臺灣博碩士論文加值系統

在組織病理學上（histopathology），型態結構（morphometric）的分析無論對於鑑別或者診斷，都有十分重要的幫助。然而，生物結構的型態極為複雜，諸如面積、周長等型態參數，在檢驗樣本的不同尺度規模之下，可能會有相當大的變異。因此，「組織的描述」對傳統幾何學（conventional Euclidean geometry）來說，一直相當棘手，例如：結腸直腸息肉（colorectal polyps）、口腔上皮損傷（epithelial lesions in the oral cavity）、乳房X光攝影片（mammograms）顯示的乳癌、腫瘤的血管增生（vasculature）。有鑑於此，「找尋客觀的方法以確實定量細胞或組織的複雜型態」是一個相當活躍的研究領域。
近來，在一些特定病理過程中細胞複雜程度的量化上，碎形分析（fractal analysis）似乎有著越來越大的影響力。現有的碎形分析工具裡，生物科學經常使用方格計數法（box-counting）來估算碎形維度（fractal dimension）。碎形維度是一項描述「空間填補程度」（degree of space-filling）的參數。根據之前的研究顯示，方格運算法求得之碎形維度在多種疾病上可作為一個有用的辨識值。然而，在抽象的數學參數與病理意義之間，尚缺乏理論基礎提供兩者關連。因此，在文獻中存有相當數量的爭議，值得且必要進一步的探討。
本研究之目的為：透過檢驗已知維度之對象，探討方格計算法的物理意義，尤其是，比較「單一物件」及「其集合構成」（ensembles）的碎形維度。結果顯示：兩者之間並不相同！對於這看似與碎形理論矛盾的結果，本研究將從數學模型上加以說明，並提出衍伸之意涵。另外，對於碎形分析應用在數位影像上的固有特性、使用時應注意的事項也有討論。

Morphometric analysis is important in the assistance of differential diagnoses in histopathology. However, for morphologically complicated biological structures, description of texture using conventional Euclidean geometry has been difficult because the estimation of shape parameters such as area or perimeter may vary significantly with the magnification at which the specimen examinations are performed. Examples include colorectal polyps, epithelial lesions in the oral cavity, breast cancer on mammograms, or tumor vasculature. The search for an objective means to reliably quantify complicated cell or tissue morphology is thus an active field of research development.
Recently, fractal analyses seem to gain on influence in quantifying the degree of cell complexity that may have been altered during certain pathological processes. Among many existing tools for fractal analysis, the box counting algorithm is frequently used in biological science to obtain the fractal dimension, a parameter that describes the extent of the space-filling property. Previous studies have demonstrated that the box counting fractal dimension is a helpful diagnostic discriminant in various diseases. However, there lacks a theoretical essence providing the linkage between this abstract mathematical parameter and the pathological meanings. A significant number of controversies hence exist in the literature, which warrants the necessity of further investigations.
The goal of this study was to explore, using examinations on illustrative objects of known geometry, the physical implications of the box counting fractal dimension. In particular, the fractal dimensions computed using box counting on single objects were compared with the results on an ensemble of the same objects. An explanation was provided for our findings in this study, and the consequent implications were addressed. Some of the inherent characteristics demanding cautions when using this method on digitized images were also discussed.

CHINESE ABSTRACT
ENGLISH ABSTRACT
1. BACKGROUND …………………………………………………………1-1
2. INTRODUCTION ………………………………………………………2-1
3. THEORY ………………………………………………………………3-1
3-1 Metric Space ………………………………………………… 3-1
3-2 Fractal Dimension ……………………………………………3-2
3-3 Box Counting Algorithm …………………………………… 3-4
3-4 Triadic Koch Curve ………………………………………… 3-7
3-5 Self-similarity ………………………………………………3-10
4. METHODS …………………………………………………………… 4-1
4-1 Generation of Single Objects and Their Ensembles……4-1
4-2 Box Counting Fractal Dimension ………………………… 4-4
4-3 Linear Regression and Correlation Coefficient……… 4-6
4-4 Postprocessing of Retinogram …………………………… 4-11
5. RESULTS …………………………………………………………… 5-1
5-1 Box Counting Fractal Dimension of Single Object…… 5-1
5-2 Box Counting Fractal Dimension of Object ensemble… 5-3
5-3 Spacing Between Individual Objects …………………… 5-6
5-4 Non-integer Divisions of the Field of View ………… 5-8
5-5 Fractal Analysis of Retina Vasculature ……………… 5-10
6. DISCUSSION …………………………………………………………6-1
6-1 Limitations of Box-counting Fractal Analysis in Digital
Images……………………………………………………………6-1
6-2 Examples in Biomedicine ……………………………………6-6
7. CONCLUSION …………………………………………………………7-1
APPENDIX ……………………………………………………………… A-1
I. Theoretical prediction of N(e) versus e for gridline
objects………………………………………………………………A-1
II. Other algorithms for fractal dimension estimation…… A-6
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