應用碎形維數為地理特徵物指標之研究

Abstract

地理特徵物，隱含著大自然界中一些複雜但有序的結構型態，這些自然現象，無法以傳統的歐幾里德幾何做適當的描述，而碎形維數可以對此提供一種量測與描述。依此概念，本文由碎形維數的觀察角度，探索地理特徵物所隱藏的訊息，及應用為量化指標的可行性。 本研究敘述了包括角規法等三種碎形維數的計算方法，以及在地理特徵分析上的適用性。並選取地理特徵中，具一維型態的河川以及二維形態的數值地形為探討對象，說明碎形維數在河川蜿蜒度以及自然地形分析、模擬與影像內插領域的應用。 研究結果發現，在同一比例尺的情況下，碎形維數與河川蜿蜒度具有相當程度的相關性(r > 0.8 )；但包含由不同比例尺之圖籍所得之河川圖徵時，兩者的相關性隨之降低。比較不同縮編方法時，使用 Douglas & Peucker 方法縮編較人工縮編所得之相關性為高。此外，在地形分析與模擬的應用方面，以變異元法計算碎形維數，可以偵測出自然地形具有方向性，而碎形模式生成的地形則沒有。以傅立葉濾波法模擬為基礎，結合由真實地形中獲取的低頻資訊，可使模擬地形保持原地形的資料特性，讓生成之DEM 能適合更多元化的應用。在影像內插方面，以碎形模式為基礎的內插方式，不論是以均方根誤差、相關係數等統計指標，或是以影像匹配為評估方式的幾何精度指標，均較傳統的雙線性內插以及立方迴旋內插的方法效果為差，本文所提出的修正模式雖較原模式有所改進，但是仍較以取樣理論所發展之方法為差。在碎形特性的保持能力方面，以碎形為基礎的內插模式則如預期的獲得較好的結果。

Geographical feature frequently possesses a chaotic and complex structure. These features can not be described with Euclidean geometry, however, fractal geometry may produce better explanation. Fractal dimension is a measure based on the self-similarity concept, and can function as an index which reflects the ruggedness of lines, surfaces, and other "irregularities" in Euclidean space. In this study, three methods are applied to compute the fractal dimension. The limitations of these methods are discussed respectively. In addition, the fractal behavior of geographical features such as a river and DEM, and the application of fractal dimension on terrain simulation and on fractal-based image interpolation, are studied. Results show that fractal dimension of river and the sinuosity are related (r>0.8). However, the correlation decreases when features from multiple scales of maps are included. The correlation between river and sinuosity by using numerical generalization is higher than by manual generalization. Regarding the application of fractal-based terrain simulation, the fractal dimensions of simulated terrains are computed via the variogram method. The outcomes show that the simulated terrains are isotropic. This finding agrees with the expectation that fractal terrains in general have no inherent global erosion features due to their isotropy and stationarity. The study has shown that the simulated model is improved by combining the low frequency of real terrain and the high frequency portion of fractal terrain generated by the Fourier filtering method. When comparing the schema of interpolation, statistic indices, such as root mean square error, average error, maximum deviation, and correlation coefficient are derived from interpolation and its reference images can be evaluated. Besides, an image matching process based on the least squares principle is also utilized for comparison. All quality measures indicate that the conventional interpolation schemes are better than the fractal-based schemes.