示了兩種方法皆比起傳統的方式快，四分樹顯示了在大結構元素下的優勢，另外，平行處理法表現了最快的效率在一般的應用上。

The morphological image processing can modify the shapes of objects very efficiently by structure elements. Thus, the morphology processing has recently been applied to industry auto-inspection and medical image processing successfully. In this thesis, we incestigate the efficient processing of morphological image processing by two approaches: quadtree approach and paralell approach.
By the quadtree decomposition, any binary image can be decomposed into black and white square blocks with some fixed size of power of 2. Thus, dilation of the whole image can be accomplished by dilating individual decomposed square blocks. On the other hand, any binary image can be presented by bit per pixel basis. Thus, we can exploit the parallel on a personal computer to speed up the set oriented morphological image processing.
Experiments have revealed that both approach are much faster than the direct method. The quadtree approach are most advantageous for large structure elements. Whereas, the parralel approach are the fastest for the usual applications.

第1章 引言 5
第2章 四分樹資料結構簡介 11
2.1 四分樹基本概念 11
2.2 影像平面座標與四分樹的對應關係 12
2.2.1 Morton Order與平面x,y座標之互換 12
2.2.2 四分樹節點在平面上的位置 14
2.3 四分樹資料結構的表示方式 15
2.3.1 指標四分樹 15
2.3.2 非指標四分樹 15
2.3.3 指標樹和字串樹的比較 16
第3章 形態影像學 18
3.1 簡介 18
3.2 黑白影像的擴張運算 18
3.2.1 基本運算定義 19
3.2.2 擴張的定義 19
3.3 黑白影像的侵蝕運算及性質 20
3.4 開放和閉鎖運算 21
第4章 四分樹上的擴張運算 24
4.1 基本概念 24
4.1.1 運用四分樹的優點 25
4.1.2 四分樹所帶來的問題 26
4.2 樹圖案的製作 27
4.2.1 運用擴張運算的交換性縮減樹圖案 28
4.2.2 以二分樹縮減四分樹的體積 32
4.2.3 探討結構元素的位移量 35
4.3 樹圖案的懸掛 39
4.4 記憶體的配置問題 43
第5章 實驗方法與比較 47
5.1 實驗說明 47
5.2 圖案對映法 47
5.3 階層式演算法 48
5.4 線上切割演算法 49
5.5 平行處理法 (Parallel processing) 52
5.5.1 輸出分解方法（The output-decomposition method） 53
5.5.2 輸入分解方法（The input-decomposition method） 54
5.6 實驗數據與結果 54
5.6.1 實驗圖形說明 55
5.6.2 實驗數據與結果 57
5.7 樹圖案的比較 60
5.8 應用於圖片的檢視 61
第6章 結論 63
參考文獻 64

1.Samet H “Applications of spatial data structures, computer graphics,image processing, and GIS.” Reading, Addison Wesley, 1990.
2.E. Kawaguchi and T. Endo, “On a method of binary picture representation and its application to data compression.” IEEE Trans. on PAMI, Jan. 1980, pp.27-35.
3.Serra J. “Image analysis and mathematical morphology.” New York：Academic Press, 1983.
4.Matheron G. “Random sets and integral geometry.” New York：John Wiley and Sons, 1975.
5.Jain A. K. “Fundamentals of digital image processing.” Prentice-Hall, Englewood Cliffs, N.J. 1989.
6.M. F. Goodchild and A. W. Grandfield, “Optimizing raster storage：an examination of four alternatives.” Proceedings of Auto-Carto 6, vol. 1, Ottawa, Oct. 1983, pp.400-407.
7.G. M. Morton “A computer oriented geodetic data base and a new technique in file sequencing.” IBM Ltd. Ottawa, Canada 1966.
8.E. Kawaguchi, T. Endo, and J. Matsunaga, “Depth-first expression viewed from digital picture processing.” IEEE Trans. on PAMI 5,4 Jul. 1983, pp.373-384.
9.Philippe Salembier, Patrick Brigger, Josep R. Casas, and Montse Pardas, “Morphological operators for image and video compression.” IEEE Trans. On Image Processing, vol. 5, no. 6, Jun. 1996.
10.C. K. Lee and S. P. Wong, “A methematical morphological approach for segmentation heavily noise-corrupted images.” Pattern Recognition, 1996, vol. 29, no. 8, pp.1347-1358.
11.En-Hui Liang and Edward K. Wong “Hierarchical algorithms for morphological image processing.” Pattern Recognition, 1993, vol. 26, no. 4, pp.511-529.