本研究提出一個能夠表達連續的與不連續的、平滑的與不平滑的、以及碎形的與非碎形的特徵物之三維物面表達及重建演算法。研究中運用具備碎形表達能力之二維的三階Daubechies尺度函數，做為描述重建模型之基底函數，並藉此組成點雲之觀測方程式，進而以最小二乘法重建點雲之物面。而為克服此平差模式所可能產生的劣態問題，本文運用由粗而細的求解策略，並應用二次分點位置之虛擬觀測量PHO (Pseudo Height Observations) 及POI (Pseudo Observations by Interpolation)，穩定平差系統之求解。

　　研究成果顯示，在二次分點位置之虛擬觀測量的輔助下，此重建法之求解系統確實能夠獲得一穩定且滿足演算法的基本假設之解。此外，我們發現： (1)附加物效應:部分重建訊號之斷面及獨立物處，產生一些不存在原始取樣訊號中之較高振幅的擺盪。(2)改正數較大的點雲幾乎都落在取樣點訊號中之斷面或獨立物處。運用(2)之特性，我們提出了一個全自動的重新給權模式，降低改正數絕對值大於兩倍點雲先驗高程精度之點雲權值。實驗顯示，配合虛擬觀測量及此重新給權模式，附加物效應已完全的被消除，且重建模型之後驗單位權中誤差可達與點雲之先驗高程精度相當之等級。以本研究之測試區為例，當尺度函數之解析力與點雲之平均取樣點間距相當時，後驗單位權中誤差皆約為±20cm之等級，此與點雲之先驗高程精度±25cm相當。
　　
　　相較於現有的物面重建法，本研究所提出之小波物面重建法能夠於重建時，將不規則的、不平滑的、及碎形的訊號特徵一併考量在內，使得重建訊號不會因所使用之訊號表達模式，產生部份細節訊號於重建後之遺失情況。其能夠在完全沒有任何資料預處理之情況下，以相當微量的人工輔助(如:給定計算參數)，全自動的進行物面重建計算，並能夠將各式的訊號特徵精確的重建出。

　　We propose a 3D surface reconstruction algorithm which is capable of describing continuous, discontinuous, smooth, rough, fractal, and non-fractal surface. We utilize the 2D Daubechies scaling function of 3rd order, which can describe fractal geometry, to write the observation equations of the point cloud. Furthermore, the linear system is solved by the least-squares adjustment and the reconstructed surface model can then be generated. To overcome the probable ill-posed problem, we employ a from-coarse-to-fine strategy and use the pseudo observations on dyadic points, PHO (Pseudo Height Observations) and POI (Pseudo Observations by Interpolation) to stabilize the linear system.

　　Our experimental results show that with assistance of the pseudo observations on dyadic points, the algorithm can yield a stable solution and can meet with our basic hypotheses. Besides, we find: (1)artifact effect: some irregular artifacts are shown around the abrupt areas of the reconstructed model. (2)points with larger residuals are largely located on abrupt areas such as walls, poles, or isolated trees. To eliminate the artifact effect, we propose a full-automated weighting model to reduce the weights of the point cloud whose absolute residuals are higher than twice the a priori height accuracy of the LIDAR data. The results reveal that by combining the pseudo observations and the weighting model, artifacts can be completely eliminated and the a posteriori standard deviation of unit weight of the reconstructed surface can reach the same level of the height accuracy of LIDAR data points. For instance, while the resolution of scaling function is about the point interval of LIDAR point cloud, the a posteriori standard deviations of unit weight of our test areas are about ±20cm and are all to the extents of the a priori height accuracy, ±25cm.

　　By comparing to the diverse currently available surface reconstruction algorithms, the proposed approach can handle irregular, non-smooth, and fractal signals quite well and significant surface features registered in the original discrete sample points can be clearly expressed in the reconstructed surface. After some computation parameters are manually given, without the need on any other data preprocessing, our reconstruction system can automatically reconstruct a precise, highly complex, and multi-resolution surface model from discrete LIDAR points.

黃國良、蔡展榮，1999。小波表達多層解析度城市幾何表面函數之方法，第十八屆測量學術及應用研討會論文集，國立宜蘭技術學院，民國88年，第551-560頁。

單維彰，1999，凌波初步，全華圖書公司，ISBN 957-21-2430-7。

蔡展榮，2004。小波基礎理論與應用-授課講義，國立成功大學。

Alexa, M., Behr, J., Cohen, D., Fleishman, S., Levin, D. and Silva, C. T., 2001. “Point set surfaces”. IEEE Transactions on Visualization 2001, pp. 21-28.

Anca, A., Gaildrat, V., Barthe, L., 2004. “Modeling and representing surfaces: Interactive modeling from sketches using spherical implicit functions”. Proceedings of the 3rd international conference on Computer Graphics, Virtual Reality, Visualization and Interaction in Africa, pp. 25-34.

Busé, L. & Galligo, A., 2004. “Using semi-implicit representation of algebraic surfaces”. IEEE Transactions on Shape Modeling Applications 2004, pp. 342-345.

Carr, J. C., Beatson, R. K., McCallum, B. C., Fright, W. R., McLennan, T. J., and Mitchell, T. J., 2003. “Representation: smooth surface reconstruction from noisy range data”. Proceedings of the 1st international conference on Computer Graphics and Interactive Techniques in Australasia and South East Asia, pp. 119-126.

Cohen, Fernand S., Ibrahim, W., and Pintavirooj, C., 2000. “Ordering and parameterizing scattered 3D data for B-spline surface approximation”. IEEE Transactions on Pattern Analysis and Machine Intelligence 2000, Vol. 22, Issue 6, pp. 642-648.

Daubechies, I., 1992. “Ten lectures on wavelets”. SIAM, Philadelphia, PA.

Ebert, David S., Musgrave, Kenton F., Peachey, D., Perlin, K., and Worley, S., 2003, “Texturing & modeling : a procedural approach”. Amsterdam, Morgan Kaufmann, ISBN: 1558608486.

He, Y. & Qin, H., 2004. “Surface reconstruction with triangular B-splines”. IEEE Transactions on Geometric Modeling and Processing 2004, pp. 279-287.

Hill, Francis S., 2000. “Computer graphics using OpenGL”, 2nd edition. ISBN: 0023548568, Prentice Hall, USA

Hoppe, H., DeRose, T., Duchamp, T., McDonald, J., and Stuetzle, W., 1992. “Surface reconstruction from unorganized points”. ACM SIGGRAPH 1992, pp. 71-78.

Hoppe, H. & Eck, M., 1996. “Automatic reconstruction of B-spline surfaces of arbitrary topological type”. Proceedings of the 23rd annual conference on Computer Graphics and Interactive Techniques, pp. 325-334.

Kim, Y., Machiraju, R., and Thompson, D., 2003. “Rough surface modeling using surface growth”. IEEE Transactions on Shape Modeling International 2003, pp. 33-42.

Lee, I. K. & Kim, K. J., 2004. “Shrinking: another method for surface reconstruction”. IEEE Transactions on Geometric Modeling and Processing 2004, pp. 259-266.

Loop, C., 1994. “Smooth spline surfaces over irregular meshes”. Proceedings of the 21st annual conference on Computer Graphics and Interactive Techniques, pp. 303-310.

Lorensen, W. E. & Cline, H. E., 1987. “Marching cubes: A high resolution 3D surface construction algorithm”. ACM SIGGRAPH 1987, pp. 163-169.

Mallat, S., 1998. “A Wavelet Tour of Signal Processing”. Academic Press, ISBN: 0-12-466605-1.

Namane, R., Boumghar, F. O., and Bouatouch, K., 2004. “Modeling and representing surfaces: QSplat compression”. Proceedings of the 3rd international conference on Computer Graphics, Virtual Reality, Visualization and Interaction in Africa, pp. 15-24.

Optech Incorporated, 2004. ALTM 3100 System Specifications, http://www.optech.ca/pdf/Specs/specs_altm_3100.pdf

Park, I. K. & Lee, S. U., 1997. “Geometric modeling from scattered 3-D range data”. IEEE Transactions on Image Processing 1997, Vol. 2, pp. 712-715.

Park, I. K., Yun, I. D., and Lee, S. U., 1999. “Constructing NURBS surface model from scattered and unorganized range data”. IEEE Transactions on 3-D Digital Imaging and Modeling 1999, pp. 312-320.

Park, I. K., Yun, I. D., and Lee, S. U., 2000. “Automatic 3-D model synthesis from measured range data”. IEEE Transactions on Circuits and Systems for Video Technology 2000, Vol. 10, Issue 2, pp. 293-301.

Peitgen, H. O. & Saupe, D., 1988. “The Science of Fractal Images”. Springer-Verlag, New York.

Pfister, H., Zwicker, M., Baar, J., and Gross, M., 2002. “Surfels: surface elements as rendering primitives”. Proceedings of the 27th annual conference on Computer Graphics and Interactive Techniques, pp. 335-342.

Prusinkiewicz, P. & Hammel, M., 1993. “A fractal model of mountains with rivers”. Proceedings of Graphics Interface '93, pp. 174-180.

Reuter, P., Tobor, I., Schlick, C., and Dedieu, S., 2003. “Representation: point-based modeling and rendering using radial basis functions”. Proceedings of the 1st international conference on Computer Graphics and Interactive Techniques in Australasia and South East Asia, pp. 111-118.

Schröder, P. & Sweldens, W., 1995. “Spherical wavelets: efficiently representing functions on the sphere”. Proceedings of the 22nd annual conference on Computer Graphics and Interactive Techniques, pp. 161-172.

Shaffer, E. & Garland, M., 2005. “A multiresolution representation for massive meshes”. IEEE Transactions on Visualization and Computer Graphics 2005, Vol. 11, Issue 2, pp. 139-148.

Strang, G. & Nguyen, T., 1996. “Wavelets and Filter Banks”. ISBN: 0-9614088-7-1. Wellesley Cambridge Press.

Sweldens, W., 1996. “Wavelets and the lifting scheme: A 5 minute tour”. Zeitschrift fur Angewandte Mathematik und Mechanik, Vol.76, pp. 41-44.

Tsay, Jaan-Rong, 2000. “A new algorithm for automatic precise reconstruction of a real object surface”. the XIXth Congress of the International Society for Photogrammetry and Remote Sensing (ISPRS), Amsterdam, The Netherlands, 16-23 July 2000, International Archives of Photogrammetry and Remote Sensing (IAPRS), Vol. XXXIII, Supplement B3, pp. 59-66.

Tsay, Jaan-Rong, 2002. “A concept and algorithm for 3D city surface modeling”. proceedings of the International Society for Photogrammetry and Remote Sensing (ISPRS) Commission III symposium, Graz, Austria, September 9-13, 2002, “Photogrammetric Computer Vision (PCV’02)”, Vol. 34 (ISSN 1682-1777), Part 3B, pp. B-283-B-286.

Wang, Q., Hua, W., Li, G., and Bao, H., 2004. “Generalized NURBS curves and surfaces”. IEEE Transactions on Geometric Modeling and Processing 2004, pp. 365-368.

Xie, H., Wang, J., Hua, J., Qin, H., and Kaufman, A., 2003. “Piecewise C/sup 1/ continuous surface reconstruction of noisy point clouds via local implicit quadric regression”. IEEE Transactions on Visualization 2003, pp. 91-98.

You, S., Hu, J., and Fox, P., 2003. “Urban site modeling from Lidar”. Proceedings of Second International Workshop on Computer Graphics and Geometric Modeling, Vol. 2668, pp. 579-588.

Zhong, D., Li, M., Wang, G., and Wei, H., 2004. “NURBS-based 3D graphical modeling and visualization of geological structures”. IEEE Transactions on Image and Graphics 2004, pp. 414-417.

Zhang, J. J. & You, L., 2001. “Surface representation using second, fourth and mixed order partial differential equations”. IEEE Transactions on Shape Modeling Applications 2001, pp. 250-256.