在中國水墨畫中，流水及飛泉是意境呈現的重要技法，而呈現水流的畫法則以勾勒、渲染或二者結合之交互應用為主，用以強調呈現水紋之流暢與連貫。本研究中，以那維斯托克方程式(Navier-Stoke equation)的物理模型來模擬此一效果，以使水流效果符合自然法則，同時利用張量場域平滑法(tensor field smoothing)，來消除因為使用者多重輸入所造成的成像缺陷，最後才得呈現平順且時序一致性的國畫動畫。在本論文中，我們提供簡單且直覺的介面來建構國畫場景及控制速度場，用以繪製出符合使用者意念的水墨河流動畫效果。在系統中，使用者可利用系統提供之介面，自行設置場景，包括山及水流中的石頭，並且設定各部水流的基本流向及速度，依據使用者的設定，本系統可以計算出穩定的速度場後，再根據速度場產生具水墨效果排列的筆觸，並可更可進一步做出此一效果的三維模擬及動畫。該系統主要貢獻在於整合那維斯托克方程式(Navier-Stoke Equation)和張量平滑演算法(tensor field smoothing)，並將其引入國畫的流水技法中，更進一步建構了以此為基底的使用者介面，讓畫者可以快速地建立心中的意境及產生所需的動態水流，且水流是平滑且呈現時序及空間一致性，避免了一般非擬真算圖(Non-photorealistic rendering)畫法在動畫中的雜訊。

Rivers and falls are important elements in Chinese ink painting. Rivers and falls are painted by strokes to sketch the lines of the water flow and ink dispersion to express the spray of the flow surface in order to present the sense of smooth waterlines and the characteristics of coherent diffusion. This thesis uses the Navier-Stoke equation to simulate the water flow in a river and fall to create physically correct flow fields. At the same time, it also applies a tensor field smoothing method to the constructed flow fields in order to reduce the possible rendering artifacts when multiple flow patterns are specified. In the last, the constructed field would be animated as Chinese ink painting with smooth and time coherence. Our system provides an intuitive user interface to let the painter construct their virtual world with mountains, plains, rocks, rivers and falls. Then, the painter can specify the flow pattern in the rivers and falls, and the system can compute the physically correct flow field in the rivers and falls according to the boundaries, obstacles and user-specified fields. The physically correct flow field is used to construct the ink strokes with our smooth field stroke algorithm. Our system can easily generate view consistent and temporally coherent 3D animated rivers and falls in a Chinese ink painting style. The main contributions of our system lies in the integration of the Navier-Stoke equation and tensor field smoothing to generate a Chinese-ink-painting river and fall animation. Furthermore, we propose an interactive system that can quickly create a Chinese ink painting world containing the animated rivers and falls with the water flow drawn with spatial and temporal coherence. The overall animation is pleasant without the flickering artifacts commonly existing in a stroke-based non-photorealistic rendering (NPR) animation.

[1] 曾肅良著, 山水畫技法, 三藝文化民國101年2月,三版一刷.
[2] ZHANG, E., HAYS, J., AND TURK, G. 2007. Interactive tensor field design and visualization on surfaces. IEEE Transactions on Visualization and Computer Graphics 13, 1, 94–107.
[3] Steve Strassmann, “Hairy Brushes”, Proceedings of ACM SIGGRAPH 86, page 225-232. 1986.
[4] Cassidy J. Curtis, Sean E. Anderson, Jonathon. E. Seims, and Kurt W. Fleischer, and David H. Salesin, “Computer-generated watercolor”, Proceedings of SIGGRAPH 97, in Computer Graphics Proceedings, Annual Conference Series, pp. 421-430, August 1997.
[5] Sheng-Wen Huang, Der-Lor Way, Zen-Chung Shih, “Physical-Based Model of Ink Diffusion in Chinese Ink Paintings”, Journal of WSCG 2003.
[6] Chu, N. S. H. and Tai, C. L., “ MoXi: Real-time Ink Dispersion in Absorbent Paper ” , ACM Transactions on Graphics (Proceedings of ACM SIGGRAPH 2005), Vol. 24, No. 3, pp. 504-511, 2005.
[7] Hsin-Yi Chiu, Chun-Chieh Hsiao, Zen-Chung Shih, “The Synthesis of Chinese Fine-Brushwork Painting for Flower”, Master thesis, National Chiao Tung University. 2000.
[8] Der-Lor Way, Zen-Chung Shih, “The Synthesis of Rock Textures in Chinese Landscape Painting”. COMPUTER GRAPHICS Forum Volume 20, Number 3, pp. C123-C131, 2001.
[9] Yi-Wei Ho, Zen-Chung Shih, “The Synthesis of Chinese Fine-Brushwork Painting for Flower”, Master thesis, National Chiao Tung University. 2005.
[10] Hsin-Yi Chang , “ The Synthesis of Lingnan School Painting for Embroidered Fish ”, Master dissertation, Computer Science and Engineering College of Computer Science National Chiao Tung University, 2006.
[11] Jun-Wei Yeh , “ Non-Photorealistic Rendering in Chinese Painting of Animals ”, Journal of System Simulation, 1220--1224, 1262 , 2002.
[12] Yung-Nien Chen, “Non-Photorealistic 3D Rendering in Chinese Painting Style ”, Master dissertation, Computer Science in Information Engineering National Cheng Kuna University, 2003.
[13] Kuan-Ming Yu, “The Simulation of Chinese Ink Style Figure Painting Techniques in 3D Game Engine ”, Master dissertation, Computer Science and Engineering College of Computer Science National Chiao Tung University, 2005.
[14] D.-L. Way, Y.-R. Lin, and Z.-C. Shih.The synthesis of trees in Chinese land-scape painting using silhouette and texture strokes. WSCG, 10(3):499–507, 2002.
[15] Der-Lor Way, “ Wrinkle Rendering of Terrain Models In Chinese Landscape Painting ”, IEICE Transactions 89-D(3), 1238-1248, 2006
[16] Doug DeCarlo and Szymon Rusinkiewicz. Highlight Lines for Conveying Shape. International Symposium on Non-Photorealistic Animation and Rendering (NPAR), August 2007.
[17] Chao-Chin Wang. “ The Generation of the Flow of Water in Chinese Ink Paintings ” Master dissertation, Computer Science and Engineering College of Computer Science National Chiao Tung University ,2007.
[18] Song-hai Zhang, Tao Chen, Yi-Fei Zhang, Shi-Min Hu, Ralph R. Martin, "Video-Based Running Water Animation in Chinese Painting Style" Science in China Series F: Information Sciences, 2009, Vol. 52, No. 2, 162-171
[19] Jos Stam, "Stable Fluids", In SIGGRAPH 99 Conference Proceedings, Annual Conference Series, August 1999, 121-128.
[20] Jos Stam, "Real-Time Fluid Dynamics for Games". Proceedings of the Game Developer Conference, March 2003.