本論文的目的是利用濾波改良動態流體作用下的草地模擬，且仍然保持即時效能的特性。我將動態的Laplacian of Gaussian filter嵌入於60x60x20 的封閉性流體網格大小的草地場景，其中每網格中的流體場的力學作用獨立運算，其受到流體場的力學作用時，會整合搖擺、形狀彎曲以及葉片扭轉這三種運動作為運動呈現，在進行高速計算力學架構時必須利用GPU繪圖處理器進行整合運算。場景中一根草的架構是由四個控制節點做出粗略模擬，我將每個控制節點受到流體場的力學作用時的三維位置做下紀錄，使用每32個場景構成的動態window構成動作訊號(Motional Signal)，對於每維度訊號加入四位元代數結構(Quaternion)的濾波的調變，其中濾波中的部分參數被時間與原始的動作決定，可使用傅立葉轉換動作訊號的二階導數計算濾波的所需參數，以得到不誇張偏離原本的物理模擬模型，且可以加強動畫動作的精華部份。後續處理過程是將計算好的物理模型取代原始的場景，使用Tessellation著色器將其粗糙模型轉變成貝氏(Bezier)與拉格朗日(Lagrange)綜合圓滑曲線畫於屏幕上。本論文的動畫效果方法並不會嚴重加大計算量，卻能加強草地被流體場影響時栩栩如生的效果，且使用者可根據濾波給定的不同參數達成不同程度的調變，簡易有效的得到需求效果。

The purpose of the thesis is to modify a simulation of grass swaying with dynamic wind sources by a dynamic filter, and the method can still maintain its original property of running real-time. We insert dynamic Laplacian of Gaussian filter into a frame of meadow with 60x60x20 fluid grid size, where the force of fluid field calculates independently in each grid. When a wind source blows, the position of each grass would be affected by force field simultaneously and perform as three types of motion: swinging, shape bending and blade twisting. To deal with the high performance in grass swaying simulation, we use GPU to accelerate the computation of a bunch of grass data.
Each coarse grass model in a frame is defined as four control points per unit. After being calculated their strengths and directions, the force for each grid change the directions of those control points on the corresponding grass, and the outcome become the simple models of all affected grasses. In order to make the original output being more alive and animated, we record three dimensions of every node positions in 31 frames to be a dynamic sliding window, and later modulate its motion signal in each dimension by applying dynamic filter. Some of the parameters in the filter is decided by time and the original position of model, thus, we utilize a Discrete Quaternion Fourier Transformation to acquire parameters in need. Finally, we replace the original motion signal by our modified output, and send it into Tessellation shader to transform a coarse model with four points into a smoothed synthesis of Bezier curve and Lagrange polynomial which describes the actual shape of grass. The method of this paper would not severely increase computation in our PC, but it enhances the detail motion of an animation, which makes the visual effects of grass simulation more lifelike. This simple and efficient way allows user achieves different consequence by giving different parameters in our dynamic filter.

Chapter 1 Introduction - p.7
Chapter 2 Related Works - p.10
Chapter 3 Modified Grass Swaying Simulation - p.12
3.1 Overview - p.12
3.2 Fluid Simulation - p.13
3.2.1 Navier–Stokes equation - p.13
3.3 Plant Simulation - p.18
3.3.1 Grass Modeling - p.18
3.4 Motion Filtering - p.21
3.4.1 Construction of Sliding Window - p.22
3.4.2 Choosing the Filter Width - p.24
3.4.3 Discrete Quaternion Fourier Transformation - p.25
Chapter 4 Results - p.28
Chapter 5 Conclusions - p.35
References - p.36

[1] Harlow, F. H., & Welch, J. E. (1965). Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Physics of fluids, 8(12), 2182.
[2] Foster, N., & Metaxas, D. (1996). Realistic animation of liquids. Graphical models and image processing, 58(5), 471-483.

[3] Stam, J. (1999, July). Stable fluids. In Proceedings of the 26th annual conference on Computer graphics and interactive techniques (pp. 121-128)
[4] Stam, J. (2003, March). Real-time fluid dynamics for games. In Proceedings of the game developer conference (Vol. 18, p. 25).
[5] Adam W. Bargteil, Tolga G. Goktekin, James F. O’brien, and John A. Strain A Semi-Lagrangian Contouring Method for Fluid Simulation in ACM Transactions on Graphics, 25(1), 2006.

[6] Bridson, R. (2008). Fluid simulation for computer graphics. CRC Press.
[7] Belyaev, S., Laevsky, I., & Chukanov, V. (2011). Real-time Animation, Collision and Rendering of Grassland. Proc. GraphiCon’2011, 1-4.
[8] QIU, H., CHEN, L. T., & QIU, G. (2013). A Novel Approach to Simulate the Interaction between Grass and Dynamic Objects. WSEAS Transactions on Computers, 12(7), 2.
[9] Fan, Z., Li, H., Hillesland, K., & Sheng, B. (2015, February). Simulation and rendering for millions of grass blades. In Proceedings of the 19th Symposium on Interactive 3D Graphics and Games (pp. 55-60). ACM.
[10] Armin Bruderlin & Lance Williams. (1995) Motion signal processing. In SIGGRAPH (1995).In Proceedings of the 22nd annual conference on Computer graphics and interactive techniques Pages 97-104.ACM
[11] Seyoon Tak, Oh-young Song, and Hyeong-Seok Ko (2000). Motion Balance Filtering. In EUROGRAPHICS Volume 19, Number 3 (2000)
[12] Jue Wang, Steven M. Drucker, Maneesh Agrawala&Michael F. Cohen (2006)
The Cartoon Animation Filter in ACM SIGGRAPH 2006 Papers Pages 1169-1173
[13] Mawardi Bahri & Surahman(2013), Discrete Quaternion Fourier Transformation in Int. Journal of Math. Analysis, Vol. 7, 2013, no. 25, 1207 - 1215
[14] Ben Kenwright (2015) Quaternion Fourier Transform for Character Motions in Eurographics